Alternative investments role in institutions’ 
strategic allocation 
Tail risk in portfolio optimization 
 
 
 
 
 
Master's thesis  
in Accounting and Finance 
 
Author: 
Johan Elfvengren 
 
Supervisor: 
Prof. Luis Alvarez 
 
3.4.2022 
Helsinki 
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The originality of this thesis has been checked in accordance with the University of Turku 
quality assurance system using the Turnitin Originality Check service.  
  
 
Master's thesis 
 
Subject: Accounting and Finance 
Author: Johan Elfvengren 
Title: Alternative investments role in institutions’ strategic allocation: Tail risk in portfolio 
optimization 
Supervisor: Prof. Luis Alvarez 
Number of pages: 93 pages + appendices 5 pages 
Date: 3.4.2022 
 
Abstract. 
Recently there has been a growing demand for alternative assets. Low yield and expected return 
environment in traditional assets and appealing performance of the alternative investments have 
been the key drivers for that development. While the different risk-return structures of alternative 
investments have been under broad interest among academics, the scarce data have driven the 
practitioners to utilize traditional portfolio construction methods with alternative investments. 
This study examines whether alternative assets enhance the efficiency of multi-asset portfolios 
when issues related to scarce and smoothed data have been taken into account and whether we 
can construct more efficient portfolios when we take the higher moments of returns into account 
in the optimization. We examine these questions from the institutional investor point of view, 
which have the most favorable characteristics to emphasize alternative investments in their 
strategic allocation. The studied asset classes are Private Equity, Private Credit, Infrastructure, 
Real Estates, Natural Resources, and Hedge Funds. The alternative assets data are aggregated 
private market fund quarterly returns from 2008 to 2021, which we desmooth to monthly returns. 
For traditional asset classes, we utilize monthly index returns. 
The results confirm that alternative assets enhance the portfolio efficiency despite the data 
adjustments and chosen optimization objective. The optimal allocations towards alternative 
investments decreased when we took the higher moments into account, which implies that while 
alternative investments still enhance the portfolio efficiency, they do not look as attractive as the 
traditional mean-variance optimization suggests. When examining portfolio construction with 
backtests from an ex-ante perspective, we found that mean-variance optimization offered more 
efficient results than mean-expected shortfall optimization due to higher moments estimation 
complexity. We also found indications that CRRA objective function could work as an alternative 
for mean-expected shortfall optimization if we want to take the higher moments into account in 
optimization. 
 
Key words: Alternative assets, Alternative investments, Strategic allocation, Expected 
Shortfall, Portfolio optimization, Institutional investors 
  
  
Pro gradu -tutkielma 
 
Oppiaine: Laskentatoimi ja Rahoitus 
Tekijä: Johan Elfvengren 
Otsikko: Alternative investments role in institutions’ strategic allocation: Tail risk in portfolio 
optimization 
Ohjaaja: Prof. Luis Alvarez 
Sivumäärä: 93 sivua + liitteet 5 sivua 
Päivämäärä: 3.4.2022 
 
Tiivistelmä 
Vaihtoehtoisten sijoitusten kysyntä on kasvanut merkittävästi viimeisen kymmenen vuoden 
aikana ja erityisesti mielenkiinto epälikvidejä vaihtoehtoisia sijoituksia kohtaan on lähtenyt 
räjähdysmäiseen kasvuun Covid-19 kriisin jälkeen. Matalan tuotto-odotuksen ympäristö 
perinteisissä omaisuuslajeissa sekä vaihtoehtoisten sijoitusten houkutteleva performanssi 
nähdään tärkeimpinä ajureina tässä kehityksessä. Vaikka vaihtoehtoisten sijoitusten poikkeavat 
riski-tuottorakenteet ovat olleet akateemikoiden laajan mielenkiinnon kohteena, niukka ja 
heikkolaatuinen tuottodata vaihtoehtoisissa sijoituksissa on ohjannut rahoitusalan toimijoita 
hyödyntämään perinteisiä portfolion rakentamismenetelmiä. 
Tämä tutkielma tarkastelee parantaako vaihtoehtoiset sijoitukset yhdistelmäportfolioiden 
tehokkuutta, kun dataan liittyvät ongelmat pyritään korjaamaan, ja voimmeko rakentaa 
tehokkaampia portfolioita ottamalla optimoinnissa huomioon tuottojen korkeammat momentit. 
Tarkastelemme näitä kysymyksiä institutionaalisten sijoittajien näkökulmasta, koska näiden 
sijoittajien ominaisuudet ovat suotuisimpia vaihtoehtoisten sijoitusten painottamiseen 
strategisissa allokaatioissa. Tutkittavia vaihtoehtoisia omaisuusluokkia ovat Private Equity, 
Private Credit, infrastruktuuri, kiinteistöt, luonnonvarat ja Hedge rahastot. Vaihtoehtoisten 
sijoitusten data on indekseiksi aggregoitua kvartaalittaista listaamattoman markkinan tuottodataa, 
joka tutkimuksessa avataan kuukausituotoiksi. Perinteisissä omaisuuslajeissa hyödynnämme 
laajoja markkinoita kuvaavia kuukausittaisia indeksituottoja. Tutkimuksessa tarkkaillaan tuottoja 
vuosilta 2008-2021. 
Tutkielmassa saadut havainnot vahvistavat aiemmissa tutkimuksissa saatuja tuloksia 
vaihtoehtoisten sijoitusten kyvystä parantaa portfolioiden tehokkuutta, huolimatta datan 
korjauksista tai optimoinnin tavoitteista. Vaihtoehtoisten sijoitusten osuus väheni optimoiduissa 
portfolioissa, kun otimme tuottojen korkeammat momentit huomioon optimoinneissa. Tämä 
viittaa siihen, että vaikka vaihtoehtoiset sijoitukset edelleen parantavat salkkujen tehokkuutta, ne 
eivät näytä niin houkuttelevilta kuin perinteisessä tuotto-varianssi optimoinnissa. Kun portfolion 
rakentamista tutkittiin simuloinnilla ex-ante näkökulmasta, havaitsimme että tuotto-varianssi 
optimointi tarjoaa tehokkaampia tuloksia kuin tuotto-odotettu alijäämä optimointi, mikä 
todennäköisesti viittaa korkeampien momenttien estimoinnin haasteellisuuteen. Löysimme myös 
viitteitä CRRA tavoitefunktion käyttökelpoisuudesta tuotto-odotettu alijäämä optimoinnin sijaan, 
kun haluamme ottaa korkeammat momentit optimoinnissa huomioon. 
 
Avainsanat: Vaihtoehtoiset sijoitukset, Strateginen allokaatio, Odotettu alijäämä, Portfolion 
optimointi, Institutionaaliset sijoittajat 
  
CONTENTS
1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Purpose of the study and research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Scope and limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 LITERATURE REVIEW .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Alternative assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Private Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Private Credit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.4 Real-Estates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.5 Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.6 Natural Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Investment vehicles and liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Institutional investors & Case Yale Endowment. . . . . . . . . . . . . . . . . . . . . . . . 32
3 THEORETICAL FOUNDATION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Portfolio construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Modern Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Mean-Variance optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Post Modern Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Mean-Expected Shortfall optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 Higher-order portfolio moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Previously utilized models, obtained results and critique . . . . . . . . . . . . . . 56
4 METHODOLOGY .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 Research strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Data & Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 CONCLUSION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
APPENDIX I
APPENDIX II
FIGURES
Figure 1 Yale Endowment fund asset allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 2 Yale Endowment fund performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 3 Portfolio optimization process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 4 Efficient frontier with riskless lending and borrowing . . . . . . . . . . . . 40
Figure 5 Efficient frontier with alternative assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Figure 6 Second order stochastic dominance rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Figure 7 Profit-Loss probability density function with VaR and ES . . . . . . 50
Figure 8 Cumulative distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 9 Histograms with distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 10 Normality observation with Henry’s line . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 11 Efficient frontiers in mean-variance space . . . . . . . . . . . . . . . . . . . . . . . . . 73
Figure 12 Asset weights in mean-variance efficient portfolios . . . . . . . . . . . . . . . 73
Figure 13 Asset weights in mean-es efficient portfolios . . . . . . . . . . . . . . . . . . . . . . 74
Figure 14 Mean-variance and mean-ES efficient frontiers. . . . . . . . . . . . . . . . . . . . 75
Figure 15 Historical simulations return indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Figure 16 Mean-variance out-of-sample weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure 17 Mean-ES p=0.90 out-of-sample weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 18 Mean-variance out-of-sample performance . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 19 Mean-ES p=0.90 out-of-sample performance . . . . . . . . . . . . . . . . . . . . . 80
Figure 20 CRRA objective function out-of-sample weights. . . . . . . . . . . . . . . . . . 81
Figure 21 CRRA objective function out-of-sample performance . . . . . . . . . . . . 81
Figure 22 Finnish Pension Funds asset allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Figure 23 Desmoothed return indices for asset classes . . . . . . . . . . . . . . . . . . . . . . . 95
Figure 24 Asset weights in efficient portfolios without alternative invest-
ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Figure 25 Efficient frontiers in mean-ES space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Figure 26 Mean-variance and mean-ES efficient frontiers without altern-
atives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Figure 27 Mean-ES out-of-sample weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Figure 28 Mean-ES out-of-sample performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
TABLES
Table 1 Investors’ views for alternative assets characteristics . . . . . . . . . . . . . 19
Table 2 Utilized data and corresponding indicator series . . . . . . . . . . . . . . . . . 65
Table 3 Descriptive Statistics and correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Table 4 In-sample and out-of-sample results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

91 INTRODUCTION
1.1 Background
Due to the current market situation where valuations are higher, and yield curves
are lower than long-run averages, institutional investors have started looking for
alternatives to increase their portfolios’ expected returns. Alternative investments
such as Private Equity (PE), Private Credit (PC), Real-estates (RE), Hedge funds
(HF), natural resources (NR), and infrastructure (IF) have faced growing demand.
Assets under management (AuM) in the alternative investments industry have
grown from 3.1 trillion dollars in 2008 to 12.5 trillion dollars in 2021, which gives
a 12% Compounded annual growth rate (CAGR) (Preqin 2021). Also, portfo-
lios’ allocations to alternative investments have risen significantly. Global pension
funds’ allocations to alternative investments have increased from 5% in 1996 to
26% in 2019 (BlackRock 2020). Net flow to alternative investments has been ar-
guably strong. Reasons for this kind of rise in allocation have been argued to be
at least; significantly lower interest rates and expected returns, alternatives’ ad-
ditional return premiums, the need created by market uncertainties for increased
diversification, and the fact that the number of listed investments have not sig-
nificantly increased from the beginning of the millennium while investable assets
are (The World Bank, 2019). In recent years, exceptional liquidity in the mar-
ket due to the quantitative easing followed by the aftermath of the COVID-19
crisis has further accelerated the shift in the institutions’ strategic allocation from
traditional asset classes to alternative ones. Alternative investments are now the
second-largest megatrend in the financial markets after the ESG investing.
The trendiness of the asset class after every major crisis raises an interesting
question. Does it require extreme market conditions for investors to find this
appealing-sounding asset class?1 Maybe. But does it give investors a simple answer
to navigate through financial markets in the coming decade? Probably not. Like
every other asset class, alternative assets have their pros and cons. However,
investors often fall into a pitfall when they try to enhance their portfolios by
adding alternative investments. Compared to conventional asset classes, risk and
return structures of alternative investments are often misunderstood. Alternative
investments typically have low volatility due to reporting biases. Even if the
volatility is low after adjusting the reporting biases, it might be obtained at the
1This asset class seems to have a pattern with dot-com bubble, financial crisis, and covid-19
crisis.
10
cost of negative skewness and higher kurtosis (Amin and Kat 2003). If an investor
is not restricted with quadratic preferences, these risks also need to be taken
into account. When we use alternative assets in a portfolio, we should also have
alternative tools to assess the risk and return of the portfolio.
Alternative investment vehicles are not new inventions in the financial markets.
The earliest examples of private equity investments are from the 19th century’s in-
dustrial revolution. The first alternative investment to infrastructure can be traced
back as long as 1862, when individuals invested in the transcontinental railroad
in the United States (Duran 2013). Nevertheless, the proliferation of alternative
investments did not begin until after the millennium. The dot-com bubble crash
reminded investors how the financial markets could be fragile. In 2002, as again in
2021, all traditional asset classes had single-digit expected returns. At the same
time, David Swensen introduced Yale Endowment fund’s success with alternative
investments in his book Pioneering Portfolio Management. The Yale Endowment
fund, which had around 60% of its allocation in alternative assets at the time, had
survived surprisingly well from the crash and had an even more lucrative outlook
ahead. This market environment led to the boom of an alternative investment era,
and the period between 2001-2007 can be seen as the golden age of alternative
assets. Eventually, in 2008 alternative assets disappointed investors when most
sub-asset classes crushed down with the market (Ilmanen 2011). Since then, the
rallying of the asset class has been substantial, but still, alternative investments are
a fairly new part of the institutions’ strategic allocation. According to the Prequin
survey2, 24% of institutional investors still do not have alternative investments in
their portfolios.
In traditional mean-variance portfolio construction, one has expected return,
volatility, and correlation as an input. Optimizing a portfolio is quite straightfor-
ward with conventional asset classes, which have relatively efficient markets and
abundant data. However, with alternative assets, one usually needs to consider
also illiquidity, opaque valuations, scarcity of data, and rigidity of implementation
(Chan 2021). This challenge investor to evaluate the risks of the portfolio from
new perspectives. For example, illiquid assets usually look better than their liquid
counterparts even if the economic factors behind these different asset classes are
largely the same. This is due to smoothed data that fools the conventional risk
indicator, namely volatility. However, if we use alternative risk indicators which
highlight the worst possible outcomes (e.g. Value at Risk and Expected Shortfall),
we perceive that investors with illiquid assets probably suffer more profound and
2Preqin surveyed 231 institutional investors in June 2021
11
sustained outcomes. Due to the stale pricing and smoothing bias of illiquid assets,
brief tail events, such as the Covid-19 drawdown, do not fully reveal these risks to
investors, but when the stress period is more extended, these risks will probably
be materialized worst than expected (Group 2021). Nevertheless, when we under-
stand the additional risk factors behind alternative investments, we can manage
and deploy them to returns (Ilmanen 2011, 239-240).
1.2 Purpose of the study and research questions
Most part of the portfolio’s return and risk contribution comes from strategic
allocation (Hoernemann, Junkans and Zarate 2005; Ibbotson and Kaplan 2000).
However, instead of this being considered a truism of finance theory, it should be
considered more about explaining the behavior of an average investor. Meaning
that tactical asset allocation and security selection can add value and have their
time and place3.(Swensen 2009) Nevertheless, the well-diversified portfolio drives
rational investors to the starting point where strategic asset allocation serves most
opportunities, reduces the degree of unreliable factors, and grounds the decision-
making framework on the stable foundation of long-term policy actions. Having
strategic asset allocation in the center of investment philosophy enables long-term
investment success for institutions’ portfolio management. (Swensen 2009, 50-55.)
Since Markowitz’s paper in the Journal of finance in 1952, the Modern Portfolio
Theory (MPT) has been one of the most studied and challenged finance literat-
ure topics. At the same time, its simplicity has made it the number one tool for
practitioners to construct portfolios and strategic allocations. Post Modern Port-
folio Theory (PMPT) suggests many improved models optimize portfolios, but
its results have remained controversial enough to keep MPT as a leading tool in
practice (Malkiel 2019, 185-224). However, when we talk about alternative invest-
ments, the MPT and its mean-variance framework become arguably too restrictive
and straightforward, not least because it assumes that investors have a quadratic
utility function and are only interested in the first two moments of the return
distribution4 . It is studied by many that alternative investments return distri-
butions differs significantly from the normal distribution which makes traditional
3The more proper way to analyze the added value of tactical asset allocation and security selection
would be to analyze attribution instead of examining contribution.
4It has been shown that investors are willing to have lower expected returns and higher volatility
in exchange for lower kurtosis and higher skewness.(Dittmar 2002; Harvey and Siddique 2000;
Mitton and Vorkink 2007; Barberis and Huang 2008.)
12
MPT optimizations inefficient (Fung and Hsieh 2000, 2001; Brooks and Kat 2002;
Popova, Popova, Morton and Yau 2006; Agarwal and Naik 2004; Jondeau and
Rockinger 2006). Also, alternative investments’ data usually have stale pricing
and appraisal smoothing biases, which must be considered before constructing a
portfolio (Cumming, Helge Haß and Schweizer 2014).
Institutional investors are usually characterized by infinite investment horizons
and future liabilities to cover with investment returns. Institutions’ infinite invest-
ment horizon supports the risk appetite. Contrarian, investment objectives usually
emphasize capital conservation and hedging against liabilities. When we also con-
sider a large absolute proportion of an asset that institutions have, enabling them
to invest in illiquid assets, it is easy to see that alternative investments are partic-
ularly interesting opportunities in institutions’ strategic allocation. Large initial
investments and long investment horizons enable institutions to carry risks and
harvest premiums that are usually beyond individual investors’ reach. This study
examines institutions’ portfolio construction from the traditional mean-variance
framework and post-modern portfolio theory’s unconventional tail risk perspect-
ive. Two different risk perspectives enable us to assess portfolios risk properties
from a broader range and examine how these properties affect the allocation ob-
tained from optimization problems. The mean-variance framework is used as a
starting point to see how alternative investments affect the portfolio’s perform-
ance when added to the optimization problem with traditional assets. After that,
we examine how optimal allocations will change when we use optimization methods
that control the tail risk. The two research questions are:
1. Can we obtain more efficient portfolios by adding alternative investments
with traditional asset classes into the investment universe?
2. Can we obtain more efficient portfolios by optimizing return to expected
shortfall instead of variance?
Efficiency is defined in this study as a risk-adjusted return, and consequently,
the efficient portfolio is the highest expected return portfolio in chosen risk level.
In this study, the risk is defined and examined as variance and expected shortfall,
focusing on the Sharpe ratio and Return on Expected Shortfall metrics. This
definition makes us focus on the efficient frontiers, whether we examine them from
a variance or expected shortfall point of view. Utilizing two efficiency metrics
allows us to examine whether we obtain more efficient portfolios in one perspective
at the expense of another efficiency perspective.
There are two reasons why we use the mean-variance framework in this study,
13
even after earlier stated drawbacks; First, it represents the standard industry ap-
proach. Its simplicity makes it the most used tool, and this way, it works as a
great benchmark. Secondly, its premise about normally distributed returns makes
it an interesting method to be compared with the expected shortfall optimiza-
tion method, which can deviate from this assumption. This way, we can quantify
how much the estimated distributions affect the allocations and the performance
of these optimal portfolios. The expected shortfall measure has brought to the
optimization problem of this study because this way, we can take the higher mo-
ments into the heart of optimization. Higher moments are engaging in this study
since they illustrate the tail risks that alternative investments are usually prone
to. If we optimize risk measure that is affected by return distributions higher mo-
ments, as the expected shortfall is, then we can manage the whole risk spectrum
and make more efficient portfolios. In addition, tail risk measures like expected
shortfall and Value at Risk are essential for institutions. Even with a long invest-
ment horizon, institutions still need to ensure that they are not endangering to
fulfill their recurring liabilities. The third reason for examining portfolios from the
expected shortfall point of view is the increasing regulatory standards. In 2016
Basel Committee added Expected Shortfall to complement Value at Risk measure
to calculate financial institutions capital requirements for the market risk.
This study tries to deep dive into alternative investments as a heterogeneous
asset class. The goal is to understand better the characteristics of different al-
ternative investments sub-asset classes and portfolio construction techniques from
different risk perspectives. These insights are helpful when building a strategic
asset allocation for institutions and help us understand the role of alternative
investments in institutions’ portfolios.
1.3 Scope and limitations
The scope of this study is to evaluate alternative investments’ suitability into insti-
tutions’ allocation in the long horizon and limit tactical asset allocation outside of
the study. The studied assets class and investor profile under examination provide
a sound basis for this scope. The illiquidity of alternative investments makes it un-
reasonable to allocate this asset class tactically due to slow operational execution
and costs. The institutions’ long-term investment horizon also rationalizes them
to focus on long-term asset-liability management and minimize the dependence on
uncertain short-term factors.
14
The idea is to view the allocation problem from a policy maker’s point of view,
where the institutions’ investment policy should guide the portfolio management
for decades. That is also why we focus on rather static allocations. There are
at least two arguments to justify this decision. First, even though alternatives
as an asset class are a heterogeneous group, most are characterized by illiquidity.
It is unreasonable to expect that investors could dynamically optimize this asset
class’s allocation very frequently in practice. For the same reason, we assume that
the investor would rebalance its portfolio once a year, even though more frequent
rebalance might be possible. The second reason is that alternative asset risk factors
and returns premiums are seen at least conceptually static and are studied to be
nearly so also in practice (Ilmanen 2011). This enables us to focus on the asset
class’s long-term return and risk structure and helps us cope with scarce data since
we do not have to consider changes in distribution when examining individual sub-
asset classes.
This study groups alternative investments as a Preqin classifies them. The
sub-asset classes are; Private Equity (PE), Private Credit (PC), Hedge funds (HF),
Real-Estate (RE), Infrastructure (IF), and Natural Resources (NR) (Preqin 2021).
These assets can be invested directly (e.g., real-estates, timber, investments in
private companies) through funds or fund of funds. The data for alternative in-
vestments is Preqin quarterly index data from 2008 to 2021, which consists of all
reported vintages. Indices are aggregated from reported funds’ money-weighted
rate of returns (IRR). The data is free from survivorship bias but suffers from back-
fill, selection, and liquidation bias which we look over in the methodology chapter.
In general, alternative investment data suffers from appraisal smoothing and stale
pricing. The data will be desmoothed, and potential lags are fixed. For traditional
asset classes, we use index data from Bloomberg, and it covers government bonds,
investment grade and high yield corporate bonds separately and developed and
emerging market equities separately. The study is conducted in U.S. dollars and
the investment universe is global.
Efficiency in research questions is defined as a risk-adjusted return. The ef-
ficiency metrics, which this study will mainly focus on, are Sharpe and Return
on Expected Shortfall (RoES). These indicators describe portfolios’ performance
from a different risk point of view. Since mean-variance optimization optimizes
Sharpe ratio and on the other hand mean-expected shortfall optimization optim-
izes RoES ratio, it can be expected that depending on the optimization model
on review, the simulated performance should show better results with the pairing
efficiency metric. That is why it is vital to examine these metrics together and
cross the optimization method. This way, we will get a more comprehensive pic-
15
ture of portfolios’ performance, which helps us evaluate alternative investments’
potential benefits in strategic allocation. In addition to the performance indic-
ators, we are interested in how the allocations will change when we change the
optimization method. When we take the higher moments into account, the alloca-
tion to alternative investments will probably decrease compared to mean-variance
optimization. Also, the sub-asset classes allocation is expected to change. Since
this study is carried out from an institutional investor perspective, it raises for
review of the institutions’ liquidity needs and expected shortfall objectives. These
factors are taken into account when conclusions of optimal allocations are made.
The results offer a robust view of why we should allocate at least some proportion
of our assets to alternative investments. The scope of this study is motivated by
the fact that the current practice in the market is focused on handling alternative
investments the same way as the traditional ones.
1.4 Structure of the thesis
This thesis can be divided into four parts: Literature review, theoretical found-
ation, empirical research, and conclusion. First, we explore earlier studies for
characteristics of alternative investments and institutional investors. Here the
main point of interest is how alternative investments’ product structures and risk
characteristics differ from traditional assets. We also get insight into which pur-
pose investors usually link different sub-asset classes. In this chapter, we also
identify the most common characteristics of an institutional investor and use the
Yale endowment fund as a case study.
After the literature review, we go through the essential theoretical background.
The reader of this thesis is assumed to be familiar with the most fundamental
theoretical framework of finance, but still, we iterate the most critical theories:
Modern Portfolio Theory (MPT) and Post Modern Portfolio Theory (PMPT).
Modern Portfolio Theory is the foundation of portfolio optimization, and Post
Modern Portfolio Theory brings important tools to deal with asymmetric returns
and tail risk. Here we also introduce the basic optimization process in the mean-
variance framework and the mean-expected shortfall framework. We introduce the
higher-order moment estimation and how these have brought to the optimization
problem. We also introduce an objective function utilized in the empirical simu-
lations as an interesting reference portfolio. After that, we go through previously
utilized models for portfolio optimization with alternative investments. The idea
16
is to highlight the pros and cons of models utilized in previous studies. This way,
we can lay the groundwork for optimization models utilized in empirical research.
The thesis continues with the methodology chapter, demonstrating how the em-
pirical study is conducted. Here we first recall the research objectives and explain
the research strategy, which contains a comprehensive review of data collection
and analysis, together with the descriptive statistics. This chapter also shows how
portfolio optimization routines and out-of-sample simulations have been carried
out.
In the end, results are analyzed and reported. Here the main interest revolves
around the changes in allocation and efficiency metrics. We go through the valid-
ity and reliability of this study, and to establish the robustness, backtests are
discussed. Finally, we bring the literature review and the empirical study together
and shortly conclude the findings.
17
2 LITERATURE REVIEW
Alternative investments are typically characterized by an inflation hedge, a weak
correlation between other assets, alternative risk premiums, lock-up periods, and
illiquidity. At the same time, institutional investors are usually characterized by
an infinite investment horizon, liabilities to cover with portfolio returns, and sig-
nificant investment assets. Institutions’ infinite investment horizon supports the
risk appetite, while investment strategies usually emphasize capital conservation
and hedging against inflation. This starting point gives institutions the ability
to utilize alternative investments. However, the question is how this should be
done allocation-wise. For example, an infinite investment period could suggest
that institutions should harvest liquidity premium by locking up to private equity
instead of the public one. Same time capital conservation and yearly cash flow
needs suggest that institutions should focus on stable inflation hedged returns like
unleveraged real-estates. To find out from which perspective institutions should
look at their strategic allocation and whether alternative investments enhance their
portfolios, we need to deep dive into alternative investments’ heterogeneous charac-
teristics. Before quantitative research, in this chapter, we first evaluate alternative
assets and institutional investors from a qualitative perspective.
2.1 Alternative assets
Ilmanen (2011) defines alternative investments as everything other than traditional
assets. Fraser-Sampson (2010) challenges the categorization of "Alternative As-
sets." The problem with the word "Alternative" is that it could be understood
as a substitute for something even though they should be recognized as comple-
ments for the traditional assets. When we talk about alternative assets, we must
know that the term is subject to unconscious prejudice. These prejudices are one
potential reason why investors, especially outside the U.S., still allocate little to
these assets (Fraser-Sampson 2010). Same time we are dealing with a very het-
erogeneous group which challenges the rationality of categorizing these assets as
one group. So why do we bring these heterogeneous assets together, and why are
they called alternatives? One sensible explanation to group these assets is the lack
of knowledge that most investors carry over these assets. The adage of famous
Warren Buffett states that one should never invest in anything they do not un-
derstand. It explains why investors and finance professionals willingly group these
18
assets as "others" to focus on the traditional expertise of publicly traded stocks
and bonds. However, alternative investments also have, as a group, recognizable
characteristics compared to traditional asset classes. They are typically illiquid,
rarely publicly traded, and exhibit higher fees. However, commodities, which are
part of the natural resources, can be seen as an exception for the properties men-
tioned above. Alternative investments are also less scalable, less transparent, and
more exposed to information asymmetries. Even though they are, as a group, great
diversifies, correlations vary a lot between different sub-asset classes. They also
have the potential to enhance returns through alternative risk premiums; illiquid-
ity premia, alternative beta premia, and potential alpha. Alternative investments
also have one common drawback: limited historical return data and questionable
quality. The historical data problems originate from the fundamental difference
in valuation. When traditional asset classes value forms at the market and the
main risk comes from the volatility of this value, at the same time, illiquid altern-
ative investments mark-to-markets base on the fund manager’s or external service
provider’s valuations. For example, Private Equity funds have barely any volatil-
ity, but the mark-to-market value represents problematic appreciation-based fair
value. The difference in asset valuation calls for different risk metrics as well.
(Fraser-Sampson 2010; Ilmanen 2011.)
The high costs usually related to alternative investments are due to the active
management which they demand. Since alternative investments have less efficient
markets and lack an investable benchmark, it is hard to find comprehensive passive
products to get exposure purely to alternative betas. From the institutional in-
vestor point of view, while one remains passive with the allocation perspective, the
underlying fund management is everything else than passive. The value created
with alternative investments is due to active investment management. However,
the importance of active portfolio management varies between different alternative
asset classes. While in real estate and natural resources, the assets themselves drive
substantial returns passively, active management plays a critical role in private
equity, private credit, hedge funds, and infrastructure. (Swensen 2009)
While categorizing heterogeneous assets is dangerous, it gives us the ability to
understand the bigger picture of multi-asset allocation. This study focuses not
on individual sub-asset classes but rather on the wider allocation problem that
portfolio managers worldwide face. Still, to get there, we need to understand
where these different sub-asset classes stand in their own and the minds of market
participants. For that reason, we go through each asset class and investors’ current
general view of them. Table 1 illustrates the general views of asset classes by
ranking their properties according to institutional investors’ reasons when making
19
an investment decision. Properties are categorized as diversification, high absolute
return, inflation hedge, high risk-adjusted return, reliable income stream, and low
correlation to other asset classes, and they are scaled from 0-100% according to
Preqin interviews (Preqin 2020).
Table 1: Investors’ views for alternative assets characteristics
This data has been collected from Preqin. They interviewed almost 400 institutional investors
and asked the main reasons for investing in alternative assets. Numbers under each asset class
represent how many percent of interviewed investors named characteristics in question to one
of their main reasons to invest in that asset class.
Characteristic PE PC HF RE IF NR
Diversification 60 65 63 68 70 66
Low correlation 15 29 38 30 35 32
High absolute return 55 28 37 26 23 26
High risk-adjusted return 42 37 29 20 28 20
Inflation hedge 4 6 2 30 25 28
Reliable income stream 7 38 0 35 28 8
When we examine this table vertically, we see that diversification is the most
important reason any sub-asset class is invested in. This strengthens the narrative
that the current market uncertainties support the record inflows to alternative
investments. Otherwise, the importance of different characteristics starts to differ
when going down the list. However, in this point, it is essential to note that all
of these mentioned categories are listed as one of the main reasons to invest in a
particular sub-asset class, some more often than others. This said it confirms that
investors expect the exact properties from alternative investments stated earlier in
this study. When we examine this table horizontally, we can rank different sub-
asset classes according to their importance for each characteristic, and this way,
we get qualitative differences between assets.
While the diversification row tells that it is the most important for IF and least
necessary for PE, the differences are minimal. However, we get more significant
differences when we examine the low correlation to other asset classes, which means
the diversification in the second-order moment. Again, it is the least important
reason for investing in PE, but after that, the rank goes as PC, RE, NR, IF, and
for HF, it is the most important reason to invest in. While PE is not seen as a good
diversifier, it is seen as an essential asset class from a returns perspective. PE ranks
in number one asset class in factors High Absolute Returns and High Risk-Adjusted
Returns. We see interesting changes in investors ’ expectations from an inflation
hedge and cash flow perspective. Here PE and HF shine with their absence, but
these are important characteristics for RE and IF investing. Even though reliable
income stream and inflation hedge are usually complementing characteristics, PC
20
is not used for inflation hedge at all, but at the same time, it is used as the most
important source of reliable income stream. With NR, the opposite is true, and
it is easy to see why it is not seen as a source of income stream, but instead an
inflation hedge since inflation usually goes fast to raw material prices. However,
with PC, it is interesting how this asset class is seen as a reliable income stream but
not an inflation hedge. While with PC, one usually gets better terms for the lender
money compared to marketable bonds, the companies funded have higher credit
risk. At the same time, PC bonds are usually with the floating rate, which should,
at least indirectly, hedge against inflation. These views lay an exciting background
for alternative assets when we examine each sub-asset class more comprehensively,
especially when discussing the descriptive statistics in methodology chapter.
2.1.1 Private Equity
The private market (equity and credit) is one of the most booming alternative in-
vestments sub-asset classes right now, not least because of low rates and tightened
bank regulations. The former has led to chasing alternative risk premiums, which
private investments typically have, and the latter has created a gap in financing
(Chan 2021). Also, Ilmanen (2011) defines Private Equity as a return enhancer
compared to other alternative assets, while Swensen (2009) points out that its
diversifying properties are limited.
As most of the alternative sub-asset classes, also private equity is a heterogen-
eous class itself. On the other end is venture capital firms which invest in start-ups
and early-stage companies usually as a target to take them public, while in other
end is buyout firms which, with the help of leverage, turn public companies to
private as a target to develop the company and maybe bring it later back to pub-
lic or sell it to a strategic investor. These different types of private equity firms
have usually focused on the specific category, but recently there have been more
and more balanced private equity funds that handle the whole spectrum. While
venture capital funds are more exposed to fundamental risks of immature busi-
nesses, leveraged buyout funds carry the risk involved to heavy leverage. Even
though these different categories have quite different risk characteristics, they still
carry the same distinguishable structure (Swensen 2009). A private equity fund
is formed from a general partner (private equity firm) and limited partners (in-
vestors). General partner works as an investment manager whose responsibility is
to find attractive opportunities, acquire them and add value by developing them.
At the same time, limited partners have first committed money to the fund. When
21
the general partner finds an investment opportunity and makes the capital call,
limited partners deploy the money. This structure enables efficient active portfolio
management, which reduces capital inefficiencies and agency problems due to the
ownership of general partners. However, this raises the limited partner’s particular
capital commitment type of illiquidity risk, where investors might need to liquidate
capital from other investments in the worst possible time in the market to cover
called capital. (Ilmanen 2011.)
The main ways to invest in private equity are a direct investment to the
firm through private equity fund presented above, funds of funds, publicly-traded
private equity firms, and secondary transactions (Kiesel, Zagst and Scherer 2010).
This study excludes publicly traded private equity firms from our review. This
way, we study components that give us the purest exposure to this market since
publicly traded private equity firms have significant public market equity beta and
give oversized correlation with the equity market. The return is calculated as a net
return to limited partners to represent the actual value added to the investor. We
also include fund of funds structure since it gives broad private market exposure,
often including funds in the different stages as well as direct investments and sec-
ondary transactions. Overall, the fund of funds structure is more diversified but
contains an extra layer of costs.
Private equity’s historical performance remains controversial despite the large
number of papers published around this topic. Ilmanen (2011) results with Cam-
bridge Associates data that private equity earned 12,4% per annum in 1986-2009
while the public equity market earned 9,2% p.a. Also Leitner, Mansour and Naylor
(2007) got complimentary results with Thomson’s Venture Economics data. Ven-
ture capital funds earned 21%, while buyout funds earned 12% from 1986 through
2006. However, most of the returns came in the 90s, and after that, the returns
have been more modest (Fraser-Sampson 2010). When we drill down deeper to the
returns, some studies conclude that private equity managers, as an average, do not
outperform the public market after fees (Phalippou and Gottschalg 2009; Kaplan
and Schoar 2005). The fees seem to be the main reason for underperformance since
with gross returns, private equity overperforms the public market (S&P 500) by
3% on average. Ilmanen (2011) interprets that PE managers are skillful enough
to bring an excess return, but the hurdle after fees is too high to deliver value to
limited partners, so fund managers are the only ones collecting the added value.
Even though private equity has a high correlation with the equity market, some
additional risk factors should provide additional premiums (e.g. illiquidity, size,
leverage) compared to the public market. For that reason, the view of the average
private equity manager being skillful is at least doubtful.
22
Since we know this asset class’s controversial performance, why does it attract
so much interest and cash inflows? One crucial factor is the significant dispersion
between a best and worst manager. In the last ten-year time frame, J.P.Morgan
Asset Management (2021) studied that the top quartile delivered 24,5% IRR after
fees while the bottom quartile delivered only 1,6% IRR. The difference is the
largest of all studied asset classes. Global public equity managers have 12,5% and
10,7% corresponding numbers for comparison. At the same time, these top-quartile
managers tend to stay as top-quartile managers, indicating performance persist-
ence and skill. More tremendous inefficiencies enable harvest alpha from private
markets while a positive circle of success keeps top performers at the top (Ilmanen
2011). It is safe to say that manager selection matters in private equity. Swensen
(2009) capsulizes that investors should avoid the private equity market in the ab-
sence of truly superior fund selection skills since only top quartile performance is
sufficient to compensate for the higher risk and illiquidity.
Nonetheless, other possible reasons for strong cash inflow to this asset class
are reporting biases and right-skewed returns, making the investors with lottery-
seeking preferences pay overprice. We will study those two factors later in the
empirical section. Since there is a lot of dispersion between funds performance
and many different data providers that collect the data mainly from general and
limited partners and not so much from the public sources, the possibility of selec-
tion bias is inevitable, which could explain the controversial results private equity
performance.
While in the U.S., the Venture Capital category has historically been a vi-
tal part of the private market and socio-economic development of the country, in
Europe, not least because of the hostile regulation and tax environment, it has less
than 20% of private equity AuM (McKinsey & Company 2021). It can be argued
that during this latest trend of private market growth, the U.S. Venture Capital
market starts to be overvalued, while in Europe, there still might be possible to
find attractive opportunities (Fraser-Sampson 2010). Overall, Venture Capital in-
vestments stumble to the same problems as private capital in general. High fees,
modest average returns, and usually terrible average risk-adjusted returns chal-
lenge the investor to re-evaluate the attractiveness of this strategy. The typical
characteristics that are different from other private capital investments are higher
operational risk, the exclusivity of the top quartile funds, and the glamour repu-
tation of high-profile start-up success stories. The Venture capital results exhibit
wide dispersion of returns. For example, in the Investment Benchmarks report
(1985-2005), the VC mean return was 3,2% p.a. while returns ranged between
721% and -100%, resulting in a 51,1% standard deviation. Since the dispersion of
23
returns is highest in this strategy, it emphasizes the importance of manager selec-
tion. However, the exclusivity brings its hurdle here, since as an investor, to get
the money to the most skillful managers, one has to been already invested earlier
funds of this manager. At the same time, most promising success stories want
to corporate only with the best VC managers, which makes the success stories
naturally flow these funds. This creates a circle of success that diverges more and
more from the overall VC market. (Swensen 2009.)
In Europe, the leverage buyout strategy has been the leading private equity
investment strategy (Fraser-Sampson 2010). It is the more criticized strategy for
the abundant management fees and modest average risk-adjusted returns since,
in many cases, it is simply financial structuring more than developing corporate
operations. It is also studied that this strategy contains a significant negative
relationship between the size of the fund and the performance (Lerner, Schoar and
Wang 2008). However, as Swensen (2009) stress, an investor cannot just pick the
small funds since those usually contain higher operational risks. Instead, to find
successful funds from leveraged buyout firms, Swensen (2009) emphasizes the funds
with lower leverage and higher operations development focus. This way, LBO can
offer significant risk-adjusted returns. The key to this is still to find skillful PE
managers, which is possible only with heavy due diligence. This recipe applies to
all private capital investments (Swensen 2009).
2.1.2 Private Credit
Private credit can be defined as higher-risk debt investments in unlisted, typically
small and medium-sized companies. PC is often used to finance corporate and real
estate transactions, projects, foreign trade, and corporate restructuring. While
many LBO private equity funds seek successful businesses with positive cash flows
to squeeze a little bit more out of them by developing the company, private credit
is usually used to distress and turnaround deals since this allows the creditor to
structure propitious loans from an investor point of view (Fraser-Sampson 2010).
From a borrower’s view, PC is usually desired private capital in family-owned
companies since it does not affect the ownership situation as PE does (Kiesel et al.
2010).
In particular, the market exists because post-financial crisis regulation has
forced banks to reduce their lending, especially for riskier opportunities. The lack
of credit supply created a gap in the market that Private Credit players have
grasped. Indeed, the market has grown from 150 billion USD in 2007 to about
24
1000 billion USD by 2021. Differences to traditional bonds are lack of credit rat-
ings, little information about counterparty, non-existent or poor secondary market,
credits that are almost always collateralized, multiply covenants, higher recovery
rates, and the effective loan term is usually three to four years. Although the
asset class has many more favorable features to the investor than in traditional
public bonds, counterparties are still interested in borrowing from private credit
investors. Reasons for this are faster loan structuring, a small number of investors,
and cost-effectiveness. These features make PC loans more flexible for changes
which can be helpful for both lender and borrower. (Kiesel et al. 2010.)
Private credit strategies can be divided into three subcategories depending on
their risk level. Namely, these categories are from the high to low risk; oppor-
tunistic credit, specialty finance, and covered senior loans. Opportunistic credit
comprises, for example, distressed debt for companies that have or have the chance
of filing for bankruptcy in the near future and mezzanine investments in debt that
subordinate to the primary debt issuance and senior to equity positions. Specialty
finance includes, for example, project finance, foreign trade financing, and invoice
factoring. Covered senior loans are usually direct loans to small and medium-sized
companies, and they are highest on the capital structure so that they will be repaid
first if the borrowing company default. (Fraser-Sampson 2010.)
2.1.3 Hedge Funds
Hedge funds, or as Swensen (2009) refer to them as Absolute return funds, are
again a very heterogeneous group that is argued not to be asset class on its own. It
covers a wide range of investment strategies, which similarities are usually limited
to the use of derivatives and arbitrage hunting from the market (Fraser-Sampson
2010). One crucial difference to other alternatives introduced in this study is
that hedge funds use mostly liquid assets. However, many times hedge funds are
illiquid for the investor. Hedge funds face less regulation than conventional funds
with traditional assets, and that way, they are more flexible in using leverage,
short selling, and derivatives. Investing in hedge funds is typically investing in the
fund managers’ skills since the idea is to find inefficiency from a relatively efficient
market creatively. The word "hedge" does not mean that the managers hedge all
or even systematic risk away but tells more about the standpoint where managers
try to isolate the wanted exposure from everything else. In other words, they
try to hedge all other risks away from the fund, but the risk exposure they have
some view (Ilmanen 2011). To go through all the different hedge fund strategies
25
and their properties, one could write a book about it, which is not the focus
of this study. However, namely to go through, there are identified at least the
following strategies; long/short, market neutral, Convertible arbitrage, statistical
arbitrage, merger arbitrage, fixed income arbitrage, global macro, event-driven,
and distressed (Fraser-Sampson 2010).
The freedom often explains the hedge funds’ positive performance even after
private equity-like fees compared to the average fund manager, which entails a
broader spectrum of opportunities and skills acquired to hedge funds with attract-
ive rewards. This would indicate that money moves from not so skillful investors
to skillful investors in a systematic and persistent way. It is also argued that
hedge funds earn rewards from market completion economic functions and illi-
quidity premium thanks to investor lock-ups. From a pessimistic point of view,
it can be argued that positive results are due to reporting biases and unobserved
risks. Since hedge funds reporting is voluntary, most return data sets have some
degree of selection, survivorship and backfill biases, making the data overstate re-
turns and understate risks. One way to decrease those biases is to use Fund of
funds data. That data, however, suffers from double-layer fees. Also, the compel-
ling risk-adjusted returns are due to chosen risk indicator, volatility. Many hedge
fund strategies contain adverse skewness and kurtosis by their nature. These cata-
strophe insurances like strategies involve, for example, being short in volatility. In
a normal market situation, they collect a steady premium from the market, but
the value is expected to collapse under market stress. (Ilmanen 2011.)
As Ibbotson, Chen and Zhu (2011) illustrated, survivorship and backfill biases
are particularly big problems with hedge funds. In their study, survivorship bias
contributed 3,1-5,2 percentage points to yearly hedge fund returns and backfill
bias 1,4-3,5 percentage points. Even so, with adjustments for survivorship and
backfill biases as well as traditional risk factors, Ibbotson et al. (2011) found that
hedge funds, on average, produced 3,0% alpha per annum in the period 1995-2009.
However, there are still selection, liquidation, lookback, and lookahead biases, that
have not been studied. Also, and more importantly, the risk adjustments do not
account for illiquidity or tail risks. Like in private equity, Swensen (2009) highlights
that investing successfully in hedge funds demands the ability to find top-quartile
managers, which again demands extraordinary resources to do due diligence to
identify those managers.
26
2.1.4 Real-Estates
Real-estates are available to investors through private real estate (direct invest-
ment, fund, or fund of funds) and publicly traded real estate investment trusts
(listed REITs). The asset class can be divided roughly into two types; commer-
cial and residential real-estates. Typically institutions hold investments in private
commercial real estate. Ilmanen (2011) identifies that real-estates are usually ad-
ded to portfolios, not because of their expected returns but their inflation hedge
and diversification properties. This finding is in line with the earlier introduced
Preqin survey. Real-estates are one of the three real assets in this study. (Ilmanen
2011.)
Francis and Ibbotson (2009) studied real estate investing compared to tradi-
tional asset classes and highlighted its ability to diversify the volatility and enhance
the returns through the available leverage. They also point out the problematic
smoothing bias in physical real-estate value time series. Indeed, real estate data
has its challenges. While REIT data is relatively clean, it correlates more with
the equity market than private real estate and, for that reason, has higher volat-
ility. The private real estate return comprises of appreciation and rental yield.
While rental cash flows are quite straightforward, calculating the appreciation can
be done through appraisal-based or transaction-based values. Appraisal-based in-
dices aggregate real estate values from the previous period5 and, for that reason,
smooths and stales the index values. This gives an erroneous picture of volatility
and correlation of the asset class. Transaction-based data tackle these problems
but, at the same time, it is very scarce data. (Ilmanen 2011.)
Real-estates can be classified according to their risk and return characteristics.
Core and Core Plus categories are the safest, including main properties in primary
markets and a low level of leverage. The Plus category differs from the former with
the modest value-add approach, wherewith slight improvements in the property
can create value. Then we have the value-added category, representing moderate
risk/return investments. These real estates need proper development to get them
class A buildings, and these type of investments usually requires 50-70% leverage.
In the "high yield" end of real estate investments, opportunistic and distressed
categories represent the high-risk, high reward investments. While the real-estate
investments can be quite heterogeneous as well, the main difference in risk/return
structure in real-estates comes from the level of utilized leverage. Data of this study
aggregates all those categories to one index, but in practice, the investor can quite
5Typically a year.
27
easily choose a preferred category from the spectrum through private real estate
fund, open-end fund, or direct investment to real estate. (Fraser-Sampson 2010.)
Depending on the used appreciation technique, the historical real total return
of commercial U.S. real-estates varies between 6% and 7.3% p.a. in different data
sets. The former is from Francis and Ibbotson (2009) study and covers the period
between 1978-2008. The latter is from NCREIF6 appraisal based data and covers
1984-2009 (Ilmanen 2011). Real total return for residential U.S. real-estates has
greater dispersion depending on reviewed data. Francis and Ibbotson (2009) get a
geometric average per annum real return of 1,7% for the 1978-2008 period, while
Ilmanen (2011) gets 7% between 1960 and 2008 by combining Davis-Heathcote
(2007) HPA data with Davis-Lehnert-Martin (2008) rental yields data. While there
is a dispersion in the reported numbers, they leave open a quation about inflation
hedge properties. By definition rental contracts are inflation hedged. Also, as
Swensen (2009) points out, the real estate market has a strong relationship between
market value and replacement cost. Since replacement costs consist largely of labor
and material costs that move with inflation, real estate appreciation also enjoys
inflation hedge properties. In empirical section we examine the real-estate return
correlation with CPI inflation to confirm these properties.
2.1.5 Infrastructure
Fraser-Sampson (2010) defines infrastructure investing as planning, funding, con-
structing, and operating public sector assets. Infrastructure is a real asset like
real estate. However, infrastructure can be divided into two distinctive opportun-
ities; primary and secondary. Primary investing means private investment pools
that are used to the long horizon projects, so the investor is not buying anything
already built but instead committing to the building projects and that way have
the opportunity to earn from the added value of creating something new. Second-
ary investments instead resemble long-term government bonds as they are already
operating real assets with long horizon steady cash flows. The former represents
private equity-like investment (higher risk and return characteristics), while the
latter is an excellent alternative for government bonds with inflation hedge proper-
ties. Even though the asset class is heterogeneous, its investments usually focus on
transportation, utilities, communications, and energy. The investments are char-
acterized by quasi-monopolistic position and inelastic demand. Again, investors
6National Council of Real Estate Investment Fiduciaries
28
can invest in infrastructure through private investment pools or publicly traded
infrastructure companies (Ilmanen 2011).
Investors need to consider specific risk characteristics when investing in infra-
structure, namely regulatory, political, and geopolitical unrest. Most importantly,
the market needs to have a low probability of getting governmental intervention
in projects and their financing (Fraser-Sampson 2010). At the same time, the gov-
ernment needs to develop, and that way, financing national infrastructure creates
investment opportunities for the infrastructure market. For example, it is not by
accident that infrastructure investments are booming right now. In addition to the
normal need for infrastructure development, the green transition moves tremend-
ous capital to infrastructure. IRENA (2021)7 approximates that over the period
to 2050, there will need to flow 131 trillion USD to energy systems to keep us in
the 1,5C pathway8. At the same time, Joe Biden’s Bipartisan Infrastructure Bill
is approved in the U.S., totaling 1,2 trillion USD, where 550 billion USD is new
capital to the infrastructure sector (Times 2021). This trend inevitably lays the
ground for lucrative return possibilities to this asset class in the future.
As with real-estates, the infrastructure uses the same categories for the con-
stituents for the index used in this study. Namely Core, Core Plus, Value-Added
and Opportunistic. The Core represents the low-risk end with secondary invest-
ments in stable countries with transparent regulations. These infrastructure op-
erations are characterized by a monopoly position, demonstrated demand, and
long-term stable cash flows that are forecastable with a low margin for error. On
the other end of the risk spectrum is opportunistic infrastructure investments.
These are primary investments that focus on capital growth completely without
existing cash flow. (Preqin 2021.)
It is important to notice that the way investors create exposure to this asset
class is not irrelevant. For example, many infrastructure ETFs have just exposure
to sectors related to infrastructures like utilities and telecommunication. However,
that does not mean that these ETFs have any exposure to actual infrastructure
projects. At the same time, their correlation with the general equity index in the
same market is usually extremely high9, which spoils the idea of diversification
which usually is one of the main reasons to invest in infrastructure in the first
place. (Fraser-Sampson 2010.)
7International Renewable Energy Agency
8Keeping the climate warming under the 1,5℃
9For example Fraser-Sampson (2010) calculated correlation of 90% between MSCI European
infrastructure index and MSCI European equities index
29
2.1.6 Natural Resources
Natural resources combine agriculture, energy, commodities, timberland, and wa-
ter (Preqin 2021). This asset class represents real assets along with earlier in-
troduced real estate and infrastructure, and it is typically characterized by low
correlation with other assets and inflation hedging properties (Ilmanen 2011).
Natural resources exposure can be created through futures markets or purchasing
physical reserves. While one gets easy exposure to commodity price movements
through future contracts, it is highly dependent on the spot price movements. As
Ilmanen (2011) denotes, future returns consist of purely collateral return, roughly
representing a risk-free rate, roll return representing ex-ante risk premium, and
spot price change representing the unexpected return. Swensen (2009) highlights
the reserves’ ability in the energy market to produce high expected returns even
without price movements. Same time investors still get exposure to the price move-
ments. Also, investing in timber through private timberland enhances the returns
through other activities in addition to changes in timber value. First of all, there
is biological growth which silvicultural practices can influence. Biological growth
usually contributes the most to the total return of the timberland. At the same
time, these are incremental revenue streams for timberland, and an increase in
land value is one significant contributor. A steady revenue stream can be gained
through ecosystem services. These include carbon sequestration, forest protection,
recreational pastimes, and leasing land for alternative energy use like wind power.
All this makes timberland investment uncorrelated with the equity market while
it is highly correlated with inflation making it a great hedge against it. (Swensen
2009.)
Again, this value creation is due to the complex and inefficient real asset world
where expertise can create value compared to the highly efficient commodity fu-
tures market. Private acquisition on energy reserves carries the same principles
as the private equity fund investment, so it is not trivial which manager runs the
operations. This study focuses on physical natural resources that can be invested
through private funds, open-end mutual funds, or direct investments in a physical
asset.
30
2.2 Investment vehicles and liquidity
Alternative assets are usually linked to be illiquid investments. While most of
the underlying assets of alternative investments are illiquid, different investment
vehicles enable investors to invest liquid or at least more liquid ways to alternative
investments. However, as Fraser-Sampson (2010) and Swensen (2009) point out,
investors should always seek illiquid private investments to get clear exposure to
alternative betas and low as possible correlation with equity and bond market.
However, liquid publicly traded alternatives can be helpful in situations where
illiquidity is a problem for the investor from a regulatory or investment objectives
point of view.
Liquid investment vehicles that invest in illiquid alternatives offering liquidity
premium to end investors have become trendy products in the asset management
industry. Namely, these vehicles are usually alternative investment mutual funds or
fund of funds. Nevertheless, as Fraser-Sampson (2010) doubts, it might be hard to
keep the fund’s liquidity in the market stress situations since the underlying assets
are still illiquid. For that reason, mutual funds usually have the right to freeze the
redemptions in case of liquidity droughts. However, here is important to stress that
even traditional assets that are generally considered liquid have turned out to be
everything else under severe market stress (Fraser-Sampson 2010). Herd behavior
can make liquidity management assets to be the most illiquid ones in certain
market drawdowns. This gives us an interesting perspective on the empirical part
of this study, where we look at the tail risks of different asset classes. If the risks of
alternative investments are primarily realized under the market stress, as suggested
in this study, and at the same time the risks of traditional asset classes are realized
both in calm market periods and under the market stress, the question arises as
to whether it makes sense for a long-term investor to invest in the public market
at all.
Investing in alternative assets ETFs means investing basket of publicly traded
stocks that operate on sectors related to alternative assets but are not purely
alternative investments. An investor can usually find a single stock from the
market which operates either with private equity, private credit, hedge funds,
real-estates, infrastructure, or natural resources. However, those usually differ
significantly from alternative investments’ characteristics, and even though they
might have some small correlation with the corresponding alternative asset class,
they are mainly driven by stock market beta. (Fraser-Sampson 2010.)
Traditionally liquidity of an asset is seen as a virtue. When an investor gets the
31
sudden need for cash, investments are helpful only when they are liquid. However,
as Swensen (2009) points out, liquidity is expensive. Having too much liquid-
ity in one’s portfolio is poor investing. It can be argued that the opportunity
cost from holding liquid assets is due to weak liability modeling. This is why
long-term investors should seek a liquidity premium for assets they will not need
in the near future. However, illiquidity can be a problem also when investing
money. Especially private equity and credit investors face problematic illiquidity
through capital commitments where the capital call can come in an unfavorable
moment (Ilmanen 2011). From the fund manager’s point of view, illiquidity en-
ables the manager to make long-term decisions necessary for investment success.
This longevity should convert higher returns for the limited partners as well. In
liquid markets, public companies’ managers and investors get feedback constantly.
This pressure might push to make decisions that are not constant with the long-
term strategy (Swensen 2009). So it is safe to say that liquidity is a double-edged
sword. It is highly dependent on the investment objectives of how much liquid
assets one should have. Planning liquidity needs accordingly and optimizing the
portfolio with that objective in mind should enhance the investment portfolio’s
performance.
Traditionally in finance, volatility is seen as negative, and liquidity is seen as
positive property. To carry illiquid assets, one should earn a liquidity premium.
According to that, alternative investments’ illiquidity should reduce the utility
obtained from their low volatility. However, low volatility is mainly due to low
liquidity with alternative investments. This might be one reason why, a little
counter-intuitively, there is some evidence that institutional investors are ready to
pay extra instead of getting a premium for illiquid assets. The idea is to smooth
return volatility away, albeit artificially (Ilmanen, Chandra and McQuinn 2019).
This challenges the existence of a liquidity premium.
Assets illiquidity is usually statistically estimated with serial correlation (Cave-
naile, Coën and Hübner 2011; Getmansky, Lo and Makarov 2004). However, we
must acknowledge that liquidity typically drains in extreme market stress condi-
tions. For that reason, using serial correlation of returns, which mostly incorporate
normal market times, is wanting estimator, but it is typically recognized as the
best available option for lack of a better one. We will examine asset classes serial
correlation in our data sets in the empirical section of this study.
32
2.3 Institutional investors & Case Yale Endowment
Goal is not to make money,
making money is how one
accomplishes the goal
Robert van den Meer
Institutions differ from individuals as an investor a lot. While individuals have
preferences that we try to model with utility functions, institutions have object-
ives that can be modeled with economic factors and liabilities. While individuals’
investment horizons and allocations rely on the life-cycle hypothesis, institutions
typically have an infinite investment horizon. At the same time, we have seen
institutions allocations evolve. While Dutch Pension funds have led the way in
increasing equity proportion in pension funds, the Yale Endowment fund has in-
creased the interest towards alternative investments as a major asset class in insti-
tutions’ portfolios (Van der Meer 2001; Swensen 2009). In this chapter, the idea
is to drill down how these differences affect asset allocation and how, on the other
hand, the institutions have evolved in their asset allocation policies.
Portfolio construction is about risk/return optimization, and like previously
discussed, risk can be defined and observed in many ways. After we have an effi-
cient set of choices, individual investors pick portfolios according to their subjective
utility function. However, institutional investors usually have some liabilities to
meet in the future that are not constant. For example, they can vary according to
inflation, demographic trends, number of participants, maturity of the workforce,
or interest rates. These factors add return and liabilities correlation to review
and eventually lead to return/liabilities optimization. Since institutions’ liabilities
can differ a lot from each other, this return/liabilities aspect in portfolio optim-
ization explains the allocation differences that institutions have. However, even
though the eventual objectives for institutions’ portfolios can vary a lot, a start-
ing point for every institution’s portfolio construction is to cover inflation, and on
top of that, institutions start to pile up different objectives. Swensen (2009) for
example, distinguish endowments and foundations investment goals due to regulat-
ory. While endowment can dampen fluctuations in the market value by stretching
the rules with asset preservation or stable allowance distribution, foundations are
usually obliged to achieve a minimum payout level to retain their public benefit
tax status10. This way, they are bound to stretch with the asset preservation.
10In the U.S. the payout level is strictly given to 5% of assets, while in Finland the payout level
33
Swensen (2009) argues that this should lead to more conservative allocations in
foundations than in endowments. (Swensen 2009; Van der Meer 2001.)
To succeed, institutions need to do the investment planning from the objective
perspective where the only concern is to meet the institution’s purpose, whatever
future liabilities that hold. However, in practice, institutions have many stakehold-
ers with different incentives. This adds complexity to the institutions’ portfolio
management and is the most common reason institutions fail to meet their liab-
ilities (Swensen 2009). However, from the risk perspective, Sortino, Satchell and
Sortino (2001) argue that every stakeholder is concerned about the expected short-
fall being more than the minimum acceptable return (MAR) to meet the liabilities.
That is also why we are interested in the expected shortfall in this study since it
is a risk metric that everyone agrees to be managed. The rest of the institutions’
investment planning lies in the fiduciaries’ hands, who have a critical job to decide
the allowance level trade-off between current and future beneficiaries (Swensen
2009).
The Van der Meer (2001) introduced the Dutch Triangle to illustrate how dutch
pension funds operate. The idea is that, at the strategic asset allocation level, every
pension fund should start with comprehensive asset-liability management (ALM)
analysis where every stakeholder’s risk-return structures are analyzed in relation
to their constraints. This way, we get constraints to the actual optimization prob-
lem, where the idea is to maximize the ratio of expected return over MAR relative
to expected shortfall under the MAR. Van der Meer (2001) argues that the Dutch
Triangle differs from traditional pension fund management by bringing the MAR
in the center of asset allocation and performance measurement. It represents the
benchmark to which all risks are related. The level of MAR represents the objective
goal to meet the institution’s purpose, while the ALM analysis brings the stake-
holders different incentives to the optimization problem as a constraint. This way,
the stakeholder’s individual risks are taken into account while the maximization
objective is based on purely institution objectives.
Another interesting and groundbreaking asset allocation in institutional port-
folio management is introduced by Swensen (2009) in the Yale Endowment fund.
Swensen (2009) was the first portfolio manager who brought alternative invest-
ments into the major role of endowments’ asset allocation. While many asset
classes he introduced were unique in the institutional portfolio, the share of the
alternative assets in the overall portfolio stays unique in the Yale Endowment fund
is defined in income tax law to be "sufficient" and is more evaluated case by case (Swensen
2009; Verohallinto 2021)
34
even today. The figure 1 shows the asset allocation development in the Yale En-
dowment fund from 2004 to 2020. As we see, traditional asset classes have never
exceeded even half of the portfolio during that period11. Honor for this excep-
tional allocation goes to David Swensen, who joined Yale University as a CIO in
1985. Even though there were alternative assets in endowment allocation already
in the ’80s, in 1989, almost three-quarters of the endowment’s assets were still
in traditional asset classes. However, in the ’90s, Swensen increased allocation to
alternative assets drastically. As Swensen (2009) states, committing less than 5%
in some particular asset class is not worth of effort since the effect on the overall
portfolio is so small, and committing more than 30% is predisposed to overconcen-
tration. So accordingly, the ideal number of asset classes is about half a dozen, as
we see in the Yale Endowment fund.
0 %
20 %
40 %
60 %
80 %
100 %
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
FI & MM EQ HF PE NR RE
Figure 1: Yale Endowment fund asset allocation
If we compare this, for example, to Finnish Pension funds, the difference is
substantial (see Appendix II). The difference can be partly explained by the dif-
ferent additional factors affecting pension fund liabilities like demographic trends,
number of participants, and workforce maturity. However, endowment and pen-
sion funds also share many similarities. Contributions in a pension fund are more
predictable but represent additional cash inflows similarly to gifts in the endow-
ment fund. They increase assets as well as liabilities for the fund. However, the
investment objectives of the two institutions are the same, preserving allowances
purchasing power in the future and preserving fund’s assets. Finnish pension funds’
investment performance has been in the recent target of interest in Finland, and
11Highest allocation to traditional asset classes is 40,5% in 2004.
35
it is argued that current investment policy might not be able to meet the future
liabilities (Andersen 2021). As previously mentioned, the last ten-year market con-
ditions have also driven these funds to seek alternative assets. At an aggregated
level, Finnish pension funds allocation to alternative investments have increased
from 14,3% in 2004 to 28,5% in 2020 (Tyoelakevakuuttajat TELA Ry 2021). The
Yale Investment Office (2020) motivates the heavy alternative investment alloc-
ation to assets’ diversifying power and return potential properties. They further
ground their heavy allocation to alternative investments by tying it to their infin-
ite investments horizon, which is, according to Yale Investment Office (2020), well
suited for illiquid, less efficient markets. Throughout the Swensen (2009) book, the
attractiveness of alternative investments is based on exploitable inefficiencies that
they offer for a skillful active manager. These inefficiencies are due to the illiquidity
of these markets. This way, the tools to manage liquidity risk offer entry to these
markets, and we are back to emphasizing the importance of financial planning.
Naturally, the interest arises in how the Yale Endowment fund has performed
in history with this unconventional asset allocation. As we see from the figure 2,
the endowment has substantially overperformed compared to wide market indices
in the observed period. We view the performance in logarithmic scale so we get
accurate effects from crisis to performance. Endowment returned 10,6% annually
between 2004 and 2021, while in the same period, global MSCI World returned
7,9%, and U.S.-based S&P500 returned 9,6%, respectively. Even though we com-
pare apples to oranges, this gives a sense of how the market-insensitive portfolio
have performed compared to the sensitive equity market. Some might even ar-
gue that comparing endowment with risky equity-only indices is unfair since the
endowment also includes conservative assets. Here is, however, important to re-
member that during that period, the endowment had 14-40% in Private Equity,
which is usually seen riskier investment than public equity. The most interesting
remarks from the figure are the timing and size of drawdowns in various crises.
The way endowment survived from the dot-com bubble in 2001 is one of the main
reasons alternative investments and especially hedge funds gained attention in the
20th century. The endowment was substantially affected by the financial crisis
along with the market. However, while MSCI world recorded -40,7% and S&P500
-37% in 2008, the endowment recorded a positive return in the same year. Instead,
in 2009 endowment recorded -24,6% while the overall market went already up. The
lag was due to the endowment’s illiquid assets and yearly reporting. This logar-
ithmic return index illustrates well how the reporting biases affect the risk metrics
of illiquid assets. A great example is the Covid-19 crisis in spring 2020, which
can not be seen from the Yale endowment return index, even though the dip was
36
severe, since the rise was so fast. The lag with alternative investments is partly due
to reporting bias and partly due to market inefficiencies, which slows the market
information to transmit asset values, called stale pricing. The infrequent mark to
market with alternative assets gives portfolio managers and the investments them-
selves peace of mind to work towards long-term success and hedge investments
from the market fluctuations. This is one of the key factors why Swensen (2009)
sees private assets so remarkable in institutions’ strategic allocation. However, in
portfolio optimization and risk management context, we need to consider these
lags to understand the potential insolvency risks.
1047,4
464,6
349,2
1998 2002 2006 2010 2014 2018
L
o
g
ar
it
h
m
ic
 S
ca
le
 R
et
u
rn
 i
n
d
ex
MSCI World Net TR USD S&P 500 TR Yale Endowment
Figure 2: Yale Endowment fund performance
37
3 THEORETICAL FOUNDATION
3.1 Portfolio construction
The Modern Portfolio Theory (MPT) introduced by Markowitz (1968) demon-
strates how to construct and select portfolios based on the expected returns of
the investments and desired risk appetite of the investor. It works as a back-
bone of portfolio construction even today. In addition to the heavy utilization
of the modern-day practical finance industry, this model works as an important
starting point for new methods and inventions in the field of portfolio optimiz-
ation (Fabozzi, Focardi and Kolm 2010). It shapes a clear framework that can
be utilized when we challenge the premises of MPT. Figure 3 represents the pro-
cess Fabozzi, Gupta and Markowitz (2002) created for MPT. In MPT, Risk &
Co-movement estimates mean volatility and correlation estimates. However, when
we interpret that part of the process to illustrate risk and co-movements more
generally, the process becomes valid for other optimization models as well. This
study utilizes this framework in mean-variance optimization and mean-expected
shortfall optimization. In the latter, we have Expected Shortfall as a risk measure,
and in addition to correlation, we also consider co-skewness and co-kurtosis.
Figure 3: Portfolio optimization process
38
3.2 Modern Portfolio Theory
Modern Portfolio Theory (MPT) is based on the idea that the investor balances
with risk and return. The return is expressed by the ex-ante expected return, and
the risk is expressed as the ex-ante expected variance of this return. Markowitz
(1968) showed that one gains diversification benefits by constructing a portfolio of
not perfectly correlated securities. This benefit is often called the only free lunch
in finance (Choueifaty and Coignard 2008). One of the major contributions of
the theory is the perception that when adding security to a portfolio, relevant for
the risk of this portfolio is the correlation between security and portfolio, not the
variance of this security. Markowitz (1968) proved this mean-variance theorem
where investors can either minimize variance (i.e., the expected risk on return
being constant) or maximize the expected return when variance is constant.
The diversification benefit means that security’s risk in the portfolio is lower
than holding it standalone. After Markowitz (1968) proved this to be accurate,
we needed a new risk definition. Sharpe (1964) introduced the Capital Asset Pri-
cing Model (CAPM) for this need. In this model, the total risk of a security
is divided into two, systematic market risk and non-systematic security-specific
risk. So the model continued to fine down the diversification benefit introduced
by Markowitz (1968). Indeed, Sharpe (1964) showed that diversification could ex-
plicitly eliminate the unsystematic risk of securities when adding securities to the
portfolio so that the overall risk of the portfolio approaches the systematic risk of
the securities. As unsystematic risk can be diversified away, the market does not
reimburse this part of the risk to the investor. Since the model assumes homogen-
eous expectations and efficient markets, the unsystematic risk is not included in
the CAPM since investors should not expect any returns from carrying idiosyn-
cratic risk. Therefore, according to the model, investors are only interested in the
systematic risk and correlation of securities with the market portfolio. The rela-
tionship between systemic risk and expected return of an asset can be presented
as the CAPM formula:
E(Ri) = rf + i(rm − rf ); (1)
where E(Ri) is expected return of an asset i, rf is the risk-free rate, rm is the return
of a market portfolio, and i is a market risk premium multiplier for security i. In
other words, i describes the systematic risk of security (Brandimarte 2017). Now
we can divide risk of an individual asset i = m + ie to two, systematic risk
(m) and unsystematic or idiosyncratic risk (ie), where m is the market risk.
39
The idea of diversifying idiosyncratic risk away by forming efficient portfolios is
the most important insight of the Modern Portfolio Theory.
We can form an efficient frontier for the best-performing portfolios at each
risk level by calculating the expected returns and their variances on all alternat-
ive portfolios in the market. The efficient frontier represents the portfolios with
the best return-to-risk ratio at each risk level and thus in which every rational
investor invests. By minimizing the function of the efficient frontier, we obtain the
minimum variance portfolio, which represents the portfolio with the lowest risk
on the efficient frontier. According to the mean-variance theorem, all portfolios
above the minimum variance portfolio are efficient when moving on the frontier.
All well-diversified portfolios belong to the efficient frontier if the capital markets
are efficient. (Sharpe 1966.)
Another important portfolio in the model is the tangent portfolio, which, under
the assumptions of the Efficient Market Hypothesis (EMH) introduced by Fama
(2021), represents the market portfolio if all possible securities have been involved
in optimizing the efficient frontier. Thus, the market portfolio includes all risky
assets with weights according to their market values (Markowitz 1968). When we
add the opportunity to lend and borrow at a risk-free rate, we can form a capital
market line (CML). Lending in risk-free rate is practically done by being long in
short-term Treasury bills and borrowing by being short in those same Treasury
bills (Elton, Gruber, Brown and Goetzmann 2009). The capital market line is
formed from various portfolios consisting of a market portfolio and a risk-free
rate investment. Assuming that the investor is only interested in the expected
return and its variance, every rational investor maximizes their utility by forming
a portfolio from the market portfolio and risk-free rate investment according to
their risk preferences (Sharpe 1964). The theoretical relationship between efficient
frontier, risk-free rate, and market portfolio are illustrated in figure 1. Here Pi
represents the market portfolio, RF the risk-free rate, and the line between them
is a Capital Market Line (CML). The line between B and C represents the efficient
frontier, and point B represents the minimum variance portfolio.
As the figure illustrates, the investors maximize their utility by selecting a
portfolio from CML, assuming that they can invest in the market portfolio and
risk-free rate. It implies that the only efficient portfolio in the efficient frontier
is the market portfolio. This is consistent with the CAPM since, under CAPM
assumptions, investors should hold the same market portfolio, and so the tangency
portfolio must be the market portfolio (Brandimarte 2017).
It is essential to point out, that Markowitz (1968) MPT does not assume joint
normality of security returns, contrary to popular beliefs. MPT is just consist-
40
Figure 4: Efficient frontier with riskless lending and borrowing
ent with the assumption, that security returns are jointly normally distributed.
However, since the optimization and the efficient frontier focuses only first two
moments of the return (mean and variance), it does not tell us anything about
risk related to higher moments. For that reason, it is an optimal optimization
method for investors having a quadratic utility function. (Fabozzi et al. 2010.)
In modern portfolio theory, the market portfolio is said by definition to include
all risky assets. However, practically the difference between the tangent portfolio
and theoretical market portfolio is the lacking alternative investments in the former
one. In theory, leaving out the alternative investments should lead us to a sub-
optimal solution where we have not utilized all the diversification benefits, and our
portfolios are susceptible to unsystematic risks which are not rewarded since it is
diversifiable. Hence, it is studied that in practice allocating between the risk-free
rate and tangent portfolio has not delivered optimal results since more efficient
portfolios can be found. The problem stems from the difference between tangent
and market portfolios. For example, the MSCI All Country World Index, which is
usually used as a tangent portfolio, is actually quite focused on a couple of large
companies as well as sectors, and for that reason, the degree of diversification is not
ultimately maximal (Choueifaty, Froidure and Reynier 2013.). However, it is often
stated that most of the idiosyncratic risk is diversified away only with 15 different
securities, and the marginal utility of adding more securities decreases drastically
after that. It is important to note that these studies usually focus only on equity
41
portfolios with less diversification benefit available than multi-asset portfolios12
(Mao 1970; Statman 1987; Elton and Gruber 1977.)
So, at least in theory, adding alternative investments to the portfolio should
raise the efficient frontier, assuming that they are not perfectly correlated with
other investments. This is demonstrated in Figure 5, which represents a practical
view of the efficient frontier. Here the line between B1 and A1 represents the
efficient frontier with traditional investments. If we include alternative investments
in the investment universe, we can move to the upper efficient frontier that the
line between B2 and A2 represents. The figure also illustrates the well-known
problem of tangent (P1) and market (P2) portfolios being unequal. If investors
cannot invest in the market portfolio due to some restrictions, they will be moving
on CML1 instead of CML2, which has a higher expected return on every level of
risk or lower expected risk on every level of return. In this framework, the next
natural question is whether this rise of an efficient frontier is exploitable. In other
words, is it remarkable enough to cover costs and other inefficiencies related to
alternative investments?
3.3 Mean-Variance optimization
Before optimizing portfolios, we need to review how we generate the inputs we
utilize. Key concepts for portfolio management are risk and return. Returns
are used in portfolio management instead of prices because they offer a complete
picture of the investment opportunities that are not affected by the amount of
average investor invests in competitive markets and because they have desirable
statistical properties compared to prices (Campbell, Lo and MacKinlay 2012).
Return of an investment can be calculated either by simple percentage returns or
logarithmic continuously compounded returns. The relation between these two
can be shown as:
rt = ln(1 + Rt) = ln

Pt +Dt
Pt−1

; (2)
where rt is continuously compounded return at time t, Rt is percentage returns
12With his theoretical model, Mao (1970) tested the percentage diversification benefits of cor-
relations () and numbers of different securities and found that if securities  = 0:5, then 17
securities achieve 90% of the available diversification benefits. If, on the other hand,  = 0:2,
the available diversification benefit is greater, and then 34 securities are required for the cor-
responding 90% relative benefit.
42
Figure 5: Efficient frontier with alternative assets
at time t, Pt is price of an instrument at time t, and Dt is the cash flow of an
instrument in period t. Continuously compounded logarithmic returns have many
desired properties compared to percentage returns. They do not have boundaries,
are easy to calculate and can be normally distributed in time series aggregation.
The continuously compounded returns beneficial time aggregation properties are
easy to see when we compare its calculation to percentage returns in Equation 3:
Ri;T =
TY
t=1
(1 + Ri;t)− 1
ri;T =
TX
t=1
ri;t;
(3)
where T means the aggregated time period. While percentage multi-period re-
turn is a product of single-period percentage returns, the logarithmic multi-period
return is the sum of single-period logarithmic returns. However, we need to use
simple percentage returns in cross-sectional aggregation since portfolio return is
a weighted average of percentage returns instead of logarithmic ones. Also, it is
important to convert returns back to simple returns at the end of the portfolio
43
optimization process since, at the end of the day, simple returns are the ones that
matter for the investor.
In MPT, the risk is defined as a standard deviation from its mean return. It is
typically calculated as a variance of returns for computational convenience instead
of standard deviation (Brandimarte 2017). This symmetric risk metric for security
can be calculated from the return sample as follows:
2 =
1
N
NX
t=1
(rt − )2; (4)
where ri is the return of security,  is the sample’s mean return, and N is the
sample size. When we start to aggregate risks at the portfolio level, covariance
comes to an important role. Covariance measures the linear dependence of two
assets’ returns. In other words, it tells how two individual asset returns move
together. It can be calculated from the return sample as follows:
ij =
1
1−N
NX
t=1
(ri;t − i)(rj;t − j): (5)
We should use a normalized version from covariance to compare different assets
relationships. Correlation is a metric for that purpose since it can get values only
between 1 and -1. Correlation can be calculated from Equations 5 and 4 as follows:
ij =
ij
ij
: (6)
Now we get to the actual mean-variance optimization, utilizing Markowitz
(1968) findings of Modern Portfolio Theory. As stated in the last section, the
diversification benefit stems from the idea that security’s risk is lower in the port-
folio context than individually. To define the portfolio risk, we are interested in
covariances of assets included in the portfolio. For that reason, we need to con-
struct the following variance-covariance (VarCov) matrix:
 =
2664
11 · · · 1N
... . . .
...
N1 · · · NN ;
3775 (7)
where 1N is the covariance between asset 1 and N. In diagonal, we have variances
of an assets since covariance of an asset with itself is variance (Fabozzi et al. 2010).
Portfolio return can be calculated in matrix context:
p = w
′; (8)
44
where w′ is transpose from a vector of individual assets weights in portfolio and
 is a vector of expected returns of assets. While portfolio return is calculated
as a weighted average of individual asset returns, the portfolio’s variance utilizes
VarCov matrix in Equation 9:
2p = w
′w: (9)
It follows that the portfolio’s variance is lower than the simple weighted average
of individual asset variance.
When we move on to the optimization problem, we must notice that we are
dealing with expectations. This means that we are interested in constructing op-
timal portfolio ex-ante. For that reason, inputs to the return vector and VarCov
matrix will play a significant role. There are many alternatives to estimate ex-
pected returns and their covariances. One can observe historical returns and use
sample mean for returns and sample covariance for covariance matrix as we intro-
duced above. However, it can be argued that past performance is not a reliable
indicator of future performance. To get more forward-looking prediction power,
one can use statistical techniques such as Bayesian and shrinkage estimator or
factor models to generate more robust inputs (Fabozzi et al. 2010). However,
here we go through the portfolio optimization problem assuming expected returns
and their covariances are given and utilize historical samples in the methodology
chapter.
In order to optimize asset weights in a mean-variance framework, one can
either maximize expected return for target variance or minimize expected variance
for target return. We go the optimization problem through with the latter setup.
Following MPT, investors have a constrained minimization problem. It is quadratic
optimization with equality constraints:
min
w
1
2
w′w
s.t. w′ = 0
w′ = 1;
(10)
where 0 is the target return, and  is a vector of ones. Since this optimization
problem can be solved with Lagrange multipliers, we get the Lagrangian function:
L = 1
2
w′w + (1−w′) + 
(0 −w′); (11)
where  and 
 are the Lagrange multipliers. Now we can take first-order conditions
from to function, which means partial derivatives with respect to each parameter:
45
L
w
= w − 
−  = 0
L

= 1−w′ = 0
L

= 0 −w′ = 0:
(12)
Solving first-order conditions with respect to weights w, we get:
w = −1+ 
−1: (13)
Combining this with constraints, we get a linear system for  and 
:
w′ = 0−1+ 
0−1 = 1
w′ = −1+ 
0−1 = 0;
(14)
which can be solved using the following matrix form: 
A B
B C
! 


!
=
 
1
0;
!
(15)
where A = 0−1; B = 0−1 and C = 0−1. Now we can combine equation
9 and equation 15 to get the variance of the portfolio:
2p = w
′w =
A20 − 2B0 + C


; (16)
where 
 = AC − B2. To obtain weights for the portfolio at the efficient frontier
for chosen return level, we solve Equation 16 with parameter 0. To get the global
minimum variance portfolio Pi presented in figure 4, we can take derivative respect
to 0 from equation 16:
d20
d0
=
2A0 − 2B


= 0; (17)
and now we get analytical solution for portfolio weights:
wg =
−1
0−1
: (18)
The basic mean-variance optimization is very sensitive to expected return in-
puts. The way expected returns are estimated can greatly impact weights obtained
from optimization (Fabozzi et al. 2010). While historical mean returns are poor
estimators for future returns, the problem even increases when we are dealing with
fat-tailed return distributions since sample mean is best linear unbiased estimator
(BLUE) only for normal tailed distributions (Ibragimov 2007).
46
3.4 Post Modern Portfolio Theory
Risk, like beauty, is in the eye of
the beholder
Leslie A. Balzer
In the 1990s, Rom and Ferguson (1994) published a paper called Post-Modern
Portfolio Theory comes of age in the Journal of Investing. This paper created
a concept called Post-Modern Portfolio Theory (PMPT) which recognized the
limitations of MPT and focused especially on defining the risk to answer better in
real world implications. In a nutshell, the idea of PMPT is to combine the long-
identified investor preferences to value upside volatility over downside volatility
to the MPT framework (Sortino et al. 2001). However, as the name of their
publication refers, this was not a particularly new perspective. Rom and Ferguson
(1994) start by pointing out that the fathers of MPT, Markowitz and Sharpe,
stated already when creating MPT, that downside risk measures would be more
appropriate to define risk than dispersion measures (variance). However, due to
the limited computational possibilities in the 1950s, they needed to base their
analysis on variance and standard deviation. Rom and Ferguson (1994) saw that
the time to implement downside risk into the MPT framework in the 1990s was
right and introduced the tools to do that in their paper. They concluded that
MPT is a special case of PMPT where applies symmetric returns and quadratic
utility function. Since then, the different asymmetric downside risk measures have
gained attention in the risk and portfolio management field, expanding the idea of
PMPT.
Rom and Ferguson (1994) defined downside risk as tied to investors’ objectives.
They introduced a concept called Minimum Acceptable Return MAR, represent-
ing the boundary between risk and opportunity. Unlike in MPT, where volatility
around mean return is seen as a risk, in PMPT, volatility below target return is
only seen as a risk. Volatility over target return is seen as a riskless opportunity
instead. This approach removes the symmetric risk measure problem, which pen-
alizes overperformance as much as underperformance. Rom and Ferguson (1994)
utilized expected returns, standard deviation, and skewness to measure downside
risk. They divided downside risk into two components, downside probability, and
average downside magnitude. Today’s downside risk measures, Value at Risk, and
Expected Shortfall can be seen to represent these components. In portfolio man-
agement, this risk definition leads investors to optimize a new efficiency metric,
47
the Sortino ratio. Since PMPT, in addition to downside risk, also focuses a lot on
investors’ objectives, it emphasizes the importance of investment planning, which
defines the strategic allocation.
Hadar and Russell (1969) introduced already in 1969 Stochastic Dominance
rules, which are grounded on the idea that expected utility theory, which MPT is
also based on, is a function of all moments of the probability distribution function,
not just the first two. These rules can assess which return distribution dominates
with any other distribution and is not affected by the assumed utility function. In
other words, it tells the investor’s choice despite one’s preferences. The first-order
stochastic dominance rule states that investors would always prefer distribution A
with the lowest return higher or the same as the highest return of distribution B,
even though its variance would be much higher. In MPT, that would not be a clear
choice since it is blind to the fact that A always does better, and it only focuses
on fitting the return/risk ratio to the investor’s preferences. The second-order
stochastic dominance rule ties the investor objectives to the risk. It states that
despite the risk preferences, an investor with MAR objective would always prefer
distribution C over A, presented in Figure 6. This is because C has more upside
potential and less downside deviation than A when we proportion the risk to the
objective MAR. Again, MPT focuses on variance and sees C way riskier than A,
and for that reason, the selection would not be apparent without the investor’s
utility function.
Figure 6: Second order stochastic dominance rule
Since the 1990s, there has been increasing interest in examining risk from a
wider perspective than just volatility. It is understood that the mean-variance
optimization incorporates only one definition of risk and does not capture the
other undesirable properties of return. While investors prefer less volatile returns
48
over more volatile returns, other things being equal, the preferences to variance
start to change when we do not have other things equal, which leads to problems
of using variance as a measure of risk. In particular, these other things refer to
skewness and kurtosis (Sortino et al. 2001). Since returns are not often normally
distributed, the mean-variance optimization model does not give desired result
(Fabozzi et al. 2010).
In his groundbreaking work for MPT, Markowitz (1968) noted that semi-
variance could be used instead of variance to cover asymmetric preferences. How-
ever, Rom and Ferguson (1994) introduced a downside risk measure that was a
general description of return probability below target, so it differed from semi-
variance by its capability to view the whole probability structure. In the case of
normal distribution and mean as a target, it did not offer any additional informa-
tion compared to semi-variance, but the idea was its capability to model risk with
asymmetric distributions and different target returns than a mean. The general
downside risk measure by Rom and Ferguson (1994) quickly created many metrics
that linked in it from different perspectives. Namely, Van der Meer (2001) spoke
about shortfall probability, risk capital, and discounted downside risk, while Forsey
(2001) talked about upside probability, downside deviation, upside potential, and
upside potential ratio. Rom and Ferguson (1997) emphasized the downside fre-
quency and average downside deviation. All of these had in common that they
described the return probability distribution and used some target return level as
a reference point. However, maybe the most utilized downside risk measure is the
Value at Risk (VaR) introduced by JPMorgan in 1994. Value at risk is the quantile
measure of the loss distribution (left tail of the return distribution). It thus indic-
ates the critical loss level that will be exceeded during the reference period with a
probability of . Portfolio’s Value at Risk can be defined as follows:
VaR1−(L) = inf{l|P (L < l) ≥ 1− }; (19)
where portfolio loss is L = −′w, and  is the probability that loss exceeds VaR.
It can also be defined with cumulative distribution function (CDF). If we have a
random profit or loss R over a holding period, and this random variable R has
continuous CDF F (r), then for 0 <  < 1, VaR is defined as the maximum
potential loss realized by R over the holding period at (1−) confidence level and
can be written as:
VaR = −F−1(): (20)
49
Thus, VaR can be interpreted as the negative of left -quantile of the distribution.
(Wong 2008.)
The Value at Risk became the standard measure for financial market risk since
Basel Committee introduced it in 1996 to determine capital requirements for banks
(Committee 1996). It is a standard tool for commercial banks, risk management
consulting firms, and other financial entities. It gained popularity among downside
risk measures since it is easy to understand, applicable, and universal after Basel
requirements. However, its simplicity has its downside.
As a Artzner, Delbaen, Eber and Heath (1999) states, VaR is not a coherent
risk measure. Risk measure is coherent if it fulfills the four axioms: subadditivity,
translation invariance, monotonicity, and positive homogeneity. While VaR satis-
fies all the other axioms, it lacks subadditivity. This flaw means that VaR does
not count the diversification benefit, which is essential in the portfolio optimization
framework. VaR has gained many critiques as a risk measure after being selected
as the official market risk measure for the banks by Basel Committee in 1996. For
example, Danielsson, Embrechts, Goodhart, Keating, Muennich, Renault, Shin
et al. (2001) stated that since VaR does not measure the shape or extent of the
distribution’s tail risk but instead estimates a particular point in the distribution,
it is misleading if the returns are not normally distributed. For that reason, VaR
models generate imprecise and fluctuating risk forecasts. Return distribution X
with light tail probabilities can have the same VaR as return distribution Y with
heavy tail probabilities (Tasche 2002).
To overcome all the previously mentioned shortcomings that VaR has, we can
utilize Expected Shortfall (ES), sometimes called Conditional Value at Risk, as a
risk measure13. Expected Shortfall generates a conditional expectation of losses
breaching VaR level. It is probability-weighted average returns beyond determined
VaR in a given confidence level. Wong (2008) presents ES with confidence level
(1− ) as follows:
ES(R) = −E(R|R < VaR) = −−1
Z VaR
−∞
rf(r)dr: (21)
where R is the expected negative return below Value at Risk on given confidence
. ES integrates all values of r, weighted by probability density function f(r),
from minus infinity to VaR, and therefore accounts for all losses beyond VaR level.
(Wong 2008.)
13While Expected Shortfall (ES) and Conditional Value at Risk are equivalent, it is important
to notice that they are defined differently (Fabozzi et al. 2010). In ES, we fix the target loss
priori, while in CVaR, VaR defines it (Brandimarte 2017).
50
To illustrate the difference between these two metrics, VaR and ES is presented
in the figure 7 with a probability density function. While VaR is one point in the
distribution, ES integrates the whole area beyond VaR. Even that small VaR does
not necessarily mean small ES; a small ES means small VaR, so for this reason
too, ES optimization covers the risks more broadly.
!"#$%&'#((
!"#$%& '$%&'($%&
!
Figure 7: Profit-Loss probability density function with VaR and ES
Expected Shortfall overcomes the following shortcomings of VaR: it accounts
for the severity of losses, owns subadditivity properties, is a coherent risk measure,
and mitigates the impact of confidence level decision. In statistical terms, to
connect VaR to investors’ probability of solvency, we need the smallest coherent
and law invariant risk measure that dominates VaR. It is shown that ES owns these
properties and better statistical properties than the first introduced downside risk
measure standard semi-deviation since it is also co-monotonically additive (Tasche
2002).
When investors can invest assets susceptible to infrequent but severe losses
(i.e., has fat tales) as some alternative investments, the problems of using VaR
as a downside risk measure increase. Using more fat-tailed assets, investors can
decrease VaR while downside risk increases in terms of ES. Yamai and Yoshiba
(2005) illustrated this problematic feature with the figure 8 of the cumulative
distribution function. Rather than minimizing VaR, we can minimize ES due to its
ability to say something about the shape of the tail and the structure of maximum
loss. At the same time, we can utilize the eligible mathematical properties that
ES has. It is important to notice also that while we minimize ES, the VaR is
necessarily low as well (Rockafellar, Uryasev et al. 2000).
51
Figure 8: Cumulative distribution
As we see, VaR and ES are downside risk metrics, just as Rom and Ferguson
(1994) introduced downside risk. They describe the probability and expected
downside, but instead of deciding the MAR level and calculating the probability
to fall under that, we first fix the loss probability in these metrics and then get
the amount of loss we can expect from holding an asset. However, in practice,  is
usually set as 1-5%, so it represents preferences to avoid near-catastrophic events,
and for that reason, it is used for capital requirement calculations for institutions.
This study tries to use ES to determine institutions’ strategic allocation.
However, also ES has its drawbacks. It depends on the chosen confidence
level and the underlying return distribution function. In portfolio optimization,
it is often criticized to be very sensitive to the chosen confidence level and this
way highly disposed to estimation errors (Lim, Shanthikumar and Vahn 2011). It
also lacks elicitability properties, which theoretically precludes the backtesting of
Expected Shortfall. However, it is shown that ES is jointly elicitable with VaR,
and that way, it could be backtested with the Value at Risk measure. The most
important property to use this measure in this study is its flexibility to be measured
from chosen probability density function, and that way, optimization is not locked
to the Gaussian premise. Estimation error also becomes a problem in downside
risk measures. Even more extensive than in dispersion risk measures since one
uses only a fraction of the data (Fabozzi et al. 2010). Neither VaR nor ES is an
exception to this. Yamai and Yoshiba (2005) showed that ES is particularly prone
to significant estimation error with fat-tailed distributions, and this is due to more
52
infrequent and larger losses that fluctuate the estimations. One way to tackle this
problem is to increase the sample size of the simulations significantly.14
One common way to estimate portfolio ES, is to assume that returns are con-
ditionally normally distributed with mean  and covariance matrix :
ESw() = −w′+
√
w′w
(z)

; (22)
where z is the  quantile of the standard normal distribution and  is the standard
normal density function. However, we can estimate ES by utilizing Cornish-Fisher
expansion to account for excess skewness and kurtosis of portfolio returns. As
Boudt, Peterson and Croux (2008) demonstrate, the Cornish-Fisher expansion of-
fers a parametric way to correct Gaussian distribution with historical return series’
multivariate moments. This asymptotic expansion enables us to estimate the ES
without explicitly defining the empirical return distribution function. Instead, we
can utilize the empirical return series to estimate higher-order moments to take
excess skewness and kurtosis into account. This way, we get the portfolio returns’
higher moments to the Expected Shortfall metric and ultimately to the optimiza-
tion problem. Now ES can be defined:
ES(w) = −w′+√m(2) × 1

[a + bk(w) + cs(w) + ds
2(w)]; (23)
where a, b, c, and d are the chosen loss level  and respectively s(w) and k(w)
are the portfolio’s skewness and excess kurtosis. We will show in chapter 3.6 how
the skewness and kurtosis are estimated in this study.
One possible way to examine ES is always by historical values. However, since
tail events are infrequent, it is shown that historical values are bad estimates for
tail risk measures. The Cornish-Fisher expansion estimate has been shown to
deliver accurate estimates for ES, and for that reason, every ES value shown in
the empirical section is calculated with function 23. The more comprehensive
definition for this estimator is shown in Appendix I. (Boudt, Carl and Peterson
2013.)
3.5 Mean-Expected Shortfall optimization
As we have now discussed the PMPT and downside risk measures, it is time to
derive how we can utilize these risk perspectives in the portfolio optimization same
14Yamai and Yoshiba (2005) increased the sample size in simulations from 10000 to 1000000,
decreasing the estimation error to third.
53
way we utilized dispersion measure variance in chapter 3.3. Now the objective is
to maximize the expected return subject to the expected shortfall, and we can
express this as:
max
w
0w
s.t. ES(w) ≤ c0;
(24)
where c0 is constant, expressing the maximum level of risk. Next, we need to
define the loss function f(w;y), where w is the N-dimensional vector of assets
weights in the portfolio and y is a random vector describing uncertain outcomes.
If random values are continuous and p(y) is the probability related to scenario y,
we can denote that the cumulative probability gives the probability of loss function
exceeding value 
	(w;y) =
Z
f(w;y)≤
p(y)dy; (25)
where we get ES by utilizing the definition 21 to following form:
ES(w) = 
−1
Z
f(w;y)≥VaR(w)
f(w;y)p(y)dy: (26)
It can be shown that the function 26 is concave and therefore has a unique
minimum. (Fabozzi et al. 2010.) However, the problem mentioned above has VaR
in the formula and is complex to utilize in practice. Therefore we follow Rockafellar
et al. (2000) method, which simplifies the direct ES optimization in the way that
we avoid incorporating VaR into our optimization problem. The idea is to utilize
the following auxiliary function in optimization:
F(w; ) =  + 
−1
Z
f(w;y)≥
(f(w;y)− )p(y)dy: (27)
This function 27 involves three important properties. First, it is a continuously
differentiable convex function that is particularly easy to optimize numerically.
Second, VaR is a minimizer of F(w; ), which means that we get the VaR simul-
taneously when minimizing ES and third and most importantly, minimum value
of F(w; ) is minimum value of ES. So the optimization problem simplifies to
the following function:
min
w;
F(w; ): (28)
From here we get optimal weights w∗ for portfolio and the corresponding VaR as
∗. However, the probability density function is hard to estimate in practice, so
we need to use sampled scenarios. In this case, we can modify the 27 function as:
54
F Y (w; ) =  + 
−1T−1
TX
i=1
max(f(w;yi)− ; 0); (29)
where we have T different scenarios for yi. When we replace max(f(w;yi) −
; 0) with variable zi we can rewrite the optimization problem with appropriate
constraints as:
min
w;
 + −1T−1
TX
i=1
zi
s.t. zi ≥ 0; i = 1; :::; T
zi ≥ f(w;yi)− i = 1; :::; T:
(30)
If f(w;y) is linear, we have a linear optimization problem that can be solved
efficiently with standard linear programming techniques. This optimization model
can be seen as an extension for the mean-variance model introduced earlier, and
it is advantageous when the underlying assets’ return distributions have skewness
and kurtosis. (Fabozzi et al. 2010.)
3.6 Higher-order portfolio moments
To consider the non-normality of the portfolio assets’ returns in the portfolio op-
timization context, we need to get higher-order moments explicitly under consid-
eration. To do so, we define coskewness and cokurtosis as we defined covariance
in chapter 3.3. The coskewness and cokurtosis matrices account for diversification
benefit through higher-order terms and are crucial when we aggregate assets’ risks
in the portfolio context. These higher-order moment tensors can be calculated as
follows. First,
ijk = E[(ri;t − i)(rj;t − j)(rk;t − k)]; (31)
where ijk is coskwness of assets i,j and k. Then,
 ijkl = E[(ri;t − i)(rj;t − j)(rk;t − k)(rl;t − l)]; (32)
where  ijkl is cokurtosis of assets i,j,k and l.
We already showed how covariance is constructed in matrix form in equation7.
Respectively we can form coskewness  and cokurtosis 	 matrices,
55
 = E[(r − r)(r − r)′ ⊗ (r − r)′];
	 = E[(r − r)(r − r)′ ⊗ (r − r)′ ⊗ (r − r)′];
(33)
where ⊗ denotes the Kronecker product. Now we can calculate portfolio return
moments for the portfolio with weights w like we did with covariance in equation
9:
m2(w) = w
′w;
m3(w) = w
′(w ⊗w);
m4(w) = w
′	(w ⊗w ⊗w):
(34)
We can define portfolio skewness and kurtosis through the portfolio return mo-
ments as follows,
s(w) =
m3(w)
m
3
2
2 (w)
(35)
k(w) =
m4(w)
m22(w)
− 3: (36)
Estimating these higher-order moment tensors is essential when we optimize
portfolios in the empirical section since we want the tail risks to be appropriately
aggregated to get robust portfolio performance evaluations for different portfolios.
(Boudt, Lu and Peeters 2015.)
Another way to utilize estimated moment tensors is to bring them directly to
the optimization problem. We introduce the objective function as a third port-
folio constructing technique utilized in this study, the Constant Relative Risk
Aversion (CRRA) utility function with Taylor Expansion. This technique is one
of the standard expected utility maximization frameworks from which the MPT
and PMPT originate. The fourth-order Taylor expansion enables us to interpret
investors’ preferences towards skewness and kurtosis intuitively, making the ob-
jective function interesting in an optimization problem with non-normal return
distributions. Since we have already defined the higher-order moment tensors, the
objective function optimization problem can be defined as:
min
w

2
m2(w)− 
(
 + 1)
6
m3(w) +

(
 + 1)(
 + 2)
24
m4(w)
s.t. w′ = 1;
(37)
where 
 is the risk aversion parameter and w is the portfolio weights. In this CRRA
objective function, investors prefer higher first and third moments and lower second
and fourth moments. It is intuitive that investors prefer higher returns and positive
56
skewness (extreme positive returns) and avoid high variance and kurtosis (extreme
deviations from mean). (Martellini and Ziemann 2010.)
Investors’ preferences towards risk aversion are studied a lot, and the consensus
often points out that investors typically have DARA, which means that investors
invest more in risky assets in absolute terms when their wealth increases. However,
in relative terms, investors usually argued to have either DRRA or CRRA. The first
one implies that investors would invest a higher proportion of their wealth to the
risky assets when their wealth increases, and the second one implies that investor
allocates the same proportion of their wealth to the risky assets when their wealth
increases (Levy 1994). While both Relative Risk aversion preferences have good
arguments, we can divide the individual and institutional investors’ preferences to
represent the different RRA preferences. While individuals’ consumption is bound
to a standard of living, which eventually has some limits, individuals start to rein-
vest this wealth surplus, increasing the relative proportion of risky assets in their
portfolio. At the same time, institutional investors are usually obligated to con-
sume a constant relative proportion of their wealth for their purpose. For example,
when endowments’ wealth increases, they offer more allowances in absolute terms
while keeping the relative outflow constant (Swensen 2009). On these grounds,
we choose the objective function to follow DARA and CRRA which function 37
satisfies.
3.7 Previously utilized models, obtained results and critique
When discussing practical portfolio optimization, we often have to answer the ques-
tion, which we value more, appropriate model specification or improved estimates.
Martellini and Ziemann (2010) propose that with low data frequency, investors
should depart from correct model specification (e.g., using gaussian premise with
non-normally distributed returns) to get lower estimation errors since the utility
from latter exceeds the former. Hitaj, Martellini and Zambruno (2011) got the
same kind of results. They showed that with sufficiently large sample size and im-
proved estimators, the higher-order optimization outperforms mean-variance op-
timization consistently from a utility perspective with hedge funds. They used
out-of-sample analysis and dynamic allocation (Hitaj et al. 2011). However, Hitaj
et al. (2011) study utilized the Constant Absolute Risk Aversion (CARA) utility
function. From institutions’ preference perspective, a more representative utility
function can be argued to be Decreasing Absolute Risk Aversion (DARA).
57
Allen, Lizieri and Satchell (2020) challenged the mean-variance framework by
improving the risk estimate accuracy leading to an increase in out-of-sample port-
folio performance from the power utility point of view. They incorporated heavy
tails, skewness, volatility clustering, and asymmetric dependencies in financial data
using univariate exponential GARCH, generalized Pareto distribution, and skewed-
t copula. Their results show a significant increase in the economic value added to
the investor portfolio. (Allen et al. 2020.)
Jondeau and Rockinger (2006) studied the opportunity cost between optimiz-
ation with mean-variance criterion and with the third and fourth moment. To do
so, they used Taylor expansion for expected utility. They showed that only with
large departure from returns’ normality will make the higher moment approxim-
ation useful. In other words, the four-moment approximation for expected utility
will add value in specific markets or asset classes, but in relatively well-diversified
or liquid markets, the mean-variance approximation is close to direct optimization.
The study shows an efficient way to optimize portfolios with non-normal return
distributions even when the number of assets increases. (Jondeau and Rockinger
2006.)
Jurczenko and Teiletche (2019) optimized cross-asset portfolio in expected
shortfall framework. Their risk parity optimization dealt with valuation, illiquid-
ity, asymmetry, and tail risks. They were interested in integrating two separate
research paths, deviation from normality in asset allocation and illiquidity on as-
set moments. To get the contribution of each asset class to the overall expected
shortfall, they decompose ES to the first derivative relative to weight. They as-
sumed expected returns to include mispricing to manage valuation risk, which they
estimated with assets carry. They used popular Cornish-Fisher (CF) expansion
to approximate assets and aggregated portfolio moments to capture asymmetry
and tail risk. To incorporate with illiquidity Jurczenko and Teiletche (2019) used
moving average properties as Getmansky et al. (2004) has used earlier. This way,
they could correct covariance, coskewness, and cokurtosis, which all suffer from
stale pricing and smoothed returns. Their study illustrates how major risk contri-
bution to traditional assets comes from volatility, while to alternative investments,
it comes from skewness, kurtosis, and illiquidity. Their study shows how taking
these risk factors into account in portfolio optimization, one can get an allocation
with the same expected Sharpe but higher expected RoES and lower expected
drawdown. (Jurczenko and Teiletche 2019.)
The natural extension for the Jurczenko and Teiletche (2019) study is to use
out-of-sample analysis to compare the model performance to other optimization
models (e.g., minimum variance). However, this analysis is limited due to the low-
58
frequency data that hamper reliable comovement estimation. This problem could
be solved by using shrinkage estimators or a factor model. In fact, Martellini and
Ziemann (2010) have shown that investors can only enhance their ex-ante utility,
compared to a mean-variance analysis, by exploiting these improved risk estimates.
Other applicable extensions are the wider asset universe and different investments
horizon which this study tries to answer.
Cavenaile et al. (2011) studied the impact of illiquidity and higher moments
on performance and optimal allocation with different hedge fund strategies. They
conducted portfolio analysis for portfolios fully consisting of different hedge fund
strategies and portfolios consisting of equity, bonds, and hedge fund indices. With
the latter portfolio analysis, they were able to evaluate the hedge funds’ diversi-
fication properties. To incorporate illiquidity in the hedge funds risk measures,
they unsmoothed the returns with two different methods, the same moving av-
erage properties by Getmansky et al. (2004), which also Jurczenko and Teiletche
(2019) used in their study, as well as Okunev and White (2003) method where
observed returns are alleged to be a weighted average of "true" returns. Even
though the technical implementation of these two methods is slightly different,
they produced very similar results15. They adjusted risk and performance meas-
ures to incorporate the higher moments of hedge fund returns. This was done by
adjusting the Value at Risk measure for non-normal skewness and kurtosis with
Cornish-Fisher expansion. They also used Bell’s utility function as a risk meas-
ure since it has an intuitive interpretation for the parameters representing higher
moments. They used Taylor series expansion as Capocci, Duquenne and Hübner
(2006) have previously done to get four first moments directly to the utility func-
tion and optimization problem. Since they used two different methods to review
illiquidity and non-Gaussian properties in the returns, they offered comprehensive
and reliable results. They showed that risk measures that do not consider higher
moments or illiquidity can underestimate hedge funds’ risks up to 80%. The diver-
sification properties showed that even when taking higher moments and illiquidity
into account, the hedge funds offer diversification benefits in a traditional equity-
bond portfolio, and that way, hedge funds can be found in optimal portfolios.
However, diversification benefits are significantly lower than what mean-variance
optimization would suggest. (Cavenaile et al. 2011.)
Cavenaile et al. (2011) study was a great example of what my study tries to
achieve. Instead of focusing on different hedge fund strategies, this study will
15For example, the smoothing parameter is estimated in the Okunev and White (2003) method
with first-order autocorrelation coefficient, while in Getmansky et al. (2004) method, it is
estimated with maximum likelihood estimation of moving average process.
59
bring a broader range of alternative asset classes to the review. Since alternative
investments are a more heterogeneous group than different hedge fund strategies,
it can be expected that illiquidity and higher moment adjustments will show even
more significant changes in performance and diversification results. The results
are likely to be parallel with Cavenaile et al. (2011) study; alternative investments
offer diversification benefits but not as much as mean-variance optimization from
smoothed returns would indicate.
60
4 METHODOLOGY
4.1 Research strategy
This empirical research aims to determine how alternative assets affect the al-
location and performance of a multi-asset portfolio. We also examine how risk
perspective affects the output of portfolio optimization. The motivation behind
that is to consider better the heterogeneous characteristics that multi-asset port-
folios and earlier describer alternative assets have. This way, the study aims to
help institutions determine their strategic allocation reflecting their objectives and
liabilities.
To improve a portfolio’s performance ex-ante, one can improve forecasts or
optimization. This study tries to improve performance by improving the optim-
ization to consider return distribution’s higher moments. It also introduces new
asset classes that should offer diversification benefits to improve performance.
We are particularly interested in the portfolio’s efficiency (return/risk ratio)
with different investment universes and risk metrics. We compare mean-variance
optimization against mean-expected shortfall optimization. The idea is to find out
how much different risk perspectives in optimization affect the allocation and ef-
ficiency of the portfolio. Mean-variance optimization works as a benchmark since
it only takes the first two moments into account. We include skewness and kur-
tosis characteristics of different asset classes in the optimization model with mean-
expected shortfall optimization. In addition to that, we also use CRRA objective
function in simulations. This objective function brings higher-order moments to
optimization from expected utility theory and works as an alternative for our risk
metric approach, working as a great benchmark. The empirical research process
goes as follows:
1. Desmoothing and cleaning up the data
2. Descriptive analysis of the clean data
3. Efficient frontier construction with different investment universes and optim-
ization objectives
4. Analyzing the static portfolios’ efficiency and allocation
5. Conducting out-of-sample backtests with three optimization objectives
6. Analyzing the historically simulated portfolio’s efficiency and allocation
61
The first step of the process tries to improve the poor-quality data that altern-
ative assets are typically prone to. In this step, the private alternative asset data
is disaggregated from quarterly to monthly. The disaggregation method reduces
the problematic stale pricing and appraisal smoothing biases. At the same time,
it forces us to make some assumptions about the underlying time series factors.
We utilize same kind of methods as Getmansky et al. (2004) and Okunev and
White (2003) where we assume Markov process or Markov process and random
walk depending on the serial correlation obtained from Generalized Least Squares
Regression. This step is not trivial for the optimization results, but as mentioned
before, it is not the main focus of this study. In the next chapter, we go more
comprehensively through the disaggregation process and also how we handle other
biases involved in the data.
The descriptive analysis focuses on different asset classes’ statistics and risk/re-
turn structures. The main focus of this analysis is the correlation of different asset
classes and the normality tests of the returns. This step also compares the data
to the investors’ expectations introduced in chapter 2.1. The efficiency metrics we
utilize throughout the study are the Sharpe ratio,
SR =
Rp −Rf
p
; (38)
and the return on expected shortfall,
RoES =
ESp−Rf
p
; (39)
where Rp is portfolio’s mean monthly return, Rf is risk-free rate which is assumed
to be zero and p is portfolio’s monthly variance. The portfolio ESp is calculated
as represented in equation 23.
In the third and fourth step, two different dimensions, risk and investment
universe are examined with two different alternatives in each dimension. Risk is
examined from variance and expected shortfall perspective, while the investment
universe is examined with and without alternative investments. Mean-variance
efficient frontiers are constructed with quadratic programming solver, solving the
problem introduced in equation 10. We utilize a linear programming solver for
mean-expected shortfall efficient frontiers, solving the problem introduced in equa-
tion 30. For these four situations, we utilize historical sample data from the whole
period of 2008-2021. The examination includes minimum risk, maximum efficiency,
and efficient frontier portfolios. The idea is to comprehensively compare the four
different situations from risk, return, and allocation perspectives. This part gives
62
insight into how alternative investment allocations evolve when moving along the
risk range.
The fifth and sixth steps’ role is to offer robustness to the results obtained
from previous steps. Before this, we have only examined results ex-post. How-
ever, by conducting backtests (historical simulation), we can examine how things
would have looked ex-ante and what results would have followed from decisions
based on ex-ante estimations. This procedure can be seen as an out-of-sample
analysis, where we examine how different optimization objectives would have per-
formed if they had been utilized in practice. While in-sample analysis examines
how portfolios should be constructed to perform well in history, the out-of-sample
analysis relies on ex-ante estimates and uses historical data to simulate the results.
We also changed the method to differential evolution heuristic optimization to get
more simulated data in the fifth step. The backtesting is performed for different
maximum efficiency portfolios with three objectives; Sharpe, RoES, and CRRA.
We utilize investment universes with and without alternative investments, and the
only constraints are fully invested and long-only portfolios. Since we have 12 years
of data, we utilize three years for training to get a sufficiently long review period.
The rolling window is locked to inception to improve the estimations over time.
Rebalancing is done once a year, consistent with the illiquid assets causing fairly
static allocations.
With the differential evolution optimization method, the idea is to repeat the
objective function to evolute towards the global maximum. Here, the investment
constraints are directly imposed by mapping the generated populations into the
feasible space, and as the optimization outcome, we get a random variable. This
algorithm performs genetic biology-inspired operations to search global optimum
and efficiently optimize non-linear optimization problems with many comovement
parameters. This way, we get the CRRA objective function also optimized effi-
ciently.
Conclusions for the two research questions, can we obtain more efficient portfo-
lios by adding alternative investments with traditional asset classes into the invest-
ment universe and by optimizing return to expected shortfall instead of variance,
are formed reflecting the empirical research with the literature review.
63
4.2 Data & Descriptive statistics
Private alternative investment data is scarce and suffers previously mentioned
biases more adversely than public benchmarks. For that reason, alternative in-
vestment studies usually focus on publicly listed alternative investments. One
distinguishing factor compared to other studies is that we use private return data
when available in this study. An essential difference between traditional and al-
ternative assets data in this study is that alternative assets’ data are aggregated
fund data and include fees. At the same time, this alternative assets’ data has
backfill, selection, and liquidation bias. These biases compensate the penalty from
the net-of-fee return calculation and make the asset class comparison meaningful.
From the six studied alternative investment sub-asset classes, five utilize Pre-
qin private fund data. These are Private Equity (PE), Private Credit (PC), Real-
Estate (RE), Infrastructure (IF), and Natural Resources (NR). The Preqin data
is quarterly index data, consisting of all reported vintages. Indices are aggreg-
ated from reported funds’ money-weighted rate of returns (IRR). However, in this
study, we view monthly returns, and for that reason, quarterly returns have been
disaggregated. This disaggregation enables us to create more data, unsmooth,
and fix the lagging returns. The disadvantages of disaggregation are that we need
to estimate these time series, increasing the uncertainty of our obtained results’
robustness. This study utilizes monthly Hedge Fund Research (HFRI) Fund of
Funds Composite index data for hedge funds. HFRI is one of the most compre-
hensive hedge fund data providers, and the fund of funds composite index ensures
the most comprehensive sample possible from a very heterogeneous asset class.
However, the fund of funds structure suffers from the double-layered fee structure.
Equities are included in the study with two separate components; developed (DM)
and emerging (EM) markets. The indices used for equities are MSCI World total
return and MSCI Emerging Markets total return indices. Government bonds are
studied through Bloomberg US Treasury Total Return index, investment-grade
corporate bonds through Bloomberg Global aggregated Corporate total return
index and high-yield corporate bonds through Bloomberg Global High Yield Cor-
porate total return index. The money market is left out of the study because of its
contradictory role in portfolio optimization in a negative rate and high inflation
environment. The study is conducted in U.S. dollars and data is collected from
the period 2008-2021.
A summary of the data used in this study is presented in table 2. This table
displays the public indicator time series utilized to disaggregate private data. To
64
disaggregate and unsmooth quarterly data to monthly data, we use Litterman
(1983) technique for asset classes whose serial correlation coefficient is greater than
zero, and Fernandez (1981) method for assets whose serial correlation coefficient
is negative. Both methods are regression-based, performing Generalized Least
Squares Regression (GLS) to the quarterly values of the private data and the
monthly indicator series. The Litterman (1983) technique combines AR(1) Markov
process and random walk and assumes that monthly residuals follow the non-
stationary process. The process is ut = u(t−1)+vt, where v is an AR(1) process vt =
v(t−1)+t, and where  is White Noise (0,). Fernandez (1981) method has  = 0,
so it follows a purely random walk. Litterman (1983) showed that his method
produced the smallest mean squared errors with time-series which had positive
autoregressive parameters. However, distributions with negative autoregressive
parameter, Fernandez (1981) and Chow and Lin (1971) methods produced better
results. This disaggregation keeps the mean of the original time series but increases
the variance of returns. Forming monthly returns adds up to the original quarterly
returns, decreasing the stale pricing of the time series.
The introduced desmoothing is done with the temporal disaggregation al-
gorithms introduced by Sax and Steiner (2013). The method is widely used to
convert low-frequency to high-frequency data in economic statistics like Gross Do-
mestic Product calculations. This method generates new high-frequency data con-
sistent with the original low-frequency data. This method aims to bring some of
the variance found in the public market to private market data, consistent with
the view that many economic factors behind public and private asset returns are
the same. Mathematically this GLS process can be presented as,
^() = [X ′C ′(XC ′)−1CX]−1X ′C ′(CC ′)−1y1; (40)
where ^ is the GLS estimator, CX is the monthly indicator series, y1 is the
quarterly value of original series, X is a n × m matrix and where m represents
the number of indicators. Critical assumption in this method is that the linear
relationship between CX and y1 also holds with X and y. The distribution matrix,
D, is the function of covariance matrix:
D = C ′(CC ′)−1; (41)
and the covariance matrix is calculated in Fernandez (1981) and Litterman (1983)
method as,
L() = 
2
 [

′H()′H()
]−1; (42)
65
Ta
bl
e
2:
U
ti
liz
ed
da
ta
an
d
co
rr
es
po
nd
in
g
in
di
ca
to
r
se
ri
es
In
di
ce
s
w
hi
ch
ha
ve
in
di
ca
to
r
se
ri
es
ar
e
co
nv
er
te
d
fr
om
qu
ar
te
rl
y
to
m
on
th
ly
da
ta
.
In
di
ce
s
w
hi
ch
do
es
no
t
ha
ve
in
di
ca
to
r
se
ri
es
ar
e
or
ig
in
al
ly
m
on
th
ly
da
ta
.
T
he
da
ta
is
co
lle
ct
ed
fr
om
pe
ri
od
20
08
-2
02
1.
A
dj
us
te
d
R
-s
qu
ar
ed
te
lls
ho
w
w
el
l
in
di
ca
to
r
se
ri
es
ex
pl
ai
ns
qu
ar
te
rl
y
in
de
x.
In
di
ca
to
r
se
ri
es
ha
ve
be
en
se
le
ct
ed
fr
om
a
br
oa
d
un
iv
er
se
of
av
ai
la
bl
e
in
di
ce
s
ba
se
d
on
th
ei
r
co
effi
ci
en
t
of
de
te
rm
in
at
io
n.
A
ss
et
cl
as
s
In
de
x
In
di
ca
to
r
se
ri
es
A
dj
us
te
d
R
-s
qu
ar
ed
R
eg
re
ss
io
n
co
effi
ci
en
t
P
E
P
rE
Q
in
qu
ar
te
rl
y
in
de
x
L
P
X
50
0
:6
1
0
:1
6
P
C
P
rE
Q
in
qu
ar
te
rl
y
in
de
x
B
lo
om
be
rg
G
lo
ba
l
H
ig
h
Y
ie
ld
C
or
po
ra
te
T
R
In
de
x
0
:8
4
0
:4
9
IF
P
rE
Q
in
qu
ar
te
rl
y
in
de
x
S&
P
G
lo
ba
l
In
fr
as
tr
uc
tu
re
In
de
x
0
:0
7
0
:0
8
N
R
P
rE
Q
in
qu
ar
te
rl
y
in
de
x
S&
P
G
SC
I
T
ot
al
R
et
ur
n
C
M
E
0
:7
0
0
:2
0
R
E
P
rE
Q
in
qu
ar
te
rl
y
in
de
x
F
T
SE
N
A
R
E
IT
0
:1
4
0
:1
2
H
F
H
F
R
I
Fu
nd
of
Fu
nd
s
C
om
po
si
te
In
de
x
E
Q
-
D
M
M
SC
I
W
or
ld
T
R
E
Q
-
E
M
M
SC
I
E
m
er
gi
ng
M
ar
ke
ts
T
R
G
O
V
B
lo
om
be
rg
U
S
T
re
as
ur
y
T
R
IG
B
lo
om
be
rg
G
lo
ba
l
A
gg
C
or
po
ra
te
T
R
in
de
x
H
Y
B
lo
om
be
rg
G
lo
ba
l
H
ig
h
Y
ie
ld
C
or
po
ra
te
T
R
in
de
x
66
where 
 denote as n× n difference matrix, H() as a n× n matrix with 1 on its
diagonal, − on its subdiagonal and 0 elsewhere. Since in Fernandez (1981) case
we had  = 0, the equation 42 simplifies:
L(0) = 
2
D: (43)
The autoregressive parameter ^ is estimated with maximization of the like-
lihood of the GLS-regression following Sax and Steiner (2013). Now we have
everything estimated for the final estimation of the monthly series,
y^ = p+Du1; (44)
where preliminary series is p = ^X, D is the distribution matrix from equation
41 and u1 is a vector containing the differences between quarterly and monthly
values. While this method can be argued to desmooth time series quite artificially,
it resembles Getmansky et al. (2004) and Okunev and White (2003) methods in
their studies. Similar methods are also seen to be utilized in practice when portfolio
management needs to give some estimates about illiquid assets’ volatility.
Indicator series for each alternative investment time series have been selected
based on their R-squared values, and means that selected public indices have the
best coefficient of determination to the private index in question. The small re-
gression coefficient and R-squared value with IF index implicates the earlier stated
difference between private and public infrastructure investment and reinforces the
perception that the only way to get exposure for the alternative betas in infra-
structure investing is through private investments rather than listed companies.
After disaggregating the data, we have 156 observations of monthly logarithmic
return data per asset class. It represents the period from 2008 to 2021, including
market pressures of the financial crisis in fall 2008 and the Covid-19 crisis in march
2020. Between those years, we had the longest bull market in history. Reference
period’s first and last years are crucial for the utilized data since that way we
get versatile return behavior for the studied asset classes and observations for the
scarce tail events.
The descriptive statistics and correlation matrix of studied asset classes are
represented in table 3. When we examine the statistics, we see that PE offers the
highest mean returns while HF has the lowest. The result with PE is not surprising
since it is studied that PE typically has 1.3-2.4 equity market beta, depending on
the utilized data (Buchner 2020). However, the annual HF return of 1.68% is
surprisingly low compared to earlier studies, which have recorded returns between
7.6%-12.3% after fees, survivorship, and backfill bias adjustments (Ilmanen 2011;
67
Ibbotson et al. 2011). Those studies included periods from 1990 to 2009, so the
returns were from different decades than this study. It is good to point out that, for
example, Ilmanen (2011) utilized the same HFRI index data as we do in this study,
so we can argue that there has been a regime shift in the HF asset class. If we
compare these statistics to Preqin’s findings of investor expectations introduced in
chapter 2, we see that the expectations for high absolute returns with PE align with
the historical data, while the HF return expectations do not get supported with
the historical data. When we move on to the risk metrics, we see that government
bonds represent the lowest risk in standard deviation and expected shortfall as
expected. From the other end of the risk spectrum, emerging markets equity has
the highest standard deviation and expected shortfall.
Another interesting part of the data is the historical efficiency metrics of dif-
ferent asset classes. In the asset allocation context, Sortino et al. (2001) conclude
that downside risk measures will always provide more correct efficiency rankings
for assets than the mean-variance framework if return estimates are reliable, which
means that we should emphasize RoES more than Sharpe. This statement brings
up the critical note that estimating the optimization inputs is not a trivial task,
and in many cases, it is even more important than utilizing an appropriate op-
timization model. However, as stated earlier, this study focuses on assessing an
appropriate optimization model and, for that reason, leaves an in-depth analysis of
estimation methods for further research. For that reason, we can examine ex-post
efficiency rankings of different asset classes to learn something from history. The
EM equity has the worst Sharpe ratio while IF has the best. If we look at RoES,
EM equity is also worst while PE is the best. The ranking with Sharpe is from
most efficient to least: IF, PE, PC, GOV, IG, HY, RE, NR, DM, HF, EM. While
with RoES, it is: PE, GOV, IF, PC, IG, NR, HY, DM, RE, HF, EM. If we compare
these two rankings, we see quite many differences, which strengthens the view that
we should also evaluate the risks above the variance. This gives the first indicators
that taking higher moments into account dilutes the attractiveness of alternative
investments since government bonds overtake the private credit and infrastructure
with RoES and at the same time RE collapsed in rankings. Statistics of alternative
investments align with investors’ expectations since PE, IF, and PC offer the best
risk-adjusted returns from alternative investments with both efficiency metrics.
In panel B, we have asset classes and inflation correlations with each other.
With IF, we have most correlations that do not significantly differ from zero.
GOV has the most negative correlations, which supports the view of asset class
being an excellent risk diversifier. Surprisingly, the best inflation hedge seems to be
with HF from a correlation perspective. However, since the mean of inflation has
68
been the same as the HF returns, 1.68% annually, HF seems to have perceived the
purchasing power but not offered any real returns during the period. As a second
and third in correlation with inflation comes NR and RE, which are expected since
materials such as timber and rental yields of real-estates usually follow inflation.
If we compare these results with investors’ expectations shown in chapter two,
the RE and NR are in line with that study. However, the inflation hedge often
connected with infrastructure was not found from observed data.
Table 3: Descriptive Statistics and correlations
Panel A describes the summary statistics for each asset class. The stats are calculated from
monthly log-returns and comprise 156 observations of each asset. ES is calculated with  =
0:95. Sharpe and RoES is calculated with Rf = 0. All Jarque Bera statistics have p < 0:01
so we can reject the null hypothesis of normally distributed returns. Panel B displays the
correlations between the asset classes and inflation. Correlation matrix also include correlation
tests and the significance is displayed with *. Correlation test null hypothesis states that there
is not linear relationship between the two variables. *) means that the p > 0:05, and we can
not reject the null hypothesis.
Panel A: Descriptive Statistics
Class Mean St.
Dev.
ES Sharpe RoES Skewness Kurtosis Max Min JB
stat.
PE 0:0085 0:0167 0:0429 0:5082 0:1703 −1:8901 8:9930 0:0514 −0:0708 324:56
PC 0:0054 0:0188 0:0529 0:2899 0:0672 −2:8783 16:7081 0:0434 −0:1167 1432:70
IF 0:0066 0:0123 0:0353 0:5368 0:1422 −2:8705 15:4881 0:0332 −0:0657 1225:26
NR 0:0020 0:0172 0:0436 0:1188 0:0433 −1:2983 5:7201 0:0312 −0:0722 91:08
RE 0:0024 0:0183 0:0600 0:1352 0:0333 −3:8114 21:4078 0:0214 −0:1141 2572:94
HF 0:0014 0:0163 0:0441 0:0860 0:0245 −1:7641 9:0172 0:0384 −0:0793 314:71
DM1 0:0051 0:0492 0:1200 0:1031 0:0380 −0:9104 5:2120 0:1203 −0:2102 52:94
EM2 0:0022 0:0652 0:1611 0:0341 0:0110 −0:9135 6:4667 0:1577 −0:3198 99:40
GOV 0:0031 0:0120 0:0212 0:2558 0:1679 0:5231 4:5450 0:0517 −0:0296 22:49
IG 0:0038 0:0204 0:0503 0:1849 0:0574 −1:0079 7:6640 0:0640 −0:0861 167:29
HY 0:0059 0:0330 0:0855 0:1801 0:0416 −1:7004 12:6917 0:1123 −0:1904 684:27
CPI 0:0014 0:0330 0:0100 −0:0182
Panel B: Correlations
PE PC IF NR RE HF DM EM GOV IG HY CPI
PE 1
PC 0:81 1
IF 0:27 0:12* 1
NR 0:60 0:62 0:38 1
RE 0:74 0:60 0:34 0:47 1
HF 0:41 0:37 −0:03* 0:37 0:36 1
DM1 0:76 0:82 0:15* 0:57 0:49 0:12* 1
EM2 0:67 0:80 0:02* 0:54 0:42 0:11* 0:87 1
GOV −0:39 −0:32 0:02* −0:37 −0:20 −0:24 −0:32 −0:24 1
IG 0:44 0:71 0:03* 0:36 0:23 0:03* 0:68 0:74 0:20 1
HY 0:66 0:93 −0:05* 0:53 0:41 0:23 0:82 0:84 −0:28 0:79 1
CPI 0:34 0:33 0:08* 0:48 0:40 0:61 0:14* 0:14* −0:37 0:02* 0:26 1
1 Equity - Developed markets.
2 Equity - Emerging markets.
When we move on to evaluate the normality of log returns, we find out that
69
every other asset class except GOV has negative skewness. This means that all
the other assets have long left tails and are prone to severe losses. PE, PC, IF
and especially RE have significant negative skewness. PE having such a negat-
ive skewness is an interesting finding, since earlier stated in the literature review,
Phalippou and Gottschalg (2009) rationalized PE investments, despite their con-
troversial performance, with lottery-seeking preferences and right-skewed returns.
One explanation is that they emphasized venture capital investments, and this
way, this heterogeneous asset class includes very different return structures, which
aggregates to the significant negative skewness, again emphasizing strategy (buy-
out vs. venture capital) and manager selection in this asset class. From a kurtosis
perspective, GOV, NR, DM, and EM are closest to the normal distribution value
of 3. PC, IF and RE are the most leptokurtic distributions (have the highest kur-
tosis), meaning they are more prone to extreme values than other asset classes.
After skewness and kurtosis statistics, it is no surprise that according to the Jarque
Bera test, we can reject the null hypothesis of normally distributed returns with
every asset class.
However, since the Jarque Bera test’s chi-squared approximation is very sensit-
ive with small samples, it is important to evaluate the normality also with graphs.
The return distributions are shown visually in figure 9. The lighter line illustrates
the normal distribution with the given mean and variance, while the darker line
shows the empirical distribution.
From these graphs, we can say that the traditional assets are closer to the
normal distribution than the alternative ones. We can still evaluate the normality
with Henry’s line in figure 10, which has 95% confidence bands, and we get the
same kind of results as from histograms. Alternative assets tend to depart from
the normality, especially in the left (negative) tail of the distribution. GOV and
EQ - DM tend to be closest to the normality.
70
PE
Returns
−0.06 −0.02 0.02
0
10
20
30
40
50
PC
Returns
D
en
si
ty
−0.10 −0.05 0.00
0
10
20
30
40
IF
Returns
D
en
si
ty
−0.06 −0.02 0.02
0
20
40
60
80
NR
Returns
−0.06 −0.02 0.02
0
10
20
30
40
RE
Returns
D
en
si
ty
−0.10 −0.06 −0.02 0.02
0
10
30
50
HF
Returns
D
en
si
ty
−0.08 −0.04 0.00 0.04
0
10
20
30
40
EQ − DM
Returns
−0.20 −0.10 0.00 0.10
0
2
4
6
8
12
EQ − EM
Returns
D
en
si
ty
−0.3 −0.1 0.0 0.1
0
2
4
6
8
10
GOV
Returns
D
en
si
ty
−0.02 0.00 0.02 0.04
0
10
20
30
40
50
IG
−0.05 0.00 0.05
0
5
10
20
30
HY
D
en
si
ty
−0.20 −0.10 0.00 0.10
0
5
10
15
20
25
Figure 9: Histograms with distribution function
71
PE
norm Quantiles
−2 −1 0 1 2
−
0.
06
−
0.
02
0.
02
PC
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
10
−
0.
05
0.
00
IF
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
06
−
0.
02
0.
02
NR
norm Quantiles
−2 −1 0 1 2
−
0.
06
−
0.
02
0.
02
RE
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
10
−
0.
06
−
0.
02
0.
02
HF
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
08
−
0.
04
0.
00
0.
04
EQ − DM
norm Quantiles
−2 −1 0 1 2
−
0.
20
−
0.
10
0.
00
0.
10
EQ − EM
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
3
−
0.
1
0.
1
GOV
norm Quantiles
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
02
0.
02
0.
04
IG
−2 −1 0 1 2
−
0.
05
0.
00
0.
05
HY
E
m
pi
ric
al
 Q
ua
nt
ile
s
−2 −1 0 1 2
−
0.
20
−
0.
10
0.
00
0.
10
Figure 10: Normality observation with Henry’s line
72
5 RESULTS
We start by optimizing portfolios in the mean-variance (Mean-Var) framework
as in chapter 3.3. In practice, we do this with quadratic programming solver
and use historical samples as input estimates for mean and variance. We are
interested in maximizing the Sharpe ratio and, in the first point, comparing the
portfolios’ efficiency when we have just traditional assets versus when we include
alternative assets to the investment universe as well. The only constraints we have
in optimization are the long-only and fully invested portfolio and the objective is
to maximize Sharpe ratio. The first and easy way to examine the differences in
portfolios’ efficiency is to construct efficient frontiers with and without alternative
investments. As we see from figure 11, it is clear that the diversification benefit
moved the efficient frontier upwards, as we expected in chapter 3. The curve rose
especially from the riskier end of the risk spectrum. If we examine the allocation of
the efficient frontiers in figure 12, we can notice that this progress is mainly driven
by the opportunity to invest in PE. It is also noticeable that the optimization
gives quite concentrated portfolios in both cases, with and without alternatives,
since efficient portfolios utilizes mainly just PE, IF and GOV with alternatives and
HY and GOV without alternatives (see Appendix II). The concentration is due
to utilizing a historical sample mean and variance as an estimate for an expected
return and variance, making the estimates susceptible to estimation error. In
reality, to make efficient ex-ante portfolios, we should utilize more diversification
across asset classes. We will examine these ex-ante portfolios later in backtesting.
In number terms, the efficiency improved from 0.366 to 0.789 ex-post Sharpe
in maximum SR portfolios and dropped the minimum Standard Deviation from
3.44% to 2.24% per annum in minimum variance portfolios. We have similar results
when examining alternative assets’ effect on the efficient frontier with ES as a risk
metric and RoES as an optimization objective. The maximum RoES improved
from 0.225 to 0.385 and the minimum ES from 1.60% to 1.25%. It is interesting
to see how the HF are included in the efficient portfolios even though it has the
worst efficiency metrics as an individual asset class. However, it happens to have
the smallest correlation with the dominating assets, PE, IF, and GOV, and for
that reason, it comes to the portfolio to offer pure diversification. This finding is
in line with the Preqin study viewed in chapter 2.
The utilized asset classes in efficient frontiers were very similar with mean-
variance and mean-expected shortfall (Mean-ES) optimization, but as we see from
figure 13, the latter is not as linear with the allocations as the former. The dif-
73
Efficient Frontiers
StdDev
M
ea
n
0.00 0.02 0.04 0.06
0.
00
0
0.
00
2
0.
00
4
0.
00
6
0.
00
8 PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Traditional assets
All assets
Figure 11: Efficient frontiers in mean-variance space
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
0.00646 0.00657 0.00696 0.00771 0.00916 0.0107 0.0125 0.0156
0.00408 0.00474 0.0054 0.00593 0.00659 0.00725 0.00791
StdDev
Mean
W
ei
gh
t
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 12: Asset weights in mean-variance efficient portfolios
74
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
0.0125 0.0126 0.0131 0.0144 0.0168 0.0203 0.0244 0.0296 0.0369
0.00382 0.00452 0.00522 0.00592 0.00662 0.00732 0.00802
ES
Mean
W
ei
gh
t
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 13: Asset weights in mean-es efficient portfolios
ference in allocation is due to the higher moments taken into account in mean-ES
optimization with Cornish-Fisher ES estimation. The improvement in portfolios
efficiency compared to mean-variance optimization (or mean-ES optimization with
normal distribution assumption) is shown in Figure 14. Even though the improve-
ment is small, it implies that taking higher moments into account improves portfo-
lios performance at least ex-post. The results look very similar to that Krokhmal,
Palmquist and Uryasev (2002) obtained from their study. Improvement is because
coskewness and cokurtosis offer diversification benefits as well. When we compare
how these higher moments affect the allocation towards alternative investments,
we see from table 4 that with mean-variance optimization, the average allocation
to alternatives is 75% in the efficient frontier, while with mean-expected short-
fall optimization, this number decreases to 69%. This implies that alternative
investments do not look as attractive when examined from the expected shortfall
perspective compared to variance. This finding confirms the initial expectations
for alternative investment optimization. However, at the same time, these efficient
frontiers confirm that adding alternative investments in the investment universe
can significantly improve portfolio efficiency, despite the non-normal return dis-
tributions. It is safe to say that alternative investments can increase returns and
decrease risks when added to a multi-asset portfolio.
75
Efficient Frontiers
ES
M
ea
n
0.00 0.05 0.10 0.15 0.20
0.
00
0
0.
00
2
0.
00
4
0.
00
6
0.
00
8 PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Mean−Var
Mean−ES
Figure 14: Mean-variance and mean-ES efficient frontiers
While mean-ES optimization revealed that, even though alternatives are im-
proving traditional portfolios, they are not as attractive as typically have seen with
mean-variance optimization. At the same time, optimization findings suggest that
mean-ES optimization leads to better results regardless of the available assets.
However, here is vital to remember that we have examined only ex-post numbers
so far, and portfolio construction is all about ex-ante decisions. For that reason,
we conduct out-of-sample simulations to put optimizations to test.
The discussed results can be found in table 4 panel A. Next, we move on to
the backtests, which results can be found from panel B. Figure 15 summarizes the
different portfolios’ backtest results. Solid lines are performance for backtested
portfolios with alternative investments in the investment universe, and dashed
lines are portfolios without alternative investments in the investment universe.
If we jump right to the most interesting results, efficiency metrics, we can eas-
ily support our earlier findings of alternative investments’ capability to work as
efficiency enhancers. No matter which objective we have in optimization, when we
add alternative investments into the investment universe, the Sharpe and RoES
ratios doubles. However, when we deep dive into different optimization object-
ives, we see how the traditional mean-variance framework outperforms the other
optimization methods, both in absolute returns and efficiency terms. This finding
76
Table 4: In-sample and out-of-sample results
Returns and standard deviations are annualized, expected shortfall is over one month period.
% Alt. stands for the share of alternative assets in the portfolio.
Panel A: In-sample portfolios
Return% StdDev% ES% SR RoES % Alt.
Without
alt.
assets
Mean-Var1 5:67 6:65 4:57 0:85 0:12 0
Max SR 4:47 3:53 1:94 1:27 0:19 0
Min Var 4:23 3:44 1:79 1:23 0:19 0
Mean-ES1 5:70 6:74 3:81 0:85 0:15 0
Max RoES 4:41 3:48 1:63 1:27 0:23 0
Min ES 4:28 3:45 1:60 1:24 0:22 0
With alt.
assets
Mean-Var1 7:53 3:19 2:30 2:36 0:30 75:01
Max SR 7:06 2:58 1:71 2:73 0:34 61:81
Min Var 4:90 2:24 1:36 2:19 0:32 54:21
Mean-ES1 7:37 3:18 2:01 2:32 0:32 68:97
Max RoES 6:29 2:47 1:36 2:54 0:39 46:06
Min ES 4:58 2:33 1:25 1:96 0:31 45:24
Panel B: Out-of-sample portfolios
Return% StdDev% ES% SR RoES % Alt.2
Without
alt.
assets
MaxSR 3:76 2:86 1:45 1:32 0:22 0
MaxRoES 3:55 3:13 1:47 1:13 0:20 0
MaxRoES90 3:58 3:05 1:74 1:17 0:17 0
MaxCRRA 3:67 2:89 1:47 1:27 0:21 0
With alt.
assets
MaxSR 6:88 2:00 1:03 3:45 0:56 50:2
MaxRoES 5:42 2:42 1:04 2:24 0:44 21:7
MaxRoES90 6:46 2:25 1:47 2:87 0:37 50:8
MaxCRRA 4:61 1:86 0:81 2:48 0:48 50:6
1 Efficient frontiers average results.
2Average weight during the review period.
77
!"
#""
##"
#$"
#%"
#&"
#'"
#("
#)"
#*"
#!"
$""
$#"
#$+$"#" #$+$"## #$+$"#$ #$+$"#% #$+$"#& #$+$"#' #$+$"#( #$+$"#) #$+$"#* #$+$"#! #$+$"$"
,-./001 ,-.20 ,-.0342 ,-.0342!" ,-./001 ,-.20 ,-.0342 ,-.0342!"
Figure 15: Historical simulations return indices
suggests that while the mean-expected shortfall optimization might look attract-
ive ex-post because we can utilize the actual realized coskewness and cokurtosis
numbers, the situation is different when we need to estimate comovements in ex-
ante. Since the number of estimated parameters increases rapidly when we move
from covariance to the third and fourth moment, the risk for estimation error
increases substantially. These out-of-sample backtests suggest that the efficiency
gained from considering higher moments does not payout since the increasing es-
timation error overtakes it. This is consistent with the Martellini and Ziemann
(2010) findings.
When we compare ES optimization with different confidence levels (p = 1−),
we recognize the sensitivity of this metric, as earlier stated (Lim et al. 2011). The
efficiency and allocation dramatically change when we utilize a 10% loss probab-
ility level instead of a 5% level. While ES with 5%  focuses mainly on GOVs
over the whole review period, ES with 10%  has significant allocations to PE, IF,
and HF. Of course, this allocation change increases the absolute returns, but with
efficiency metrics, the portfolio performs better with SR but worse with RoES.
When we increase the loss probability level, the optimization is not so sensitive for
the fat-tailed distributions, and the weight of the alternative assets in the portfolio
increases. At the same time variance of returns decreases. However, when we ex-
amined asset classes individually, it is noticeable that alternative investments did
not have higher ES than traditional assets. Instead, lower variance and comove-
ment properties attract alternative assets to the portfolio when we allow overall
riskier assets with 10% loss probability. While the SR increased and the asset
allocation developed closer to levels we obtained from MaxSR optimization (see
78
figures 16 and 17), at the same time, the RoES efficiency metric went in the op-
posite direction. This means that the higher returns came at the cost of even
higher risks. For example, when we examine the drawdown metric in figures 18
and 19, we can see that MaxRoES90 optimizations have more severe drawdowns
than MaxSR optimization. This finding suggests that the mean-ES optimization
did the opposite of expected results. We argued that mean-ES optimization could
fit for institutions that are ready to carry high volatility portfolios if they can
at the same time decrease maximum drawdowns through expected shortfall. In
our backtests, expected shortfall optimization did not offer that feature, and the
mean-variance framework did the best job from that viewpoint.
Mean−Variance Weights
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
V
al
ue
Dec 10 Dec 12 Dec 14 Dec 16 Dec 18 Dec 20
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 16: Mean-variance out-of-sample weights
The observations about chosen loss probability level bring the problematic fea-
ture of having ES as an objective in optimization. While in the mean-variance
framework, one can create an efficient frontier and pick a portfolio according to
the risk preferences from that frontier, the subject has one risk level to decide.
With ES, the investor needs to pick the initial risk level by choosing the prob-
ability of loss  and, after creating an efficient frontier, decide another risk level,
preferred expected shortfall from formed efficient frontier. These problems with ES
highlight the practicality of variance as a risk metric and assumption of normally
distributed returns.
When we finally examine the CRRA objective function out-of-sample perform-
79
Mean−ES p=0.90 Weights
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
V
al
ue
Dec 10 Dec 12 Dec 14 Dec 16 Dec 18 Dec 20
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 17: Mean-ES p=0.90 out-of-sample weights
Jan
2011
Jan
2012
Jan
2013
Jan
2014
Jan
2015
Jan
2016
Jan
2017
Jan
2018
Jan
2019
Jan
2020
Dec
2020
Cumulative Return 2011−01−01 / 2020−12−01
0.2
0.4
0.6
0.8
Monthly Return
−0.01
 0.00
 0.01
 0.02
Drawdown
−0.020
−0.015
−0.010
−0.005
Mean−Var Optimization Performance
Figure 18: Mean-variance out-of-sample performance
80
Jan
2011
Jan
2012
Jan
2013
Jan
2014
Jan
2015
Jan
2016
Jan
2017
Jan
2018
Jan
2019
Jan
2020
Dec
2020
Cumulative Return 2011−01−01 / 2020−12−01
0.2
0.4
0.6
0.8
Monthly Return
−0.02
−0.01
 0.00
 0.01
 0.02
Drawdown
−0.030
−0.025
−0.020
−0.015
−0.010
−0.005
Mean−ES p=0.90 Optimization Performance
Figure 19: Mean-ES p=0.90 out-of-sample performance
ance, we can notice the smallest drawdown levels in figure 21 but at the same
time also the most negligible absolute return. The allocation with alternative in-
vestments is most comprehensive with MaxCRRA, it contains significant level HF
throughout the review period, and the allocation also stays relatively stable in
the review period (see figure 20). It also has the slightest standard deviation and
expected shortfall with the alternative investments in the investment universe.
However, its modest returns leave it third in SR and second in RoES. Overall,
MaxCRRA optimization performance is relatively stable, and it could work as a
tool when we try to create small drawdown portfolios. It would be interesting to
study how it behaves when we increase the risk level. This study utilized the risk
aversion parameter 
 = 1, since we wanted to describe long investment horizon
institutions that can bear high levels of risks in their investment portfolios.
Overall, the backtests revealed that the mean-variance framework, in all its
simplicity, offers efficient results while the mean-ES framework was unable to de-
liver the hoped results from maximum drawdown and efficiency perspective. At
the same time, CRRA objective function optimization offered promising results
to incorporate higher moments effectively in the optimization. For that reason,
it could work as an alternative for the mean-variance framework when optimizing
multi-asset portfolios with alternative investments.
81
CRRA Weights
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
V
al
ue
Dec 10 Dec 12 Dec 14 Dec 16 Dec 18 Dec 20
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 20: CRRA objective function out-of-sample weights
Jan
2011
Jan
2012
Jan
2013
Jan
2014
Jan
2015
Jan
2016
Jan
2017
Jan
2018
Jan
2019
Jan
2020
Dec
2020
Cumulative Return 2011−01−01 / 2020−12−01
0.1
0.2
0.3
0.4
0.5
Monthly Return
−0.010−0.005
 0.000 0.005
 0.010 0.015
Drawdown
−0.015
−0.010
−0.005
CRRA Optimization Performance
Figure 21: CRRA objective function out-of-sample performance
82
6 CONCLUSION
The alternative assets appear as a safe haven from the recent markets turmoils cre-
ated by the Covid-19 crisis and the Russian war against Ukraine. At the same time,
the long-term outlook for traditional asset classes’ expected returns has uniformly
moderated. These uncertainties have further accelerated the long-lived trend of
alternative investments getting a share of the investors’ wallets. While the shift in
strategic allocations has been remarkable, the practice of constructing portfolios
has not significantly changed from the traditional MPT mean-variance framework.
The research presented in this thesis examines whether it is naive to think of
alternative assets as an answer to increasing portfolios’ expected returns and hedge
against market turmoils. This research also takes a stand if we should develop how
we construct the portfolios when we add new ingredients to the problem. From
these standpoints, we got the research questions of this study: Can we obtain
more efficient portfolios by adding alternative investments with traditional asset
classes into the investment universe, and can we obtain more efficient portfolios
by optimizing return to expected shortfall instead of variance?
We examined these questions from an institutional investor perspective be-
cause they have particular characteristics that make these research questions most
relevant. First, the long investment horizon enables institutional investors to har-
vest liquidity premium to a greater extent than other investors. The investment
horizon reduces the limitations for illiquid assets and enables a greater share of
alternative assets in the portfolio. At the same time, we can assume that insti-
tutions’ large investable assets, in absolute terms, do not create constraints for
available assets. However, we need to emphasize that the framework utilized in
the empirical section is applicable as a starting point for individual investors as
well. With individual investors, we need to consider some extra practicalities to
add to the problem. These things are available assets with small investable assets
and liquidity concerns on the life cycle hypothesis. So the institutional investor
perspective simplifies the study to focus better on the research questions.
As we found in the literature review, different alternative investments were
profiled as standalone asset classes for different purposes. At the same time, how
exposure was created to different asset classes mattered. All the alternative in-
vestments were seen as great diversifiers, but still, there were differences between
different asset classes. While PE was characterized as a return enhancer, real
assets were utilized for diversification. RE and IF were seen as great cash flow
producers and inflation hedges. When we compare these views to the findings we
83
obtained from this study, we can confirm that our data supported the view for
PE as a return enhancer, real-assets as diversifiers, and RE as an inflation hedge.
However, we also found that HF worked as a great diversifier and the best diver-
sifier for the portfolios we constructed. We did not find the inflation hedge from
IF, and the HF performance was inconsistent with the investors’ view. When we
studied different investment vehicles to get the alternative asset class exposure, the
earlier studies emphasized the importance of private market vehicles, particularly
for IF, RE, and NR. The reasoning behind those views was that these investments
had distinctly different risk and return characteristics compared to publicly traded
substitutes. This difference was well demonstrated in the empirical section where
we desmoothed the private investment data with the public one. RE and especially
IF had the worst coefficient of determination emphasizing the different character-
istics of private investments than public ones in these asset classes. We did not
find that quality from NR, emphasizing that the private market NR index still
holds a significant amount of commodity funds.
In the empirical section, we put the characteristics mentioned above in the
portfolio context, and we started to examine whether alternative investments cre-
ate value in the portfolio and whether the way we optimize portfolios affects that.
When we added alternative assets in the investment universe, we saw an increase
in portfolio efficiency regardless of our optimization objective. This improvement
was confirmed with both efficiency metrics, SR and RoES, and the results were
also confirmed in backtests. However, the proportion of alternative investments in
efficient portfolios differed depending on the optimization objective. In the mean-
variance framework, the proportion of alternative investments was systematically
higher than in the mean-expected shortfall framework. This finding supports the
view that alternative investments are not so attractive when higher moments are
taken into account. At the same time, we saw that alternative investments stan-
dalone ES did not significantly differ from traditional assets. This finding suggests
that the comovements of higher moments did not offer as good diversification as
covariance did. However, as we saw in the out-of-sample backtest portfolios, the
chosen confidence level of ES significantly affects the portfolio constituents. For
that reason, the introduced findings can not be generalized to mean-ES optimiza-
tion but instead illustrate the results with a particular confidence levels 95% and
90%. For further research would be interesting to examine portfolio construction
comprehensively with a wide range of ES confidence levels.
When we changed the optimization method from mean-variance framework to
mean-expected shortfall framework, we improved return on expected shortfall in
the ex-post situation. However, the efficiency did not improve from the Sharpe
84
ratio perspective, and even more importantly, in the ex-ante situation with his-
torical simulations, the efficiency performance decreased significantly with both
metrics. The possible explanation for poor backtest performance is the increas-
ing estimation error with mean-ES optimization due to the increasing number of
estimated parameters. This problem is further accelerated with the less available
data in the simulation compared to ex-post examination. These findings suggest
that the mean-variance framework should be the preferred optimization method
for all its simplicity, even with alternative investments. However, it is essential to
state that before any of these optimizations, we improved the smoothed data of
alternative investments, which already in itself improves the mean-variance optim-
ization. Indeed it would be a good topic for further research to examine how this
data desmoothing affects the obtained portfolios.
We also wanted to examine an alternative way to take higher moments into
account in portfolio optimization, and for that reason, we introduced the CRRA
objective function in the ex-ante examination. This method produced promising
preliminary results from a risk management perspective since it produced the smal-
lest drawdown and other risk metric levels in the review period while maintaining a
large allocation to alternative assets. However, since the result highly depends on
the chosen risk aversion parameter, this would be an interesting topic for further
research with more comprehensive examination with different parameter levels.
Let us summarize the study through research questions. Can we obtain more
efficient portfolios by adding alternative investments with traditional asset classes
into the investment universe? Yes, we can. Despite the desmoothing of the data
and chosen risk perspective, alternative investments enhance portfolios efficiency.
Can we obtain more efficient portfolios by optimizing return to expected shortfall
instead of variance? Ex-post yes if we value expected shortfall more than vari-
ance as a risk metric. Ex-ante no without more comprehensive data and better
comovement estimates.
We can consider at least the following if we discuss what these results imply in
a broader portfolio management picture. It can be argued that with alternative
investments, the heterogeneous nature of the asset classes and dispersion in port-
folio managers’ performance reduces the rationality of wasting too much effort for
portfolio optimization. Instead, investors should focus on manager selection and
finding skills from the private market. The allocations obtained from optimization
problems should more likely see as a confirmation that there should be alternat-
ive investments in the portfolio, but the exact weights are secondary. Since the
estimation errors increase with scarce data and higher moments, one might utilize
85
some rule of thumb16 to obtain as efficient portfolios as with complex optimiza-
tion models. We need to remember that we never invest for the sake of investing,
but instead to achieve some other objectives. Simple as it may sound and still
often ignored, determining the goal for one’s investments is the most crucial part
of investing (Sortino et al. 2001). This again emphasizes the asset and liability
management introduced in chapter 2.3. While this study’s results do not offer a
straightforward solution for a superior way to construct a portfolio, it will hope-
fully make investors critically re-evaluate their risk perspectives and investment
universes for constructing the strategic allocation for their portfolios.
16Like the rule of half a dozen asset classes in the Yale Endowment fund (Swensen 2009).
86
REFERENCES
Agarwal, V. – Naik, N. Y. (2004) Risks and portfolio decisions involving hedge
funds. The Review of Financial Studies, vol. 17 (1), 63–98.
Allen, D., Lizieri, C. – Satchell, S. (2020) A comparison of non-gaussian var es-
timation and portfolio construction techniques. Journal of Empirical
Finance, vol. 58, 356–368.
Amin, G. S. – Kat, H. M. (2003) Stocks, bonds, and hedge funds. The journal of
portfolio Management, vol. 29 (4), 113–120.
Andersen, T. M. (2021) Elakkeiden riittavyys ja kestavyys: arvio suomen elake-
jarjestelmasta.
Artzner, P., Delbaen, F., Eber, J.-M. – Heath, D. (1999) Coherent measures of
risk. Mathematical finance, vol. 9 (3), 203–228.
Barberis, N. – Huang, M. (2008) Stocks as lotteries: The implications of probability
weighting for security prices. American Economic Review, vol. 98 (5),
2066–2100.
BlackRock (2020) Alternative investments in modern portfolios. https:
//www.blackrock.com/institutions/en-us/insights/
portfolio-design/alternatives-in-modern-portfolios, [On-
line, accessed 23.10.2021].
Boudt, K., Carl, P. – Peterson, B. (2013) Asset allocation with conditional value-
at-risk budgets. The Journal of Risk, vol. 15, 39–68.
Boudt, K., Lu, W. – Peeters, B. (2015) Higher order comoments of multifactor
models and asset allocation. Finance Research Letters, vol. 13, 225–
233.
Boudt, K., Peterson, B. G. – Croux, C. (2008) Estimation and decomposition of
downside risk for portfolios with non-normal returns. Journal of Risk,
vol. 11 (2), 79–103.
Brandimarte, P. (2017) An Introduction to Financial Markets: A Quantitative
Approach. John Wiley & Sons.
87
Brooks, C. – Kat, H. M. (2002) The statistical properties of hedge fund index
returns and their implications for investors. The Journal of Alternative
Investments, vol. 5 (2), 26–44.
Buchner, A. (2020) The alpha and beta of private equity investments. Available at
SSRN 2549705.
Campbell, J. Y., Lo, A. W. – MacKinlay, A. C. (2012) The econometrics of finan-
cial markets. princeton University press.
Capocci, D., Duquenne, F. – Hübner, G. (2006) Diversifying using hedge funds:
a utility-based approach. Working paper HEC-ULG Management
School.
Cavenaile, L., Coën, A. – Hübner, G. (2011) The impact of illiquidity and higher
moments of hedge fund returns on their risk-adjusted performance
and diversification potential. The Journal of Alternative Investments,
vol. 13 (4), 9–29.
Chan, e. a., Boyd (2021) A better way to build private market portfolios. Black-
Rock.
Choueifaty, Y. – Coignard, Y. (2008) Toward maximum diversification. The
Journal of Portfolio Management, vol. 35 (1), 40–51.
Choueifaty, Y., Froidure, T. – Reynier, J. (2013) Properties of the most diversified
portfolio. Journal of investment strategies, vol. 2 (2), 49–70.
Chow, G. C. – Lin, A.-l. (1971) Best linear unbiased interpolation, distribution, and
extrapolation of time series by related series. The review of Economics
and Statistics, 372–375.
Committee, B. (1996) The amendment to the capital accord to incorporate market
risk.
Cumming, D., Helge Haß, L. – Schweizer, D. (2014) Strategic asset allocation and
the role of alternative investments. European Financial Management,
vol. 20 (3), 521–547.
Danielsson, J., Embrechts, P., Goodhart, C., Keating, C., Muennich, F., Renault,
O., Shin, H. S. et al. (2001) An academic response to basel ii.
88
Dittmar, R. F. (2002) Nonlinear pricing kernels, kurtosis preference, and evid-
ence from the cross section of equity returns. The Journal of Finance,
vol. 57 (1), 369–403.
Duran, X. (2013) The first us transcontinental railroad: Expected profits and gov-
ernment intervention. The Journal of Economic History, vol. 73 (1),
177–200.
Elton, E. J. – Gruber, M. J. (1977) Risk reduction and portfolio size: An analytical
solution. The Journal of Business, vol. 50 (4), 415–437.
Elton, E. J., Gruber, M. J., Brown, S. J. – Goetzmann, W. N. (2009) Modern
portfolio theory and investment analysis. John Wiley & Sons.
Fabozzi, F. J., Focardi, S. M. – Kolm, P. N. (2010) Quantitative equity investing:
Techniques and strategies. John Wiley & Sons.
Fabozzi, F. J., Gupta, F. – Markowitz, H. M. (2002) The legacy of modern portfolio
theory. The journal of investing, vol. 11 (3), 7–22.
Fama, E. F. (2021) Efficient capital markets a review of theory and empirical work.
The Fama Portfolio, 76–121.
Fernandez, R. B. (1981) A methodological note on the estimation of time series.
The Review of Economics and Statistics, vol. 63 (3), 471–476.
Forsey, H. (2001) The mathematician’s view: Modelling uncertainty with the three
parameter lognormal. In Managing Downside Risk in Financial Mar-
kets, 51–58, Elsevier.
Francis, J. C. – Ibbotson, R. G. (2009) Contrasting real estate with comparable
investments, 1978 to 2008. The Journal of Portfolio Management,
vol. 36 (1), 141–155.
Fraser-Sampson, G. (2010) Alternative assets: investments for a post-crisis world.
John Wiley & Sons.
Fung, W. – Hsieh, D. A. (2000) Performance characteristics of hedge funds and
commodity funds: Natural vs. spurious biases. Journal of Financial
and Quantitative analysis, vol. 35 (3), 291–307.
Fung, W. – Hsieh, D. A. (2001) The risk in hedge fund strategies: Theory and evid-
ence from trend followers. The review of financial studies, vol. 14 (2),
313–341.
89
Getmansky, M., Lo, A. W. – Makarov, I. (2004) An econometric model of serial
correlation and illiquidity in hedge fund returns. Journal of Financial
Economics, vol. 74 (3), 529–609.
Group, T. P. S. (2021) What’s the worst that should happen? Alternative Think-
ing.
Hadar, J. – Russell, W. R. (1969) Rules for ordering uncertain prospects. The
American economic review, vol. 59 (1), 25–34.
Harvey, C. R. – Siddique, A. (2000) Conditional skewness in asset pricing tests.
The Journal of finance, vol. 55 (3), 1263–1295.
Hitaj, A., Martellini, L. – Zambruno, G. (2011) Optimal hedge fund allocation
with improved estimates for coskewness and cokurtosis parameters.
The Journal of Alternative Investments, vol. 14 (3), 6–16.
Hoernemann, J. T., Junkans, D. A. – Zarate, C. M. (2005) Strategic asset allocation
and other determinants of portfolio returns. The Journal of Wealth
Management, vol. 8 (3), 26–38.
Ibbotson, R. G., Chen, P. – Zhu, K. X. (2011) The abcs of hedge funds: Alphas,
betas, and costs. Financial Analysts Journal, vol. 67 (1), 15–25.
Ibbotson, R. G. – Kaplan, P. D. (2000) Does asset allocation policy explain
40, 90, or 100 percent of performance? Financial Analysts Journal,
vol. 56 (1), 26–33.
Ibragimov, R. (2007) Efficiency of linear estimators under heavy-tailedness: convo-
lutions of -symmetric distributions. Econometric Theory, vol. 23 (3),
501–517.
Ilmanen, A. (2011) Expected returns: An investor’s guide to harvesting market
rewards. John Wiley & Sons.
Ilmanen, A., Chandra, S. – McQuinn, N. (2019) Demystifying illiquid assets: Ex-
pected returns for private equity. The Journal of Alternative Invest-
ments, vol. 22 (3), 8–22.
IRENA (2021) Wirkd energy transitions outlook. URL: https://www.irena.org/
publications/2021/Jun/World-Energy-Transitions-Outlook.
Jondeau, E. – Rockinger, M. (2006) Optimal portfolio allocation under higher
moments. European Financial Management, vol. 12 (1), 29–55.
90
J.P.Morgan Asset Management (2021) Guide to Alternatives Q4 2021.
URL: https://am.jpmorgan.com/us/en/asset-management/adv/
insights/market-insights/guide-to-alternatives/, acquired
December 11th 2021.
Jurczenko, E. – Teiletche, J. (2019) Expected shortfall asset allocation: A multi-
dimensional risk-budgeting framework. Journal of Alternative Invest-
ments, vol. 22 (2), 7–22, URL: www.scopus.com, cited By :1.
Kaplan, S. N. – Schoar, A. (2005) Private equity performance: Returns, persist-
ence, and capital flows. The journal of finance, vol. 60 (4), 1791–1823.
Kiesel, R., Zagst, R. – Scherer, M. (2010) Alternative investments and strategies.
World Scientific.
Krokhmal, P., Palmquist, J. – Uryasev, S. (2002) Portfolio optimization with con-
ditional value-at-risk objective and constraints. Journal of risk, vol. 4,
43–68.
Leitner, C., Mansour, A. – Naylor, S. (2007) Alternative investments in perspect-
ive. PREEF Research, vol. 9, 9–26.
Lerner, J., Schoar, A. – Wang, J. (2008) Secrets of the academy: The drivers
of university endowment success. Journal of Economic Perspectives,
vol. 22 (3), 207–22.
Levy, H. (1994) Absolute and relative risk aversion: An experimental study.
Journal of Risk and uncertainty, vol. 8 (3), 289–307.
Lim, A. E., Shanthikumar, J. G. – Vahn, G.-Y. (2011) Conditional value-at-risk
in portfolio optimization: Coherent but fragile. Operations Research
Letters, vol. 39 (3), 163–171.
Litterman, R. B. (1983) A random walk, markov model for the distribution of time
series. Journal of Business & Economic Statistics, vol. 1 (2), 169–173.
Malkiel, B. G. (2019) A random walk down Wall Street: the time-tested strategy
for successful investing. WW Norton & Company.
Mao, J. C. (1970) Essentials of portfolio diversification strategy. Journal of Fin-
ance, 1109–1121.
Markowitz, H. M. (1968) Portfolio selection. Yale university press.
91
Martellini, L. – Ziemann, V. (2010) Improved estimates of higher-order comoments
and implications for portfolio selection. The Review of Financial Stud-
ies, vol. 23 (4), 1467–1502.
McKinsey & Company (2021) A year of disruption in the private
markets. URL: https://www.mckinsey.com/industries/
private-equity-and-principal-investors/our-insights/
mckinseys-private-markets-annual-review, acquired December
11th 2021.
Van der Meer, R. (2001) The dutch view: developing a strategic benchmark in
an alm framework. In Managing downside risk in financial markets,
26–40, Elsevier.
Mitton, T. – Vorkink, K. (2007) Equilibrium underdiversification and the pref-
erence for skewness. The Review of Financial Studies, vol. 20 (4),
1255–1288.
Okunev, J. – White, D. R. (2003) Hedge fund risk factors and value at risk of
credit trading strategies. Available at SSRN 460641.
Phalippou, L. – Gottschalg, O. (2009) The performance of private equity funds.
The Review of Financial Studies, vol. 22 (4), 1747–1776.
Popova, I., Popova, E., Morton, D. – Yau, J. (2006) Optimal hedge fund alloca-
tion with asymmetric preferences and distributions. Available at SSRN
900012.
Preqin (2020) Preqin investor outlook: Alternative assets h1 2020.
Preqin (2021) The past, present, and future of the industry. URL: https:
//www.preqin.com/academy/lesson-1-alternative-assets/
past-present-future-of-the-alternative-assets-industry.
Rockafellar, R. T., Uryasev, S. et al. (2000) Optimization of conditional value-at-
risk. Journal of risk, vol. 2, 21–42.
Rom, B. M. – Ferguson, K. W. (1994) Post-modern portfolio theory comes of age.
Journal of Investing, vol. 3 (3), 11–17.
Rom, B. M. – Ferguson, K. W. (1997) Using post-modern portfolio theory to im-
prove investment performance measurement. Journal of Performance
Measurement, vol. 2 (2), 5–13.
92
Sax, C. – Steiner, P. (2013) Temporal disaggregation of time series.
Sharpe, W. F. (1964) Capital asset prices: A theory of market equilibrium under
conditions of risk. The journal of finance, vol. 19 (3), 425–442.
Sharpe, W. F. (1966) Mutual fund performance. The Journal of business,
vol. 39 (1), 119–138.
Sortino, F. A., Satchell, S. – Sortino, F. (2001) Managing downside risk in financial
markets. Butterworth-Heinemann.
Statman, M. (1987) How many stocks make a diversified portfolio? Journal of
financial and quantitative analysis, vol. 22 (3), 353–363.
Swensen, D. F. (2009) Pioneering portfolio management: An unconventional ap-
proach to institutional investment, fully revised and updated. Simon
and Schuster.
Tasche, D. (2002) Expected shortfall and beyond. Journal of Banking & Finance,
vol. 26 (7), 1519–1533.
The World Bank, (2019) Listed companies, total. URL: https://data.
worldbank.org.
Times, F. (2021) House passes joe biden’s $1.2tn bipartisan infrastruc-
ture bill. Financial Times, URL: https://www.ft.com/content/
a588078c-8dec-4bd4-8f99-d7f0ff61ed05.
Tyoelakevakuuttajat TELA Ry (2021) Elakevarojen sijoitustoiminnan ke-
hitys. https://www.tela.fi/elakevarojen-sijoittaminen/
elakevarojen-maara/varojen-kehittyminen/, [Online, accessed
23.10.2021].
Verohallinto (2021) Verotusohje yleishyodyllisille yhteisoille. https:
//www.vero.fi/syventavat-vero-ohjeet/ohje-hakusivu/47999/
verotusohje-yleishyodyllisille-yhteisoille3/, [Online,
accessed 19.1.2022].
Wong, W. K. (2008) Backtesting trading risk of commercial banks using expected
shortfall. Journal of Banking & Finance, vol. 32 (7), 1404–1415.
Yale Investment Office (2020) The yale endowment 2020. https://investments.
yale.edu/reports, [Online, accessed 23.10.2021].
93
Yamai, Y. – Yoshiba, T. (2005) Value-at-risk versus expected shortfall: A practical
perspective. Journal of Banking & Finance, vol. 29 (4), 997–1015.
94
APPENDIX I
The Cornish-Fisher estimation for ES introduced in chapter 3.4 is derived as Boudt
et al. (2013) propose. The second order Cornish-Fisher expansion of the quantile
function around the Gaussian quantile function equals:
gt = z +
1
6
(z2 − 1)st +
1
24
(z3 − 3z)kt −
1
36
(2z3 − 5z)s2t ; (45)
with z = −1(). Now the ES for the portfolio in month t is given by:
ESt() = −w′t +√m2tdt(); (46)
where:
dt() =
1

{(gt) + 1
24
[I4 − 6I2 + 3(gt)]kt+
1
6
[I3 − 3I1]st + 1
72
[I6 − 15I4 + 45I2 − 15(gt)]s2p};
(47)
and where:
Iq =
8>>>>><>>>>>:
q
2X
i=1
 Q q
2
j=1 2jQi
j=1 2j
!
g2it(gt) +
0@ q2Y
j=1
2j
1A(gt) for q is even
qX
i=0
 Qq
j=0 2j + 1Qi
j=0 2j + 1
!
g2i+1t (gt) +
 
qY
j=0
2j + 1
!
(gt) for q is odd
(48)
respectively q∗ = (q − 1)=2 and (·) and (·) are the standard normal density
and cumulative distribution functions.
95
APPENDIX II
0 %
20 %
40 %
60 %
80 %
100 %
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
FI & MM EQ Alternatives
Figure 22: Finnish Pension Funds asset allocation
Figure 23: Desmoothed return indices for asset classes
96
0.
0
0.
4
0.
8
0.00992 0.0105 0.0123 0.0148 0.0179 0.0212 0.0247 0.0283 0.0319
0.00352 0.00381 0.00411 0.0044 0.00469 0.00498 0.00527 0.00556 0.00585
StdDev
Mean
W
ei
gh
t
EQ − DM
EQ − EM
GOV
IG
HY
0.
0
0.
4
0.
8
0.016 0.0177 0.0214 0.0271 0.0343 0.0423 0.051 0.0597 0.0686
0.00357 0.00385 0.00414 0.00443 0.00471 0.005 0.00528 0.00557 0.00586
ES
Mean
W
ei
gh
t
EQ − DM
EQ − EM
GOV
IG
HY
Figure 24: Asset weights in efficient portfolios without alternative investments
Efficient Frontiers
ES
M
ea
n
0.00 0.05 0.10 0.15 0.20
0.
00
0
0.
00
2
0.
00
4
0.
00
6
0.
00
8 PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Traditional assets
All assets
Figure 25: Efficient frontiers in mean-ES space
97
Efficient Frontiers
ES
M
ea
n
0.00 0.05 0.10 0.15 0.20
0.
00
0
0.
00
2
0.
00
4
0.
00
6
EQ − DM
EQ − EM
GOV
IG
HY
Mean−Var
Mean−ES
Figure 26: Mean-variance and mean-ES efficient frontiers without alternatives
Mean−ES Weights
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
V
al
ue
Dec 10 Dec 12 Dec 14 Dec 16 Dec 18 Dec 20
PE
PC
IF
NR
RE
HF
EQ − DM
EQ − EM
GOV
IG
HY
Figure 27: Mean-ES out-of-sample weights
98
Jan
2011
Jan
2012
Jan
2013
Jan
2014
Jan
2015
Jan
2016
Jan
2017
Jan
2018
Jan
2019
Jan
2020
Dec
2020
Cumulative Return 2011−01−01 / 2020−12−01
0.2
0.4
0.6
Monthly Return
−0.01
 0.00
 0.01
 0.02
Drawdown
−0.025
−0.020
−0.015
−0.010
−0.005
Mean−ES Optimization Performance
Figure 28: Mean-ES out-of-sample performance