Turun kauppakorkeakoulu •  Turku School of Economics 
 
 
 
 
 
 
 
 
 
 
MARKET ANOMALIES AND TACTICAL 
ASSET ALLOCATION 
 
Utilising market anomalies in multiple asset class portfolios 
with the Black−Litterman model 
 
 
 
 
 
 
 
 
Master´s Thesis 
in Accounting and Finance 
 
 
Author: 
Paavali Pouttu 
 
Supervisors: 
Prof. Mika Vaihekoski 
M.Sc. Mika Hannula 
 
 
22.07.2015 
Turku 
 
 
  
   
TABLE OF CONTENTS 
1! INTRODUCTION ....................................................................................................... 7!
1.1! Background ..................................................................................................... 7!
1.2! Purpose of the study and research questions .................................................. 9!
1.3! Scope and limitations .................................................................................... 10!
1.4! Structure of the thesis ................................................................................... 11!
2! THEORETICAL FOUNDATION ............................................................................ 13!
2.1! Modern portfolio theory ............................................................................... 13!
2.1.1! Return and risk of the portfolio ........................................................ 13!
2.1.2! Asset diversification and the efficient frontier ................................. 15!
2.1.3! Risk-free asset and market portfolio ................................................ 16!
2.2! Mean-variance optimisation in practice ........................................................ 18!
2.2.1! Description of the optimisation process ........................................... 18!
2.2.2! Computing mean-variance efficient portfolios ................................ 20!
2.2.3! Issues on the inputs of mean-variance framework ........................... 22!
2.3! The Black−Litterman Model ........................................................................ 23!
2.3.1! Role in the asset allocation problem ................................................ 23!
2.3.2! The equilibrium portfolio and implied equilibrium returns ............. 25!
2.3.3! Incorporating investors’ views ......................................................... 26!
2.3.4! Uncertainty of the views and tau coefficient .................................... 28!
2.4! Market anomalies .......................................................................................... 30!
2.4.1! Definition and persistence on the markets ....................................... 30!
2.4.2! Value effect ...................................................................................... 31!
2.4.3! Momentum effect ............................................................................. 34!
2.4.4! Long-term reversal ........................................................................... 36!
2.4.5! Idiosyncratic volatility ...................................................................... 37!
3! DATA AND METHODOLOGY .............................................................................. 40!
3.1! Test assets ..................................................................................................... 40!
3.2! Descriptive statistics ..................................................................................... 40!
3.3! Building anomaly-based factors ................................................................... 47!
3.3.1! Defining the anomalies in different asset classes ............................. 47!
3.3.2! Trading algorithm for the factors ..................................................... 50!
3.4! Portfolio optimisation in the Black−Litterman framework .......................... 52!
3.4.1! Parameters for the model .................................................................. 52!
3.4.2! Optimal portfolios based on market anomalies ................................ 54!
3.5! Measuring portfolio performance ................................................................. 55!
 4! RESULTS OF THE EMPIRICAL STUDY .............................................................. 57!
4.1! Zero-cost portfolios ...................................................................................... 57!
4.2! Black−Litterman portfolios .......................................................................... 67!
4.3! Robustness checks ........................................................................................ 69!
5! CONCLUSION ......................................................................................................... 73!
REFERENCES ................................................................................................................ 76!
APPENDIX! CODE FOR THE EMPIRICAL STUDY ........................................ 81!
 
  
LIST OF FIGURES 
Figure 1! The Efficient Frontier ............................................................................... 15!
Figure 2! The MPT Investment Process (Fabozzi, Gupta, Markowitz 2002, 8) ...... 18!
Figure 3! Combining market equilibrium returns and 
investor’s views (Idzorek 2005) ......................................................... 24!
Figure 4! Return distributions of different asset classes against the normal 
distribution from the time period of 3.1.2000−1.4.2014. ................... 47!
Figure 5! The portfolio construction algorithm ........................................................ 51!
Figure 6! Performance of long-only strategies by asset classes; values 
presented with a logarithmic scale ...................................................... 63!
Figure 7! Performance of anomaly based Black−Litterman portfolios 
presented on a logarithmic scale ......................................................... 69!
 
 
LIST OF TABLES 
Table 1! Equities: Descriptive statistics ................................................................... 41!
Table 2! Government bonds: Descriptive statistics .................................................. 43!
Table 3! Currencies: Descriptive statistics ............................................................... 44!
Table 4! Commodities: Descriptive statistics ........................................................... 45!
Table 5! Properties of zero-cost portfolios ............................................................... 58!
Table 6! Results for long-only portfolios ................................................................. 60!
Table 7! Results for long-short portfolios ................................................................ 65!
Table 8! Results of multiple asset class portfolios with 
the Black−Litterman strategy ............................................................. 68!
Table 9! Robustness of test portfolios ...................................................................... 70!
Table 10! Equality of Sharpe ratios .......................................................................... 72!
  
7 
 
7 
1 INTRODUCTION 
1.1 Background 
In efficient markets, all relevant information should already be included in asset prices 
and thus investors should not be able to gain excess returns with respect to common risk 
factors by analysing the past. However, occasionally some consistent patterns appear to 
violate the efficient market hypothesis. The hypothesis argues that the current asset 
prices reflect all relevant information, thus, making it impossible to outperform the 
market with timing or excellent asset picking skills (see, e.g., Fama 1970). Some of the-
se patterns, also known as anomalies, tend to persist from year to another even though 
they should disappear after becoming publicly known. If such anomalies exist, utilising 
them within investment strategies could provide superior returns to those of a bench-
mark, e.g., the market portfolio. 
Asness, Moskowitz, and Pedersen (2013) study the existence of the value and mo-
mentum effects across different markets and asset classes, i.e., the relation between asset 
returns and book-to-market ratio and the one between asset returns and their recent cu-
mulative returns, respectively. Their results show that value and momentum strategies 
are not only profitable when considered separately but also when combined together. In 
addition, these phenomena are significant across different asset classes. Significance of 
value factor acquires strong empirical evidence in many different studies (see e.g. Fama 
and French 1993). Accordingly, the momentum strategy, first analysed by Jagadeesh 
and Titman (1993), is shown to realise significant abnormal returns. However, there 
seems to be contradictory results when it comes to applying these factors in an interna-
tional context. While Asness et al. (2013) report significant value and momentum 
premia across different markets, it is also claimed that the Fama−French three-factor 
model is not applicable for international portfolios (Griffin 2002; Mirza & Afzal 2011; 
Fama & French 2012). 
Momentum effect is shown to be strong with the past 3−12 month returns but with a 
longer time period the phenomenon is shown to revert, i.e., the past winners become 
losers and vice versa. This anomaly is known as a long-term reversal. One of the first 
who discovered the long-run reversal effect were DeBondt and Thaler (1985) showing 
that the long term losers change their course becoming winners in the future. These two 
effects rest on performance of assets, however, there is another phenomenon that leads 
to mispricing of assets, which is based on unsystematic or idiosyncratic risk. There are 
number of studies on idiosyncratic volatility effect claiming that either high or low vola-
tility of the error terms of an underlying asset pricing model results in a higher return-
risk ratio in the future. The effect is interesting because theory and empirical studies 
8 
 
8 
argue how idiosyncratic volatility should affect asset returns (see, e.g., Merton 1987 or 
Ang, Hodrick, Xing, & Zhang 2009). All things considered, there is plenty of debate on 
these different anomalies but it would be appealing to observe them together from the 
perspective of the modern portfolio theory. 
The common idea behind successful investing is diversification. Allocating inves-
tor’s wealth over different assets results in lower risk based on the low correlation be-
tween asset returns. This means that an investor is not dependent only on the perfor-
mance of one asset. However, a relevant question is, how should this diversification be 
done in an optimal way. Markowitz is generally considered as the founding father of 
modern portfolio theory. Markowitz (1952) argues that one can construct a portfolio 
that minimises variance—considered as the risk of the investment—with respect to the 
desired expected return. The mean-variance optimisation framework provided by mod-
ern portfolio theory is widely used by practitioners as it addresses the main question of 
how an investment portfolio should be diversified. However, the mean-variance optimi-
sation tends to be very sensitive to the expected returns, which raises concerns about the 
accuracy of the predicted returns. This means that investors who are willing to utilise 
strategies based on anomalies in this framework are likely to face the problem of corner 
solutions and instability of the optimal weights. 
As a promising solution to the problem, investors could express their views with re-
spect to a benchmark, namely the market portfolio. One method that allows investors to 
approach the asset allocation problem from this viewpoint is the Black−Litterman mod-
el. This promising approach to enhance mean-variance optimisation was introduced by 
Black and Litterman (1992). The fundamental advantage of the model is that investors 
are able to include their views on asset returns in the optimisation problem, thus making 
the investment decision more flexible. In addition, a tempting feature of the model is 
that the output, i.e., expected returns, can be adapted to the familiar mean-variance 
framework. In this thesis, the Black−Litterman model is used as a tool in order to in-
clude the views based on market anomalies in the portfolio optimisation. There are sev-
eral reasons that make the model superior to the traditional mean-variance method or 
volatility weighted long-short portfolios (see, e.g., Asness et al. 2013). Firstly, the port-
folios given by the Black−Litterman model are realistic in terms of weights. Secondly, it 
provides a flexible way to translate the observed phenomenon into a relative view with-
out a need to express an absolute forecast to each asset. Additionally, the method makes 
it possible to incorporate more than one anomaly simultaneously in the analysis still 
allowing the familiar optimisation procedure.  
The method used here is inspired by Fabozzi, Focardi, and Kolm (2010, 385−393) 
although, within this thesis, the method is taken to the next level. In other words, inves-
tors will be allowed to add different views on different asset classes. Additionally, dif-
ferent weighting methods are allowed when translating anomalies into model specific 
9 
 
9 
views. At first, the assets under each asset class are ranked according to the factor in 
question. For example, with momentum, the ranking is based on 12-month cumulative 
returns. Unlike in Fabozzi et al., the ranked assets are divided into groups (see, e.g., 
Asness et al. 2013) in order to more clearly separate the two extremes. Then, a long-
short portfolio is constructed in which the weight of each asset depends on the group it 
belongs to. Using historical returns of this portfolio the expected return can be estimat-
ed, e.g., as a simple mean. This scalar is then fed into the Black−Litterman framework 
as a view, i.e., this would be the relative return by which the group of the ‘winners’ out-
performs that of the ‘losers’. Finally, when all the parameters for the model are estimat-
ed, rest of the analysis is conducted as a conventional out-of-sample study where the 
performance of such a strategy is analysed through different portfolio performance 
measures. 
1.2 Purpose of the study and research questions 
The primary goal of this research is to examine whether market anomalies can be profit-
ably utilised within portfolio strategies for multiple asset classes. The anomalies used in 
the study, are the value, momentum, long-term reversal, and the idiosyncratic volatility 
anomaly. Based on these market anomalies, a portfolio of stocks, government bonds, 
currencies, and commodities is optimised and studied, out of sample. An interesting 
question to answer is how the portfolio performance is influenced by information given 
by a simple long-short portfolio. 
Different anomalies have been studied a lot and some of them have become com-
monly used risk factors in finance literature, value and momentum being examples. 
However, most of the studies concentrate purely on the stock market, which does not 
benefit those investors who are seeking returns over different asset classes, e.g., institu-
tional investors. On the other hand, finance theory provides sophisticated tools for effi-
cient portfolio allocation. The Black−Litterman model is a good example of such an 
intuitive tool but yet, practical solutions to utilise theoretical findings are quite rare. 
Within this thesis, the purpose is to show that market anomalies can be incorporated 
into portfolio optimisation as views. 
Going into the details, the actual research question is, can investors enhance their 
portfolio performance by including anomaly-based views into the optimisation process. 
More specifically, if an anomaly-based zero-cost portfolio yields a superior risk-return 
ratio, is it profitable to consider a historical mean return as a source of information, and 
thus, as a view in portfolio optimisation. If so, then such a strategy should provide sig-
nificant returns and risk-return ratio compared to benchmarks. Additionally, it is essen-
tial to investigate whether there is a persistent return premium offered by the market 
10 
 
10 
anomalies. Finally, it is worth studying whether it is effective to simultaneously incor-
porate different views from all asset classes into a cross-asset context. 
The comparison of each portfolio strategy is done with several measures. The return-
risk relation is measured with the Sharpe ratio and also its components mean return 
(with statistical significance level) and standard deviation. In order to show whether the 
strategies truly provide superior returns, it is crucial to compare the returns to those ex-
plained by the Fama−French three-factor model. In other words, it is studied whether 
the test portfolios create statistically significant positive alpha. Understanding the chal-
lenges of a multifactor regression in a world of multiple asset classes, the Sharpe ratio 
will be given great importance. That is why robustness of the reported Sharpe ratios is 
assessed with respect to underlying parameters. Observing out-of-sample performance 
ensures a realistic analysis. This means that return of the next period purely depends on 
decisions based on information prior to each portfolio rebalancing. 
1.3 Scope and limitations 
The theoretical framework in this thesis is the modern portfolio theory (MPT), intro-
duced by Markowitz (1952). It provides the concept of mean-variance optimisation and 
supports the general understanding of the benefits of diversification. In this thesis MPT 
is approached from a more practical perspective, namely, the theory provides a concep-
tual framework and tools for dealing with the actual subject. Especially concepts, such 
as, efficient frontier, minimum variance portfolio, tangency portfolio and capital market 
line will be central. There are number of extensions to the basic mean-variance frame-
work. One of these is the Black−Litterman model, the target of which is to provide an 
intuitive way to set expected returns for the optimisation process. 
The scope of the empirical study is to cover a wide range of investable assets from 
different asset classes with a geographical focus. Here, the analysed asset classes are 
stocks, government bonds, currencies, and commodities. Country indices are chosen for 
stocks and government bonds for the interest rate. In order to improve reliability of the 
results, it is essential to investigate a long time period. Hence, a 20-year long time series 
is examined to conduct the empirical study. This requirement naturally sets limitations 
to the data sets. For all assets this long data was not available. This also raises a ques-
tion about selection bias. For example, collecting stock prices of individual companies 
over 20 years would automatically leave out those companies that have gone bankrupt-
cy. Only the strong old enterprises existing over such a long period of time would be 
included in the sample. One important reason for using country level indices has been to 
avoid this bias. 
11 
 
11 
There are several theories that are linked to MPT. On the other hand the MPT is 
based on utility theory and utility function. It is assumed, however, that the reader is 
familiar with the basics of this theory, and therefore certain topics are not discussed. In 
addition, there are some (not that obvious) theoretical concepts that require a brief dis-
cussion before stepping into certain essential topics. For example, market equilibrium 
return will be discussed within the Black−Litterman model, which is tightly linked to 
the Capital Asset Pricing model of Sharpe (1964). Another important theory is associat-
ed with the desire of making future predictions based on historical data. According to 
the Efficient Market Hypothesis, future returns cannot be predicted from historical pric-
es thus making it impossible to earn excessive returns with solutions discussed in this 
paper (Fama 1970). Hence, this theory must not be left without consideration, and 
moreover, some contrary evidence should be presented. 
Another important note, considering this thesis, is related to the concept of the mar-
ket portfolio. Market portfolio is a wide concept that may include many kinds of in-
vestments, e.g., stocks, real estates, bonds, currencies, commodities, derivatives, etc. In 
this thesis, different asset classes, such as, stocks, bonds, currencies, and commodities 
are to be covered but obviously not the whole investable universe. Even this wide selec-
tion of asset classes sets some major challenges when it comes to defining the market 
portfolio. It is not realistic to have reliable time-series data on relative market values of 
these market portfolio constituents. This means that even a limited estimate of the mar-
ket portfolio structure is hard to retrieve. Due to this limitation, the market portfolio will 
be considered as an equal weighted portfolio of all test assets. 
When discussing the anomalies, it should be remembered that there are plenty of dif-
ferent anomalies related to, e.g., fundamentals, calendar events, and technical trading 
rules. The value effect is an example of the first category and the momentum effect be-
longs to the latter one. In addition to these two, long-term reversals as close cousins of 
the momentum and idiosyncratic volatility anomaly are discussed within this thesis. 
Only cross-sectional anomalies are chosen because they are easy to utilise in portfolio 
strategies. For example, calendar anomalies are ignored because they do not tell which 
assets should be preferred to others. 
1.4 Structure of the thesis 
This thesis is divided into three entities: Theoretical foundation and literature review, 
description of the data and empirical methods, and results of the empirical study. Theo-
retical discussion covers essentials of the modern portfolio theory and leads then to a 
more important question of how to incorporate investor’s views into efficient portfolio 
optimisation. More precisely, the Black−Litterman model and its parameters will be 
12 
 
12 
discussed thoroughly from both practical and theoretical point of view. The other half of 
the theoretical discussion covers four different anomalies, namely, value, momentum, 
reversal, and idiosyncratic volatility. These effects will be defined and their theoretical 
importance is justified with the existing literature. Here, the currently known empirical 
findings will be compared and assessed in the light of different theories that explain the 
existence of these anomalies. 
In the empirical part, the research data is studied and analysed. In addition, the meth-
ods used in measuring different anomalies will be discussed and justified. More im-
portantly, it is explained how to construct simple zero-cost portfolios based on these 
anomalies, and moreover, how to turn them into investor’s views. In addition, it will be 
shown how to incorporate these views into the Black−Litterman model and how the 
model parameters are calibrated in this thesis. For those readers who are interested in 
how the study was actually conducted, a programming code for R is provided in the 
appendix of this thesis. 
Finally, results of the empirical study are reported and analysed. This covers analysis 
of both long-only and long-short portfolios and, of course, performance of the optimised 
portfolios. Based on the findings, robustness of the test portfolios will be assessed 
which is highly important in terms of implementing a strategy in real life. 
13 
 
13 
2 THEORETICAL FOUNDATION 
2.1 Modern portfolio theory 
2.1.1 Return and risk of the portfolio 
When discussing investing it is necessary to be familiar with two key-terms: return and 
risk. Return on a stock can be calculated by using either percentage (simple) returns or 
continuously compounded (log) returns. Continuously compounded return at time ! can 
be denoted by !! and percentage returns by !!. The link between these two can be 
shown as 
 !! = ln 1+ !! = ln !! + !!!!!! ! , (1) 
where !! and !! are continuously compounded and percentage returns (respectively) at 
time !, and ! and ! are price and dividend at time ! or at the previous period, ! − 1. 
The natural logarithm of the price (including a possible dividend) at time ! is denoted 
by !!. Continuously compounded returns (or log-returns) are commonly used by re-
searchers because of their statistical properties. As can be seen from the Equation 1, the 
logarithmic return of one period is the remainder of two serial logarithmic prices. This 
procedure enables the returns of different periods to be summed together and thus, 
makes it simple to compute cumulative returns of longer periods of time. 
The advantage of continuously compounded returns over the simple ones is illustrat-
ed in the following two equations: 
 !!,! = !!"!!!!   
 !!,! = (!!" + 1)− 1!!!!  (2) 
While percentage cumulative return is a sum of products of simple returns, the corre-
sponding log-return can be calculated as a sum of shorter period log-returns. Another 
advantage of continuously compounded return is in its simplicity to calculate mean re-
turns. With percentage returns, this is given by a geometric mean, whereas logarithmic 
returns allow the use of an arithmetic average. It is obvious that log-returns provide an 
14 
 
14 
easier approach to return computations, but it should be noticed that the case is slightly 
different when dealing with portfolios. The return on a portfolio is the weighted average 
of the simple return on each asset. However, the same does not apply to log-returns. To 
be precise, they need to be transformed back to simple returns first in order to compute 
weighted sums. (Tsay 2005, 2−5; Rasmussen 2003, 9−11.) 
As a concept, risk is not as unambiguous as return and thus, it needs to be explained. 
It could be easily assumed that investors prefer higher returns. Since returns are often 
assumed symmetrically distributed, the return is most likely equal to the mean return. 
Naturally, only negative changes should be considered as risk (downside risk) from the 
investor’s point of view, but because of the symmetrical distribution both gains and 
losses have equal probability. Hence variance or equivalently its square root, standard 
deviation (or volatility) is generally used as a measure of risk. (Elton et al. 2011, 
46−49.) 
When adding more securities to a portfolio, computing variance becomes less 
straightforward. In addition to individual variances, the correlation between the returns 
also becomes essential. Since returns are dependent on each other, it affects their co-
movement, namely, covariance. If, for example, positive returns occur simultaneously, 
the correlation between these returns is high (maximum +1). In the opposite case where 
two return series are the mirror image of each other, the correlation is −1, which is the 
minimum. As mentioned, covariance between two individual assets describes how their 
returns are moving together. When computing covariance of two assets, the correlation 
is multiplied with the product of standard deviations of the both return series. The fol-
lowing formula presents the role of correlation in covariance: 
 !!" = !!"!!!!, (3) 
where !!" is the covariance between assets ! and !, !!" is the correlation and ! denotes 
the standard deviation. (Elton et al. 2011, 54.) 
In order to analyse the risk of all the securities together, a formula for the variance of 
the portfolio needs to be constructed. To conclude what is already discussed above, it 
can be said that the variance of the portfolio consists of both variance of each asset and 
covariance terms. However, it can be shown that covariance of an asset with itself 
equals the variance of the asset. Hence, variance of the portfolio can be expressed as 
 !!! = !!!!!!" !!!!!!!!! , (4) 
where the variance of the portfolio !!! is the weighted sum of the covariance (and vari-
ance) coefficients. (Markowitz 1952, 81.) 
15 
 
15 
2.1.2 Asset diversification and the efficient frontier 
The concept of efficient portfolios is based on the idea that investors are trying to max-
imise the expected return on a portfolio, while avoiding variance, which can be seen as 
a risk that should be minimised (Markowitz 1952, 77). When more assets are added to 
the portfolio the variance decreases as a consequence of the smaller correlation. As dis-
cussed above, the variance of the portfolio can be divided into two components: vari-
ances of individual assets and covariance terms. The first, the individual variances, can 
be diversified away but part of the risk remains; this is the average covariance, and 
eventually the market’s volatility. (Elton et al. 2011, 59−60; Rasmussen 2005, 90−91.) 
The essential outcome of Markowitz’s (1952) portfolio theory is that it gives a tool 
for constructing portfolios in an efficient way, i.e., it provides a way to determine mean-
variance efficient portfolios. This theory allows investors to utilise two different ap-
proaches (Markowitz 1952, 82): 
1. Minimum risk with desired level of expected return, or 
2. the highest possible expected return with minimum risk. 
For each level of risk (volatility) given there is a maximum expected return. Alternative-
ly, for each desired level of return there is a minimum achievable volatility. Finally, this 
results in the efficient frontier, on which each point is an efficient portfolio. 
 
Figure 1 The Efficient Frontier 
Each different asset combination produces a different result in the return-volatility 
space. However, no portfolio can sit on the left side of the frontier shown in Figure 11. 
Even if the final placement of a portfolio is on the frontier line, this does not necessarily 
                                                
1
 There are also other important terms related to the efficient frontier, such as a tangency portfolio, risk-
free return, and a capital market line. These concepts will be discussed later in this chapter.  
16 
 
16 
mean that it is efficient. By definition, the efficient frontier is a range from the mini-
mum-variance portfolio ! to the portfolio ! with the maximum-variance. Once these 
two portfolios are found the entire efficient frontier can be derived as their linear com-
bination. (Rasmussen 2003, 111−114.) 
The shape of the efficient frontier indicates how much one can benefit from diversi-
fication. This means that correlation between the assets determines the shape of the 
curve of the portfolio possibilities. This can be understood by observing a portfolio of 
two (risky) assets, where short sales are not allowed. In this case, the curve will be 
drawn between two extreme portfolios, both of which consist of either asset A or B. If 
the correlation between these two assets equals one, the assets are alike, and thus, the 
efficient frontier is a straight line between the assets A and B. Accordingly, one cannot 
benefit from diversification. Another extreme case is when the correlation is perfectly 
negative (! = −1). In this theoretical situation, investors would be able to find the op-
timal weights for both assets, which would result in a portfolio of zero risk. In this case, 
the curve would actually be a line from A to the expected returns axis and from there to 
asset B. In real life, however, the correlation is almost always something between zero 
and one. This leads to the portfolio possibilities curve being more likely to resemble 
Figure 1. (Elton et al. 2010, 68−77.) 
When adding more securities to the portfolio, investors obtain more diversification 
opportunities. The efficient frontier still looks the same but it shifts to the left. This is 
very understandable. As mentioned before, the risk becomes lower when more securi-
ties are added to the portfolio. Thus, any combination of a few assets cannot be as effi-
cient as the optimal combination of a larger number of assets. Most likely none of the 
securities alone lie on the efficient frontier but the optimal combination of them offers 
the mean-variance efficient portfolio. Again, no higher expected return can be obtained 
for a certain risk level and, respectively, no lower risk can be obtained for the desired 
expected return. (Rasmussen 2003, 113, 125−126.) 
2.1.3 Risk-free asset and market portfolio 
In the above, the discussion was limited to cases of portfolios with risky assets only. 
Hence, their returns were uncertain, and in order to compute the riskiness of the entire 
portfolio, one had to be aware of the correlation among the returns. Usually, however, 
there is a risk-free asset available. Basically the idea is that one can borrow and lend at a 
risk-free interest rate. Lending here means that investors obtain, e.g., by buying a short-
term government bond, a risk-free rate of return. On the other hand, there is the possi-
bility to borrow at a risk-free rate, which gives investors an opportunity to purchase 
17 
 
17 
more securities than what they can afford. Since the asset is risk-free, it is clear that the 
return is certain and thus, the standard deviation is zero. (Elton et al. 2011, 84−85.) 
The risk-free asset changes the problem of constructing efficient portfolios. In this 
case the task is to find the optimal allocation of two assets, namely, the risk-free asset 
and a mean-variance efficient portfolio. As the risk-free return must have a standard 
deviation equal to zero, it can be proven that the covariance coefficient (see equation 
2.3) will equal zero. Take for example, a given portfolio B (see Figure 1) from the effi-
cient frontier. If there is a risk-free asset available, the risk of the whole portfolio results 
only from portfolio B: 
 σ! = X!! , (5) 
where ! denotes the proportion of portfolio B in the mixed portfolio and !!is the stand-
ard deviation of portfolio B. 
As highlighted previously, the return on a portfolio is the weighted average of returns 
on individual assets. The weights in the portfolio are assumed to sum up to one, which 
implies that the proportion invested in the risk-free asset, must be 1− !. By rearranging 
the previous equation, the weight for portfolio B can be solved as X = σ!/!!. Now, the 
expected return on the portfolio can be expressed as follows: 
 ![!!] = !! + ![!!]− !!!! !!!, (6) 
where !! is the risk-free rate, ![!!] is the expected return on portfolio B and !! is the 
standard deviation of the whole portfolio. (Elton et al. 2011, 85−86.) 
Analysis of the equation above reveals that return on the portfolio is a linear function 
of standard deviation of the portfolio. The risk-free rate is a constant and the ratio of 
expected excess returns and standard deviation of portfolio B is the slope. This slope is 
also known as the Sharpe ratio, which changes along the efficient frontier and reaches 
the maximum above the minimum-variance portfolio (Rasmussen 2005, 198). Earlier 
efficiency was defined as a relation between return and risk. In this sense maximising 
the Sharpe ratio seems rational. If riskless lending and borrowing is possible, a rational 
investor should maximise this slope and finally find the tangent drawn from !! to the 
efficient frontier. The portfolio, at the point where the tangent touches the frontier, is 
called the tangency portfolio or maximum Sharpe portfolio (see portfolio B in Figure 1). 
(Elton et al. 2011, 86−87, 100−101.) 
If the whole asset universe is considered, the tangency portfolio is actually the same 
as the market portfolio. Now the weight for each asset is based on their market capitali-
sation. In fact, this portfolio will serve as a reference point offering the market equilib-
rium returns when introducing the Black−Litterman model. The tangent between the 
18 
 
18 
risk-free asset and the market portfolio is known as the capital market line (CML). 
Eventually, investors are able to choose any point on the CML by mixing the risk-free 
asset and the tangent portfolio. It is clear that the other portfolios on the efficient fron-
tier are not efficient anymore because investors are now able to obtain higher expected 
return without taking any additional risk. (Elton et al. 2011, 282−283; Sharpe 1964; 
Lintner 1965.) 
2.2 Mean-variance optimisation in practice 
2.2.1 Description of the optimisation process 
In the beginning of this chapter, the main features of modern portfolio theory were in-
troduced. When implementing the theory in real world portfolio management there are 
several phases that are essential in the process. First of all, the basic inputs for the 
mean-variance model need to be well determined. Practitioners need forecasts for the 
expected returns on each asset in the portfolio. There are a number of different predict-
ing models for the returns available, some of which will be discussed in this thesis. Es-
timates for volatility and correlation terms are also crucial in order to obtain accurate 
results from the optimisation. In addition, usually the portfolio manager has some limi-
tations in terms of making investment decisions. These limitations need to be added as 
constraints to the optimisation software. (Fabozzi, Focardi & Kolm 2010, 316.) The 
investment process in its entirety can be seen in following figure: 
 
Figure 2 The MPT Investment Process (Fabozzi, Gupta, Markowitz 2002, 8) 
Expected(
Return(Model(
Constraints(On(
Por6olio(Choice(
Vola9lity(&(
Correla9on(
Es9mates(
Investor(Objec9ves(
PORTFOLIO(
OPTIMIZATION(
RiskHReturn(Efficient(
Fron9er(
Op9mal(Por6olio(
19 
 
19 
One approach to expected returns is to consider historical mean returns as a predic-
tion for the future returns. If the returns are assumed to be distributed normally, the pre-
diction holds. Normality of returns is a good approximation indeed, but does not really 
reflect the real world. From real life observations, it can be seen that return distribution 
often has excess kurtosis and positive skewness. The first means that the distribution 
has fat tails, which implies that extreme values have higher probability than in the nor-
mal distribution. The latter means that there is more probability mass on the right tail of 
the distribution. These statistics can vary depending on whether daily or monthly returns 
are used, or whether the analysis includes returns on individual securities or, e.g., on 
indices. (Tsay 2005, 14−15, 17−19.) Another important point related to the normality of 
returns has to do with the length of the return series. If returns from a long period of 
time are observed, the historical mean is a more precise predictor of the future return. 
However, the more years taken into the analysis, the more changes in economic condi-
tions become involved. Including these in the calculations may be irrelevant. (Elton et 
al. 2011, 238.) 
Due to the facts deliberated above, it can be argued that the historical mean is a fairly 
poor predictor of the future return. Financial literature offers many different models for 
return forecasting, e.g., the capital asset pricing model, multifactor models, and different 
time series models. This thesis proposes the Black−Litterman approach, in which the 
expected returns are calculated with the market equilibrium returns and the views of 
analysts. The market equilibrium returns are derived from capital asset pricing theory, 
and the views are dependent on the analyst. In this paper, a simple long-short trading 
strategy is introduced to create factors that can be further used to impose analysts’ 
views. 
The variance of the portfolio is another key component in the mean-variance optimi-
sation. In this analysis the portfolio manager has to deal with both variance of individual 
stocks and correlation between the assets or asset classes. Again, one approach is to 
compute historical sample variance and correlation. From statistical perspective, the 
longer the sample period the more accurate the prediction. Nevertheless, the problem is 
that the economic conditions change over time, which changes the characteristics of the 
returns. (Elton et al. 2011, 90.) 
Before running the optimisation, investors may have to add constraints. In the sim-
plest case, the optimisation is subject to a constraint that the portfolio should be fully 
invested. In addition, they may want to achieve a certain expected return on the portfo-
lio, or they may not be able to sell the securities short. Restrictions may be comprised of 
institutional requirements or investors’ objectives. Fabozzi et al. (2010) divide the pos-
sible constraints into linear and quadratic, nonlinear, combinatorial, or integer con-
straints. Linear and quadratic constraints are the most common ones and they can be 
efficiently solved by using straightforward quadratic optimisation algorithms. More 
20 
 
20 
complex constraints require more sophisticated algorithms, which obviously makes the 
optimisation less practical. (Fabozzi et al. 2010, 327−333.) 
2.2.2 Computing mean-variance efficient portfolios 
Up to this point, the different inputs concerning mean-variance framework have been 
discussed, but not so much the quadratic optimisation itself; therefore this problem will 
be approached using matrix notations. In addition, general solutions for minimum vari-
ance portfolio and tangency portfolio will be given. In the portfolio, there are N securi-
ties, the optimal weights of which are not yet known. Weights are denoted by !×1 vec-
tor !,2 the elements of which are the weights of individual securities: !! = (!!,!!,… ,!!). By default, portfolios are always assumed to be fully invested. 
This can be expressed as follows: 
 !!! = 1 (7) 
In the formula, the transposed vector of weights, !! is multiplied by !×1 unit vector. 
In the beginning of this thesis, covariance was presented as an important part of 
computing variance of the portfolio. When covariance for each security pair and vari-
ance for each security is computed, an !×! covariance matrix can be constructed: 
 Σ = !!! ⋯ !!!⋮ ⋱ ⋮!!! ⋯ !!!  (8) 
It is already known that the covariance of a security with itself is equal to its variance, 
i.e., !!! = !!!. Hence, the values on the diagonal are variances. By utilising the covari-
ance matrix, the variance of the whole portfolio can finally be computed as: 
 !!! = !′Σ! (9) 
It was already pointed out that the expected return on a portfolio is the weighted av-
erage of simple returns on individual assets. This, denoted by !!, can be calculated as a 
product of the transposed vector of weights, !! and the vector of expected returns,!! as 
follows: 
                                                
2 
To avoid misunderstandings vectors are denoted by highlighted and scalars by normal letters (or num-
bers). By default, all vectors are written in a horizontal direction, i.e., as !"1. Transpose of a vector 
would be indicated with an apostrophe on the right corner. 
21 
 
21 
 !! = !!! (10) 
(Fabozzi et al. 2010, 317.) 
A usual setup in the mean-variance problem is that the variance is minimised, subject 
to some constraints. As already discussed, there is always at least one constraint, the 
budget restriction, according to which the weights must sum up to one. If the optimal 
allocation should reach a certain expected return on the portfolio, it is common to add 
this as a constraint as well. Now the optimisation problem can be expressed as follows: 
 min! 12!′Σ!  
 s.t. !!! = !!  
 !!! = 1  (11) 
Finally, by using Lagrange multipliers, the problem can be interpreted in the following 
linear form: 
 L(!,!,!) != 12!!Σ!+ ! !!!− !! + ! !!!− 1 !, (12) 
where ! and ! are the Lagrange multipliers that allow the constraints to be added to the 
same equation with the objective function. The problem can be solved by taking partial 
derivatives with respect to the variables, and then by solving the first-order conditions. 
This kind of analytical solution is possible only when using equality constraints. In the 
previous, equation the weights can result in negative values since the individual weights 
are not constrained. In practice this means that short selling is allowed. If they need to 
be ruled out, inequality constraints come into question. For example, the long-only rule 
(!! ≥ 0, ! = 1,… ,!) is an inequality constraint that requires numerical optimisation. 
It is not relevant to go any deeper in the mathematics but instead the general solu-
tions for the global minimum-variance and tangency portfolio will be given. Both of 
these solutions can be derived from the function presented above. One should note, that 
in tangency portfolio the expected excess returns (!− !!!) are taken in consideration. 
The solution for the global minimum variance portfolio is: 
 ! = Σ!!!!′Σ!!!, (13) 
and correspondingly, tangency portfolio acquires the following expression: 
22 
 
22 
 ! = Σ!!(!− !!!)!′Σ!!(!− !!!)!!, (14) 
where Σ!! is an inversed covariance matrix. (Fabozzi 2010, 318−319, 323.) 
2.2.3 Issues on the inputs of mean-variance framework 
Jobson and Korkie (1980) have studied the key inputs, mean return, and variance, in 
Markowitz’s mean-variance framework. They tested the normality assumption of re-
turns with different sample sizes by comparing theoretical values and simulated (Monte 
Carlo) values with each other. The results prove that the estimation bias is the ratio of 
the number of assets and sample size, i.e., !/!. Variance especially, in the portfolio did 
not seem to be comparable with the simulated value even when the sample size grew to 
1,000. As already mentioned, such a large number of returns is not relevant, and there-
fore, it will make the use of the sample values less accurate. 
Another interesting finding was that quality of the estimators clearly improved when 
mean excess returns were relatively high compared to the covariance terms among the 
assets. Correspondingly, small element values in the vector of expected (mean) returns 
together with relatively high off-diagonal elements in the sample covariance matrix re-
sulted in weak estimates for the mean-variance analysis. However, t securities with the-
se “good” properties tend to acquire unreasonably large weights in the optimisation pro-
cess, which leads to errors in portfolio weights (Michaud 1989). 
Another problem is that the optimal portfolio weights are very unstable when the in-
puts change. The main reason for this is the ill-conditioned covariance matrix, which 
appears when using historical data. As an inversed covariance matrix is used in the op-
timisation process, ill-conditioning3 of the matrix makes the results extremely unstable. 
In some cases, when the number of assets exceeds the sample size, the covariance ma-
trix is not even invertible (Ledoit & Wolf 2004). In the worst case, the mean-variance 
optimiser becomes an “estimation-error maximiser” rather than anything else. Actually, 
it often emerges that equal allocation outperforms mean-variance efficient portfolios. 
(Michaud 1989.) 
                                                
3
 Ill-conditioning of a matrix is related to its condition number. Condition number determines how accu-
rate and stable results an inversed matrix produces in linear problem solving. 
23 
 
23 
2.3 The Black−Litterman Model 
2.3.1 Role in the asset allocation problem 
Black and Litterman were working at Goldman Sachs at the time when they published 
the first paper on the model in the company’s internal issue in 1990. In 1992 the method 
entered the academic world as their paper, Global Portfolio Optimization, was pub-
lished in Financial Analysts Journal. In this context, the importance of Black’s academ-
ic publications is worth mentioning. Among his other work with asset pricing theory, 
the most famous achievement is the Black−Scholes option pricing equation published 
with Scholes (1973). Motivation of Black and Litterman when creating the model was 
driven by the problem that quantitative asset allocation models were very complex, and 
furthermore, they tended to result in unstable portfolios with remarkable short positions. 
A reason behind this is that portfolio optimisation models are highly sensitive to ex-
pected returns. Even long-only rule does not solve the problem since it leads to solu-
tions where small-cap markets have extremely large weights, and on the other hand, 
many securities result in almost zero weight. Accordingly, this is an issue that would 
make such a model practically useless. 
The Black−Litterman model approaches the allocation problem from the perspective 
of expected returns suggesting that the global CAPM equilibrium (see e.g., Sharpe 
1964; 1974; Lintner 1965) should be used as a reference point when estimating the ex-
pected returns. The market portfolio represents the equilibrium where all investors have 
neutral views and which equilibrates the global demand and supply. This equilibrium 
consists only of long positions, which makes it appealing as a benchmark. Eventually, 
investors are allowed to tilt these market equilibrium weights towards their views con-
trolled by the level of confidence. Hence, the excess returns are expected to be provided 
by the public information of the market and an insight into the companies or markets, 
which investors may believe they have. In other words, the model combines two differ-
ent sources of information. Finally, the familiar mean-variance model is fed by the out-
put of the model, namely, the vector of expected returns conditional on investors’ 
views. (Black & Litterman 1992.) 
The ultimate purpose of the model is to produce expected returns, which behave well 
in the optimisation framework. As a process the Black−Litterman model is relatively 
straightforward despite the mathematical presentation. Shortly described, the model 
consists of three entities: Vector of implied equilibrium returns, investor’s views, and 
uncertainty of views. The first one requires an estimate of investor’s risk aversion, a 
covariance matrix of the returns, and the market capitalisation weights. Since the ap-
proach considers both equilibrium returns and views uncertainty, they are handled with 
24 
 
24 
probability distributions (Black & Litterman 1992). The following figure illustrates the 
process and illustrates the role of different inputs: 
 
Figure 3 Combining market equilibrium returns and investor’s views (Idzorek 
2005) 
As presented in the figure, the means and variances of the distributions are the inputs 
shortly introduced above. Some of them are very intuitive and simple to calculate 
whereas some are more theoretical. For example, defining the market portfolio may not 
be easy since it is supposed to cover the whole investable universe. Additionally, quan-
tifying uncertainty of views requires more theoretical approach. 
After defining all the parameters a complete expression of the mean of the posterior 
return distribution, i.e., a combined return vector, acquires the following formula: 
 ! ! = τΣ !! + !!Ω!!! !! τΣ !!!+ !!Ω!!! !, (15) 
where Σ is a covariance matrix of excess returns, τ is a scaling factor, ! is a matrix to 
weight investor’s views, Ω is a diagonal covariance matrix of the views, and ! denotes 
the actual views (Idzorek 2005). The components and composition of the formula will 
be discussed in the following sub sections. 
Risk%Aversion%
Coefficient%
%
Market%Capitalisa4on%
Weights%
Uncertainty%of%Views%Covariance%Matrix% Views%
Implied%Equilibrium%Return%Vector%
Prior%Equilibrium%Distribu4on% View%Distribu4on%
Combined%Distribu4on%
25 
 
25 
2.3.2 The equilibrium portfolio and implied equilibrium returns 
The basic principles of the market portfolio, considered as a benchmark in this context, 
are already discussed within this thesis. It was explained that every investor should pick 
the portfolio from the efficient frontier, which maximises the Sharpe ratio. Assuming 
that investors are able to borrow and lend at a risk-free rate of interest, all with equal 
terms, they should be able to move from this point along the capital market line by mix-
ing the maximum Sharpe ratio portfolio and the risk-free asset in their portfolios. It is 
also assumed that all investors have similar expectations concerning the expected re-
turns, variance, and correlation structure among the assets in the portfolio. Even though 
these assumptions are not realistic, they set the frames for the theory. Under these cir-
cumstances all investors finds the same market portfolio. (Sharpe 1964.) 
When discussing the equilibrium, it can be understood as a portfolio that equilibrates 
the global market when investors have equal views, i.e., the global market portfolio in 
which each asset is weighted based on its capitalisation. To be precise, the market port-
folio contains the whole investable universe including equities, bonds, currencies, etc., 
which is why defining the market portfolio may be challenging if not impossible. Addi-
tionally, in the original paper (Black & Litterman 1992), the equilibrium portfolio is 
partly currency hedged based on universal hedging constant. According to so-called 
“Siegel’s paradox”, investors can benefit from bearing part of the currency risk. This 
ratio, the universal hedging constant, is discussed in Black (1989). 
Regarding the global market portfolio with given market capitalisation weights, the 
expected returns can be found implicitly utilising Markowitz’s mean-variance frame-
work. The idea is introduced in Sharpe (1974) where the essential finding relies on the 
fact that the market portfolio already holds expectations of the investors, which can be 
imputed from asset weights. In the context of the Black−Litterman model, the implied 
market equilibrium returns are calculated using the following equation for a reverse 
optimisation: 
 ! = !Σ!!"#!, (16) 
where the implied equilibrium returns, !, is calculated as a product of a risk-aversion 
parameter, !, the covariance matrix, Σ, and the market capitalisation weights denoted by !!"#. The term ‘reversed’ here means that the vector of returns given by the formula 
provides an optimal portfolio when plugged into the Markowitz framework without 
constraints. 
The other multipliers have been discussed earlier within this thesis except the risk-
aversion parameter. This can be solved as ! = ! !! − !! /σ!! , which defines the 
amount of additional risk that investors are willing to take when the expected excess 
26 
 
26 
return increases. Note that the formula of the risk-aversion is almost similar to the one 
of Sharpe ratio; instead of variance, the market risk premium is divided by volatility of 
the market in case of the Sharpe ratio. (Rachev et al. 2008, 96, 143.) 
The vector of implied equilibrium returns represents the prior estimate of the mean 
returns of a multivariate return distribution. To clarify the idea of the prior equilibrium 
distribution, normally distributed expected returns could be used as a starting point. In 
the portfolio world, a vector of expected returns, ! ! , can be assumed to follow a mul-
tivariate normal distribution with mean, !, and covariance, Σ: 
 ! ! ∼ ! !, Σ  (17) 
However, the market is not in the equilibrium all the time, which is why an estimate is 
needed, namely, the vector, !, given by the reversed optimisation. This estimate is a 
combination of the actual means and errors, ε, which are normally distributed with 
mean, 0, and variance, Σ!. Since the estimate is affected by the views given by the mar-
ket, it should be considered more reliable than simple mean returns and, thus, Σ! < Σ. 
The proportionality of the covariance matrices of the estimate and the returns (Σ!and Σ) 
is indicated by τ. This parameter is also understood as uncertainty of the estimate. Final-
ly, the prior distribution is expressed as 
 ! ∼ ! !, τΣ  . (18) 
The ’tau’ coefficient is rather theoretical and there are several different approaches sug-
gested in various papers. This problem will be discussed later within this section. (Wal-
ters 2011; Cheung 2010.) 
2.3.3 Incorporating investors’ views 
Allocating the funds according to the market portfolio can be interpreted as having no 
views at all. However, usually investors do have some kind of insight into the assets 
they consider investing in. The Black−Litterman model introduces two different ap-
proaches to incorporate these views, namely, the investors may have either absolute or 
relative views on the performance of the assets. An absolute view means that the inves-
tor has a specific estimate for the return on a particular asset. Accordingly, a relative 
view stands for a usual case when, e.g., one asset class is expected to outper-
form/underperform another by X per cent. 
In the mathematical expression, the views are added using matrix notations. The as-
sets that the views apply to are defined by matrix P with dimensions, k!×!n. In the ma-
trix, k represents the number of views, whereas n is the number of assets. Both absolute 
27 
 
27 
and relative views are included in the same matrix with a small difference. Assume 
three assets, A, B, and C, in a portfolio. The investor may believe, for example, that the 
asset A will yield 3 % and the asset B will outperform the asset C by 1.5 %. In this case, 
the matrix, P is composed as 
 ! = 1 0 00 0.5 −0.5  . (19) 
Since there are three assets and two views, the dimensions are 2!×!3. An essential rule is 
that when adding an absolute view, the element equivalent to the particular view and 
asset has to equal one. Accordingly, in case of relative view, the entire row needs to 
sum to zero. There are also alternative ways to approach the relative view setting. One 
possibility is to give equal weights to the outperforming assets, and correspondingly, to 
the underperforming ones as done in the example above. This means that on each row, 
the outperforming assets get a value of one divided by the number of the outperforming 
securities related to a particular view. The same principle applies to the underperform-
ing securities (see Satchell & Scowcroft 2000). However, Idzorek (2005) prefers so-
called market capitalisation scheme where the assets related to the view in question are 
weighted as market capi / market capview total, where ‘view total’ means the total market 
capitalisation of either outperforming or underperforming assets. Again, the weights are 
positive or negative if the corresponding assets are expected to outperform or underper-
form, respectively. The intuition behind this approach is that now the final changes in 
portfolio weights will be proportional to market capitalisations of the assets. 
As presented above, the matrix, P weights the views across the assets. The expected 
excess returns, accordingly, are included in a k!×!1 vector !, where k represents the 
number of views. In the example, given above, it was expected that the asset A will 
yield 3 % and the asset B will outperform the asset C by 1.5 %. In this case, the views 
would be presented as 
 ! = 3%1.5%  . (20) 
It should be noted that each asset is not necessarily included in the ! vector but the en-
tries are given by the views instead, whether they are absolute or relative. In the exam-
ple there are only two views but, however, assets B and C are both affected by the se-
cond view. 
In addition to the views themselves, investors are asked to set the uncertainty of the 
views that is given by a covariance matrix of the views. It is assumed that the views are 
independent, and therefore, uncorrelated. This means that the covariance matrix is diag-
onal consisting of the variances of the views: 
28 
 
28 
 Ω = !! 00 !!  (21) 
Presumably, the values for this matrix are difficult to estimate intuitively. Black and 
Litterman (1992) do not provide any solution for the matrix either, which makes it ra-
ther problematic. However, there are several ways to approach this problem, which will 
be discussed next. The expressed views and uncertainty of the views finally define the 
distribution of the views. In an asset space, views are distributed as 
 ! ∼ ! P!!!, !′Ω!!! !!  . (22) 
Considering the views only, the distribution would be determined by a mean, ! and 
variance, Ω. The P!matrix, however, translates the views in accordance with the assets. 
(Walters 2011.) 
2.3.4 Uncertainty of the views and tau coefficient 
Both an uncertainty matrix Ω and a scalar τ are in the model to cope with confidence 
issues. The uncertainty matrix reflects variances of the views whereas τ is understood as 
a confidence of the prior equilibrium. The shrinkage between the prior distribution and 
views is a function of these two factors, which requires that they should be considered 
together. They are also highly abstract concepts, which makes them difficult to calcu-
late. In general, the principle is that a higher confidence is related to a greater departure 
from the market equilibrium portfolio. 
As already discussed, an uncertainty matrix of views, Ω is a k!×!k diagonal matrix the 
elements of which consist of variances of the error terms of individual views. This 
means that value zero can be interpreted as 100 % confidence on a particular view and 
vice versa. The views can be written as !+ !, where the error term vector is independ-
ent, random, and it follows a normal distribution with mean zero and a covariance ma-
trix, Ω. One way to calculate the diagonal values of Ω is to start with variance of the 
view portfolios. For each view (k = 1…K), the variance of the view portfolio is p!Σp′!, 
where p! is a row (vector) of the P matrix subject to a view k and Σ is a covariance ma-
trix of excess returns. He and Litterman (1999) suggest a link between Ω and τ so that 
variance of a view portfolio equals !!/τ. According to this approach, the elements of Ω 
acquire the following expression: 
29 
 
29 
 Ω = (!!! Σ!!)τ 0 00 ⋱ 00 0 (!!! Σ!!)τ  (23) 
The same expression can be also written as Ω = !"#$ ! τΣ !′ . This approach miti-
gates the estimation process since it makes the scalar τ irrelevant regarding the final 
combined vector of excess returns. It is also the most used method in literature. (Idzorek 
2005.) 
Another fairly intuitive way to quantify the variance of the views is to utilise confi-
dence intervals. Assume an investor who has estimated that a mean return of an asset is 
5%. The investor is 68% confident that the mean falls onto a range of 4 %−6 %. Now, it 
is possible to calculate the standard deviation of this view by using an assumption of the 
normal distribution. In the standardized normal distribution, probability of 68 % corre-
sponds 0.47 standard deviations. Considering both tails of the distribution, this falls 
within 1 standard deviation of the mean and thus, variance for the view can be translated 
into (12 %). 
In addition, there are some other alternative ways to estimate the variance of the error 
term. When a factor model is used to estimate mean excess returns a natural method is 
to calculate the variance of the error term of the regression. It is typically assumed that 
residuals are independent and identically distributed (i.i.d.), which leads to the fact that 
only diagonal values (variances) will be included in the view uncertainty matrix. On the 
other hand, Idzorek (2005) suggests that there would be also other factors affecting in-
vestors’ confidence on their views. He calculated so-called Implied Confidence Levels 
by combining market capitalised weights and unconstrained weights based on 100% 
confidence and user specified confidence levels, i.e., !!"#, !!""%, and !, respectively. 
This approach utilises the weight differences of these three different weight vectors to 
estimate an implied confidence level. This is done by dividing individual values of ! − !!"#  by the corresponding differences from !!""% − !!"# . (Walters 2011.) 
In connection with a distribution of the market equilibrium returns, the variance of 
the prior mean is proportional to the covariance matrix of the asset returns. This propor-
tionality is expressed with the constant, τ, and thus the covariance matrix of the estimate 
as τΣ (see equation 18). Hence, τ may be understood to indicate the uncertainty of the 
prior distribution given by CAPM (He & Litterman 1999). In Black and Litterman 
(1992), the constant is only defined to be close to zero meaning that the expected excess 
returns are fairly stable. However, there has been a lot of discussion on a reasonable 
value of the ‘tau’ coefficient. For example, Satchell and Scowcroft (2000) suggest that τ 
should be set to one because investors’ return forecasts and equilibrium excess returns 
are equal on average. Michaud, Esch, and Michaud (2013) acknowledge the combined 
effect of τ and Ω by adjusting τ so that the model results in investable portfolios. The 
30 
 
30 
smaller the scalar is the more the final portfolio approaches the benchmark portfolio, 
which is constrained by definition. Following the method of He and Litterman (1999), 
however, leads to a situation where the constant becomes completely irrelevant because Ω is proportional to τ (see equation 23). 
When τ is relevant in the final Black−Litterman equation one commonly used meth-
od is to approach the scalar from a statistical viewpoint. Now, the sample size, T (length 
of return series), and possibly the number of assets, N can be used in the estimation. 
Thus, the scalar τ can be calculated either as 
 τ = 1!!!!  
or τ = 1! − !!!, (24) 
where the former formula is so called the maximum likelihood estimator, and the latter 
one is the best quadratic unbiased estimator. This approach is based on statistics and it 
is commonly used when a covariance matrix is estimated from a sample of return. 
Since the variance of the prior mean is assumed to be proportional to the covariance 
matrix of the returns as τΣ, the scalar can be also calibrated with respect to a confidence 
interval. With given confidence level, estimated mean, and observed standard deviation, 
it is possible to define a confidence interval for the most likely mean values. As a con-
sequence of a higher standard deviation, the confidence interval becomes wider as τ 
increases. This allows setting the scalar so that the interval becomes reasonable and a 
good prior estimate of the mean. (Walters 2011.) 
2.4 Market anomalies 
2.4.1 Definition and persistence on the markets 
When markets are efficient all relevant information should be included in asset prices, 
which should make it ineffectual to pursue excess returns consistently. Studies on mar-
ket efficiency rely strongly on the work of Fama (1970) where he introduces three 
forms of efficiency: weak form, semi-strong form, and strong form. Weak form of mar-
ket efficiency states that historical price or return patterns do not provide any relevant 
information for investors profit wise. Accordingly, semi-strong form realises when pub-
licly available information like fundamental factors cannot be utilised in order to make 
31 
 
31 
excess returns. The strictest requirement for efficient markets is set by the strong form 
claiming both either private and public information, held by any group of investors, 
should be included in asset prices. Finding evidence against the efficient market hy-
pothesis may be a sign of inefficiency although, usually the underlying asset pricing 
model is the one that should be put in test. 
Generally, financial markets can be considered efficient and nowadays even more so 
as computerised trading makes arbitrage opportunities to disappear before they are even 
noticed. However, sometimes there are return patterns appearing on the market that vio-
late the efficient market hypothesis and cannot be explained by asset pricing models, 
like CAPM. Such anomalous returns can be based on historical performance, fundamen-
tal characteristics, or some other technical trading rules. By the nature of the anomalies, 
Keim (2008) categorises them into cross-sectional and time series anomalies. All the 
anomalies discussed within this thesis belong to the first group but, of course, momen-
tum and reversal effects are based on time series characteristics. 
Even though, anomalies seem to provide superior trading opportunities there is one 
common problem. Once they are studied and become publicly known they usually cease 
to exist, i.e., they are quickly arbitraged away. This also supports the efficient market 
theorem. However, some of the anomalies have been reported to show strong evidence 
still after a long time since they were found. Rather than a persistent inefficiency, this 
should most likely be considered as a consequence of an inadequate asset pricing model 
used as a benchmark. After the basic CAPM some of the anomalies have been included 
in better explaining asset pricing models. For example, Fama and French (1992; 1993) 
introduce a three-factor model that, in addition to market, explains asset returns with 
size and value factors. This means that small cap stocks with high book-to-market ratio 
(value) outperform the market. Additionally, Carhart (1997) improves the model by 
adding a fourth factor, momentum to explain the continuing rise of the past winners. In 
order to investigate the returns provided by the strategies suggested within this thesis, 
the three-factor model is used as a benchmark. (Keim 2008; Schwert 2003.) 
2.4.2 Value effect 
A phenomenon where securities with high value outperform those with lower value is 
called value effect. Stocks may be classified as value and glamorous stocks or value and 
growth stocks. A theoretical explanation to the value effect is that assets with a higher 
risk of financial distress also provide a higher expected rate of return (Fama & French 
1993). Typically, value is assessed with respect to market valuation, which reveals 
whether an asset is under- or overvalued. There has been a number of different studies 
investigating the relation between value, calculated with different measures, and cross-
32 
 
32 
sectional returns. First, Stattman (1980) reports a positive relation between cross-
sectional US stock returns and book-to-market (BE/ME) ratio. Book-to-market ratio is 
also considered as value measure in Fama and French (1992), however, they also test if 
earnings-to-price (E/P) ratio explains average cross-sectional returns. The results show 
that the positive relation between positive E/P ratio and average returns is mostly due to 
the fact that high E/P firms often have a high book-to-market equity ratio, i.e., there 
seem to be multicollinearity issues in the regression. In addition to BE/ME and E/P, 
Babameto and Harris (2008) consider also dividend yield and cash-to-price ratio when 
constructing value portfolios. Nevertheless, book-to-market ratio appears to be the most 
used measure to characterise value in literature. 
All the value indicators, discussed above, are accounting based measures that are ap-
plicable with stocks. However, the value effect is reported to exist in other asset classes 
as well, which means that alternative value measures are required. Asness et al. (2013) 
estimate value for commodities, currencies, and bonds as a negative five-year return. 
This method is supported by Gerakos and Linnainmaa (2012) who suggest that BE/ME 
could be replaced by market value when calculating a value measure for stocks. They 
argue that book and market values covary and that an increment in a book value yields 
even a higher increment in the market value. As a consequence, the BE/ME ratio starts 
to decrease. In addition, cross-sectional Fama−MacBeth4 regressions show that the 
market value of equity captures both past stock returns and net issuance, which makes a 
negative five-year return a good estimate for value. Also Fama and French (1996) report 
positive correlation between book-to-market measure and negative five-year return. 
Asness et al. (2013) apply this method to different asset classes with slight nuances. 
More specifically, they calculate the spot price five years ago as an average of the spot 
price from 5.5 to 4.5 years ago. Finally, the negative five-year return becomes !" !!!! − !! , where index ! represents years. For each asset class, the basic principle is 
the same but the calculation methods differ slightly in practice. The equation, presented 
above, can be straight used to estimate the value measure for commodities. For currency 
exchange rates, the return is adjusted with Consumer Price Index (CPI) and thus, the 
final measure consists of five-year change in Purchasing Power Parity (PPP). The cal-
culation is better explained by Accominotti and Chambers (2014)5 and it can be written 
as  
                                                
4 Fama−MacBeth regression is used to test risk factors of asset pricing models across multiple assets. The 
method was introduced by Fama and MacBeth (1973). 
5
 Accominotti and Chambers (2014) calculate undervaluation of currencies so that currencies with small 
values are considered undervalued. Here, the measure should be comparable with book-to-market ratio. 
Hence, equation (25) is a negative form of that introduced by Accominotti and Chambers. 
33 
 
33 
 !"#$% = !!!!! − !!! − !!!!! − !!!!!"# + !!! − !!!"#  , (25) 
where !!! denotes the natural logarithm of spot exchange, !!! and !!!"# denote respective-
ly the natural logarithms of foreign and domestic price indices (measured with CPI); 
again, the index ! corresponds to years. Accordingly, for bonds the negative five-year 
return can be calculated with changes in ten-year bond yields. In addition, Asness et al. 
suggest three alternative value measures for bonds, namely, five-year change in yields 
of ten-year bonds, yield of a ten-year bond minus five-year inflation forecast (i.e., the 
real bond yield), and term spread (difference between a ten-year and short bond yields). 
Ultimately, they report that even more reliable risk premia arise when these different 
measures are combined. 
In case of stocks, the definition of value is quite intuitive to understand as the market 
price can be evaluated against the book value. For other asset classes, this might not be 
as straightforward. At least, the definition is not really discussed in anomaly related 
studies. However, the measures used here give some guide. For example, fair value for 
currencies can be measured with PPP, which is assessed with foreign and domestic CPI 
changes. Additionally, using steepness of the yield curve (term spread) to analyse value 
for a bond can be well explained. The term spread indicates steepness of the yield curve. 
When the yield curve is steep, bond investors benefit from positive roll-down effect, 
assuming that no parallel shifts of the curve are not seen. Now, the price of the bond 
would increase as the maturity comes closer. 
When it comes to portfolio strategies, a typical way to utilise such an anomaly is to 
construct a zero-cost long-short portfolio where, in this case, undervalued assets are 
bought and overvalued assets are sold. The same principle is behind the multifactor 
models, e.g., the value factor, HML in the Fama−French three-factor model is a portfo-
lio where returns of small value stocks are deducted from those of high value stocks. 
However, different methods have been used to construct these portfolios. Firstly, assets 
can be divided in number of portfolios based on their ranked values. Secondly, these 
portfolios can be weighted in different ways. For example, Fama and French (1993) 
create the value factor by dividing both small and big stocks (related to the size factor) 
into three different value weighted portfolios based on their book-to-market rank. Final-
ly, an HML (high-minus-low) portfolio is constructed by subtracting the returns of these 
two extreme portfolios one from another. Value portfolios are usually weighted by mar-
ket capitalisation (see, e.g., Babameto & Harris 2008; Chaves & Arnott 2012; Asness et 
al. 2013). In addition to capitalisation-weighting, Asness et al. construct factors where 
assets of a long-short portfolio are weighted based on their ranks so that assets with 
highest value (or momentum) acquire the largest weight and vice versa. On the other 
hand, the ones in the middle have weight close to zero. Moreover, equal weighted re-
34 
 
34 
turns in high or low value portfolios are used by, e.g., Hjalmarsson (2011), although 
instead of three portfolios, he divides the assets into deciles, which is to emphasise the 
effect of the most extreme characteristics. This is obviously technically more simple 
approach but it surely serves the purpose when it comes to results. 
2.4.3 Momentum effect 
There is strong evidence that those assets with superior short-term past returns will con-
tinue outperforming the past losers in the near future. Such a momentum effect is prov-
en to be significant not only for stocks but also for various different asset classes. In this 
case, the meaning of ‘short-term’ is crucial since long-term reversal effect (discussed 
next) has exactly the opposite effect on future returns. The work of Jagadeesh and Tit-
man (1993) claims that a relative strength trading strategy where recent winner stocks 
over the past 3 to 12 months are selected provides abnormal returns in the next 12 
months following the time of portfolio formation. However, after these 12 months the 
stocks start losing this return up to the following 24 months. Jagadeesh and Titman re-
port the highest annual return when stocks are selected based on the past six-month re-
turn and held for the next six months. 
It has been shown that momentum returns cannot be explained by common risk fac-
tors. Hence, returns realised by momentum effect cannot be a result of systematic risk. 
Delayed stock price reactions to the risk factors could possibly result in lead-lag effects 
causing the profitability of such a momentum strategy. For example, Lo and MacKinlay 
(1990) report that higher expected returns provided by a contrarian strategy where re-
cent winners are sold and losers bought (related to reversals but theoretically close to 
momentum) are not solely due to negative serial dependence in individual stocks, i.e., 
market overreaction. Instead, they document that the phenomenon is a consequence of 
positive cross-autocorrelation among asset returns. This so-called lead-lag effect means 
that positive returns of large capitalisation stocks imply good performance of small-cap 
firms in the future. 
However, Jagadeesh and Titman (1993) report significantly negative relation be-
tween relative strength portfolio returns (based on momentum effect) and lagged 
squared returns of a value-weighted portfolio. This strongly supports a hypothesis ac-
cording to which relative strength returns would not be contributed by lead-lag effect. In 
addition to Jagadeesh and Titman (1993), also DeLong, Shleifer, Summers, and Wald-
mann (1990) suggest that momentum effect would be caused by investors who make 
prices to overreact by buying recent winners and selling the losers. They call the inves-
tors following such a strategy positive feedback traders. The abnormal returns, however, 
are only created in short run because investors tend to overestimate the long-term pro-
35 
 
35 
spects thus eventually causing return reversals. This is not far from the explanation of 
Vayanos and Woolley (2013) who introduce an institutional theory of momentum. They 
explain continuingly falling prices so that funds impacted by a negative shock start los-
ing investors gradually. These outflows make the fund managers to sell the underlying 
assets, which causes a continuing decline in asset prices and therefore, their expected 
returns. 
Momentum effect has proven its significance against common risk factors. Fama and 
French (1996) study whether different anomalies provide abnormal returns that cannot 
be explained with their three-factor model. Unlike all the other competing anomalies, 
the short-term (-12 to -2 months) momentum portfolio generates significant positive 
alpha, which can be considered anomalous from the viewpoint of the model. Addition-
ally, Carhart (1997) considers one-year momentum as a common risk factor and pre-
sents his four-factor model to explain equity returns. He improves the three-factor mod-
el of Fama and French by introducing the four-factor model, which includes one-year 
momentum factor in addition to market, size, and value factors. Carhart uses equal-
weighted decile portfolios in constructing the factor-mimicking portfolio based on elev-
en-month returns lagged one month. The results indicate that the four-factor model re-
sults in lower pricing error than CAPM or three-factor model in explaining cross-
sectional variation in stock returns. In fact, the results are to a great extent supported by 
those discussed above. When researching performance of mutual funds, buying the 
winners of the last year and selling the losers yields higher expected returns in the fol-
lowing year. However, after one year the continuance tends to disappear. After all, Car-
hart does conclude that transaction costs cut a great deal of the returns when momentum 
strategy is followed. 
In the momentum literature, it is a standard to exclude the most recent month in port-
folio formation. The reason for such a procedure is to avoid one-month reversal, which 
implies an inversion in price movements. This is possibly a consequence of liquidity 
issues or microstructure related factors making the momentum measures biased (Asness 
et al. 2013). It has been reported that illiquid stocks often cause return reversals. How-
ever, the effect might be also dependent on firm size as well since small firm stocks are 
more likely to be illiquid. (Tang & Zhang 2014.) A vital microstructure related issue is a 
bid-ask spread leading to weaker momentum returns. Keim (1989) suggests that sys-
tematic trading patterns (such as momentum) lead to wider bid-ask spreads, which may 
introduce errors when calculating returns. 
Despite of the general approach to skip the latest month, Asness et al. (2013) remind 
that some asset classes suffer little from illiquidity. Therefore, they suggest that the 
momentum effect could be even stronger when the most recent observations are includ-
ed in the estimation window. However, this does not remove the fact that bid-ask spread 
leads to biased estimates if the most recent prices are taken into account. Additionally, 
36 
 
36 
Tang and Zhang (2014) find that winner stocks tend to show reversal in a short run but 
the loser ones tend to continue perform poorly, i.e., these two groups behave in different 
ways. This may also have an impact on how the recent winners and the losers should be 
treated. 
Explanations to momentum in the finance literature can be divided into three groups: 
risk and characteristics-based explanations, bias related explanations, and explanations 
related to transaction costs. The first group of explanations is based on results suggest-
ing that momentum is higher among small firms and those bearing higher financial 
risks. Investors’ underreaction to news is a clear example of behavioural biases causing 
the momentum effect. Underreaction is often the greater the less analysts are following 
a firm. Transaction costs, again, are usually seen as a drawback for momentum strate-
gies. (Menkhoff et al. 2012.)  
2.4.4 Long-term reversal 
If the momentum effect is a continuing pattern in asset returns reversal is the opposite. 
After a period of continuous decline or growth the market recognises the mispricing and 
the direction tends to change, i.e., a reversal occurs. Return reversals are often explained 
with investors’ overreaction to fundamentals. After an overreaction, either positive or 
negative, the prices are ought to reverse back towards their fundamental values. For this 
reason, the effect is originally discussed as stock market overreaction. In finance litera-
ture, two types of reversals are observed: long-term and short-term reversals. The dis-
cussion, however, is often related to long-term return reversals since short-term rever-
sals are usually led by biases concerning bid-ask spreads or liquidity issues (see, Keim 
1989; Asness et al. 2013). Recognising these patterns could possibly provide appealing 
trading strategies for investors who make decisions based on historical performance of 
assets, against the efficient market hypothesis. 
From institutional perspective, the (long-term) reversal effect is explained with ra-
tional investors who make market prices to deviate from fundamental values. When 
asset prices drop suddenly due to a negative shock investors tend to withdraw their 
shares from investment funds. This causes pressure for the portfolio managers to start 
selling assets. Eventually, these outflows push the prices below fundamental values, 
which finally increases expected returns thus making the assets tempting. (Vayanos & 
Woolley 2013.) Such an overreaction is an elementary factor explaining the success of 
reversal strategies. Comparison of past winner and loser portfolios shows that the loser 
portfolios yield considerably higher returns than the winner portfolios, regardless of the 
fact that the former one carries less systematic risk. This result is found to be significant 
with a formation period of 36 months and a test period with the same length. Being that 
37 
 
37 
the cumulative abnormal returns (over the market) are significantly positive to the loser 
portfolios confirms that return reversal is explained by general overreaction. It is also 
found that the effect is stronger for loser portfolios meaning that overreaction is not 
symmetric. (DeBondt & Thaler 1985.) 
Another factor explaining return reversals is leverage. Chopra, Lakonishok, and Rit-
ter (1992), referring to past literature, explain that negative returns result in higher betas 
for equity, which increases expected returns. Since the equity beta is a function of asset 
risk and leverage, continuous negative returns decreases the level of equity thus increas-
ing the beta. In addition, they highlight that the size effect might be behind reversals. 
However, Tang and Zhang (2014) stress that liquidity issues would be the actual reason. 
Small firms are only associated with small liquidity and that is why they seem to come 
to the fore in practical studies. 
Despite of the clear evidence, there are also major drawbacks regarding the persis-
tence of the long-term reversal effect. Jordan (2011) stresses that long-term reversals 
only exist when transaction costs and return-risk adjustments are ignored. Risk adjust-
ments are made with a conditional CAPM, which allows time varying alpha. However, 
McLean (2010) argues that the mispricing is strongly affected by idiosyncratic risk, i.e., 
the volatility of the residuals resulting from an asset-pricing model (e.g., Fama−French 
three-factor model). Testing with stock return data shows that securities with higher 
idiosyncratic risk demonstrate stronger reversal effect. However, results of this study do 
not support such a proposition. Moreover, the findings remain significant when transac-
tion costs are taken into account. 
2.4.5 Idiosyncratic volatility 
The standard Capital Asset Pricing model emphasizes that diversification results in port-
folios where only systematic risk is carried and idiosyncratic risk is eliminated. Techni-
cally speaking, idiosyncratic risk yields when an asset-pricing model fails in explaining 
asset returns, i.e., the regression returns error terms. Idiosyncratic volatility, i.e., the 
volatility of the error terms, and its importance in the asset pricing has been discussed in 
finance literature. It is often argued that idiosyncratic risk is highly relevant factor when 
it comes to explaining stock returns. Many empirical studies report a high idiosyncratic 
risk associated with low returns. This effect has been reported, not only in country level, 
but also in an international context for stock returns (Ang, Hodrick, Xing, & Zhang 
2009.) 
From theoretical point of view, Merton (1987) shows that idiosyncratic risk cannot 
be fully eliminated due to market frictions, such as, limited access to market infor-
mation. In a world of perfect information, investors would hold mean-variance efficient 
38 
 
38 
portfolios, which would yield a mean-variance efficient market portfolio. In reality, 
however, market frictions (e.g., costly information, short selling restrictions etc.) make 
it unrealistic to diversify firm specific risk completely away. For this purpose, Merton 
(1987) derives the capital market equilibrium with incomplete information. The model 
predicts that alpha is a function of different characteristics, such as idiosyncratic volatil-
ity, which has a positive impact on alpha. This would be intuitive, as taking additional 
risk should indeed be rewarded with higher expected return. More recent support for 
this view is given by Goyal and Santa-Clara (2003) who find a significantly positive 
relation between average firm specific variance (lagged for one month) and market re-
turns. They suggest that background risk would partly explain their results as holding 
non-traded assets (i.e., human capital or private businesses) increases background risk 
to investors’ portfolio decisions. This means that increasing risk in non-traded assets 
makes investors to demand higher premium in order to hold traded assets. 
All in all, a positive relation between idiosyncratic risk and expected returns seems to 
have a link to expected returns. However, empirical studies often suggest that this rela-
tion is negative. For example, Ang et al. (2006) find that monthly stock returns are neg-
atively associated with one month lagged idiosyncratic volatilities. It is known that pe-
riods of high market volatility are strongly related to increasing downside risk. Thus a 
logical explanation to their finding would be that stocks with high idiosyncratic volatili-
ty would have a high exposure to market aggregate volatility. However, Ang et al. 
(2006) show that this does not completely explain the phenomenon. Moreover, the re-
sults are robust to other known factors such as size, book-to-market, leverage, liquidity, 
volume, turnover, bid-ask spreads, coskewness as well as dispersion in the characteris-
tics of analysts’ forecasts. As a counterargument, Fu (2009) addresses that the results 
can be explained by return reversal and small size of the firms. The negative associa-
tion, however, is given more support in Ang et al. (2009). In addition to U.S. stock data, 
they include stock markets worldwide and manage to repeat the results, out-of-sample. 
Fink, Fink and He (2012) approach the effect of idiosyncratic risk from both theoret-
ical and practical viewpoints. They estimate idiosyncratic risk using exponential general 
autoregressive conditional heteroskedasticity model (EGARCH) and test whether idio-
syncratic volatility is associated with expected returns, in and out-of-sample6. The in-
sample results prove that accurate estimates of idiosyncratic volatility do have signifi-
cantly positive relation with expected returns, which supports the theory discussed in 
                                                
6
 In-sample and out-of-sample data are important concepts when back-testing with historical data. In-
sample data are the initial or already “known” information used as an input of a test. In this context, out-
of-sample data are considered “unknown” at the time of fitting the model. As more iteration is done, the 
out-of-sample data become in-sample, and thus input of further optimisations. 
39 
 
39 
Merton (1987). However, out-of-sample tests with EGARCH forecasts of idiosyncratic 
volatility show that such a relation cannot be approved. 
Finding the true effects of volatility premium in asset pricing is still under investiga-
tion. A number of empirical studies have been conducted during the recent years and the 
results remain somewhat contradictory. Some report a positive relation between idio-
syncratic risk and expected returns and some findings support the opposite view. In 
many studies, the relation has been shown to be flat as well. In relating discussion, 
however, it is commonly agreed that idiosyncratic volatility is a risk factor that should 
be priced. Although, some argue that the effect is driven by other factors. 
40 
 
40 
3 DATA AND METHODOLOGY 
3.1 Test assets 
The main source of data for this thesis is Thomson Reuters Datastream and the empiri-
cal study is conducted by using R programming language. In order to conduct a reliable 
empirical study, data are collected from the year 1989 to 2014, i.e. over 25 years. Over-
all, daily observations of stock prices from different markets, government bonds, and 
spot rates of currencies and commodities are used covering a time period of 
31.3.1989−1.4.2014. 
In order to avoid selection bias, indices from different markets and asset classes are 
used. Stocks are presented by MSCI country total return indices covering USA, UK, 
Finland, Sweden, Norway, Denmark, Germany, France, Italy, Spain, Japan, Hong 
Kong, Singapore, Korea, Australia, Chile, Mexico, Brazil, Canada, and New Zealand, 
quoted in US Dollars. Bonds include ten-year government bond indices of the following 
11 countries: Australia, Austria, Canada, France, Germany, Ireland, Japan, Netherlands, 
Sweden, UK, and USA. Currencies form a group of six spot exchange rates with respect 
to US Dollar: Australia, Denmark, Germany, Singapore, Switzerland, and UK. The last 
group, commodities, includes spot prices of the following commodities: wheat, cocoa, 
corn, cotton, crude oil, nickel, aluminium, copper, gold, and silver. In addition, the S&P 
500 index is considered as a proxy of the market portfolio in multifactor regressions. 
For some asset, there are some additional data required in determining under or over 
valuation. Term spread is used when calculating value for bonds, which means that 
yields of long- and short-term government bonds, 10 and 3 years respectively, need to 
be downloaded. Also currencies require economic variables when defining value, name-
ly, Consumer Price Index (CPI) data. Generally, CPI is available for each country on 
monthly basis except for Australia the CPI of which is announced quarterly. However, 
non-daily data are matched with daily time series by filling the gaps with previous val-
ues available. A key inspiration in terms of data and methodology is the work of Asness 
et al. (2013), which certainly has an impact on the asset classes and markets chosen for 
this thesis. 
3.2 Descriptive statistics 
Descriptive statistics are calculated for all daily total return series covering the time 
period of 3.1.2000−1.4.2014 (see, Table 1). In total, this results in 3717 daily observa-
tions per asset, which is a comprehensive sample of the capital market returns over the 
41 
 
41 
past two decades. There are two reasons not to report the statistics for the entire down-
loaded dataset. Firstly, trading period starts from 11th July 1994, which is 5.5 years from 
the beginning of the data. This is because estimating the value anomaly requires sub-
stantial sample size. Secondly, even though, all the time series are daily observations 
the quality of the data is still not perfect for all asset classes. In order to make the results 
more comparable, the sample is limited to start from the beginning of year 2000. 
The data is described with the following measures: mean, standard deviation, Sharpe 
ratio, skewness, and kurtosis. Mean and standard deviation are annualised with an as-
sumption of 250 trading days per year. This procedure is made to provide a better un-
derstanding of the real performance of the assets. Kurtosis is reported as excess kurtosis 
in relation to the normal distribution. More specifically, the normal distribution has a 
kurtosis of three and thus the excess kurtosis in this case would be zero. 
Table 1 Equities: Descriptive statistics 
This table reports the basic statistics to describe the distributions of the underlying MSCI country total 
return indices. Mean, standard deviation, Sharpe ratio, skewness, and kurtosis are reported on each 
benchmark index. Also the first-order autocorrelation AR(1) is reported in order to assess if the returns 
follow a random walk. Daily continuously compounded returns are used in calculating the values cover-
ing data over 3.4.2000−1.4.2014. Each time series thus equals 3717 daily observations. Mean, standard 
deviation, and Sharpe ratio are annualised assuming 250 trading days per year. Excess kurtosis is calcu-
lated to measure kurtosis over the normal distribution, where kurtosis equals. At the bottom, average of 
all statistics are calculated to give a better overview of the asset class. 
 
Country 
 
Mean (%) 
 
Stdev (%) 
 
Sharpe 
 
Skewness 
 
Kurtosis 
 
AR(1) 
Equities 
USA 1.566 20.706 0.076 -0.177 7.734 -0.079 
UK 3.859 22.534 0.171 -0.163 9.151 -0.024 
Finland 0.143  34.404 0.004 0.196 11.505 0.008 
Sweden 7.009 31.633 0.222 0.414 8.221 0.014 
Norway 10.558 30.705 0.344 -0.360 7.598 -0.002 
Denmark 11.408 23.534 0.485 -0.059 7.748 0.023 
Germany 5.515 27.092 0.204 0.220 6.962 0.002 
France 4.431 26.187 0.169 0.127 6.489 -0.015 
Italy 2.793 27.054 0.103 0.225 7.642 0.007 
Spain 6.136 27.943 0.220 0.164 6.634 0.024 
Japan -0.312 22.637 -0.014 -0.261 4.693 -0.071 
Hong Kong 6.060 21.804 0.278 -0.309 10.430 0.013 
Singapore 5.976 22.352 0.267 -0.731 17.329 0.019 
Korea 9.451 34.910 0.271 -0.544 13.521 0.029 
Australia 10.531 24.547 0.429 -0.719 9.831 0.021 
Chile 9.112 21.397 0.426 -0.229 12.836 0.117 
Mexico 11.415 27.001 0.423 0.102 10.383 0.079 
Brazil 12.108 36.491 0.332 -0.075 7.791 0.087 
Canada 8.432 23.254 0.363 -0.416 12.417 0.043 
New Zealand 9.376 21.876 0.429 -0.513 5.489 0.044 
Average 6.778 26.403 0.260 -0.155 9.220 0.017 
 
42 
 
42 
Starting from stock returns, the first and an expected observation is that during the 
sample period the annual return has been positive in almost all countries. The only 
country making an exception is Japan, where the annual mean was -0.312 %. The best 
performing market was Brazil with 12.108 % annual rate of return. While the average 
annual mean was roughly half of that in in Brazil, this could be considered quite an out-
standing result over fourteen years of time. However, high returns usually come with a 
great risk. The annual standard deviation in Brazilian stock market was 36.491 %, 
which is the highest value in the group, the average being 26.403 %. 
Risk-wise there are no outliers to mention but Sharpe ratio gives a better comprehen-
sion on how the risk has been compensated. For example, the European and the US 
stock markets have not been much of a competition for most of the developing countries 
like Chile, Mexico, and Brazil when comparing the risk-return relationship. Neverthe-
less, Denmark manages to outperform all the other countries with a Sharpe ratio of 
0.485. Compared to Brazil, for instance, the annual mean return is not quite as high but 
a substantially lower volatility makes the return a better compensation on the risk taken. 
Stock returns are often reported to show negative skewness and excess kurtosis. The 
former one means that the left tail of the distribution is longer and there is more proba-
bility mass on the right side of the mean. Accordingly, excess kurtosis can be seen as fat 
tails of the distribution. A good interpretation to this combination is that the returns 
have a positive tendency but, from time to time, considerably higher wins or losses are 
reported than expected by the normal distribution. The stock returns used in this study 
seem to follow the same rule. The average skewness being -0.155 and excess kurtosis 
9.220 the shape of stock returns can be quite well pictured. Regardless of the average, 
not all of the markets are negatively skewed. An interesting observation is that big Eu-
ropean countries, such as, Germany, France, Italy, and Spain all appear with positive 
skewness. In addition, among stock returns, Singapore with the most negative skewness 
also has the fattest tails. As will be noticed below, this seems to be a general feature for 
some other asset classes as well. 
In addition to the basic descriptive statistics, the Jarque−Bera test was run for all 
markets. As expected, the normal distribution was rejected in almost all cases at a sig-
nificance level less than one per cent. This means that none of the return series are nor-
mally distributed. Even though, the normal distribution is assumed in many economic 
models (also in the MPT), it is commonly known that this is not the case. 
Descriptive statistics on government bond returns show that they are less volatile 
than equity markets. Even if any superior returns are not seen, the average annualised 
mean return exceeds that of reported on equities over the same time period. The table 
below describes the return distributions of the eleven government bond indices studied 
within this thesis. 
43 
 
43 
Table 2 Government bonds: Descriptive statistics 
This table reports the descriptive statistics on government bond returns. For each country, total returns of 
10-year benchmark bond index are used. The reported statistics are same for all the asset classes. 
 
Country 
 
Mean (%) 
 
Stdev (%) 
 
Sharpe 
 
Skewness 
 
Kurtosis 
 
AR(1)  
Bonds 
Germany 8.082 11.351 0.712 0.315 3.498 0.008 
France 8.200 11.744 0.698 0.234 2.915 0.014 
Australia 8.793 14.047 0.626 -0.394 9.966 -0.055 
Japan 2.616 11.385 0.230 0.326 3.330 0.008 
UK 5.905 10.572 0.558 0.091 3.146 0.033 
USA 5.511 7.780 0.708 0.084 2.674 -0.003 
Sweden 7.655 12.926 0.592 0.249 2.937 -0.018 
Netherlands 8.435 11.454 0.736 0.290 3.255 0.005 
Ireland 8.382 14.051 0.597 0.310 8.917 0.095 
Canada 8.049 10.355 0.777 0.111 3.473 -0.060 
Austria 8.358 11.618 0.719 0.223 2.963 0.016 
Average 7.271 11.571 0.632 0.167 4.279 0.004 
 
As reported in Table 2, government bond returns seem to have much lower standard 
deviation than equity returns. All the means are relatively close to the average and there 
does not seem to be significant variation over the standard deviations either. Due to 
much lower average standard deviation compared to equities (26.403 %), the Sharpe 
ratio is higher. 
If stock returns were mostly negatively skewed, the tails of bond returns, on average, 
are tilted to the positive side. The only difference is Australia, the skewness of which 
was -0.394. The distributions are not either as leptokurtic as those of equity returns. The 
average excess kurtosis of government bond returns was 4.279 and there are no real 
outliers within this sample. The most leptokurtic distributions were those of Australia 
and Ireland with values 9.666 and 8.917, respectively. Additionally, as mentioned 
among stock returns, the most leptokurtic deviation comes with the one having the most 
negative skewness. Finally, to make the comparison with the normal distribution, 
Jargue-Bera test was run for all the bond return series as well. The results prove that in 
all cases, the normal distribution is rejected at a significance level less than one per cent. 
Currency returns of six different currency pairs are described in Table 3. Common 
features of the distributions are not as obvious as for those reported earlier. The average 
mean and standard deviation are lower than those of equities or government bonds but 
deviation among the figures is more prominent. Also the shapes of the distributions dif-
fer a lot from the asset classes described earlier as any common features do not seem to 
44 
 
44 
exist. However, six currency pairs would most probably not represent the whole popula-
tion very well. 
Table 3 Currencies: Descriptive statistics 
This table reports the descriptive statistics on currency returns. For each currency, daily spot exchange 
rates have been used to calculate the returns. All the spot price series are quoted per USD to show the 
returns from a dollar investor’s point of view. The reported statistics are same for all the asset classes. 
 
Country 
 
Mean (%) 
 
Stdev (%) 
 
Sharpe 
 
Skewness 
 
Kurtosis 
 
AR(1) 
Currencies 
Germany 3.671 14.365 0.256 0.007 -0.230 -0.080 
Australia 2.526 13.581 0.186 -0.282 6.965 -0.009 
UK 0.221 9.532 0.023 -0.050 3.900 0.029 
Denmark 2.212 10.307 0.215 -0.057 2.305 0.009 
Switzerland 4.139 11.116 0.372 -0.209 6.561 0.011 
Singapore 1.941 5.282 0.367 0.057 4.897 -0.025 
Average 2.452 10.697 0.237 -0.089 4.066 -0.011 
 
Currencies differ from the other asset classes as exchange rates are affected by the 
riskless interest rates of both countries of the currency pair. US Dollar, as a quote cur-
rency, is paid USD Libor interest rate, whereas foreign interbank offered rate is paid on 
the base currency. In real life, currency returns should be adjusted with these rates. 
However, the effect is not likely to be significant, which is why currency returns are 
simply calculated from the spot prices, just like for any other asset within this study. 
The average mean return per annum is 2.452 %. which clearly below the average 
among the other asset classes. However, a relatively low standard deviation in average 
results a risk-return ratio close to that of equities. Skewness and kurtosis among ex-
change rates vary quite a lot, which makes it challenging to find similarities. Neverthe-
less, the sign seems somewhat clear. 
According to the sample used here, currency returns are usually negatively skew and 
the tails are heavy. This finding is also documented in other FX-related studies (see, 
e.g., Cotter 2005). It should be noted, however, that the sign of skewness depends on 
how the exchange rate is presented. Here, the currency pairs are reported as US dollars 
per one unit of the underlying currencies. For example, Cotter (2005) uses the other 
expression. In any case, normality of currency returns can be ruled out quite unambigu-
ously. The only currency close to normal is the German Mark/Euro. In Jargue-Bera test 
Germany fails in rejecting the normal distribution at 5 % significance level (p-value = 
0.071). 
Commodity returns, calculated from daily spot prices, result in the highest annual 
mean returns in this paper. There is some dispersion across the commodities but the 
45 
 
45 
means are rather high indeed. Aluminium has the lowest mean, 0.527 % per annum, but 
all the others have means at least 4.530 % (nickel) while the maximum is 42.717 % 
(cotton). However, these returns do not come without cost. The average standard annu-
alised standard deviation is 57.721 %, and even if cotton were excluded the average 
would be 36.563 %. Due to volatile prices, the compensation is rather poor risk-return 
wise. The average Sharpe ratio across the commodities is 0.248, which is somewhat 
close to the average Sharpe ratio of stocks and currencies. As shown in Table 4, all in-
dividual Sharpe ratios are close to the average. 
Table 4 Commodities: Descriptive statistics 
This table reports the descriptive statistics on commodity returns. For each commodity, daily spot prices 
have been used to calculate the returns. The reported statistics are same for all the asset classes. 
 
Country 
 
Mean (%) 
 
Stdev (%) 
 
SR 
 
Skewness 
 
Kurtosis 
 
AR(1) 
Commodities 
Wheat 13.686 59.128 0.231 -0.111 0.643 -0.156 
Cocoa 9.773 33.407 0.293 -0.221 11.146 -0.077 
Corn 15.832 59.307 0.267 -0.043 -1.021 -0.212 
Cotton 42.717 248.137 0.172 0.010 -1.479 -0.746 
Crude oil 10.290 35.914 0.287 -0.124 3.470 0.003 
Nickel 4.530 38.193 0.119 -0.130 3.384 -0.010 
Aluminium 0.527 22.743 0.023 -0.245 2.160 -0.046 
Copper 8.939 28.479 0.314 -0.078 3.949 -0.061 
Gold 9.971 19.039 0.524 -0.178 5.380 -0.005 
Silver 8.348 32.859 0.254 -0.913 7.402 -0.016 
Average 12.461 57.721 0.248 -0.203 3.503 -0.133 
 
For all commodities, the return distributions are negatively skewed, which means 
that observations are packed more to the right side of the mean. This can be a good 
thing from an investor’s point of view, as the returns tend to be positive. However, ex-
cess kurtosis increases the risk of extreme values in most cases. Especially, the return 
distribution of cocoa is quite unique when it comes to heavy tails. However, corn and 
cotton represent the other side with kurtosis lower than in the normal distribution. Nev-
ertheless, Jargue-Bera test proves that none of the commodity return distributions are 
normal. In all cases, the null hypothesis can be easily rejected. 
Randomness of the asset returns was tested with first-order autocorrelation denoted 
by AR(1). As reported, there is little or no serial dependence in daily return. This means 
that the previous day’s return cannot be used in predicting return of the following day. 
This shows some evidence that daily returns follow a random process. For each asset 
class, the results are quite similar and no major outliers are seen within asset classes. 
46 
 
46 
The highest absolute AR(1) was reported for cotton (-0.746) which no doubt challenges 
the randomness assumption. However, the average first-order autocorrelation among 
commodities was -0.133, and thus, not very high. Considerably lower autocorrelations 
among the other asset classes imply that the returns are truly random. The average first-
order autocorrelation for stocks, government bonds, currencies, and commodities are 
0.017, 0.004, -0.011, and -0.133, respectively. Visual observation also reveals that the 
asset class matters in terms of the sign of autocorrelation. Among stock returns, autocor-
relation is usually positive whereas it negative for each commodity. 
In addition to descriptive statistics, further investigation was made by analysing ag-
gregated return histograms of all four asset classes (see, Figure 4). The aggregated re-
turn series were created taking the average of individual daily log-returns. Also, the 
normal distribution is placed over each histogram. This provides a good comprehension 
of the nature of sample density curves. A highly important detail to note is that the his-
tograms are not directly comparable with the descriptive statistics tables. 
The aggregated return series behave like asset baskets containing each underlying as-
set with equal weight. This means that the correlation structure of these assets will also 
be captured. When visually analysing the shapes of the curves, it is important to under-
stand that skewness and kurtosis are actually coskewness and cokurtosis of the underly-
ing assets. For example, when correlation across the assets is weak, kurtosis might be 
much lower than what could be expected by looking at individual distributions (see, 
e.g., currencies). However, the purpose of the graphs is to shed light on how the asset 
classes behave risk-return wise in relation to each other. 
47 
 
47 
 
Figure 4 Return distributions of different asset classes against the normal distribu-
tion from the time period of 3.1.2000−1.4.2014. 
The histograms above reveal the increasing tail risk that leptokurtosis brings. For ex-
ample, return distributions of both stocks and commodities are spikey and also negative-
ly skewed. In both cases, there are more extreme events than the normal distribution 
predicts. Moreover, most of these events occur on the left side of the curve. In addition, 
as mentioned above, the distribution of currency returns is not as spikey as could be 
expected. One possible reason for this could be a negative correlation across the curren-
cy pairs, which is obviously good in terms of portfolio diversification. 
3.3 Building anomaly-based factors 
3.3.1 Defining the anomalies in different asset classes 
As the theoretical background shows, defining anomalies might not always be unam-
biguous, especially when the observations go beyond stock markets. Some anomalies, 
such as momentum or long-term reversal, can be easily defined for other asset classes as 
well. However, the definition of value or idiosyncratic volatility is not as straightfor-
ward. All the four anomalies (value, momentum, long-term reversal, and idiosyncratic 
0"%"
5"%"
10"%"
15"%"
20"%"
25"%"
30"%"
(5.00"%" (3.00"%" (1.00"%" 1.00"%" 3.00"%" 5.00"%"
Stocks 
0"%"
5"%"
10"%"
15"%"
20"%"
25"%"
30"%"
35"%"
40"%"
(4.00"%" (3.00"%" (2.00"%" (1.00"%" 0.00"%" 1.00"%" 2.00"%" 3.00"%" 4.00"%"
Bonds 
0"%"
5"%"
10"%"
15"%"
20"%"
25"%"
30"%"
35"%"
40"%"
45"%"
50"%"
(4.00"%" (3.00"%" (2.00"%" (1.00"%" 0.00"%" 1.00"%" 2.00"%" 3.00"%" 4.00"%"
Currencies 
0"%"
5"%"
10"%"
15"%"
20"%"
25"%"
(4.00"%" (3.00"%" (2.00"%" (1.00"%" 0.00"%" 1.00"%" 2.00"%" 3.00"%" 4.00"%"
Commodities 
48 
 
48 
volatility) will be next given definitions in the contexts of stocks, bonds, currencies, and 
commodities. 
Starting from value anomaly, the common method for almost all asset classes is the 
negative five years return. The method follows mainly the guideline introduced by As-
ness et al. (2013). However, the general value measure for stocks, namely, book-to-
market ratio, was not used due to data availability. Instead, the negative five-year return 
is applied to stocks based on the findings of Gerakos and Linnainmaa (2012). As was 
more specifically explained earlier, the negative of five years return and book-to-market 
ratio seems to have a strong positive correlation over time. About the exact calculation 
of this measure, the price five years ago is here calculated as an average of the prices 
5.5 and 4.5 years ago, according to Asness et al. (2013). This simple version was also 
applied to commodities but the two other asset classes need further clarification. 
Value measure for government bonds is the simplest of all asset classes. It is calcu-
lated as a difference between long and short-term yields of a treasury note, which im-
plies the steepness of the yield curve. This term spread is actually calculated using ten 
and three years yields. As discussed earlier, Asness et al. (2013) suggest two other value 
measures where bond yields are associated as well. One of these includes five-year in-
flation forecast and the other one compares the change in a bond’s yield during five 
years. Also a combination of different methods has been proposed. Nevertheless, a sim-
ple term-spread is chosen for this study for sake of simplicity and data availability is-
sues. More complex estimate is required for currencies as the Purchasing Power Parity 
is considered as the driver of value. The formula 25 demonstrates how it is calculated 
and this method is also used in this study. The intuition behind this method is that nega-
tive five years return is adjusted with the corresponding Consumer Price Index values. 
In other words, the value is based on the negative five years return and five-year change 
in the difference of the base and quote currency CPIs. Within this thesis USD is always 
the quote currency and its pair is the base currency. For example, the exchange rate of 
GBP/USD quotes how many USDs (quote) is needed for one GBP (base). 
Momentum is calculated the same way for all asset classes. Of course, the parameter 
values can be changed and the same setting might not be optimal for all asset classes. 
The benchmark in this thesis is the past twelve-month cumulative (logarithmic) return 
but it can be also longer. Jagadeesh and Titman (1993) suggest that momentum effect 
(for stocks) is strong when recent winners over the past 3 to 12 months are selected. 
Ranking the assets based on the cumulative returns is easy but there is an issue that 
should be considered. In momentum literature, the standard is to exclude the most re-
cent month from the calculations in order to avoid liquidity issues. This is also taken 
into account within this thesis. Considering 20 trading days per month, the assets are 
49 
 
49 
sorted on past t-240 7 through t-20 days when forming momentum portfolio. Addition-
ally, six and three-month momentum portfolios are analysed. 
The principle of selecting assets based on long-term reversal is similar to momen-
tum; only the expected outcome is the opposite. The prices of long-term winners are 
expected to decline in the future and vice versa. The formation period, however, is quite 
ambiguous when having both momentum and reversal in the same analysis. For exam-
ple, McLean (2010) sorts stocks for reversal (momentum) portfolio on past returns from 
t-36 through t-7 months (t-6 through t-1 months). The reason for this approach is that 
the same sample should not be included in the same formation period. Hjalmarsson 
(2010) uses this same principle when comparing momentum and reversal anomalies. 
Here, such a one-month cap is not necessary as daily logarithmic returns are used. How-
ever, since momentum is based on returns from t-240 days this must be the back line for 
long-term reversal. Within this thesis, the benchmark formation period for long-term 
reversal portfolios is t-36 through t-12 months. In addition, two other formation periods, 
t-48 and t-24 through t-12 moths, are compared. 
Idiosyncratic volatility is the fourth anomaly discussed within this thesis. The defini-
tion of idiosyncratic volatility (or risk) is rather intuitive. It is the unsystematic risk that 
cannot be explained with the underlying asset pricing model. However, a relevant ques-
tion is, which asset pricing model to use. Besides, how to cope with the fact, that differ-
ent asset classes are analysed in the same study? In order to prevent the theoretical 
framework from expanding too much, some kind of line had to be drawn. Idiosyncratic 
volatility here is captured by Fama−French three-factor model regressions (see Ang et 
al. 2009). More specifically, it is calculated as a standard deviation of the residuals of a 
regression run individually for each asset. For this purpose, daily US stock market fac-
tors are downloaded from Kenneth French’s website. The decision to use this data in 
particular is justified by the idea that US stock market is likely to affect the most other 
markets in the world, even beyond asset classes. In addition, if market, value, and size 
factors together manage to capture majority of the asset risk, the error terms add value 
for the analysis.  Of course, the main reason for such a compromise was availability of 
the data. 
                                                
7
 A general assumption in this thesis is that there are 250 trading days per year. In average, however, the 
number of trading days per month is 20. In momentum and reversal calculations, the changing parameter 
is the number of months, i.e., this yields 240 observations for a year. However, this is not a problem since 
days per year and days per month are never used in the same context. 
50 
 
50 
3.3.2 Trading algorithm for the factors 
The empirical part of this thesis was mainly conducted using R which is a free software 
environment for statistical computing. For those who are interested, the code is freely 
available in the appendix of this paper. In order to construct the factors, functions were 
created to rank the assets based on the underlying anomaly. The basic idea regarding the 
factors is that two portfolios are formed, one following a long-only and another one a 
short-only strategy. In fact, the short-only portfolio is treated as the long-only one, alt-
hough, the final returns are multiplied by minus one to transform them into short posi-
tion returns. Finally, the factor is considered as a long-short portfolio, which is a combi-
nation of these two, i.e., factor returns are calculated as a difference of the two portfolio 
returns. 
Before running a portfolio back-testing procedure, it is essential to define some basic 
parameters regarding the portfolio construction, and also those behind the anomalies. 
From the back-testing point of view, the relevant parameters are the estimation window 
and portfolio rebalancing period. The former parameter is often defined by the anomaly, 
e.g., the window for a momentum portfolio is usually twelve months. Rebalancing or 
update period defines how often the portfolio weights are readjusted according to the 
strategy. The default value for this parameter is 20 days, i.e., portfolios are rebalanced 
every month. Also the anomalies need further specifications. This is important when 
different settings are compared. For example, momentum and reversal strategies can be 
based on different lengths of historical returns. On the other hand, idiosyncratic volatili-
ty anomaly has two different approaches. One says that the assets with high idiosyncrat-
ic risk should be over weighted, and the other one says just the opposite. That is why 
this parameter should be left open for researcher’s decision. Additionally, there are 
some general anomaly-linked parameters that affect the portfolio formation. The first 
decision to make is, in how many percentiles the assets should be sorted based on the 
anomaly in question. By default, there are three percentiles within this study, i.e., the 
top third is represents the long-only portfolio and the bottom third is included in the 
short-only portfolio. This is also the setting that Asness et al. (2013) use. However, dif-
ferent percentiles are used in different papers. The other question is, how should these 
sub-portfolios be weighted. Often, the portfolios are equal weighted but different ap-
proaches are utilised as well, e.g., weighting based on volatility or ranking according to 
the effect in question. 
In most academic papers, the sub-portfolios are equal weighted. However, the origi-
nal groups can be further divided in smaller groups. For example, Fama and French 
(1993) use this method when building value and size factors. Take, e.g., the size factor 
where the stocks are first sorted into three equal weighted groups based on size. Then 
both small and big size groups are sorted again by value to low, medium and high. Fi-
51 
 
51 
nally, they are left with six portfolios that they use to mimic the risk factor. Asness et al. 
(2013) build factors by weighting the assets in proportion to the rank based on value or 
momentum. Then, McLean (2010) tries different weighting schemes, such as, equal, 
value, idiosyncratic risk and inversed idiosyncratic risk weights. The number of differ-
ent approaches in the academic literature supports the idea of comparing different 
weighting schemes for the portfolios. In this study, the portfolios are weighted in pro-
portion to the ranks based on the underlying anomaly or according to either idiosyncrat-
ic risk or reversed idiosyncratic risk. Also, equal weights are applied. 
The portfolio algorithm is rather simple and takes into account the fact that logarith-
mic returns are problematic when calculating return on portfolio. Logarithmic returns 
are used because of their statistical properties, but they must not be weighted. To avoid 
this, the algorithm calculates the quantity of each asset in the portfolio given the 
weights, value of the portfolio, and price of each asset. Following this logic, portfolio 
weights can be calculated when the number of each asset, their prices, and value of the 
portfolio are known. Logarithmic returns are only used as input for the models. This 
connection can be illustrated with the following figure: 
 
Figure 5 The portfolio construction algorithm 
Here, vt is a vector including the number of assets held in the portfolio at time t. Portfo-
lio weights are denoted by vector wt, portfolio value by scalar PVt , and price vector at 
time t by Pt. Obviously, when rebalancing a portfolio, the weights are produced with a 
model that uses daily returns as an input. However, when the portfolio is not rebalanced, 
the weights can be calculated from the other known parameters as the unit prices 
change. Also, the number of each asset in the portfolio remains unchanged when re-
balancing is not done. The value of portfolio is updated after each iteration by multiply-
!! → !1!!"!!!
 
Rebalance( !! = !!! ∘ !!!"!  
!"! = !!!!! !!!
 

((
Por,olio(value(
Daily(
returns(
Por,olio(weights(
!! = !"!!! ∘ 1!! 
Number(of(assets(
52 
 
52 
ing the number of each asset by the corresponding price. Mathematically speaking, this 
element-wise product is denoted by ○ which is known as Hadamard product. In the be-
ginning, the value of portfolio is one hundred and the first rebalancing is done based on 
data prior to the investment period. 
The same principal is followed next as the Black−Litterman model and mean-
variance based optimisation are applied. The only changing part is the actual rebalanc-
ing process. 
3.4 Portfolio optimisation in the Black−Litterman framework 
3.4.1 Parameters for the model 
The model itself is simple to configure and use as a part of portfolio optimisation. How-
ever, some of its parameters are abstract and there are number of different approaches to 
solve them. Probably the most problematic parts of the model are the market portfolio 
and, especially, its weights, uncertainty of the views, and the tau parameter, which indi-
cates the uncertainty of the prior mean. In addition, the risk aversion parameter, ! is not 
straightforward to estimate. In relating literature, different methods are discussed to 
solve these issues, some more thoroughly than others. Here the methods have been cho-
sen based on the existing research papers and also their importance regarding the main 
focus of the thesis. Since efficient trading strategies are more of interest, no thorough 
comparison of minor details will be necessary. 
Starting from the market portfolio, the data already defines the accuracy that can be 
realistically considered. In a stock portfolio study, it would be rather simple to use the 
weights of a major stock market index, such as, S&P 500. However, in this study not 
only market indices are used but also different asset classes, which makes it much more 
difficult to estimate their weights in an unobservable market portfolio. For this reason, 
an equally weighted portfolio (1/N) is used as a benchmark. Nevertheless, it should be 
noted that an investor with neutral or no views would eventually consider an equal 
weighted portfolio optimal. This is due to the inversed optimisation, which is to calcu-
late the implied equilibrium returns. 
For the implied equilibrium returns, it is also crucial to estimate the risk aversion pa-
rameter, !. By default, this is the ratio of expected excess return and variance of the 
market. However, it is often considered constant over the investment horizon. Curtillet 
and Dieudonne (2007) calculated risk aversion parameters for different asset classes and 
found that the variation can be relatively high. For commodities, they estimated risk 
aversion of 1.8 whereas the value for investment grade bonds was 20. In many articles, 
53 
 
53 
the parameter is set to 3, which is appropriate for the global stock market. Within this 
thesis, the benchmark is also 3 but the robustness tests indicate that performance is not 
really sensitive to risk aversion. 
The views themselves are the core of the model, although, it is the investor’s respon-
sibility to express them. Nevertheless, weighting them and defining uncertainty of the 
views is important from theoretical perspective. In the Black−Litterman model, there 
are two ways of expressing views; they can be either absolute or relative. The nature of 
this study supports the latter, i.e., the question is, how much the winners will outperform 
the losers. For this reason, the rows of the P-matrix must sum up to zero. In this view 
weighting scheme, the expected winners are given positive weights and vice versa. 
However, the weighting method slightly differs from that usually seen in relating arti-
cles. Instead of placing either minus one or one depending on which group the asset 
belongs to, the weighting follows the same method that is applied to the underlying fac-
tor. For example, momentum view is applied to stocks. At the time of rebalancing, the 
past winners and losers are identified and long-short weights are given accordingly. In 
this case, the first row of the P-matrix, i.e., the view for stocks, acquires exactly these 
portfolio weights. All the non-stock elements of the first row are given zero weight. As 
a result, the expected performance of stocks can be traced from this specific combina-
tion. The same principal is followed on all asset classes as indicated below. 
 ! = !!! ⋯ !!" 0 0 0 0 0 0 0 0 00 0 0 !!! ⋯ !!" 0 0 0 0 0 00 0 0 0 0 0 !!"! ⋯ !!"# 0 0 00 0 0 0 0 0 0 0 0 !!! ⋯ !!"  (26) 
Here, views are expressed individually on each asset class, stocks, bonds, currencies, 
and commodities, respectively. Each row of the matrix will sum to zero. The essential 
benefit of this method is in its ability to include different views in the same portfolio, 
consisting of many asset classes. 
Uncertainty of the views can be solved in different ways as already discussed earlier. 
This uncertainty is indicated with omega matrix, which can be understood as a covari-
ance of the views. Since the out-of-sample study covers over twenty years, it is clear 
that a simple and adaptive method is well justifiable. In this light, the method suggested 
by He and Litterman (1999) serves the purpose better than well. If the variance of a 
view k can be written as (p!Σp′!)τ, the result is that the scalar τ, eventually, becomes 
irrelevant. Moreover, as highlighted by Idzorek (2005), this method is the most used in 
relating literature. 
54 
 
54 
3.4.2 Optimal portfolios based on market anomalies 
The first step with market anomaly based investing was to create portfolios simply 
based on the properties of the assets. Relying on the anomalies, the assets have been 
ranked in proportion to past performance, value, or what ever the underlying anomaly 
might be. As a long-short trading strategy, this yields a factor that can provide relevant 
information also for a more diversified portfolio strategy. In this study, these asset class 
specific factors are crucial when it comes to expressing views on portfolios. 
The portfolio construction algorithm is discussed in connection to the factors. The 
logic is well explained by Figure 5, although, rebalancing the portfolio is where the 
Black−Litterman model and mean-variance optimisation come into picture. When it 
comes to rebalancing the portfolio the market portfolio should be considered as a 
benchmark. As already noted, equal weights give a starting point for every rebalancing 
procedure. Accordingly, implied equilibrium returns are calculated as a product of the 
benchmark weights, market risk aversion parameter, and the covariance matrix of the 
underlying asset returns. Since estimation of the covariance matrix is not in the focus of 
this study, a basic sample covariance is estimated. However, it is to be considered that 
covariance matrix is highly important when it comes to portfolio optimisation. 
Incorporating investors’ views is what makes the Black−Litterman model unique. 
Given the factors built prior to optimisation, it is possible to express views based on 
their historical performance. The view itself expresses the rate of return on which the 
winners are expected to outperform the losers, i.e., it is the expected return of the under-
lying factor. Within this thesis, the expected return of the following investment period 
equals the mean return of the previous month. The idea of expressing the view as an 
expected return of the corresponding long-short portfolio is suggested by Fabozzi et al. 
(2010). However, they do not provide any method to estimate this return. Babameto and 
Harris (2008), who test Black−Litterman strategy based on value and momentum, sug-
gest an appealing method to estimate the expected return of a long-short portfolio. Their 
solution to the problem is regression. They predict momentum returns with the US term 
spread and value with the aggregate book-to-market ratio of the market. 
The use of the prior mean as an expected return can be explained with number of 
reasons. First, this method is not trying to predict the return of any instrument. Instead, 
the prior mean indicates, how much the winner group can be expected to outperform the 
losers. The allocation that is expected to achieve this result is solved at the time of re-
balancing. Of course, simplification and focus of the study are also relevant reasons. 
Trying to predict the return on four different anomalies in multiple asset classes would 
go beyond the scope of this study. 
From the perspective of portfolio optimisation, some general parameters and assump-
tions are worth mentioning. In a simple mean-variance study, the in-sample period is 
55 
 
55 
unambiguous since predetermined length of data is used as an input of the optimisation. 
In this case, however, the concept is not as clear. For the covariance matrix, a sample of 
250 trading days is always used as input. However, strategies based on historical per-
formance, such as, momentum and reversal, require samples with different lengths. 
Moreover, there are a large variety of different combinations with all possible parame-
ters. This is why the comparison is mostly limited to the different ways to utilise the 
anomalies. When it comes to portfolio optimisation process itself, holding period is 
probably the one that is still essential to investigate. This is because in the reality trans-
action costs due to frequent rebalancing tend to eat all the benefits gained by the optimi-
sation. 
 
3.5 Measuring portfolio performance 
The algorithm described earlier is a tool to back-test performance of portfolios follow-
ing different strategies. In the basic setting, all the four asset classes are included in the 
portfolio. Then different anomalies are applied to the assets depending on which asset 
class they belong to. In order to attach the views into the portfolio optimisation, the 
Black−Litterman model is to be utilised. After running the code, the final product is a 
time series of the portfolio value from 11th Jul 1994 to 1st Apr 2014. The next step is to 
evaluate whether or not the strategy pays off. Does the strategy provide abnormal re-
turns or what is the return vis-à-vis the taken risk compared to other strategies? 
Probably the most intuitive measure to compare performance of different investment 
strategies is the Sharpe ratio. The ratio indicates how well the risk is compensated by 
return. Hence, adjusting returns with risk makes it easier to compare different portfolios 
together. As defined by Sharpe (1994), this reward-to-variability ratio can be written as 
 !"! = !! − !!σ! !, (27) 
Where Ri and Rf denote return of the asset i and the risk-free rate return, respectively. 
Standard deviation of the asset i, i.e., the risk measure is denoted by σ!. The original 
expression characterises the return-risk ratio ex ante, which is why the expected excess 
return is calculated. As an ex post performance measure, the realised returns and stand-
ard deviation will be used. In addition, the risk-free rate of return is ignored within this 
study. 
Since daily results are investigated, the Sharpe ratio will also acquire a daily form. 
Usually it is more intuitive to study annualised values, which is also done here. A daily 
Sharpe ratio can be annualised as follows: 
56 
 
56 
 !"!.! = 250!!"!.! (28) 
This expression assumes 250 trading days per year, which is the assumption made in 
this thesis. 
When using the Sharpe ratio as a performance measure, it is important to be aware of 
its flaws. Given excess kurtosis and skewness often reported on, e.g., stock returns may 
cause biased measures. The biggest problem relates to standard deviation and its ability 
to reflect risk. When returns are not normally distributed it may not be unambiguous to 
tell what is the effect of additional volatility risk-return wise. Especially, assets with 
different distributions might become less comparable. However, the Sharpe ratio is still 
often used in measuring the performance of a portfolio both in academic and business 
world. This is why it is the most central performance measure also in this study. 
A more sophisticated model is needed to investigate if a portfolio provides abnormal 
returns with respect to known risk factors. Abnormal returns or alpha is measured with 
the Fama-French three-factor model, which tries to explain returns with three common 
risk factors: market, size, and value. The regression is defined as 
 !!" − !!" = !! + !!" !!" − !!" + !!"!"#! + !!"!"#! + !!" , (29) 
where excess returns are explained with the three risk factors. Market factor is simply 
the excess return of the market. The size factor, i.e. small minus big, captures the expo-
sure to the size premium, which should be gained by investing in small capitalisation 
stocks. The value factor, i.e. high minus low, measures the premium earned by prefer-
ring high book-to-market stocks to the low ones. In terms of abnormal return, alpha is 
the key value. Significantly positive alpha can be interpreted as the investor’s ability to 
gain excess return beyond the risk factors, assuming the factor exposures are fully cap-
tured. Finally, the error term of the regression (for a given asset at a given time) is de-
noted by !!". 
Because of availability of the factors, daily risk factors on the US stock market have 
been used within this thesis. The factors have been downloaded from French’s website. 
In order to analyse the significance of alpha, also according t-values are reported. Since 
cumulative returns are used in calculating the t-values, the autocorrelation needs to be 
controlled by using Newey-West (1987) standard errors. 
57 
 
57 
4 RESULTS OF THE EMPIRICAL STUDY 
4.1 Zero-cost portfolios 
Before applying the anomalies into the portfolio optimisation, zero-cost portfolios are 
analysed with respect to individual anomalies. This is to see if they provide significant 
return premia and thus appealing factors to be utilised in portfolio construction. In addi-
tion, long-only strategies are tested because it is important to see whether more sophis-
ticated portfolio optimisation adds value in comparison with a simple anomaly strategy. 
As discussed earlier, there are plenty of different combinations when comparing dif-
ferent strategies. In addition to the four asset classes and anomalies, the assets can be 
sorted to different numbers of groups, there are different weighting methods, and differ-
ent lengths of data included in estimation. In total 124 portfolios are back-tested and the 
most relevant ones are taken to the next round when analysing Black−Litterman based 
portfolio strategy. 
The first step in analysing the anomalies is to construct portfolios that follow a sim-
ple long-only strategy where assets are selected based on an individual anomaly. For 
each portfolio, the input parameters are selected from Table 5. In other words, each in-
dividual portfolio includes only one asset class, the allocation of which is determined by 
one anomaly with given parameters. Say, a stock momentum factor is to be built. For 
each month, the portfolio is rebalanced and assets are divided in, e.g., three groups ac-
cording to the momentum effect. For momentum, cumulative return over, e.g., twelve 
months (minus the previous month) is assessed. Within the high momentum group, 
weights could be given, e.g., by idiosyncratic volatility where either high or low vola-
tility indicates the ranking order. Following this logic, different combinations have been 
used when constructing factors for different asset classes and anomalies. 
  
58 
 
58 
Table 5 Properties of zero-cost portfolios 
When constructing zero-cost or long-short portfolios based on anomalies, combinations of different pa-
rameters can be compared. A single strategy is always for one asset class and anomaly only. When sorting 
assets based on the anomaly they can be allocated in different numbers of groups or percentiles. For a 
long-short portfolio, the first and the last percentiles are considered, and for a long-only portfolio, the last 
or the “winner” one. Within these groups, the assets can be equally weighted, or they can be weighted 
based on ranking or idiosyncratic volatility. If the latter is used, assets with either high (1) or low (0) 
idiosyncratic volatility will be given more weight. For momentum and reversal strategies, different com-
binations of historical returns are used. This determines how many months are selected and how many 
months is the lag from the present time. 
Asset class Anomaly Groups From (months) 
Until 
(months) Weights 
IVOL 
(high/low) 
Stocks Momentum 2 12 1 Equal 1 
Bonds Reversal 3 6 1 Rank 0 
Currencies Value 5 3 1 IVOL 
 
Commodities IVOL 
 
24 12 
  
   
36 12 
  
   
48 12 
  
 
Later when the Black−Litterman model is applied, the portfolios will be long only. 
This is why it is relevant to see if a simple strategy could outperform such a sophisticat-
ed model. As will be seen in Table 6 and Table 7, it is not obvious that a beneficial 
long-only strategy would be a sign of a significant risk factor. For example, each one of 
the anomalies seems to fit very well for government bonds. However, results of the 
long-short strategy reveal that the long-only portfolio does not have superior returns 
compared to an opposite strategy. This might indicate that the anomaly would not add 
value in the Black−Litterman context. 
After constructing the long-only portfolios, returns of long-short portfolios are ana-
lysed accordingly. These results should shed more light on the question of the positive 
premium these anomalies might provide. As already discussed, these portfolios are con-
sidered as potential risk factors, which are to be utilised in a Black−Litterman based 
portfolio strategy. 
Given the parameters of Table 5, 124 portfolios are back-tested. However, in order to 
compare different anomalies across the asset classes, portfolios with the basic settings 
were analysed in the first place. This initial configuration yields 44 portfolios with the 
following parameters. In value portfolios, the assets are ranked based on their five-year 
negative return that, more precisely, comes from the average of the past 5.5 and 4.5-year 
prices. As discussed earlier, the value measure is slightly different between the asset 
classes; however, the calculation remains the same in all configurations. In momentum 
portfolios, assets are ranked according to the past 12-month cumulative returns less the 
returns of the previous month. Accordingly, past 36 less the most recent 12 months are 
59 
 
59 
of interest in the reversal strategy. For idiosyncratic volatility strategy, the assets with 
lower idiosyncratic volatility are selected in the portfolio. Idiosyncratic volatility is 
measured against the three-factor model. 
For each portfolio, the following weighting schemes are considered in the initial set-
ting. In all cases, the assets are divided into three groups after sorting them on the 
anomaly in question. The middle group is always ignored and, in long-only portfolios, 
only the upper third is selected. These assets are then weighted according to three dif-
ferent methods: equal, rank, and idiosyncratic volatility (IVOL). In equal weighting, all 
assets are given the same weight. Rank based weighting means that those assets having 
a bigger exposure to a specific anomaly have larger weights in the portfolio. In fact, the 
ranking procedure is already done when selecting assets in the portfolio. Finally, IVOL 
weighting is exactly the same as in the IVOL strategy. However, the principle in the 
basic setting is exactly the opposite: those assets with higher IVOL are given larger 
weights. The opposite weighting is also tested but this parameter was chosen for the 
initial setting in order to see how both high and low idiosyncratic volatility affects the 
returns. 
 
  
60 
Ta
bl
e 
6 
R
es
ul
ts
 fo
r l
on
g-
on
ly
 p
or
tfo
lio
s 
Th
is
 ta
bl
e 
re
po
rts
 a
nn
ua
lis
ed
 m
ea
n 
w
ith
 t-
va
lu
e,
 s
ta
nd
ar
d 
de
vi
at
io
n,
 S
ha
rp
e 
ra
tio
, a
nd
 3
-f
ac
to
r a
lp
ha
 w
ith
 a
 re
sp
ec
tiv
e 
t-v
al
ue
 (N
ew
ey
-W
es
t c
or
re
ct
io
n 
ap
pl
ie
d)
 fo
r l
on
g-
on
ly
 p
or
tfo
lio
s. 
M
om
en
tu
m
 is
 b
as
ed
 o
n 
pa
st
 1
2 
to
 
1 
m
on
th
s 
an
d 
re
ve
rs
al
 o
n 
pa
st
 3
6 
to
 1
2 
m
on
th
s. 
In
 IV
O
L 
po
rtf
ol
io
 (w
ei
gh
tin
g)
, a
ss
et
s 
w
ith
 lo
w
 (h
ig
h)
 id
io
sy
nc
ra
tic
 v
ol
at
ili
ty
 a
re
 o
ve
rw
ei
gh
ed
. S
ta
tis
tic
al
 s
ig
ni
fic
an
ce
 is
 in
di
ca
te
d 
w
ith
 s
ta
rs
: *
**
, *
*,
 a
nd
 *
 re
fe
rr
in
g 
to
 p
-
va
lu
es
 <
0.
00
1,
 <
0.
01
, a
nd
 <
0.
05
, r
es
pe
ct
iv
el
y.
 In
 th
is
 ta
bl
e,
 o
nl
y 
th
e 
ba
si
c 
se
tti
ng
 is
 te
st
ed
. O
th
er
 c
om
bi
na
tio
ns
 a
re
 c
on
si
de
re
d 
in
 fu
rth
er
 a
na
ly
si
s. 
Po
rtf
ol
io
s 
ar
e 
co
ns
tru
ct
ed
 w
ith
 ro
lli
ng
 w
in
do
w
 e
st
im
at
io
n 
w
he
re
, a
fte
r 
ea
ch
 m
on
th
, h
is
to
ric
al
 o
bs
er
va
tio
ns
 a
re
 u
se
d 
fo
r r
eb
al
an
ci
ng
 th
e 
po
rtf
ol
io
. I
n 
to
ta
l, 
da
ily
 o
bs
er
va
tio
ns
 c
ov
er
 th
e 
tim
e 
pe
rio
d 
of
 1
1.
7.
19
94
−1
.4
.2
01
4.
 
 
 
V
al
ue
 
 
M
om
en
tu
m
 
 
R
ev
er
sa
l 
 
IV
O
L 
  
  
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
St
oc
ks
 
M
ea
n 
(%
) 
8.
34
9 
8.
52
2 
9.
17
6 
 
13
.2
40
 
13
.3
90
 
13
.6
00
 
 
8.
81
2 
8.
23
3 
7.
61
7 
 
7.
82
3 
7.
84
7 
 
(t-
st
at
) 
(1
.9
46
 ) 
(1
.9
75
 *
) 
(2
.0
06
 *
) 
 
(3
.0
76
 *
*)
 
(3
.0
83
 *
*)
 
(2
.8
95
 *
*)
 
 
(2
.0
16
 *
) 
(1
.8
66
 ) 
(1
.8
46
 ) 
 
(2
.1
27
 *
) 
(2
.1
56
 *
) 
 
St
de
v 
(%
) 
19
.4
69
 
19
.5
76
 
20
.7
55
 
 
19
.5
29
 
19
.7
05
 
21
.3
13
 
 
19
.8
33
 
20
.0
23
 
18
.7
24
 
 
16
.6
90
 
16
.5
14
 
 
Sh
ar
pe
 
0.
42
9 
0.
43
5 
0.
44
2 
 
0.
67
8 
0.
68
0 
0.
63
8 
 
0.
44
4 
0.
41
1 
0.
40
7 
 
0.
46
9 
0.
47
5 
 
A
lp
ha
 (%
) 
3.
09
7 
3.
21
4 
4.
45
8 
 
7.
80
4 
7.
92
8 
8.
34
4 
 
3.
16
2 
2.
41
2 
0.
58
0 
 
1.
45
1 
1.
38
1 
 
(t-
st
at
) 
(0
.8
51
 ) 
(0
.8
76
 ) 
(1
.1
15
 ) 
 
(2
.1
28
 *
) 
(2
.1
30
 *
) 
(2
.0
14
 *
) 
 
(0
.8
85
 ) 
(0
.6
88
 ) 
(0
.1
94
 ) 
 
(0
.5
79
 ) 
(0
.5
69
 ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
B
on
ds
 
M
ea
n 
(%
) 
7.
65
0 
7.
57
7 
8.
12
2 
 
8.
66
5 
8.
68
2 
8.
83
7 
 
8.
87
5 
8.
35
4 
8.
54
8 
 
6.
71
6 
6.
69
6 
 
(t-
st
at
) 
(3
.9
8 
**
*)
 
(3
.9
4 
**
*)
 
(4
.0
09
 *
**
) 
 
(4
.2
9 
**
*)
 
(4
.2
85
 *
**
) 
(4
.2
79
 *
**
) 
 
(4
.5
15
 *
**
) 
(4
.1
54
 *
**
) 
(4
.4
59
 *
**
) 
 
(4
.3
51
 *
**
) 
(4
.4
06
 *
**
) 
 
St
de
v 
(%
) 
8.
72
1 
8.
72
4 
9.
19
1 
 
9.
16
4 
9.
19
3 
9.
36
9 
 
8.
91
7 
9.
12
4 
8.
69
7 
 
7.
00
4 
6.
89
5 
 
Sh
ar
pe
 
0.
87
7 
0.
86
8 
0.
88
4 
 
0.
94
6 
0.
94
4 
0.
94
3 
 
0.
99
5 
0.
91
6 
0.
98
3 
 
0.
95
9 
0.
97
1 
 
A
lp
ha
 
7.
76
6 
7.
68
3 
8.
07
3 
 
8.
77
5 
8.
79
2 
8.
89
0 
 
9.
02
7 
8.
49
7 
8.
78
0 
 
7.
08
7 
7.
10
4 
 
(t-
st
at
) 
(3
.8
57
 *
**
) 
(3
.8
16
 *
**
) 
(3
.7
83
 *
**
) 
 
(4
.2
09
 *
**
) 
(4
.2
03
 *
**
) 
(4
.1
97
 *
**
) 
 
(4
.3
83
 *
**
) 
(4
.0
72
 *
**
) 
(4
.3
61
 *
**
) 
 
(4
.2
50
 *
**
) 
(4
.3
22
 *
**
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
C
ur
re
nc
ie
s 
M
ea
n 
(%
) 
1.
64
2 
1.
67
6 
1.
69
3 
 
1.
45
9 
1.
43
1 
1.
46
5 
 
2.
80
8 
2.
95
1 
2.
69
8 
 
0.
40
3 
0.
45
4 
 
(t-
st
at
) 
(0
.8
67
 ) 
(0
.8
86
 ) 
(0
.8
57
 ) 
 
(0
.8
17
 ) 
(0
.7
96
 ) 
(0
.7
73
 ) 
 
(1
.6
07
 ) 
(1
.6
51
 ) 
(1
.5
96
 ) 
 
(0
.3
09
 ) 
(0
.3
58
 ) 
 
St
de
v 
(%
) 
8.
59
4 
8.
58
2 
8.
96
4 
 
8.
10
1 
8.
15
4 
8.
59
6 
 
7.
92
7 
8.
10
7 
7.
67
0 
 
5.
93
1 
5.
75
8 
 
Sh
ar
pe
 
0.
19
1 
0.
19
5 
0.
18
9 
 
0.
18
0 
0.
17
5 
0.
17
0 
 
0.
35
4 
0.
36
4 
0.
35
2 
 
0.
06
8 
0.
07
9 
 
A
lp
ha
 (%
) 
0.
97
7 
0.
99
7 
1.
03
5 
 
0.
99
0 
0.
97
2 
1.
04
0 
 
2.
21
4 
2.
36
0 
2.
08
8 
 
-0
.3
44
 
-0
.3
08
 
 
(t-
st
at
) 
(0
.5
02
 ) 
(0
.5
15
 ) 
(0
.5
14
 ) 
 
(0
.5
41
 ) 
(0
.5
28
 ) 
(0
.5
38
 ) 
 
(1
.2
58
 ) 
(1
.3
24
 ) 
(1
.2
28
 ) 
 
(-
0.
25
4 
) 
(-
0.
23
4 
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
C
om
m
od
iti
es
 
M
ea
n 
(%
) 
12
.9
22
 
13
.4
47
 
11
.9
06
 
 
9.
68
3 
9.
70
0 
9.
67
7 
 
10
.7
20
 
8.
34
4 
10
.8
33
 
 
3.
42
3 
3.
59
0 
 
(t-
st
at
) 
(2
.2
12
 *
) 
(2
.2
30
 *
) 
(1
.5
28
 ) 
 
(1
.7
68
 ) 
(1
.7
52
 ) 
(1
.4
38
 ) 
 
(1
.9
32
 ) 
(1
.5
62
 ) 
(2
.2
66
 *
) 
 
(0
.9
94
 ) 
(1
.0
69
 ) 
 
St
de
v 
(%
) 
26
.4
99
 
27
.3
63
 
35
.3
62
 
 
24
.8
43
 
25
.1
25
 
30
.5
40
 
 
25
.1
79
 
24
.2
41
 
21
.6
94
 
 
15
.6
20
 
15
.2
34
 
 
Sh
ar
pe
 
0.
48
8 
0.
49
1 
0.
33
7 
 
0.
39
0 
0.
38
6 
0.
31
7 
 
0.
42
6 
0.
34
4 
0.
49
9 
 
0.
21
9 
0.
23
6 
 
A
lp
ha
 (%
) 
10
.0
37
 
10
.5
74
 
9.
08
8 
 
7.
50
3 
7.
55
0 
7.
24
9 
 
8.
07
2 
5.
72
7 
8.
21
6 
 
2.
03
6 
2.
24
3 
 
(t-
st
at
) 
(2
.1
56
 *
) 
(2
.2
24
 *
) 
(1
.6
00
 ) 
 
(1
.5
14
 ) 
(1
.5
02
 ) 
(1
.3
04
 ) 
 
(1
.7
63
 ) 
(1
.2
39
 ) 
(1
.9
24
 ) 
 
(0
.6
12
 ) 
(0
.6
89
 ) 
  
61 
The first observation when analysing the results is that different anomalies behave 
differently in different asset classes. All strategies seem to provide positive returns in a 
long run but some strategies leave the others far behind; at least, when observing one 
asset class only. A good example would be momentum among stocks. Momentum strat-
egy for stocks yields Sharpe ratios above 0.6 regardless of the weighting scheme while 
the Sharpe ratio of the other strategies vary between 0.4 and 0.5. This is an impressive 
result considering the average Sharpe ratio among the stock indices, which was 0.26. 
An interesting and somewhat surprising observation was that an important factor behind 
this result was lower volatility provided by momentum strategy. In addition, momentum 
strategy yields superior and statistically significant three-factor alpha of 7.8−8.3 per 
cent per annum. 
Momentum strategy suits well for other asset classes as well, especially for govern-
ment bonds. However, it is never the best anomaly to be considered for the other asset 
classes. Another noticeable difference can be found in commodity strategies. Value 
strategy with either equal or ranking based weighting yields clearly better risk-return 
ratio than any other strategy for commodities. Additionally, value effect is the only one 
that is capable of producing significantly positive alpha among commodities. 
Government bonds are interesting as an asset class since all anomaly-based long-only 
strategies seem appealing for them. While the average Sharpe ratio among the bond 
indices is 0.632 and the maximum is 0.777 (Canada), all strategies provide a Sharpe 
ratio greater than 0.868. Even more surprising is that alpha is positive and statistically 
significant for all bond strategies. These findings separate government bonds from all 
other asset classes. One very likely reason for the results can be found from the bond 
indices themselves. They have not only a tendency to increase in value but they are also 
not very volatile. In fact, as the graph below shows, different strategies do not make a 
big difference between bond returns. Of course, the sample may have an important im-
pact on these findings. First, government bonds are usually not as risky as assets driven 
by corporate risk. As a result, the volatility is also considerably lower. On the other 
hand, interest rates have been mostly declining through the 21st century, which has 
made bonds look superior compared to many other asset classes. 
The reversal strategy is profitable in all asset classes but it only provides significantly 
positive alpha for government bonds. In addition, it usually does not shine next to the 
other strategies. However, it seems to fit especially well for currencies. Sharpe ratios in 
currencies are not very high in general but still reversal strategy manages to provide a 
tempting compensation on risk. Sharpe ratios of individual currencies are in a range of 
0.023−0.372 while the average is only 0.237. Reversal strategy, however, yields Sharpe 
ratios varying between 0.352 and 0.364, depending on weighting. This is clearly a better 
ratio than what any other strategy provided. 
  
62 
Overweighting assets with low idiosyncratic volatility yields somewhat disappoint-
ing results in comparison with the other strategies, especially, when it comes to mean 
returns or alpha. Risk-return wise, the most potential asset class for IVOL strategy is 
bonds where the results are at least competitive. The most evident and common result is 
that the strategy produces the lowest standard deviation in each asset class. That certain-
ly makes it important strategy in terms of portfolio allocation. 
Another interesting finding is that weighting methods have little impact on the results 
within the anomalies. In general, the results are usually in the same ballpark when com-
paring the strategies among different asset classes. For example, momentum strategy for 
stocks yields Sharpe ratios of 0.678, 0.680, and 0.638 when the assets in the winner 
group are weighted by equal weight, ranking, or IVOL, respectively. However, the 
number of sub groups seems to have a greater effect. Take, e.g., equally weighted stock 
momentum portfolios with different number or sub group. When the stock indices are 
sorted into 2, 3, and 5 groups, the most extreme winner portfolio provides Sharpe ratios 
of 0.662, 0.678, and 0.704, respectively. This obviously indicates that momentum effect 
is strong among stocks. In other words, the risk adjusted return is higher for those mo-
mentum portfolios including recently poorly performed stocks. 
In addition to static performance measures, it is relevant to analyse the dynamic be-
haviour of different strategies throughout the whole investment period. The following 
four graphs plot the portfolio performance of stocks, bonds, currencies, and commodi-
ties when different long-only strategies are applied. For each strategy, weighting meth-
od is selected according to the biggest Sharpe ratios showed in Table 6. 
  
63 
 
Figure 6 Performance of long-only strategies by asset classes; values presented 
with a logarithmic scale 
The graph strengthens the understanding that different anomalies behave in a differ-
ent manner in each asset class. While momentum is the most profitable for stocks, it is 
not as outstanding in the other asset classes. As the values are presented with a loga-
rithmic scale, movements of the lines remain comparable regardless of the initial value 
of the portfolio. As a consequence, it could be observed that even though, idiosyncratic 
volatility anomaly is not outstandingly beneficial, the returns are not very volatile ei-
ther. However, no strategy manages to avoid the collapse caused by the fall of Lehman 
Brothers. Nevertheless, the most distinct graph is the one of government bonds. It clear-
ly shows that all strategies are highly profitable for the bond indices and that the differ-
ence between them is not very strong. 
Strong movements emphasize the nature of some anomalies. For example, momen-
tum, despite of its great profitability, is reported to crash from time to time when the 
market is stressed and highly volatile (Daniel & Moskowitz, 2013). When looking at 
Figure 6, the strongest declines are seen during 2008, indeed. In fact, the biggest month-
ly shortfall was realised in September−October 2008 for stock momentum when the 
portfolio lost 44 % of its value. Knowing the negative skewness of momentum returns 
would make it important to adjust the model under a certain type of market environ-
ment. However, these considerations are left for further investigation. Additionally, re-
markable shortfalls are reported for other asset classes and anomalies as well. It also 
100#
200#
400#
800#
1600#
19
94
#
19
96
#
19
98
#
20
00
#
20
02
#
20
04
#
20
06
#
20
08
#
20
10
#
20
12
#
Stocks 
100#
200#
400#
800#
19
94
#
19
96
#
19
98
#
20
00
#
20
02
#
20
04
#
20
06
#
20
08
#
20
10
#
20
12
#
Bonds 
70#
140#
19
94
#
19
96
#
19
98
#
20
00
#
20
02
#
20
04
#
20
06
#
20
08
#
20
10
#
20
12
#
Currencies 
50#
100#
200#
400#
800#
1600#
19
94
#
19
96
#
19
98
#
20
00
#
20
02
#
20
04
#
20
06
#
20
08
#
20
10
#
20
12
#
Commodities 
50#
100#
200#
400#
800#
1600#
19
94
#
19
95
#
19
96
#
19
97
#
19
98
#
19
99
#
20
00
#
20
01
#
20
02
#
20
03
#
20
04
#
20
05
#
20
06
#
20
07
#
20
08
#
20
09
#
20
10
#
20
11
#
20
12
#
20
13
#
Commodi&es)
Value Momentum Reversal IVOL 
  
64 
seems that those of momentum strategies do not seem to occur more frequently. For 
example, within the stock strategies, there are eleven individual months for value, mo-
mentum, and reversal strategy when the drop has been over -10 %. 
Many of the strategies discussed previously were highly profitable. If the anomalies 
were strong it would be natural to assume that going short with an opposite strategy 
would yield even more superior returns. However, making such an assumption will 
probably cause a disappointment. Short strategies rarely seem to be profitable, which 
makes long-short strategies to suffer. Anyhow, some convincing results are reported, 
which become important when applying the anomalies to the Black−Litterman model. 
Table 7 presents respective results for long-short portfolios. The findings are im-
portant in finding significant risk premia to be utilised in expressing views. A signifi-
cant return in the table can be shortly interpreted as how many per cent a single long-
only strategy outperforms a corresponding short strategy. This is relevant information 
when deciding to what extent certain assets should outperform some others. Except the 
zero-cost aspect, all strategies follow the same principles as the long-only portfolios. In 
addition, the reported values are fully comparable with those in Table 6. 
 
  
65 
Ta
bl
e 
7 
R
es
ul
ts
 fo
r l
on
g-
sh
or
t p
or
tfo
lio
s 
Th
is
 ta
bl
e 
re
po
rts
 a
nn
ua
lis
ed
 m
ea
n 
w
ith
 t-
va
lu
e,
 s
ta
nd
ar
d 
de
vi
at
io
n,
 S
ha
rp
e 
ra
tio
, a
nd
 3
-f
ac
to
r a
lp
ha
 w
ith
 a
 re
sp
ec
tiv
e 
t-v
al
ue
 (N
ew
ey
-W
es
t c
or
re
ct
io
n 
ap
pl
ie
d)
 fo
r l
on
g-
sh
or
t p
or
tfo
lio
s. 
M
om
en
tu
m
 
is
 b
as
ed
 o
n 
pa
st
 1
2 
to
 1
 m
on
th
s 
an
d 
re
ve
rs
al
 o
n 
pa
st
 3
6 
to
 1
2 
m
on
th
s. 
In
 IV
O
L 
po
rtf
ol
io
 (w
ei
gh
tin
g)
, a
ss
et
s 
w
ith
 lo
w
 (h
ig
h)
 id
io
sy
nc
ra
tic
 v
ol
at
ili
ty
 a
re
 o
ve
rw
ei
gh
ed
. S
ta
tis
tic
al
 s
ig
ni
fic
an
ce
 is
 in
di
-
ca
te
d 
w
ith
 s
ta
rs
: *
**
, *
*,
 a
nd
 *
 r
ef
er
rin
g 
to
 p
-v
al
ue
s 
<0
.0
01
, <
0.
01
, a
nd
 <
0.
05
, r
es
pe
ct
iv
el
y.
 I
n 
th
is
 ta
bl
e,
 o
nl
y 
th
e 
ba
si
c 
se
tti
ng
 is
 te
st
ed
. O
th
er
 c
om
bi
na
tio
ns
 a
re
 c
on
si
de
re
d 
in
 f
ur
th
er
 a
na
ly
si
s. 
Po
rtf
ol
io
s 
ar
e 
co
ns
tru
ct
ed
 w
ith
 ro
lli
ng
 w
in
do
w
 e
st
im
at
io
n 
w
he
re
, a
fte
r e
ac
h 
m
on
th
, h
is
to
ric
al
 o
bs
er
va
tio
ns
 a
re
 u
se
d 
fo
r r
eb
al
an
ci
ng
 th
e 
po
rtf
ol
io
. I
n 
to
ta
l, 
da
ily
 o
bs
er
va
tio
ns
 c
ov
er
 th
e 
tim
e 
pe
rio
d 
of
 
11
.7
.1
99
4−
1.
4.
20
14
. 
 
 
V
al
ue
 
 
M
om
en
tu
m
 
 
R
ev
er
sa
l 
 
IV
O
L 
 
 
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
IV
O
L 
 
Eq
ua
l 
R
an
k 
St
oc
ks
 
M
ea
n 
(%
) 
-0
.9
28
 
0.
34
8 
0.
98
8 
 
6.
11
4 
5.
86
7 
6.
88
4 
 
-0
.3
92
 
-1
.1
79
 
-1
.9
26
 
 
-6
.7
20
 
-6
.4
40
 
 
(t-
st
at
) 
(-
0.
28
3 
) 
(0
.1
07
 ) 
(0
.2
72
 ) 
 
(1
.7
83
 ) 
(1
.7
58
 ) 
(1
.8
48
 ) 
 
(-
0.
13
1 
) 
(-
0.
40
8 
) 
(-
0.
54
9 
) 
 
(-
2.
09
3 
*)
 
(-
2.
02
1 
*)
 
 
St
de
v 
(%
) 
14
.8
62
 
14
.7
00
 
16
.4
58
 
 
15
.5
60
 
15
.1
40
 
16
.8
99
 
 
13
.5
52
 
13
.1
25
 
15
.9
29
 
 
14
.5
68
 
14
.4
58
 
 
Sh
ar
pe
 
-0
.0
62
 
0.
02
4 
0.
06
0 
 
0.
39
3 
0.
38
8 
0.
40
7 
 
-0
.0
29
 
-0
.0
90
 
-0
.1
21
 
 
-0
.4
61
 
-0
.4
45
 
 
A
lp
ha
 (%
) 
-0
.0
23
 
0.
81
1 
2.
33
6 
 
6.
57
7 
6.
20
4 
8.
00
4 
 
-0
.2
58
 
-1
.1
32
 
-2
.9
85
 
 
-7
.2
09
 
-7
.2
01
 
 
(t-
st
at
) 
(-
0.
00
8 
) 
(0
.2
68
 ) 
(0
.7
11
 ) 
 
(2
.0
76
 *
) 
(2
.0
29
 *
) 
(2
.3
42
 *
) 
 
(-
0.
09
2 
) 
(-
0.
42
2 
) 
(-
0.
91
1 
) 
 
(-
2.
36
3 
*)
 
(-
2.
46
5 
*)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
B
on
ds
 
M
ea
n 
(%
) 
-0
.3
15
 
-0
.0
49
 
0.
64
0 
 
2.
26
3 
2.
28
3 
2.
64
9 
 
2.
23
8 
1.
83
6 
1.
71
4 
 
-2
.4
09
 
-1
.3
05
 
 
(t-
st
at
) 
(-
0.
18
2 
) 
(-
0.
02
8 
) 
(0
.3
66
 ) 
 
(1
.0
38
 ) 
(1
.0
36
 ) 
(1
.2
17
 ) 
 
(1
.0
92
 ) 
(0
.9
07
 ) 
(0
.8
28
 ) 
 
(-
1.
28
 ) 
(-
0.
67
5 
) 
 
St
de
v 
(%
) 
7.
86
3 
8.
03
7 
7.
94
1 
 
9.
88
7 
9.
99
2 
9.
87
3 
 
9.
30
2 
9.
17
9 
9.
38
7 
 
8.
54
1 
8.
77
3 
 
Sh
ar
pe
 
-0
.0
40
 
-0
.0
06
 
0.
08
1 
 
0.
22
9 
0.
22
8 
0.
26
8 
 
0.
24
1 
0.
20
0 
0.
18
3 
 
-0
.2
82
 
-0
.1
49
 
 
A
lp
ha
 (%
) 
0.
00
1 
0.
24
8 
0.
71
6 
 
2.
63
0 
2.
61
5 
2.
87
1 
 
2.
45
8 
2.
05
5 
2.
07
4 
 
-1
.2
33
 
-0
.2
91
 
 
(t-
st
at
) 
(0
.0
00
 ) 
(0
.1
52
 ) 
(0
.4
23
 ) 
 
(1
.1
78
 ) 
(1
.1
86
 ) 
(1
.2
94
 ) 
 
(1
.2
56
 ) 
(1
.0
73
 ) 
(1
.0
65
 ) 
 
(-
0.
74
1 
) 
(-
0.
16
3 
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
C
ur
re
nc
ie
s 
M
ea
n 
(%
) 
1.
20
1 
1.
20
8 
1.
38
9 
 
1.
59
1 
1.
27
1 
1.
46
0 
 
2.
62
6 
2.
74
9 
2.
41
6 
 
-1
.3
77
 
-1
.0
43
 
 
(t-
st
at
) 
(0
.6
31
 ) 
(0
.6
13
 ) 
(0
.7
18
 ) 
 
(0
.7
55
 ) 
(0
.5
9 
) 
(0
.6
85
 ) 
 
(1
.4
19
 ) 
(1
.4
61
 ) 
(1
.2
44
 ) 
 
(-
0.
68
6 
) 
(-
0.
53
5 
) 
 
St
de
v 
(%
) 
8.
63
6 
8.
93
7 
8.
77
5 
 
9.
56
0 
9.
77
7 
9.
67
8 
 
8.
39
7 
8.
53
6 
8.
81
3 
 
9.
10
9 
8.
85
2 
 
Sh
ar
pe
 
0.
13
9 
0.
13
5 
0.
15
8 
 
0.
16
6 
0.
13
0 
0.
15
1 
 
0.
31
3 
0.
32
2 
0.
27
4 
 
-0
.1
51
 
-0
.1
18
 
 
A
lp
ha
 (%
) 
0.
82
3 
0.
88
7 
1.
06
3 
 
1.
74
9 
1.
44
6 
1.
68
2 
 
2.
47
6 
2.
59
8 
2.
20
4 
 
-1
.8
62
 
-1
.5
95
 
 
(t-
st
at
) 
(0
.5
04
 ) 
(0
.5
22
 ) 
(0
.6
37
 ) 
 
(0
.9
4 
) 
(0
.7
62
 ) 
(0
.8
93
 ) 
 
(1
.5
77
 ) 
(1
.6
48
 ) 
(1
.3
45
 ) 
 
(-
1.
12
9 
) 
(-
1.
01
2 
) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
C
om
m
od
iti
es
 
M
ea
n 
(%
) 
7.
80
0 
5.
19
5 
6.
89
1 
 
0.
71
9 
2.
62
6 
-2
.8
30
 
 
4.
83
2 
1.
37
5 
3.
59
5 
 
-1
1.
86
5 
-9
.6
56
 
 
(t-
st
at
) 
(1
.1
33
 ) 
(0
.7
21
 ) 
(0
.8
08
 ) 
 
(0
.0
86
 ) 
(0
.3
09
 ) 
(-
0.
37
 ) 
 
(0
.6
55
 ) 
(0
.1
75
 ) 
(0
.4
9 
) 
 
(-
1.
63
3 
) 
(-
1.
52
7 
) 
 
St
de
v 
(%
) 
31
.2
24
 
32
.6
95
 
38
.7
10
 
 
37
.9
33
 
38
.6
06
 
34
.7
24
 
 
33
.4
74
 
35
.6
69
 
33
.3
20
 
 
32
.9
58
 
28
.6
92
 
 
Sh
ar
pe
 
0.
25
0 
0.
15
9 
0.
17
8 
 
0.
01
9 
0.
06
8 
-0
.0
81
 
 
0.
14
4 
0.
03
9 
0.
10
8 
 
-0
.3
60
 
-0
.3
37
 
 
A
lp
ha
 (%
) 
6.
74
3 
4.
00
4 
5.
70
4 
 
1.
41
2 
3.
56
8 
-2
.5
94
 
 
4.
31
9 
0.
91
5 
2.
98
8 
 
-1
0.
63
4 
-8
.1
53
 
 
(t-
st
at
) 
(1
.2
17
 ) 
(0
.7
14
 ) 
(0
.8
95
 ) 
 
(0
.1
9 
) 
(0
.4
65
 ) 
(-
0.
42
 ) 
 
(0
.6
87
 ) 
(0
.1
31
 ) 
(0
.4
99
 ) 
 
(-
1.
89
8 
) 
(-
1.
49
6 
) 
66 
 
The first observation when comparing the results with those of long-only portfolios is 
that the level of statistical significance is extremely lower. While all long-only strategies 
for bonds provided significantly positive alpha, the zero-cost strategies are never as ap-
pealing. The most obvious reason for this is that going short on government bonds is not 
profitable. 
Momentum strategy, in average, appears as the most profitable, especially for stocks. 
Regardless of weighting method, all momentum strategies provide significantly positive 
alpha. Of all strategies and asset classes, momentum strategy for stocks is ultimately the 
most effective zero-cost strategy. Furthermore, the highest mean return and Sharpe ratio 
is achieved when the stocks with high idiosyncratic volatility are overweighed. In fact, 
the results show that high IVOL predicts higher returns, at least for stocks. When low 
IVOL strategy is applied, the stock portfolio provides noticeably negative alpha at a 
confidence level less than 0.005. However, just an opposite strategy seems well profita-
ble. When high IVOL strategy, with three sub groups and rank weighting, is applied to 
stocks, the Sharpe ratio becomes 0.391 per annum and alpha is nearly significantly posi-
tive with a p-value of 0.064. 
As discussed, idiosyncratic volatility divides opinions in finance literature. The re-
sults of all IVOL strategies here support more the findings of Merton (1987) or Goyal 
and Santa-Clara (2003). High idiosyncratic volatility, indeed, seems to have a positive 
impact on alpha. In fact, the results show that this connection exists within each asset 
class. For all asset classes, except currencies, the Sharpe ratio of low IVOL strategy is 
more negative than it is positive for other strategies. This is a strong evidence for exist-
ence of high IVOL effect although the alphas are not significant. 
The results on value effect are note quite as expected. Considering the work of Fama 
and French (1992, 1996), the absence of value premium is surprising. Of course, it cru-
cial to note that their value measure is applicable for stocks although, they show (1996) 
that negative five-year return is linked to book-to-market value. When different anoma-
lies are compared among asset classes, the negative five-year return realises the highest 
Sharpe ratio for commodities. Yet, significant value premium is not found. However, it 
seems that value premium whether it is significant or not, remains rather predictable. 
This is shown in a portfolio context when the value effect is applied alone to the 
Black−Litterman model. 
When different anomalies are compared, it is again obvious that they behave differ-
ently in each asset class. For example, value strategy yields appealing risk-return ratio 
for stocks in comparison to other asset classes. On the other hand, the same strategy for 
bonds or stocks provides even negative mean return with certain weighting methods. 
Reversal is another good example. For stocks, it can never be considered profitable as it 
almost always provides a negative premium. However, all the other asset classes usually 
benefit from the strategy, especially currencies. 
67 
 
67 
4.2 Black−Litterman portfolios 
When utilising different anomalies in a multiple asset class portfolio an important ques-
tion is, which anomaly should be considered for each asset class. One approach is to 
make the decision based on the Sharpe ratios realised from the factors analysed earlier. 
Portfolios containing multiple asset classes are optimised based on these factors with 
ultimate high Sharpe ratios. If all 124 factors are compared, the best Sharpe ratios are 
provided by the following factors: S_MOM_5_12_1_R, B_REV_2_36_12_E, 
FX_REV_3_36_12_IV_0, and C_REV_3_48_12_E. In short, only momentum or rever-
sal factors are considered for the Black−Litterman framework. This portfolio will be 
called Sharpe when analysing the results. The marking above defines the parameters 
selected for each factor, and it always consists of the following parts: Asset class, 
anomaly, number of groups, past months and those recent months ignored (for momen-
tum and reversal), weighting method, and 1 or 0 depending on whether high or low idi-
osyncratic volatility is overweighed. When IVOL is the actual anomaly, it is specifically 
indicated if high IVOL strategy is considered. 
Even the most tempting factor is not necessarily a proof of a predictable return pre-
mium. In that case a factor might not be beneficial in the Black−Litterman framework. 
Because of this, one interesting approach is to apply the same anomaly to all asset clas-
ses. Here, all the anomalies are considered with the basic settings discussed earlier. 
In addition to the Black−Litterman portfolios, three different benchmarks are ana-
lysed. Market portfolio is a natural choice for a benchmark, which in this study, is 
Standard & Poor’s 500 total return index. Another relevant benchmark is 1/! portfolio 
since it is often reported to outperform more sophisticated asset allocation strategies. 
Additionally, this is considered as a benchmark in the Black−Litterman framework be-
cause market portfolio is rather abstract when different asset classes are considered. The 
third competing portfolio is the global minimum variance portfolio with a short selling 
constraint. Since the mean-variance framework plays an important role behind the 
Black−Litterman model, it is relevant to see whether a more sophisticated model is 
worth the effort. 
  
68 
 
Table 8 Results of multiple asset class portfolios with the Black−Litterman 
strategy 
The table reports annual means, standard deviations, Sharpe ratios, and 3-factor alphas of different in-
vestment strategies. The Black−Litterman portfolios are based on anomalies, such as, value, momentum, 
reversal, and idiosyncratic volatility. The Sharpe portfolio mixes different anomalies depending on asset 
class specific Sharpe ratios. All BL portfolios are tangency portfolios, and thus, optimised to maximise 
the Sharpe ratio. The market is represented by S&P 500 index. Equal weight and minimum variance port-
folios are the other two benchmarks. All portfolios are long only and they are rebalanced after every 20 
trading days. Alpha is estimated with 3-factor model and the t-values are Newey-West corrected to con-
trol autocorrelation. 
11.7.1994− 
1.4.2014 
Sharpe VAL MOM REV IVOL Market 1/N GMV 
mean (%) 9.366 9.200 6.867 7.620 6.951 7.966 7.961 2.916 
(t-value) (4.240***) (4.241***) (2.91**) (3.506***) (2.856**) (3.357***) (3.345***) (3.502***) 
stdev (%) 10.023 9.841 10.708 9.860 11.044 10.764 10.799 3.778 
Sharpe 0.934 0.935 0.641 0.773 0.629 0.740 0.737 0.772 
alpha (%) 7.916 7.593 4.909 5.923 4.665 5.079 5.067 2.402 
(t-value) (3.575***) (3.739***) (2.180*) (2.863**) (1.978*) (2.452*) (2.438*) (2.621**) 
 
An impressive result is that all strategies provide significantly positive alpha. Risk-
return wise the Sharpe portfolio meets the expectations since it provides a highly com-
petitive Sharpe ratio of 0.934. Additionally, the annual mean and alpha are the highest 
of all strategies. The Sharpe portfolio yields in average 9.366 % per annum while the 
abnormal return is 7.916 % p.a. The alpha is also statistically highly significant. How-
ever, value anomaly is the most surprising when applied to all asset classes in the 
Black−Litterman strategy. It yields the highest Sharpe ratio of 0.935, which together 
with the Sharpe portfolio competes in its own league. For all other strategies, the ratio is 
considerably lower. 
Performance of the value strategy is outstanding considering that it does not quite 
stand out when comparing the factors. A potential reason for this performance is that the 
effect remains rather stable and thus predictable. Even if a simple long-short strategy 
does not perform well, predictability might add value in the Black−Litterman frame-
work. This is because the view is defined based on past month’s cumulative return, 
whether it is positive or negative. 
Even though the first two strategies are superior, all the different Black−Litterman 
portfolios are promising indeed. The remaining three strategies, momentum, reversal, 
and IVOL, provide somewhat similar results. Sharpe ratios of the strategies vary be-
tween 0.629 and 0.773. They also perform well against the competing benchmarks, alt-
hough, 1/N portfolio is really impressive in its simplicity. It provides very similar re-
sults with the three last mentioned strategies and even outperforms some of them in, 
e.g., Sharpe ratio or alpha. A good comprehension on how the different strategies per-
form is given by the following graph where out-of-sample performance of the strategies 
can be compared on a log scale. 
69 
 
69 
 
Figure 7 Performance of anomaly based Black−Litterman portfolios presented on 
a logarithmic scale 
Visual observation strengthens the perception given by Table 8. The figure clearly 
shows the superior out-of-sample performance of Sharpe and value strategies. In addi-
tion, sensitivity to macroeconomic events is well demonstrated by the lines. For exam-
ple, the latest financial distress launched by the fall of Lehman Brothers on 15th Sep-
tember 2008 seems to cause a drastic decline on most investment strategies. However, 
as the figure reveals, the Sharpe portfolio performs surprisingly well during those chaot-
ic times compared to most other strategies. 
4.3 Robustness checks 
In terms of real world implementations, it is critical to examine how sensitive the port-
folios are to changes in the underlying parameters. This analysis would raise whole new 
research questions but that goes beyond the scope of this study. However, a relevant 
question here is, are the results reliable when the setting is not exactly as suggested in 
the first place. In this analysis, the two best performing portfolios are tested, i.e., the 
Sharpe and value portfolio. These two are compared to 1/N portfolio, which obviously 
stands out as hard benchmark to win. 
Robustness of the portfolios is tested against different parameters. First, length of the 
investment period is an important parameter since it is closely related to transaction 
costs. Second, for some investors selling short is an opportunity, which deserves to be 
tested. Third, the Black−Litterman model has number of different parameters that might 
have a strong impact on the final allocation. Here, the effect of the risk aversion pa-
100#
200#
400#
800#
199
4 
199
5 
199
6 
199
7 
199
8 
199
9 
200
0 
200
1 
200
2 
200
3 
200
4 
200
5 
200
6 
200
7 
200
8 
200
9 
201
0 
201
1 
201
2 
201
3 
Sharpe 
Value 
1/N 
Reversal 
Momentum 
IVOL 
Market 
GMV 
70 
 
rameter, lambda, is assessed as it has a noticeable role in defining implied equilibrium 
returns. The default value for this parameter is three as discussed earlier. 
In order to assess the magnitude of the change, the Sharpe ratios are compared to 
each other. More specifically, a paired test of equality of Sharpe ratios is conducted to 
see if the Sharpe ratios differ statistically significantly from that of the 1/N portfolio. In 
the analysis, the method of Leung and Wong (2008) is followed and executed by R 
package SharpeR. 
Table 9 Robustness of test portfolios 
This table reports robustness of Sharpe and value portfolios to changing parameters in strategy implemen-
tation and over different time periods. Sharpe portfolio is a tangency portfolio where BL views consist of 
momentum for stocks and different versions of reversal for the rest of the asset classes. For the value 
portfolio, BL views are formed by value with the basic settings. The two portfolios provided the highest 
Sharpe ratios in the actual out-of-sample study. These portfolios are represented by the first benchmarks 
denoted by Basic. The other benchmark portfolio is the 1/N where equal weights are allocated when re-
balancing. Robustness is tested with three types of variables: length of rebalancing period, release of short 
selling restriction, and risk aversion parameter (lambda) of the BL model. Also, pre and post-Lehman 
periods are investigated separately. The breaking point is 15th Sep 2008. The strategies are back-tested 
using a rolling estimation window, where the whole data set covers the period of 11.7.1994−1.4.2014. 
 
 
Rebalancing Short  Risk Aversion Benchmark 
  5 days 60 days allowed  1 5 10 Basic 1/N 
PANEL A: Sharpe portfolio       
Total mean (%) 8.822 7.750 7.024  9.176 9.190 8.987 9.366 7.961 
 stdev (%) 10.337 10.124 24.730  10.569 9.647 9.128 10.023 10.799 
 Sharpe 0.853 0.766 0.284  0.868 0.953 0.985 0.934 0.737 
           Pre-Lehman mean (%) 10.037 7.437 12.027  10.442 9.925 9.657 10.272 8.588 
 stdev (%) 10.020 10.130 25.525  10.575 9.469 8.804 9.912 8.533 
 Sharpe 1.002 0.734 0.471  0.987 1.048 1.097 1.036 1.007 
           Post-Lehman mean (%) 5.715 8.551 -5.765  5.937 7.312 7.274 7.051 6.357 
 stdev (%) 11.216 10.226 22.976  10.677 10.182 9.990 10.404 15.139 
  Sharpe 0.510 0.836 -0.251  0.556 0.718 0.728 0.678 0.420 
PANEL B: Value portfolio                
Total mean (%) 9.436 8.942 10.362  8.969 9.167 8.748 9.200 7.961 
 stdev (%) 9.801 9.789 23.626  10.222 9.606 9.391 9.841 10.799 
 Sharpe 0.963 0.913 0.439  0.877 0.954 0.932 0.935 0.737 
           Pre-Lehman mean (%) 9.361 8.830 10.339  9.528 9.667 9.166 9.709 8.588 
 stdev (%) 9.455 9.336 24.718  10.061 8.971 8.464 9.410 8.533 
 Sharpe 0.990 0.946 0.418  0.947 1.078 1.083 1.032 1.007 
           Post-Lehman mean (%) 9.628 9.228 10.422  7.541 7.887 7.679 7.899 6.357 
 stdev (%) 10.670 10.923 21.031  10.692 11.101 11.458 10.906 15.139 
  Sharpe 0.902 0.845 0.496  0.705 0.710 0.670 0.724 0.420 
 
Profitability of the Sharpe portfolio is evident regardless of the observation period. 
With the original parameters, the strategy yields highly competitive Sharpe ratio, which 
is always higher than that of 1/N portfolio and most of the test portfolios. The results are 
rather stable both before and after the fall of the investment bank Lehman Brothers on 
71 
 
71 
15th September 2008. This shows that the strategy could be profitable in different in-
vesting environments. 
Rebalancing period becomes an important parameter, especially in terms of transac-
tion costs. Initially, portfolios are rebalanced every 20 trading days, i.e., once a month. 
More frequent trading has only a limited effect on the risk-return ratio of Sharpe portfo-
lio. In fact, both before and after the fall of Lehman Brothers, weekly rebalancing has 
even weakened the Sharpe ratio. On the other hand, quarterly rebalancing (60 trading 
days) provides competitive results as well. Especially after the Lehman bankruptcy, less 
frequent trading even improves risk-return ratio of the Sharpe portfolio. Nevertheless, 
the central finding is that returns of Sharpe portfolio are robust to different trading fre-
quencies. 
Same observation can be made about the value portfolio. Regardless of the holding 
period, the Sharpe ratios are always close to that of the original portfolio. Additionally, 
excluding pre-Lehman period, the Sharpe ratios are always higher than what 1/N portfo-
lio is able to provide. An interesting observation, however, is that the length of re-
balancing period has an opposite effect on the results compared to the Sharpe portfolio. 
Here, more frequent trading provides overall slightly higher Sharpe ratios than quarterly 
rebalanced portfolios. However, the difference is small. 
When short selling is allowed, the results collapse. For both Sharpe and value portfo-
lio, before and after the financial distress started in 2008, the Sharpe ratios are signifi-
cantly lower in comparison with the initial portfolio setting or the 1/N portfolio. In all 
cases, this is mostly due to variance which more than doubles in contrast to the original 
portfolio. 
Risk aversion measures the marginal reward an investor requires for additional risk. 
In the Black−Litterman framework, a higher risk aversion parameter implies greater 
expected excess returns. Even though, risk aversion would be assumed to affect the out-
of-sample returns the results prove otherwise. When the parameter is either increased or 
decreased from the initial three, mean return and standard deviation remain relatively 
stable. The same observation applies to both Sharpe and value portfolio, regardless of 
the investment period in focus. 
In order to statistically prove robustness of the portfolios, equality of Sharpe ratios 
was tested. All test portfolios were compared to the original one and equality of their 
Sharpe ratios were tested following the method of Leung and Wong (2008). In short, 
their method tests performance equality of multiple funds assuming independent and 
identically distributed (i.i.d.) returns. In order to cope with autocorrelation and het-
eroskedasticity issues, covariance matrix of Sharpe ratios is estimated using so-called 
HAC matrix proposed by Andrews (1991). Table 10 shows the p-values of Hotelling T2 
statistics. 
72 
 
Table 10 Equality of Sharpe ratios 
This table reports the p-values of Sharpe equality test results. Equality of Sharpe ratios is tested following 
the method of Leung and Wong (2008). Autocorrelation and heteroskedasticity is controlled by HAC 
covariance matrix proposed by Andrews (1991). Practical test is conducted using an R package, SharpeR. 
Sharpe and value portfolios are tangency portfolios where portfolio weights are solved by utilising the 
Black−Litterman model. Performance of the original portfolios is tested against those with modified input 
parameters, ceteris paribus. To test the robustness of a portfolio, rebalancing periods, short selling con-
straint, and risk aversion parameter (lambda) in the BL model are readjusted. 
,
Rebalancing, Short,selling, Risk,Aversion,
  5 days 60 days allowed 1 5 10 
Sharpe portfolio ,      
Total 0.418 0.158 0.001 0.174 0.607 0.556 
Pre-Lehman 0.758 0.006 0.010 0.198 0.736 0.464 
Post-Lehman 0.480 0.613 0.018 0.395 0.665 0.808 
Value portfolio ,      
Total 0.702 0.813 0.005 0.267 0.588 0.966 
Pre-Lehman 0.673 0.407 0.003 0.050 0.171 0.566 
Post-Lehman 0.230 0.497 0.510 0.891 0.874 0.746 
 
The p-values are in line with the findings made earlier. Both Sharpe and value portfolio 
are robust to changes in various input parameters. For almost all pairwise tests the p-
value is fundamentally above any reasonable rejection level. This means that the Sharpe 
ratios are not likely to differ from that given by the original portfolio. In addition, the 
results are mostly insensitive to the observation period. 
Even though a vast majority of the p-values are high, a few interesting findings are 
made where the null can be rejected. After seeing the results of unrestricted portfolios, it 
is not a surprise that the Sharpe ratio is statistically different from the original long-only 
portfolio. However, in two cases the difference is significant at 5 % level when observ-
ing a specific timeframe. Firstly, the Sharpe portfolio yields 0.302 lower risk-return 
ratio before the fall of Lehman when rebalanced quarterly. This difference is highly 
significant with p-value of 0.006. Secondly, when risk aversion of Value portfolio is set 
to 1, the Sharpe ratio before the financial crisis is 0.085 units lower at 5 % level of sig-
nificance. When testing equality of multiple Sharpe ratios (excluding those of uncon-
strained portfolios) within these specific time periods, the results of F-statistics support 
this same finding. Before September 2008, there are one or more portfolios the Sharpe 
ratio of which is significantly different from the others. For Sharpe portfolios, the p-
value is 0.017. Accordingly, the p-value for Value portfolios is 0.103. Most probably, 
the portfolios making the significant difference are exactly the two mentioned above. 
Within other observation periods the p-values were drastically higher. 
73 
 
73 
5 CONCLUSION 
The primary objective of this thesis was to combine anomalies of asset pricing theory 
with applications of the modern portfolio theory. Whenever there is a persistent market 
anomaly, additional information should be applied to the portfolio level. The link be-
tween these two fields was made with the Black−Litterman model, which allows inves-
tors to set their own views as a part of the optimisation problem. The results speak for 
themselves. Market anomalies, such as value, momentum, long-term reversal, and idio-
syncratic risk, provide valuable information to optimal asset diversification. Moreover, 
the findings extend the literature as the results can be utilised across different asset clas-
ses. 
The subject was studied by investigating out-of-sample performance of different 
anomaly-based portfolio strategies. The first step was to construct factors on different 
anomalies among stocks, bonds, currencies, and commodities. These factors were built 
by creating long-short portfolios, where underlying assets were selected to the portfolio 
based on a specific anomaly. For value and momentum, the methodology followed As-
ness et al. (2013) to a great extent. Reversal effect was measured in a similar manner 
with momentum, based on the past returns (see, e.g., McLean 2010). Idiosyncratic vola-
tility was captured from three-factor regression as suggested by Ang et al. (2009). When 
it came to creating the portfolios, different weighting schemes were considered to assess 
the power and robustness of the anomaly. Out-of-sample returns of these portfolios 
were studied to find if there is a significant positive premium. The second and the most 
important step of this study was to generate views based on return premium of the fac-
tors and apply them to portfolio optimisation. The objective was that the most profitable 
anomaly for each asset class could be considered in the same optimisation procedure. 
The main findings can be divided into two groups. First, market anomalies seem to 
provide tempting return patterns, however, they seldom result in significantly positive 
return premium within different asset classes. Second, persistent return premium gives 
valuable information to investors to be utilised in multi asset class portfolio optimisa-
tion. According to the results, stocks demonstrate 12-month momentum with signifi-
cantly positive three-factor alpha. The findings are strongly supported by the existing 
literature. Additionally, high idiosyncratic volatility is positively related to stock returns 
on country level. The effect is even stronger when momentum portfolio is scaled by 
idiosyncratic volatility. The rest of the anomalies are not as powerful as significantly 
positive alphas are not found. However, Sharpe ratios indicate that different anomalies 
might offer valuable information to portfolio optimisation. The second part puts this 
hypothesis into a test when the best factors are chosen from the 124 test portfolios simp-
ly based on Sharpe ratio. The factors are further investigated as a source of information. 
74 
 
The most outstanding finding of this thesis is that anomaly based factors include val-
uable information to enhance efficient portfolio diversification. When the highest 
Sharpe ratios for each asset class are picked from the 124 factors and applied to the 
Black−Litterman model, the final portfolio results in superior risk-return combination. If 
alphas are ignored, the highest Sharpe ratios are provided by momentum strategy for 
stocks and long-term reversal for the rest of the asset classes. When historical perfor-
mance of these factors is used to generate views for different asset classes, the result is a 
tangency portfolio that outperforms all the benchmarks, i.e., minimum variance and 1/N 
portfolios. Competing single anomaly strategies are also appealing, especially value 
strategy, which basically performs as well as the previously mentioned Sharpe strategy. 
In addition, these results are robust to real life implementation issues, such as, rebalanc-
ing frequency, trading period, and investor’s risk aversion. However, when short selling 
is allowed, poorer results are shown, which might actually be good news for most inves-
tors in terms of implementation. 
The scope of this thesis is wide in order to shed light on understanding the character-
istics of market anomalies as a part of portfolio strategy. The current literature, if it co-
vers both market anomalies and different asset classes (see, e.g., Asness et al. 2013), 
does not suggest how to utilise these characteristics in real life applications. On the oth-
er hand, when these effects are investigated in portfolio context, the study is often lim-
ited to equities only (see, e.g., Babameto & Harris 2008). However, as a coin has two 
sides, this thesis has some major drawbacks as well. First of all, a wide angle unfortu-
nately often requires simplifications or that some fields have to be examined superficial-
ly. Here, this can be seen in the depth that individual anomalies are studied. When four 
asset classes and four different anomalies are in focus, it is not of interest to go too deep 
into specifics. In addition, the Black−Litterman model has various different parameters, 
the calibration of which would have both practical and theoretical importance. Instead, 
as the main focus of this thesis was to give a broad comprehension on the benefits of 
anomaly based portfolio strategies across the asset classes, some of the details had to be 
treated more or less as given. 
This thesis surely answers to many practical needs regarding optimal portfolio diver-
sification. However, there are questions that are worth a more detailed examination. On 
the other hand, it would be interesting to see how the methods, utilised within this the-
sis, could be taken to the next level. As an example, developing more exact measures 
for the anomalies would make the results more reliable. For example, value for corpo-
rate bonds, within this thesis, is explained with the shape of the yield curve. An im-
portant step would be to study more thoroughly, how should different anomalies be un-
derstood within each asset class. This would extend the current literature, which to a 
great extent relies on stocks. 
75 
 
75 
Additionally, there is still no common opinion on the link between idiosyncratic risk 
and stock returns. Here idiosyncratic risk results support some theoretical frameworks 
but are contradictory to some of the latest literature. The question becomes even more 
attractive in terms of defining idiosyncratic risk on other asset classes. In general, when 
new risk factors themselves are of focus, more theoretical testing would be recommend-
ed, e.g., testing with Fama-MacBeth regression. 
When it comes to taking anomaly based portfolio optimisation to the next level, 
some further investigation on predicting factor returns will be needed. For example, 
Babameto and Harris (2008) already apply forecasting regressions for value and mo-
mentum whereas here the expected future premium is simply calculated from historical 
returns. If there was a reliable forecast for this premium, investors could dynamically 
set their views based on the most appealing anomaly whenever it is time to rebalance 
the portfolio. It is obvious that even the most effective market anomaly does not work 
consistently in all market environments. 
76 
 
REFERENCES 
Accominotti, Oliver − Chambers, David (2014) Out-of-sample evidence on the returns 
to currency trading. Centre for economic policy research discussion pa-
pers, No.DP9852, London. 
Andrews, Donald W K. (1991) Heteroskedasticity and autocorrelation consistent covar-
iance matrix estimation. Econometrica, Vol. 59 (2), 817–858. 
Ang, Andrew − Hodrick, Robert J. − Xing, Yuhang − Zhang, Xiaoyan (2006) The 
cross-section of volatility and expected returns. Journal of Finance, Vol. 
61 (1), 259−299. 
Ang, Andrew − Hodrick, Robert J. − Xing, Yuhang − Zhang, Xiaoyan (2009) High idi-
osyncratic volatility and low returns: International and further U.S. evi-
dence. Journal of Financial Economics, Vol. 91 (1), 1–23. 
Asness, Clifford S − Moskowitz, Tobias J. − Pedersen, Lasse H. (2013) Value and mo-
mentum everywhere. Journal of Finance, Vol. 68 (3), 929−985. 
Babamento, Elton − Harris Richard T. (2008) Exploiting predictability in the returns to 
value and momentum investment strategies: A portfolio approach. Work-
ing paper. University of Exeter. <http://papers.ssrn.com/sol3/papers.cfm? 
abstract_id=1302772>, retrieved 3.4.2014 
Black, Fischer (1989) Universal hedging: Optimizing currency risk and reward in inter-
national equity portfolios. Financial Analysts Journal, Vol. 45 (4), 16−22. 
Black, Fischer − Litterman, Robert (1992) Global portfolio optimization. Financial An-
alysts Journal, Vol. 48 (5), 28−43. 
Black, Fischer − Scholes, Myron (1973) The pricing of options and corporate liabilities. 
Journal of Political Economy, Vol. 81 (3), 637−654. 
Carhart, Mark M. (1997) On persistence in mutual fund performance. Journal of Fi-
nance, Vol. 52 (1), 57−82. 
Chaves, Denis B. − Arnott, Robert D. (2012) Rebalancing and the value effect. Journal 
of Portfolio Management, Vol. 38 (4), 59−73. 
Cheung, Wing (2010) The Black−Litterman model explained. Journal of Asset Man-
agement, Vol. 11 (4), 229−243. 
Chopra, Navin − Lakonishok, Josef − Ritter, Jay R. (1992) Measuring abnormal per-
formance: Do stocks overreact. Journal of Financial Economics, Vol. 31 
(2), 235−268. 
Cotter, John (2005) Tail behaviour of the euro. Applied Economics, Vol. 37 (7), 
827−840. 
77 
 
77 
Curtillet, Jean-Christophe − Dieudonné, Mathieu (2007) Sector-specific optimum asset 
allocation: an example for non-life insurers. Journal of Asset Management, 
Vol. 7 (6), 404−411. 
Daniel, Kent D. − Moskowitz, Tobias J (2013) Momentum crashes. Working paper. 
Columbia University and University of Chicago. 
<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2371227>, retrieved 
7.6.2015. 
DeBondt, Werner F. M. − Thaler, Richard (1985) Does the stock market overreact? 
Journal of Finance, Vol. 10 (3), 793−805. 
DeLong, James B. − Shleifer, Andrei − Summers, Lawrence H. − Waldmann, Robert J. 
(1990) Positive feedback investment strategies and destabilizing rational 
speculation. Journal of Finance, Vol. 45 (2), 379−395. 
Elton, Edwin J. − Gruber, Martin J. − Brown, Stephen J. − Goetzmann, William N. 
(2011) Modern portfolio theory and investment analysis. 8th ed. John 
Wiley & Sons Inc., Hoboken, NJ. 
Fabozzi, Frank J. − Focardi, Sergio M. − Kolm, Petter N. (2010) Quantitative equity 
investing: Techniques and strategies, John Wiley & Sons Inc., Hoboken, 
NJ. 
Fama, Eugene (1970) Efficient capital markets: A review of theory and empirical work. 
Journal of Finance, 25 (2), 383–417. 
Fama, Eugene F. − MacBeth, James D. (1973) Risk, return, and equilibrium: Empirical 
tests. Journal of Political Economy, Vol. 81 (3), 607−636. 
Fama, Eugene F. − French, Kenneth R. (1992) The cross-section of expected stock re-
turns. Journal of Finance, Vol. 47 (2), 427−465. 
Fama, Eugene F. − French, Kenneth R. (1993) Common risk factors in the returns on 
stocks and bonds. Journal of Financial Economics, Vol. 33 (1), 3−56. 
Fama, Eugene F. − French Kenneth R. (1996) Multifactor explanations of asset pricing 
anomalies. Journal of Finance, Vol. 51 (1), 55−84. 
Fama, Eugene F. − French, Kenneth R. (2012) Size, value, and momentum in interna-
tional stock returns. Journal of Financial Economics, Vol. 105 (3), 
457−472. 
Fink, Jason D. − Fink, Kristin E. − He, Hui (2012) Expected Idiosyncratic Volatility 
Measures and Expected Returns. Financial Management (Wiley-
Blackwell), Vol. 41 (3), 519−553. 
Fu, Fangjian (2009) Idiosyncratic risk and the cross-section of expected stock returns. 
Journal of Financial Economics, Vol. 91 (1), 24−37. 
78 
 
Gerakos, Joseph − Linnainmaa, Juhani (2012) Decomposing value. Working paper. 
University of Chicago. < http://faculty.chicagobooth.edu/joseph. 
gerakos/PDFs/GerakosLinnainmaa_DecomposingValue.pdf>, retrieved 
3.6.2014 
Goyal − Santa-Clara (2003) Idiosyncratic risk matters! Journal of Finance, Vol. 58 (3), 
975−1007. 
Griffin, John M. (2002) Are the Fama and French factors global or country specific? 
Review of Financial Studies, Vol. 15 (3), 783−803. 
He, Guangliang − Litterman, Robert (1999) The intuition behind Black−Litterman mod-
el portfolios. Investment management research, Goldman, Sachs & Com-
pany. 
Hjalmarsson, Erik (2011) Portfolio diversification across characteristics. Journal of In-
vesting, Vol. 20 (4), 84−88. 
Idzorek, Thomas M. (2005) A step-by-step guide to the Black−Litterman model. Ibbot-
son Associates. 
Jagadeesh, Narasimhan − Titman, Sheridan. (1993) Returns to buying winners and sell-
ing losers: Implications for market efficiency. Journal of Finance, Vol. 48 
(1), 65−91. 
Jobson, J. D. − Korkie, Bob M. (1980) Estimation for Markowitz efficient portfolios. 
Journal of The American Statistical Association. Vol. 75 (371), 544-554. 
Jordan, Steven J. (2012) Time-varying risk and long-term reversals: A re-examination 
of the international evidence. Journal of International Business Studies, 
Vol. 43 (2), 123−142. 
Keim, Donald B. (1989) Trading patterns, bid-ask spreads, and estimated security re-
turns: The case of common stocks at calendar turning points. Journal of 
Financial Economics, Vol. 25 (1), 75−97. 
Keim, Donald B. (2008) Financial market anomalies. In: The New Palgrave Dictionary 
of Economics. 2nd edition, eds. Steven N. Durlauf − Lawrence E. Blume. 
Ledoit, Oliver − Wolf, Michael (2004) A well-conditioned estimator for large-
dimensional covariance matrices. Journal of Multivariate Analysis, Vol. 
88 (2), 365−411. 
Leung, Pui-Lam − Wong, Wing-Keung (2008) On testing the equality of multiple 
Sharpe ratios, with application on the evaluation of iShares. Journal of 
Risk, Vol. 10 (3), 15–30. 
Lintner, John (1965) The valuation of risk assets and the selection of risky investments 
in stock portfolios and capital budgets. Review of Economics and Statis-
tics, Vol. 47 (1), 13−37. 
79 
 
79 
Lo, Andrew W. − MacKinlay, A. Craig (1990) When are contrarian profits due to stock 
market overreaction? Review of Financial Studies. Vol. 3 (2), 175−205. 
Markowitz, Harry (1952) Portfolio selection. Journal of Finance, Vol. 7 (1), 77-91. 
McLean, David R. (2010) Idiosyncratic risk, long-term reversal, and momentum. Jour-
nal of Financial & Quantitative Analysis. Vol. 45 (4), 883−906. 
Menkhoff, Lukas – Sarno, Lucio − Schmeling, Maik − Schrimpf, Andreas (2012) Cur-
rency momentum strategies. Journal of Financial Economics, Vol. 106 
(3), 660−684. 
Merton, Robert C. (1987) A simple model of capital market equilibrium with incom-
plete information, Journal of Finance 42 (3), 483−510. 
Michaud, Richard O. (1989) The Markowitz optimization enigma: Is ‘optimized’ opti-
mal? Financial Analysts Journal, Vol. 45 (1), 31−42. 
Michaud, Richard O. − Esch, David N. − Michaud, Robert O. (2013) Deconstructing 
Black−Litterman: How to get the portfolio you already knew you wanted. 
Journal of Investment Management, Vol. 11 (1), 6−20. 
Mirza, Nawazish − Afzal, Ayesha (2011) Size and value premium in international port-
folios: Evidence from 15 European countries. Finance a Uver: Czech 
Journal of Economics & Finance, Vol. 61 (2), 173−190. 
Newey, Whitney K. − West, Kenneth D. (1987) A simple, positive semi-definite, het-
eroskedasticity and autocorrelation consistent covariance matrix. Econo-
metrica, Vol. 55 (3), 703–708. 
Rachev, Svetlozar T. − Hsu, John S. J. − Bagasheva, Biliana S. (2008) Bayasian meth-
ods in finance. John Wiley & Sons Inc., Hoboken, NJ. 
Rasmussen, Mikkel (2003) Quantitative portfolio optimisation, asset allocation and risk 
management, Palgrave Macmillan. Hampshire. 
Satchell, Stephen − Scowcroft, Alan (2000) A demystification of the Black−Litterman 
model: Managing quantitative and traditional portfolio construction. Jour-
nal of Asset Management, Vol. 1 (2), 138−150. 
Sharpe, William F. (1964) Capital asset prices: A theory of market equilibrium under 
conditions of risk. Journal of Finance, Vol. 19 (3), 425–442. 
Sharpe, William F. (1974) Imputing Expected Security Returns From Portfolio Compo-
sition. Journal of Financial & Quantitative Analysis, Vol. 9 (3), 463−472. 
Sharpe, William F. (1994) “The Sharpe Ratio”. Journal of Portfolio Management, Vol. 
21 (1), 49−58. 
Schwert, William G. (2003) Anomalies and market efficiency. In: Handbook of the eco-
nomics and finance, eds. George M. Constantinides − Milton Harris − 
René M. Stultz, 939−972. Elsevier. B.V., Netherlands. 
80 
 
Stattman, Dennis (1980) Book values and stock returns. The Chicago MBA: A Journal 
of Selected Papers 4, 25−45. 
Tsay, Ruey S (2005) Analysis of financial time series. 2nd ed. John Wiley & Sons, Inc., 
Hoboken, NJ. 
Vayanos, Dimitri − Woolley, Paul (2013) An institutional theory of momentum and 
reversal. Review of Financial Studies, Vol. 26 (5), 1087−1145. 
Walters, Jay (2011) The Black−Litterman model in detail. Working paper. Boston Uni-
versity. <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1314585>, 
retrieved 3.4.2014 
81 
 
81 
APPENDIX CODE FOR THE EMPIRICAL STUDY 
# LOAD DATA FOR EMPIRICAL STUDY 
============================================================= 
wdpath <- "/directory/path/" 
setwd(wdpath) 
srcfile<-"Functions.r" 
source(srcfile) 
file<-"Price_data.csv"  
ff3<-"FF3_factors.csv" # Fama-French 3-factor data from Kenneth French's website 
values_file<- "Value.csv" # Daily value estimates for each asset 
 
rawdata=read.csv(file, header=T, sep=";", dec=".", na.strings="N/A") 
ff3=read.csv(ff3, header=T, sep=";", dec=".", na.strings="N/A") 
values=read.csv(values_file, header=T, sep=";", dec=".", na.strings="N/A") 
val= values[,2:48] 
 
P<-rawdata[,2:(dim(rawdata)[2]-1)] # Asset prices 
PM<-rawdata[,(dim(rawdata)[2])] # Market portfolio prices (MSCI World) 
ff3_fact <- ff3[,(2:5)] 
 
T<-dim(P)[1] 
N<-dim(P)[2] 
 
RE=rbind(rep(0,N),data.frame(diff(log(data.matrix(P))))) #log return matrix 
RM=rbind(rep(0,1),diff(log(data.matrix(PM)))) #log-return vector of market returns 
#____________________________________________________________ 
 
# FUNCTIONS FOR EMPIRICAL STUDY 
============================================================= 
library(quantmod) 
library(quadprog) 
library(fGarch) 
library(tseries) 
library(lmtest) 
library(sandwich) 
library(MASS) 
library(SharpeR) 
 
# Function for covariance matrix ******* 
# {sample} or {SI} "single-index" 
covMat = function( 
  xx, 
  RMark, 
  N=N, 
  method="sample" ) {  
  if(method=="sample"){covMat<- cov(xx)}  
   
  if(method=="SI"){ 
    betas<-rep(0,N) 
82 
 
    residuals<-rep(0,N) 
      for (k in 1:N){ #run market regressions, pick betas and residual terms 
        k.lm= lm(xx[,k]~RMark) 
        betas[k]=coeftest(k.lm, vcov=NeweyWest)[2,1] # first coefficient 
        residuals[k]=summary(k.lm)$'sigma'^2 
      } 
    Mvar=as.numeric(var(RM)) 
    covMat<-Mvar*(betas%*%t(betas))+diag(residuals) #Define sinlge-index matrix 
  } 
   
   
  covMat <- matrix(covMat,nrow=ncol(xx),ncol=ncol(xx)) 
} 
#____________________________________________________________ 
# Calculate Efficient frontier 
 
effFrontier = function (averet, rcov, nports = 20, shorts=T, wmax=1) { 
  mxret = max(abs(averet))*1.8 
   
  mnret = 0 
  n.assets = ncol(averet) 
  reshigh = rep(wmax,n.assets) 
  if( shorts ) 
  { 
    reslow = rep(-wmax,n.assets) 
  } else { 
    reslow = rep(0,n.assets) 
} 
  min.rets = seq(mnret, mxret, len = nports) 
  vol = rep(NA, nports) 
  ret = rep(NA, nports) 
  weights = matrix(rep(NA, n.assets*nports),ncol = n.assets, nrow = nports) 
  for (k in 1:nports) 
  { 
    port.sol = NULL 
    try(port.sol <- portfolio.optim(x=averet, pm=min.rets[k], covmat=rcov, 
                                    reshigh=reshigh, reslow=reslow,shorts=shorts),silent=T) 
    if ( !is.null(port.sol) ) 
    { 
      vol[k] = sqrt(as.vector(port.sol$pw %*% rcov %*% port.sol$pw)) 
      ret[k] = averet %*% port.sol$pw 
      weights[k,] = port.sol$pw    } 
  } 
   
  return(list(vol = vol, ret = ret, weights=weights)) 
} 
# ____________________________________________________ 
# Max Sharpe Ratio Portfolio function ******************************** 
 
maxSharpe = function (averet, rcov, shorts=T, wmax = 1){ 
  optim.callback = function(param,averet,rcov,reshigh,reslow,shorts) 
83 
 
83 
  { 
    port.sol = NULL 
    try(port.sol <- portfolio.optim(x=averet, pm=param, covmat=rcov, 
                        reshigh=reshigh, reslow=reslow, shorts=shorts), silent = T) 
    if (is.null(port.sol)) { 
      ratio = 10^9 
    } else { 
      m.return = averet %*% port.sol$pw 
      m.risk = sqrt(as.vector(port.sol$pw %*% rcov %*% port.sol$pw)) 
      ratio = -m.return/m.risk 
      assign("w",port.sol$pw,inherits=T) 
    } 
    return(ratio) 
  } 
  ef = effFrontier(averet=averet, rcov=rcov, shorts=shorts, wmax=wmax, nports = 200) 
 
  n = ncol(averet) 
  reshigh = rep(wmax,n) 
  if( shorts ) { 
    reslow = -reshigh 
  } else { 
    reslow = rep(0,n) 
  } 
  max.sh = which.max(ef$ret/ef$vol) 
   
  w = rep(0,ncol(averet)) 
  xmin = optimize(f=optim.callback, interval=c(ef$ret[max.sh-1], upper=ef$ret[max.sh+1]), 
                  averet=averet,rcov=rcov,reshigh=reshigh,reslow=reslow,shorts=shorts) 
  
  m.return= averet %*% w 
  m.risk = sqrt(as.vector(w %*% rcov %*% w)) 
  ratio = m.return/m.risk 
  return(list(Sharpe = ratio, ret = m.return, vol=m.risk, weights=w)) 
} 
# __________________________________________________________________ 
 
# Global Minimum Variance Portfolio ***************************************************** 
 
GMV = function(covar, 
      N = ncol(covar), 
      shorts=FALSE){ 
   
  zeros <- array(0, dim = c(N,1)) 
   
  if(shorts==TRUE){ 
  # Optimization to generate minimum variance portfolio without short selling constraint 
    aMat  <- t(array(1, dim = c(1,N))) 
    res <- solve.QP(covar, zeros, aMat, bvec=1, meq = 1)} else { 
   
  # Optimization with short selling constraint (i.e. non-negative weights) 
    aMat <- cbind(t(array(1, dim = c(1,N))), diag(N)) 
84 
 
    b0 <- as.matrix(c(1, rep.int(0,N))) 
    res <- solve.QP(covar, zeros, aMat, bvec=b0, meq = 1)} 
   
  return(res)} 
# ___________________________________________________________________ 
# Value function ************************************************************************* 
 
value=function( 
  xx, 
  vals, 
  ff3_load=NA, 
  percentiles=3, 
  weights="equal",    # {equl,rank,IVOL} 
  method=method, 
  high_IVOL=1,  # 0 if bigger weight for assets with low idiosyncratic risk 
  showOutput=TRUE){ 
   
  t <- NROW(xx) 
  cols <- NCOL(xx) 
   
  sumReturns <- colSums(xx[(t-249):(t),]) # Cumulative return for all assets 
  no_rets<- 250 # To calculate mean returns 
   
  rankVec<-as.vector(rank(vals)) # Rank by value factors (Small values first) 
  chunk <- function(xx,n) {split(xx, factor(sort(rank(xx)%%n)))} #function to split a vector 
   
  subSets<-chunk(1:cols,percentiles) 
   
  # Create vectors for the high and low value assets 
  H_Group<-eval(parse(text=paste(sep="", "subSets$'", percentiles-1, "'"))) # High value ranks 
  L_Group<-subSets$'0' # Low value ranks 
   
  L1<-length(H_Group) 
  L2<-length(L_Group) 
  L<-min(L1,L2) # Vector lengths will be adjusted according to the shorter one 
   
  H_Group <- H_Group[(L1-L+1):L1] 
  Hvalue<-match(H_Group,rankVec) # Which assets we pick 
  HvalueRets<-rep(NA,L) 
  HvalueVals<-rep(NA,L) 
  for (c in 1:L) {HvalueRets[c]<-sumReturns[Hvalue[c]]} # Vector of winners returns 
  for (c in 1:L) {HvalueVals[c]<-vals[Hvalue[c]]} 
  winMeans <- HvalueRets/no_rets # Vector of winner mean returns  
   
  L_Group <- L_Group[(L2-L+1):L2] 
  Lvalue<-match(L_Group,rankVec) # Which assets we pick 
  LvalueRets<-rep(NA,L) 
  LvalueVals<-rep(NA,L) 
  for (c in 1:L) {LvalueRets[c]<-sumReturns[Lvalue[c]]} # Vector of losers returns 
  for (c in 1:L) {LvalueVals[c]<-vals[Lvalue[c]]} # Vector of losers values 
   
85 
 
85 
  losMeans <- LvalueRets/no_rets # Vector of loser mean returns 
   
  # Weighting method 
  wHigh<-rep(NA,L) 
  wLow<-rep(NA,L) 
  if(weights=="equal"){      # EQUAL WEIGHTING 
    wHigh<-rep(1/L,L) 
    wLow<- -(rep(1/L,L)) 
  } 
  if(weights=="IVOL"){      # WEIGHTING BY IDIOSYNCRATIC RISK 
    # Portfolio is weighted based on idiosyncratic risk 
    IVOL <- IR(xx, RMark,ff3_load, method=method, high_IVOL=high_IVOL)$"IVOL" 
    wHigh <- IVOL[Hvalue] 
    wHigh <- wHigh/sum(wHigh) 
     
    wLow <- 1/IVOL[Lvalue] 
    wLow <- -wLow/sum(wLow) 
     
  } 
  if(weights=="rank"){      # RANK WEIGHTING 
    # Weights calculated according to ranks 
    # Assets with high value should get the biggest weight 
     
wHigh<-H_Group 
    scale <- 1/(sum(wHigh)) # weights need to be scaled to 100 % 
     
    wHigh<-wHigh*scale 
    # Assets with low value should get the biggest short position 
    wLow <- L_Group 
    scale <- 1/(sum(wLow)) 
    wLow <- -wLow*scale # Low value weights are presented as absolute values 
  } 
   
  # Count weights for all assets   
  weights <- rep(0,cols) 
  for(i in 1:L){weights[Hvalue[i]] <- wHigh[i]} # set High value weights 
  for(i in 1:L){weights[Lvalue[i]] <- wLow[i]}  # set Low value weights 
 
  if(showOutput==TRUE){ 
    return(list(high=Hvalue,winReturns=winMeans,HighValueWeights=wHigh, 
                low=Lvalue,losReturns=losMeans,LowValueWeights=wLow,allWeights=weights)) 
  }  
} 
 
#  Momentum ************************************************************************* 
momentum=function( 
  xx, 
  RMark=NA, 
  ff3_load=NA, 
  percentiles=3, 
  pastMonths=12, 
86 
 
  monthsTill=1, 
  weights="equal",    # {equl,rank,IVOL} 
  method=method, 
  high_IVOL=1, # 0 if big weight for assets with low idiosyncratic risk 
  showOutput=TRUE){ 
   
  t <- NROW(xx) 
  cols <- NCOL(xx) 
   
# Cumulative return for all assets   
sumReturns <- colSums(xx[(t-(pastMonths)*20):(t-monthsTill*20),])  
  no_rets<- length(xx[(t-(pastMonths)*20):(t-monthsTill*20),]) # To calculate mean returns 
  rankVec<-as.vector(rank(sumReturns)) # Rank by past performance (winners --> big values) 
  chunk <- function(xx,n) {split(xx, factor(sort(rank(xx)%%n)))} #function to split a vector 
  subSets<-chunk(1:cols,percentiles) 
   
  # Create vectors for the winners and losers 
  H_Group<-eval(parse(text=paste(sep="", "subSets$'", percentiles-1, "'"))) # Winner ranks 
  L_Group<-subSets$'0' # Loser ranks 
   
  L1<-length(H_Group) 
  L2<-length(L_Group) 
  L<-min(L1,L2) # Vector lengths will be adjusted according to the shorter one 
   
  H_Group <- H_Group[(L1-L+1):L1] 
  winners<-match(H_Group,rankVec) # Which assets we pick 
  winRets<-rep(NA,L) 
   
  sumReturns2 <- colSums(xx[(t-249):t,]) # Cumulative one year return of all assets 
  for (c in 1:L) {winRets[c]<-sumReturns2[winners[c]]} # Vector of winners returns 
  winMeans <- winRets/250 # Vector of winner mean returns 
   
  L_Group <- L_Group[(L2-L+1):L2] 
  losers<-match(L_Group,rankVec) # Which assets are picked 
  losRets<-rep(NA,L) 
  for (c in 1:L) {losRets[c]<-sumReturns2[losers[c]]} # Vector of losers returns 
  losMeans <- losRets/250 # Vector of loser mean returns 
   
  # Weighting method 
  wWin<-rep(NA,L) 
  wLos<-rep(NA,L) 
  if(weights=="equal"){      # EQUAL WEIGHTING 
    wWin<-rep(1/L,L) 
    wLos<- -(rep(1/L,L)) 
  } 
  if(weights=="IVOL"){      # WEIGHTING BY IDIOSYNCRATIC RISK 
    # Portfolio is weighted based on idiosyncratic risk 
    IVOL <- IR(xx, RMark, ff3_load, method=method,  high_IVOL=high_IVOL)$"IVOL" 
     
    wWin <- IVOL[winners] # Idiosyncratic risks of the winner assets 
    wWin <- wWin/sum(wWin) 
87 
 
87 
    
    wLos <- 1/IVOL[losers] # For losers, biggest short position for the lowest idiosyncratic risk 
    wLos <- -wLos/sum(wLos) # scaled to 100% 
  } 
  if(weights=="rank"){      # RANK WEIGHTING 
    # Weights calculated according to ranks 
    # Past winners should get the biggest weight 
     
    wWin<-H_Group 
    scale <- 1/(sum(wWin)) # weights need to be scaled to 100 % 
    wWin<-wWin*scale 
    # Assets with low value should get the biggest short position 
    wLos <- L_Group 
    scale <- 1/(sum(wLos)) 
    wLos <- -wLos*scale # Low value weights are presented as absolute values 
  } 
   
  # Count weights for all assets   
  weights <- rep(0,cols) 
  for(i in 1:L){weights[winners[i]] <- wWin[i]} # set winner weights 
  for(i in 1:L){weights[losers[i]] <- wLos[i]}  # set loser weights 
   
if(showOutput==TRUE){ 
    return(list(winners=winners,winReturns=winMeans,winWeights=wWin, 
                losers=losers,losReturns=losMeans,losWeights=wLos,allWeights=weights)) 
  }} 
 
# ________________________________________________________________ 
# Reversal *************************************************************************** 
 
# Works with the same method as momentum and so it is utilised 
# The only difference is that the final weight vector will be inversed 
 
reversal = function( 
  xx, 
  RMark=NA, 
  ff3_load = NA, 
  percentiles=3, 
  pastMonths=36, 
  monthsTill=12, 
  weights="equal",    # {equl,rank,IVOL} 
  method=method, 
  high_IVOL=1) {      # 0 if big weight for assets with low idiosyncratic risk 
   
  rev <- momentum(xx=xx, RMark=RMark, ff3_load=ff3_load, percentiles=percentiles,  
                  pastMonths=pastMonths, monthsTill=monthsTill, weights=weights ,                                
high_IVOL=high_IVOL, 
                  method=method) 
  # Here winReturns (and losReturns) are 1-year mean returns of the long term losers (winners) 
  return(list(winners=rev$winners,winReturns=rev$losReturns,winWeights=rev$winWeights, 
              losers=rev$losers,losReturns=rev$winReturns, 
88 
 
      losWeights=rev$losWeights,allWeights= -1*(rev$allWeights))) 
} __________________________________________________________________ 
 
# Idiosyncratic Risk ********************************************************************* 
 
IR <- function(xx, 
               RMark=NA, 
               ff3_load=NA, 
               assets=(1:dim(xx)[2]), 
               method = "markMod",  #{markMod, ff3Mod} 
               high_IVOL=1) 
{ 
 
  e <- rep(NA,length(assets)) 
   
  for(i in assets){ 
    if(method=="markMod"){ 
    # Market model version     
         y <- xx[,i] 
         x <- RMark 
         fit <- lm(y~RMark) 
         e[i] <- (summary(fit)$'sigma')^2 # variance of residuals 
        } 
    if(method=="ff3Mod"){ 
      # Fama-French 3-factor model 
      MKT<-ff3_load[,1] 
      SMB<-ff3_load[,2] 
      HML<-ff3_load[,3] 
      y<-xx[,i] 
      fit <- lm(y~MKT+SMB+HML) 
       
      e[i] <- sd(residuals(fit)) 
    } 
 
if(high_IVOL==0){e<-1/e} # If we want a big weight for assets with low idiosyncratic risk 
   
  return(list("IVOL"=e, "errors"=err)) 
  } 
} 
# __________________________________________________________________ 
 
# Idiosyncratic Risk Weights ************************************************************* 
 
IVOL=function( 
  xx, 
  RMark=RMark, 
  ff3_load=ff3_load, 
  method=method, 
  percentiles=3, 
  weights="equal",    # {equal,rank} 
  high_IVOL=1, # 0 if big weight for assets with low idiosyncratic risk 
89 
 
89 
  showOutput=TRUE){ 
   
  t <- NROW(xx) 
  cols <- NCOL(xx) 
   
  values <- IR(xx=xx, 
               RMark=RMark, 
               ff3_load=ff3_load, 
               method=method, 
               assets=(1:dim(xx)[2]), 
               high_IVOL=high_IVOL)$"IVOL" 
   
  rankVec<-as.vector(rank(values)) # Rank by past performance (winners --> big values) 
   
  sumReturns <- colSums(xx[(t-249):t,]) # Cumulative return for all assets 
  no_rets<- 250 # To calculate mean returns 
   
  chunk <- function(xx,n) {split(xx, factor(sort(rank(xx)%%n)))} #function to split a vector 
  subSets<-chunk(1:cols,percentiles) 
   
  # Create vectors for the high and low weights 
  H_Group<-eval(parse(text=paste(sep="", "subSets$'", percentiles-1, "'"))) # High ranks 
  L_Group<-subSets$'0' # Low ranks 
   
  L1<-length(H_Group) 
  L2<-length(L_Group) 
  L<-min(L1,L2) # Vector lengths will be adjusted according to the shorter one 
   
  H_Group <- H_Group[(L1-L+1):L1] 
  highValues<-match(H_Group,rankVec) # Which assets are picked 
  highRets<-rep(NA,L) 
   
  for (c in 1:L) {highRets[c]<-sumReturns[highValues[c]]} # Vector of high IVOL risk returns 
  
 # Here high IVOL assets are considered winners/losers depending on settings 
 winMeans <- highRets/no_rets  
   
  L_Group <- L_Group[(L2-L+1):L2] 
  lowValues<-match(L_Group,rankVec) # Which assets are picked 
  lowRets<-rep(NA,L) 
  for (c in 1:L) {lowRets[c]<-sumReturns[lowValues[c]]} # Vector of low idiosyncratic risk returns 
  losMeans <- lowRets/no_rets # Vector of low idiosyncratic risk mean returns 
   
  # Weighting method 
  wHigh<-rep(NA,L) 
  wLow<-rep(NA,L) 
  if(weights=="equal"){      # EQUAL WEIGHTING 
    wHigh<-rep(1/L,L) 
    wLow<- -(rep(1/L,L)) 
  } 
   
90 
 
  if(weights=="rank"){      # RANK WEIGHTING 
    # Weights calculated according to ranks 
    # Assets with the highest idiosyncratic risk should get the biggest weight 
     
    wHigh<-H_Group 
    scale <- 1/(sum(wHigh)) # weights need to be scaled to 100 % 
    wHigh<-wHigh*scale 
     
    # Assets with low value should get the biggest short position 
    wLow <- L_Group 
    scale <- 1/(sum(wLow)) 
    wLow <- -wLow*scale # Low value weights are presented as absolute values 
  } 
   
  # Count weights for all assets   
  weights <- rep(0,cols) 
  for(i in 1:L){weights[highValues[i]] <- wHigh[i]} # set high IVOL weights 
  for(i in 1:L){weights[lowValues[i]] <- wLow[i]}  # set low IVOL weights 
   
   
  if(showOutput==TRUE){ 
    return(list(High_Values=highValues,winReturns=winMeans,highWeights=wHigh, 
            Low_Values=lowValues,losReturns=losMeans,Low_Weights=wLow,allWeights=weights)) 
   } 
} 
# ____________________________________________________________________ 
# Function for the Black-Litterman model **************** 
B_L = function( 
  xx, 
  xx_market, 
  exp_market, 
  mWeights, 
  PMat,  
  QVec, 
  sigma, 
  lambda=NA 
){ 
  var_m <- var(xx_market) 
  if(lambda==0){lambda <- exp_market/var_m} # Risk aversion parameter 
   
  T <- NROW(xx) 
  N <- NCOL(xx) 
  tau <- 1/(T-N) 
  nviews <- NROW(PMat) 
   
   
  Phi <- (lambda * sigma) %*% mWeights  # Market equilibrium returns 
  omega <- PMat %*% (tau*sigma) %*% t(PMat) 
 
  for(r in 1:nviews){ # Diagonalise the omega matrix 
    for(c in 1:nviews){ 
91 
 
91 
      if(r != c){omega[r,c]=0}  
    }} 
   
  rets <- ginv(ginv(tau*sigma)+t(PMat)%*%ginv(omega)%*%PMat) %*% 
    (ginv(tau*sigma)%*%Phi+t(PMat)%*%ginv(omega)%*%QVec) 
   
  return(list(Returns=rets,OmegaMatrix=omega,Phi=Phi,lambda=lambda,tau=tau)) 
} 
# ********************************************************* 
 
# OUT-OF-SAMPLE STUDY 
============================================= 
 
#### FACTOR BUILDING ##### 
dates<- rawdata[beg:T,1] 
# Choose assets 
all <- 1:47 
stocks <- 1:20 
bonds <- 21:31 
FX <- 32:37 
commodities <- 38:47 
assets <- all # {choose one from above} 
anomaly <- "momentum"  # {momentum, reversal, value, IVOL} 
weight_method <- "rank" # {equal, rank, IVOL} 
IVOL_method <- "ff3Mod" #{markMod, ff3Mod} 
high_IVOL <- 1 # {1=bigger weight for high idiosyncratic risk assets, 0=opposite} 
perc <- 3 # percentiles 
pastMonths <- 12   # Beginning of the estimation window for momentum and reversal 
monthsTill <- 1 
 
update<-20  #How often the portfolio is rebalanced 
yrDays <- 250 
beg <- 1377 # Value data available from 11.7.1994 (row 1377) 
fact_win <- beg-1 # Estimation window for factor building 
len <- T-beg+1 
 
Hseries<-numeric(0) 
Lseries<-numeric(0) 
XH<-100 # starting wealth for the 'high' porfolio 
XL<-100 # starting wealth for the 'low' porfolio 
vH=(rep(0,N))  # Denotes the amount of each asset held in the portfolio 
vL=(rep(0,N)) 
 
#START ITERATION 
 
for(i in beg:T){ 
 
  XH=sum(vH*P[i,]) 
  XL=sum(vL*P[i,]) 
   
92 
 
  if(XH==0){XH=100} 
  if(XL==0){XL=100} 
  wH=vH*P[i,]/XH # weights for both portfolios change due to price changes 
  wL=vL*P[i,]/XL 
   
  if (any(seq(beg,T,update)==i)==TRUE) { #Determines if it is time to rebalance 
   
    fact_rets <- RE[(i-fact_win):i,assets] 
    RMark <- RM[(i-fact_win):i] 
    ff3_load <- ff3_fact[(i-fact_win):i,] 
 
    # Select anomaly 
     
    if (anomaly=="momentum"){values<-momentum(fact_rets,percentiles=perc,RMark=RMark,  
      ff3_load=ff3_load, 
                                              pastMonths=pastMonths, monthsTill=monthsTill, 
                                              method=IVOL_method, weights=weight_method, 
                                              high_IVOL=high_IVOL)} 
    if (anomaly=="reversal"){values<-reversal(xx=fact_rets, ff3_load=ff3_load, percentiles=perc, 
                                              pastMonths=pastMonths, monthsTill=monthsTill, 
                                              weights=weight_method,method=IVOL_method, 
                                              high_IVOL=high_IVOL)} 
    if (anomaly=="value"){values<-value(xx=fact_rets, vals=val[i,assets],ff3_load=ff3_load, 
                                        percentiles=perc, weights=weight_method,  
                                        method=IVOL_method, high_IVOL=high_IVOL)} 
    if (anomaly=="IVOL"){values<-IVOL(fact_rets, RMark=RMark,ff3_load=ff3_load, percentiles=perc, 
                                      weights=weight_method, method=IVOL_method, 
                                      high_IVOL=high_IVOL)} 
     
    # Create two different weights vectors: 1) High and 2) Low portfolio 
    wH<-as.vector(values$'allWeights') 
    wL<-(-wH) 
     
    # Only High (Low) group weights are taken into account 
    for(c in 1:length(wH)){if(wH[c] < 0){wH[c] <- 0}} 
    for(c in 1:length(wL)){if(wL[c] < 0){wL[c] <- 0}} 
     
    startCol <- assets[1] 
    endCol <- assets[length(assets)] 
     
    wH <- c(rep(0,startCol-1), wH, rep(0,N-endCol)) # weight vector can be a subset of all assets 
    wL <- c(rep(0,startCol-1), wL, rep(0,N-endCol))  
     
    vH <- wH*XH/P[i,] 
    vL <- wL*XL/P[i,] 
  } 
   
  Hseries <- rbind(Hseries,XH) 
  Lseries <- rbind(Lseries,XL) 
   
  len<-length(Hseries) 
93 
 
93 
   
  if(i%%30==0) print(paste(sep="", round((i-beg+1)/(T-beg)*100,2)," %", " // length= ",len, 
                           " // i = ",i))    # report every 30th loop passes 
} 
 
# STOP ITERATION  
 
# Translate premium to percentage returns 
FactRets <- (Hseries[2:len]/Hseries[1:(len-1)]-1)- 
          (Lseries[2:len]/Lseries[1:(len-1)]-1) 
 
# Create factor index 
Factor<-c(100,rep(NA,len-1)) 
  for(i in 2:len){ 
      Factor[i]<-Factor[i-1]*(1+FactRets[i-1]) 
  } 
# Factors are saved for further analysis (”Factor_Returns.csv”) 
_________________________________________________________________ 
 
# BACK-TEST BLACK-LITTERMAN PORTFOLIOS 
================================================================= 
 
# Load factors as source of views 
FR_file <- "Factor_Returns.csv" 
FR_raw <- read.csv(FR_file, header=T, sep=";", dec=".", na.strings="N/A") 
FR<-FR_raw[,2:(dim(FR_raw)[2])] 
fact_var_file <- "Factor_Variables.csv" # Variables/parameters are manually defined for the factors 
fact_var <- read.csv(fact_var_file, header=T, sep=";", dec=".", na.strings="N/A") 
fact_var <- fact_var[,1:8] 
 
# Asset class columns 
# Equity indices: columns: 1-20 
# Bond indices: columns: 21-31 
# Currencies: columns: 32-37 
# Commodities: columns: 38-47 
# Market portfolio: column 48 
 
#Variables 
strategy<-"BL" # Set strategy {1/N, sample, BL,} 
no_views <- FALSE # If true, P-matrix will be changed to zero 
shorts <- FALSE    # Constraints {FALSE = long-only, TRUE = short-sales allowed} 
update <- 20  # How often the portfolio is rebalanced 
X <- 100 # Starting wealth 
beg <- 1377 # Beginning of trading period (value data starts from row 1377 ) 
yrDays <- 250 
fact_win <- beg # Estimation window for factor building 
window <- 250 # Estimation window for the covariance matrix 
lambda <- 10 # Market risk aversion parameter 
IVOL_method <- "ff3Mod" #{markMod, ff3Mod} 
 
# Choose assets 
94 
 
all <- 1:47 
stocks <- 1:20 
bonds <- 21:31 
FX <- 32:37 
commodities <- 38:47 
  assets_1 <- stocks 
  assets_2 <- bonds 
  assets_3 <- FX 
  assets_4 <- commodities 
 
pfolio_assets <- all # {all or assets} The assets in the portfolio 
n <- length(pfolio_assets) # number of assets in the portfolio 
 
# Reference factors for all asset classes 
fID_1 <-"S_VAL_3_R"    #Stock factor 
fID_2 <-"B_VAL_3_R"    #Bond factor 
fID_3 <-"FX_VAL_3_R"   #FX factor 
fID_4 <-"C_VAL_3_R"    #Commodity factor 
 
Factor_1 <- c(rep(0,beg),FR[,fID_1]) 
Factor_2 <- c(rep(0,beg),FR[,fID_2]) 
Factor_3 <- c(rep(0,beg),FR[,fID_3]) 
Factor_4 <- c(rep(0,beg),FR[,fID_4]) 
 
# Anomaly specific variables 
anomaly_1 <- fact_var[match(fID_1,fact_var[,1]),"anomaly"]  # {momentum, reversal, value, IVOL} 
weight_method_1 <- fact_var[match(fID_1,fact_var[,1]),"weight_mtd"] 
 # {equal, rank, IVOL} Weights for views 
IVOL_method_1 <- "ff3Mod" # {markMod, ff3Mod} How idiosyncratic volatility is calculated 
high_IVOL_1 <- fact_var[match(fID_1,fact_var[,1]),"high_IVOL"] 
 # {1=bigger weight for high idiosyncratic risk assets, 0=opposite} 
perc_1 <- fact_var[match(fID_1,fact_var[,1]),"perc"] 
pastMonths_1 <- fact_var[match(fID_1,fact_var[,1]),"From"] 
monthsTill_1 <- fact_var[match(fID_1,fact_var[,1]),"Till"] 
 
anomaly_2 <- fact_var[match(fID_2,fact_var[,1]),"anomaly"]  # {momentum, reversal, value, IVOL} 
weight_method_2 <- fact_var[match(fID_2,fact_var[,1]),"weight_mtd"] 
 # {equal, rank, IVOL} Weights for views 
IVOL_method_2 <- "ff3Mod" # {markMod, ff3Mod } How idiosyncratic volatility is calculated 
high_IVOL_2 <- fact_var[match(fID_2,fact_var[,1]),"high_IVOL"] 
 # {1=bigger weight for high idiosyncratic risk assets, 0=opposite} 
perc_2 <- fact_var[match(fID_2,fact_var[,1]),"perc"] 
pastMonths_2 <- fact_var[match(fID_2,fact_var[,1]),"From"] 
monthsTill_2 <- fact_var[match(fID_2,fact_var[,1]),"Till"] 
 
anomaly_3 <- fact_var[match(fID_3,fact_var[,1]),"anomaly"]  # {momentum, reversal, value, IVOL} 
weight_method_3 <- fact_var[match(fID_3,fact_var[,1]),"weight_mtd"] 
 # {equal, rank, IVOL} Weights for views 
IVOL_method_3 <- "ff3Mod" # {markMod, ff3Mod } How idiosyncratic volatility is calculated 
high_IVOL_3 <- fact_var[match(fID_3,fact_var[,1]),"high_IVOL"] 
 # {1=bigger weight for high idiosyncratic risk assets, 0=opposite} 
95 
 
95 
perc_3 <- fact_var[match(fID_3,fact_var[,1]),"perc"] 
pastMonths_3 <- fact_var[match(fID_3,fact_var[,1]),"From"] 
monthsTill_3 <- fact_var[match(fID_3,fact_var[,1]),"Till"] 
 
anomaly_4 <- fact_var[match(fID_4,fact_var[,1]),"anomaly"]  # {momentum, reversal, value, IVOL} 
weight_method_4 <- fact_var[match(fID_4,fact_var[,1]),"weight_mtd"] 
 # {equal, rank, IVOL} Weights for views 
IVOL_method_4 <- "ff3Mod" # {markMod, ff3Mod } How idiosyncratic volatility is calculated 
high_IVOL_4 <- fact_var[match(fID_4,fact_var[,1]),"high_IVOL"] 
 # {1=bigger weight for high idiosyncratic risk assets, 0=opposite} 
perc_4 <- fact_var[match(fID_4,fact_var[,1]),"perc"] 
pastMonths_4 <- fact_var[match(fID_4,fact_var[,1]),"From"] 
monthsTill_4 <- fact_var[match(fID_4,fact_var[,1]),"Till"] 
 
v=(rep(0,n))  # Denotes the amount of each asset held in the portfolio 
Xseries<-numeric(0) # Portfolio value performance 
Wseries<-matrix(NA,1,n) # Time series of portfolio weights 
lambdas <- numeric(0) # Risk aversion 
colnames(Wseries)<-colnames(P[,pfolio_assets]) 
 
# START ITERATION *************************************************************** 
 
BL_errors=0 # Count when BL model doesn't work 
skips=0 # Count when cannot optimise 
k=4 #number of views 
 
for(i in beg:T){ 
  X=sum(v*P[i,pfolio_assets]) 
  if(X==0){X=100} 
  w=v*P[i,pfolio_assets]/X 
   
  if (any(seq(beg,T,update)==i)==TRUE) { #Determines if it's time to rebalance 
     
    if(strategy=="1/N"){w=rep(1/n,n)} else { 
            R <- RE[(i-window+1):i,pfolio_assets] 
            RMark <- RM[(i-window+1):i] 
            covar <- covMat(R, RMark=RMark, 
                      method="sample" ,  # {"sample" or "SI"} 
                      N=n) # covariance matrix 
            covar <- ginv(ginv(covar))} 
  # Do generalised inverse twise to return a truly inversable matrix 
 
    # Call the portfolio optimisation procedure 
    if(strategy=="sample") { # Minimum variance portfolio 
 # If error then solve GMV portfolio       
 w1 <- tryCatch(as.vector(GMV(covar, shorts=shorts)$'solution'), 
             error=function(err) NULL, 
                     warning=function(warn) NULL) 
       
      if(!is.null( w1 )){w<-w1} else {w<-w # If still no solution is found then skip optimising 
            skips <- skips +1}             
96 
 
    } 
   
    # B-L starts _________________________________________ 
    if(strategy=="BL") { # Maximum Sharpe portfolio with Black-Litterman model 
       
      rets <- RE[(i-fact_win):i,pfolio_assets] 
      RMark <- RM[(i-fact_win):i] 
      ff3_load <- ff3_fact[(i-fact_win):i,] 
 
      # Benchmark weights 
      mweights <- rep(1/n, n) # Use 1/N as benchmark 
       
      # P-Matrix reset 
      PMat <- matrix(ncol=n, nrow=k, rep(0, n*k)) # Create an empty P-matrix 
 
      # Q-Vector reset 
      QVec <- rep(0, k)  
       
      for(j in 1:k){ # Define each view 
       
        # Select anomaly 
        Factor <- eval(parse(text=toString(paste(sep="","Factor_",j)))) 
        Factor <- Factor[(i-window+1):i] 
        assets <- eval(parse(text=toString(paste(sep="","assets_",j)))) 
        anomaly <- eval(parse(text=toString(paste(sep="","anomaly_",j)))) 
        weight_method <- eval(parse(text=toString(paste(sep="","weight_method_",j)))) 
        high_IVOL <- eval(parse(text=toString(paste(sep="","high_IVOL_",j)))) 
        perc <- eval(parse(text=toString(paste(sep="","perc_",j)))) 
        pastMonths <- eval(parse(text=toString(paste(sep="","pastMonths_",j)))) 
        monthsTill <- eval(parse(text=toString(paste(sep="","monthsTill_",j)))) 
 
 
          if (anomaly=="momentum") 
 {values< momentum(rets[,assets],percentiles=perc,RMark=RMark, ff3_load=ff3_load, 
                                                  pastMonths=pastMonths, monthsTill=monthsTill, 
                                                  method=IVOL_method, weights=weight_method, 
                                                  high_IVOL=high_IVOL)} 
          if (anomaly=="reversal") 
 {values<-reversal(xx=rets[,assets], ff3_load=ff3_load, percentiles=perc, 
                                                  pastMonths=pastMonths, monthsTill=monthsTill, 
                                                  weights=weight_method,method=IVOL_method, 
                                                  high_IVOL=high_IVOL)} 
          if (anomaly=="value") 
 {values<-value(xx=rets[,assets], vals=val[i,assets],ff3_load=ff3_load, 
                                                  percentiles=perc, weights=weight_method,  
                                                  method=IVOL_method, high_IVOL=high_IVOL)} 
          if (anomaly=="IVOL") 
 {values<-IVOL(rets[,assets], RMark=RMark,ff3_load=ff3_load, percentiles=perc, 
                                                  weights=weight_method, method=IVOL_method, 
                                                  high_IVOL=high_IVOL)} 
         
97 
 
97 
          # P-Matrix 
          allWeights <- values$'allWeights' 
 
          for(c in assets){ 
            # When an anomaly is applied only to one asset class the weights have to match... 
            # ...with the whole portfolio (if all asset classes are considered) 
            PMat[j,c] <- allWeights[c-min(assets)+1] 
          } 
       
          if(no_views==TRUE){PMat <- matrix(ncol=n, nrow=k, rep(0, n*k))} 
 
          # Q-vector 
          QVec[j] <- mean(Factor) # How much the "winners" win the "losers" p.d. on average 
 
      }# BL parametres ready 
 
        # Black-Litterman returns 
      BL <- B_L(R, RMark, exp_MRet[i], mweights, PMat, QVec, covar, lambda=lambda) 
      BL_Rets <- t(as.vector(BL$'Returns')) 
      lam <- BL$"lambda" 
 
      # Max Sharpe portfolio         
      SR <- tryCatch(maxSharpe (BL_Rets, covar, shorts=shorts), 
                     error=function(err) NULL, 
                     warning=function(warn) NULL) # Error handler returns 0 if no solution is found 
 
      if(!is.null(SR)){ w <- as.vector(SR$'weights') }else { 
        w1 <- tryCatch(as.vector(GMV(covar, shorts=shorts)$'solution'), 
  # If error then solve GMV portfolio 
                       error=function(err) NULL, 
                       warning=function(warn) NULL) 
         
        if(!is.null( w1 )){w<-w1} else {w<-w # If still no solution is found then skip optimising 
                                        skips <- skips +1}  
        BL_errors <- BL_errors+1} # Count errors when Black-Litterman model won't work 
    } 
   
    v=w*X/P[i,pfolio_assets] 
  } 
  Xseries=rbind(Xseries,X) 
  Wseries=rbind(Wseries,w) 
  lambdas=rbind(lambdas,lam) 
  len<-length(Xseries) 
  if(i%%10==0) print(paste(sep="", round((i-beg+1)/(T-beg)*100,2)," %", " // length= ",len, 
                           " // i = ", i, 
                           " // BL_errors = ", BL_errors, 
                           " // skips = ", skips))    # report every 10th loop passes 
} 
#---END ITERATION---#