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Optimal Portfolios with Traditional and Alternative Investments:
An Empirical Investigation
Edwin O. Fischer*
Susanne Lind-Braucher**
November 2009
* Professor at the Institute for Finance at the Karl-Franzens-University Graz, Univer-
sitätsstrasse 15/G2, A-8010 Graz, Austria, edwin.fischer@uni-graz.at
** Research Assistant at the Institute for Finance at the Karl-Franzens-University Graz,
Universitätsstrasse 15/G2, A-8010 Graz, Austria, susanne.lind-braucher@uni-graz.at
2
Optimal Portfolios with Traditional and Alternative Investments:
An Empirical Investigation
Abstract
This paper empirically investigates the diversification effects on a traditional portfolio by in-
troducing alternative investments (hedge funds, managed futures, real estate, private equities
and commodities). This paper is the first attempt to incorporate a variety of risk measures
(Volatility, Value at Risk and Conditional Value at Risk) as the objective function for the
portfolio optimization and different estimates for the expected return (historical estimates,
robust Bayes–Stein estimates, CAPM estimates and Black–Litterman estimates). Further-
more, the alternative risk measures are additionally modified for the skewness and the kurto-
sis ((modified) VaR and (modified) CVaR). In this manner, the influences of the higher mo-
ments on the asset allocation can also be examined in connection with different risk measures
and various estimators for expected returns.
Formulation of the Problem
The last months especially have shown that investors’ confidence in capital markets has suf-
fered drastically. The turbulence on capital markets was accompanied by enormous breaks in
prices. Therefore, private as well as institutional investors looked for alternative investments,
to which less attention had been paid until then. The amount of alternative investments, which
include hedge funds, managed futures, real estate, private equities and commodities, has al-
ready increased by over 13 percent on average in Europe (JP Morgan Asset Management
[2007]). Therefore, capital assets have to be chosen, weighted and combined in the right way
to achieve preferably high returns on investments by taking low risks at the same time. This
can be carried out by an asset allocation of different types of investments. Investors who do
not want to take high risks try to choose their investments in such a way that, in the case of a
loss on stock markets, as it happened e.g. in the actual subprime crisis, their portfolio remains
widely unaffected.
To find out how such astonishing events affect different investments, the three worst months
of the international stock markets of the last ten years are pulled up and analyzed. During this
period, the world stock markets registered the strongest losses in October 2008 with -20.71%
3
(see Figure 1). This is the worst crisis since World War II and is still ongoing, having started
with the US real estate crisis in 2007. Since the middle of 2008, the real estate crisis has also
encroached on the real economy and this is still continuing. That has an effect on other asset
classes such as private equities (-38.82%), real estate (-32.33%) and commodities (-9.31%).
This was the reason that the world stock markets also had the third worst performance of the
last ten years in November 2008 (see Figure 3). The stock market registered a loss of -
14.33%, private equities even -38.23%, real estate -25.25% and commodities -15.90%. The
world economy was therefore in an unstable constitution, which continued until February
2009. The situation on the international capital markets was alarming, as shown by the nega-
tive performances of all investments in February 2009 (see Figure 2). However, especially
managed futures and bonds were excluded from that decrease. Therefore, managed futures are
much more stable than stocks, especially in times of crises (Schneeweis et al. [2001, 2002]);
and this can also be seen in a more precise overview of the rankings of different investments
by annual returns (see Figure 4). Figure 4 shows which types of investments have been losers
and winners in the last ten years. One may not wonder that real estate performed well until
2006, but since 2007, when the real estate crisis started, it has rather been one of the losers.
Looking at managed futures and hedge funds, the result shows that they achieved better per-
formance than stocks, especially in times of crisis.
Hitherto published empirical studies , e.g. Kaiser et al. [2008]; Kat [2007]; Lee and Stevenson
[2005]; Maurer and Reiner [2002]; Schweitzer [2008]) show positive effects of portfolio di-
versification if alternative investments, e.g. hedge funds or real estate, are added to a tradi-
tional portfolio of stocks and bonds. In summer 2002, Schneeweis et al. published an empiri-
cal study, which shows that the portfolio efficiency increases with hedge funds. However, in
such an analysis, it should be considered that the empirical distribution of some alternative
investments, for instance hedge funds, is not always normally distributed (see also Kat [2003,
2004]). In that case, it is necessary for the portfolio optimization to take the skewness and the
excess kurtosis into consideration (see also Kat [2005]). One alternative is to replace volatility
with other measures of risk, e.g. the modified Value at Risk, which is calculated by using the
Cornish–Fisher expansion (Favre and Galeano [2002]). The modified VaR allows us to meas-
ure the portfolio with non-normally distributed assets like hedge funds or real estate and to
compute optimal portfolios by minimizing the modified VaR at a given confidence level. A
further alternative is the Conditional Value at Risk (CVaR), showing some advantages com-
4
pared with the Value at Risk (Acerbi and Tasche 2002; Artzner et al. 1999). Many researchers
have extensively criticized the use of the VaR as a measure of risk. Therefore, Artzner et al.
[1999] suggest using the CVaR as a risk measure, because the CVaR is a coherent measure of
risk and using the CVaR ensures that the return–risk optimization of a portfolio is always
solvable (Rockafellar and Uryasev 1999). The CVaR can also be extended for skewness and
excess kurtosis. In this case of the modified CVaR (mCVaR), the impacts of these higher
moments are considered.
All the mentioned publications are based on historical return estimators. Other publications,
e.g. Black and Litterman [1992]; He and Litterman [1999]; Jorion [1986, 1990]; Kan and
Zhou [2007]; Litterman and the Quantitative Resources Group [2003]) show that in the port-
folio analysis the results are never optimal with parameters’ uncertainty. However, alterna-
tives exist for the return estimators for the case of parameter uncertainty: our study focuses on
the Bayes–Stein return estimators, the CAPM return estimators and the Black–Litterman re-
turn estimators.
The applications of global portfolio optimization in this paper proceed in the following steps.
First, the optimal portfolios with traditional and alternative investments with historical return
estimators as well as with alternative return estimators (Bayes–Stein estimator, CAPM esti-
mator and Black–Litterman estimators) are calculated. Then, the volatility is replaced by the
alternative risk measures ((m)VaR, (m)CVaR). Therefore, the paper combines the best-known
risk measure, also modified for the skewness and the excess kurtosis, with different estimators
for returns. In this manner, influences of the higher moments to the asset allocation can also
be examined in connection with different risk measures and varied estimators for expected
returns.
Our paper is organized as follows. First, an overview and analysis of the data are given. Then,
the classical Markowitz and Tobin portfolio models, risk and performance measures and re-
turn estimators are described in chapter 3. In chapter 4, the optimal global asset allocations
with traditional and alternative investments are empirically investigated and we analyze the
impacts of the return estimators, the risk measures and the non-normality of returns on the
optimal portfolios. Furthermore, we take a look at the two deep declines of global stock mar-
5
ket prices in the last decade (the dot.com crisis and the subprime crisis) and compare for these
global stock market shocks the performances of the optimal global portfolios (chapter 5).
Database and Analysis
For this study, performance indices (see Table 1) act as a database within a ten-year period of
observations from April 1, 1999 to April 1, 2009. This study is from the view of an investor
investing in USD. An extensive analysis in EUR is presented by Fischer and Lind-Braucher
[2009]. For the calculations, we use monthly historical data, which were taken from the
Thomson Financial Datastream.
Figure 5 shows the historical performances of the asset classes of the last ten years, which
yields a first impression of the asset classes among each other. It is hardly astonishing that the
chosen asset classes show different value developments. The historical performance shows
that, especially in the last month, stocks, commodities, real estate and private equities have
suffered greatly from the losses. Bonds, as well as hedge funds and managed futures, had a
rather sideward proceeding phase. Nevertheless, much more vital is how the asset classes be-
have towards each other. This is expressed by a correlation coefficient
ij
ρ
, which can have
values from -1 to +1. The higher the coefficient is, the more synchronous the development of
the asset classes is in the same direction. However, vice versa, at a value of -1, a negative de-
pendency exists; in that case, the development is accurately contrary. The correlation coeffi-
cients of this study (see Table 2) show a clear result: e.g. stocks and managed futures with a
correlation coefficient of –0.102 show rather independent values. That is also valid for man-
aged futures and private equities (-0.088). Also, in that case, the correlation coefficients are
negative. Stocks and private equities show a very high positive correlation of 0.868. This
positive correlation indicates a synchronous development of these asset classes.
For a more differentiated analysis, the applied asset classes undergo a statistical analysis. Ta-
ble 3 shows an overview of the returns and volatilities of the examined asset classes. The av-
erage returns and volatilities per year and month are calculated for all the alternatives of in-
vestment by the historical database. One can recognize that commodities as well as managed
futures as representatives of alternative investments show the highest average returns with
8.98% and 7.73% per year. Private equities (31.12%) and commodities (20.67%) range re-
spectively in the most risky investment classes with annual volatility. Furthermore, higher
6
moments (skewness and excess kurtosis) are calculated, which give information about the
(normal-) distribution attributes of returns. Additionally, all the asset classes are checked by
the Jarques/Bera Test (Bera and Jarques [1987]) for their normal distribution.
The considerations of statistical analysis as well as the correlation matrix are not sufficient for
making statements, if a certain asset class should be considered for a global portfolio. For that
reason, the portfolio optimization is used for an optimal asset allocation.
Models and Measures
Portfolio Optimization Models
A fundamental milestone of finance is the classic portfolio theory of Markowitz [1952]. This
theory combines the best possible combination of investment alternatives for establishing an
optimal portfolio, which considers the preferences of the investor concerning risk and return.
The expected portfolio return
)(
P
rE
can be shown as a weighted average of the expected re-
turns of the asset classes
)(
j
rE
, whereas
j
x
depicts the weight proportions
N
as the number
of assets:
∑
=
⋅=
N
jjjP
xrErE
1
.)()(
(1)
The risk of a portfolio conforms to the standard deviation with the covariance
jiijij
σσρσ
=
:
∑∑
= =
⋅=
N
j
N
ijiijP
xxr
1 1
.)(
σσ
(2)
Those combinations, which show the lowest risks for a given expected return or achieve the
highest expected returns for a given risk, are called efficient. The set of portfolios that can be
called efficient is pictured in the so-called efficient frontier.
In this study, we analyze two portfolios: the portfolio with the lowest risk (Minimum Risk
Portfolio MRP) and the portfolio with the highest (modified) Sharpe Ratio (Maximum Rela-
7
tive Performance Portfolio MRPP). The Minimum Variance Portfolio MVP is the portfolio
that shows the lowest risk, measured by volatility. Mathematically, this portfolio can be calcu-
lated by minimizing the risk without restricting the expected return (Elton et al. [2003]):
Minimum Variance Portfolio: (3)
min!)(
1 1
→=
∑∑
= = ij
N
i
N
jjip
xxr
σσ
s.t.
.0
1
1
≥
=
∑
=
j
N
jj
x
x
The Portfolio Theory of Markowitz was extended by Tobin [1958] by additionally viewing a
riskless investment. The Maximum Relative Performance Portfolio MRPP (Tangency Portfo-
lio) can be calculated by maximizing the Sharpe Ratio (Sharpe [1994]):
Maximum Relative Performance Portfolio: (4)
max!
)( )( →
−
=
P
P
rrrE
SR
σ
s.t.
,0
1
)(
)()(
1
1 1
1
≥
=
=
⋅=
∑
∑∑
∑
=
= =
=
j
N
jj
ij
N
i
N
jjip
j
N
jjP
x
x
xxr
xrErE
σσ
where
=
r
riskless interest rate.
Alternative Risk and Performance Measures
From these optimization problems, based on the classical portfolio theory, portfolio shares
can be calculated for all the asset classes. This implies that the volatility is used as the risk
8
measure. If alternative risk measures (VaR und CVaR) are used instead of the volatility, only
the objective function will change in the optimization problem. The asset allocation in this
study is also carried out on the basis of further risk measures, so that the effects of alternative
risk measures on the asset allocation can be analyzed.
(Conditional) Value at Risk
The Value at Risk (VaR) is the maximum deficit that is not exceeded by a given “security
probability”
α
in a certain period of time (confidence level
α
−
1
= 95%). Formally, the VaR
is defined as follows (Favre and Galeano [2002]):
α
α
zFrVaR
j
rj
==
−
)()(
1
)(
(5)
where
(.)F
= distribution function.
The Conditional Value at Risk (CVaR), also known as Expected Shortfall or Expected Tail
Loss, has been regarded more and more in recent years in theory and practice. The CVaR is
interpreted as a quantile reserve plus an Excess Reserve and is the expected loss under the
VaR (Favre and Galeano [2002]):
)).(()(
jjjj
rVaRrrErCVaR −<=
(6)
In the case of normally distributed returns, the VaR and the CVaR are calculated as follows
(Favre and Galleano [2002]):
Value at Risk:
)()()(
jjj
rzrErVaR
σ
α
⋅+=
(7)
with
α
z= quantile of the standard normal distribution
Conditional Value at Risk:
⋅+−= )(
)(
)()(
jjj
r
z
rErCVaR
σ
α
ϕ
α
(8)
with
(.)
ϕ
= density function of the standard normal distribution.
9
If an alternative risk measure (VaR or CVaR) is used instead of the volatility, it is possible to
adapt the calculation of the Sharpe Ratio to these risk measures. Analytically, the modified
Sharpe Ratios are described as follows (Favre and Galeano [2002]):
r
VaR
rrE
mSR
P
VaR
+− −
=)( (9)
.
)(
CVaR
rrE
mSR
P
CVaR
−−
=
(10)
Two further measures, which are of great interest in risk analysis, are “Maximum Drawdown”
and “Time under Water” (López de Prado and Peijan [2004]). “Maximum Drawdown” is the
maximum value deficit that occurs by assessing an investment. By using the term “Time un-
der Water,” we mean the time period that is needed until possibly occurring deficits are offset
after assessing an investment. In Figure 3, both terms are demonstrated.
Modified (Conditional) Value at Risk
The portfolio theories by Markowitz and Tobin use the volatility as the risk measure and are
therefore based on the hypothesis of normally distributed returns of the asset classes. Regard-
ing the Jarques/Bera statistics (see Table 3), attention is drawn to the fact that only managed
futures are consistent with this hypothesis, whereas all the other asset classes are non-
normally distributed. The only alternative for calculating the risk and performance measures
is seen by using alternative risk measures. With the help of the Cornish–Fisher expansion
(Cornish and Fisher [1938]), alternative risk measures (modified VaR and modified CVaR)
can be defined, which also consider the skewness
S
and the excess kurtosis
K
(Gregoriou
and Gueyie [2003]). These risk measures can be calculated by (Favre and Galeano [2002]):
modified VaR:
)()()(
jCFjj
rzrErmVaR
σ
⋅+=
(11)
modified CVaR:
⋅+−= )(
)( )(
)()(
j
CF
CF
jj
r
zN z
rErmCVaR
σ
ϕ
(12)
with
(.)
ϕ
= density function of the standard normal distribution
(.)N= distribution function of the standard normal distribution
10
and with the Cornish–Fisher expansion:
(
)
(
)
(
)
.52
36
1
3
24
1
1
6
1
2332
SzzKzzSzzz
CF
⋅−⋅⋅−⋅−⋅+⋅−+=
αααααα
Additionally, modified performance measures can be defined on the basis of the Cornish–
Fisher expansion (Gregoriou and Gueyie [2003]):
r
mVaR
rrE
mSR
P
mVaR
+− −
=)(
(13)
.
)(
mCVaR
rrE
mSR P
mCVaR −
−
=
(14)
A more detailed overview of the previously described risk and performance measures of the
single asset classes of the empirical examination is given in Table 4 for
%5
=
α
. As a riskless
interest rate, the 1-month USD-LIBOR of April 1, 2009 with the amount of 0.49% p.a. was
used.
Alternative Estimators for the Expected Returns
It is already known from literature that the classical portfolio theory is based on certain theo-
ries, which are evaluated very critically by several scientists. Some of the criticisms are
(Jorion [1992]):
• Bad out-of-sample performance and
• Sensitive results concerning changes in input parameters. Even marginal changes in
the expectation of returns can cause huge changes in the optimal weights of portfolios.
Moreover, in practice, the expected returns are very often simply calculated from the histori-
cal returns. From that point of view, the portfolio optimization has to be applied in a different
way to obtain consistent and confident structures of asset allocation for the portfolio optimiza-
tion.
Robust Bayes–Stein estimators
In literature, the use of robust estimators, so-called “Bayes–Stein” shrinkage estimators
(Jorion [1986]), is proposed as an alternative to historical returns. The basic principle of these
11
estimators is that a global, asset classes specific return average is the basis for all the asset
classes, to which asset classes specific risk premiums are added. This is demonstrated in the
following equation (Jorion [1986]):
).()()1()(
.MVP
hist
j
BS
j
rEwrEwrE ⋅+⋅−=
(15)
hist
j
rE )(
stands for the historical return and
)(
MVP
rE
for the return of the MVP without short-
sale restrictions. The MVP is calculated on the basis of the classical portfolio theory of
Markowitz (3) without short-sale restrictions. The so-called Shrinkage Factor
w
reduces the
elements of the originally expected returns in dependency from the volatility. It can be calcu-
lated, if we assume that
T
is the number of periods of estimation, N the number of assets, 1
stands for the vector of ones and
1−
Σ
is the inverse of the variance covariance matrix (Jorion
[1990]):
( )
( ) ( )
1)()(1)()(2
2
1MVPjMVPj
rErETrErEN
N
w−Σ
′
−++
+
=
−
(16)
where
.
ij
σ
=Σ
Since the variance covariance matrix
Σ
is not known in practice, it is replaced by (Kan and
Zhou [2007]):
.
2
1
ˆij
N
T
T
σ
−−
−
=Σ (17)
For the MVP and for the Shrinkage Factor, the following values are calculated:
.33.0
..%25.3)(
=
=
w
aprE
MVP
The historical returns and expected returns, which were calculated with the help of the Bayes–
Stein estimators, are compared in Table 5. From this table, we can recognize the influence the
12
factor
w
has on the calculation of the expected returns: high historical returns, e.g. commodi-
ties (8.98%), managed futures (7.73%), hedge funds (6.78%) and real estate (4.00%), drop
with the calculations of Bayes–Stein. Lower historical returns, like private equities (-5.54%),
stocks (-1.65%) and bonds (1.72%) rise. Therefore, the Shrinkage Factor causes the estimated
returns to shrink to the middle (high returns fall and low returns rise).
CAPM estimators
A further alternative to the historically based return estimators is the returns of the Capital
Asset Pricing Model (CAPM) (Sharpe [1964]):
[
]
44 344 21
premiumrisk
jM
CAPM
j
rrErrE
β
⋅−+= )()( (18)
with
)(
M
rE
= expected capital market return
rrE
M
−)(
= market risk premium
j
β
= Beta Factor =
.
M
j
jM
σ
σ
ρ
⋅
For the calculation of expected returns according to the CAPM, the expected return
)(
M
rE
and the risk
M
σ
of the market portfolio are required. In the market portfolio, all the assets are
included in proportion to their market caps. The values of the market caps date from April 1,
2009 and are based on Thomson Financial Datastream and on the homepage of the used
hedge funds index (see http://www.hedgeindex.com). The value of commodities (10%) was
calibrated following Idzorek [2006]. A more detailed overview of the empirical market caps,
Beta Factors and expected CAPM returns is given in Table 6. For the calculation of the ex-
pected capital market returns, we use the following value:
..%79.2)( aprE
M
=
This value was selected in such a way that the resulting risk premium on the international
stock markets corresponds with the usually assumed 4–5% p.a. (Damodaran [2006]).
13
Black–Litterman estimators
As a third and commonly used alternative for estimating return parameters in practice, the
Black–Litterman Model (Black and Litterman [1992]) is used for the optimization of the port-
folios. The idea of these estimators is that the expected returns of the market equilibrium are
connected with individual subjective return forecasts (views). With this model, economically
better supported returns and above all more stable portfolio weights can be deduced. These
expected returns of the market equilibrium are, comparable with the Capital Asset Pricing
Model (CAPM), deduced from the market portfolio, which depicts the benchmark. If the ac-
tual market capitalizations of the single asset classes are known, the implied returns of the
market equilibrium are calculated through Reverse Optimization. These expected returns of
the market equilibrium
)(
implizit
j
rE
serve as neutral reference returns, which are combined with
subjective return views for the Black–Litterman estimators. The advantage of the Black Lit-
terman estimators is that neutral return expectations are combined with subjective return fore-
casts. Thereby, a consistent adaption of the return expectation to subjective market evalua-
tions is carried out. The formulation is very flexible, so opinions of experts about expected
returns can be integrated. The greatest difficulty, however, is that experts do not only have to
define the directions and the heights of returns, but they also have to quantify the quality of
the views. The evaluations of the experts may be specified as absolute and as relative, but it is
not required to submit a forecast for each asset class. If a forecast is submitted for every asset
class, a “one” is entered in the diagonal element of the matrix
P
. The remaining elements
contain only zeros. In this empirical examination, the matrix
P
was chosen in such a way that
an absolute forecast was submitted for each asset class. The matrix
P
can formally be defined
as (Idzorek [2005]):
.
,1,
,11,1
=
nkk
n
pp
pp
P
L
MOM
L
(19)
For our study with seven asset classes, the forecasts matrix can be depicted as follows:
14
.
1000000
0100000
0010000
0001000
0000100
0000010
0000001
=P
Next, it can be considered how much trust an investor gives to his subjective returns. This is
achieved with the variance covariance matrix
Ω
expressing the confidence in the views. In
this study, it is assumed to be 90%. Simplifying, it is assumed in the Black–Litterman estima-
tors that errors of forecasts are spread independently. In this case,
Ω
is a diagonal matrix of
zeros in all non-diagonal positions. Now the variance covariance matrix of the expected re-
turns is proportional to the historical variance covariance matrix
Σ
. The variance covariance
matrix can be assumed as known and is estimated historically. The trust that an investor has in
the benchmark can be measured by the parameter
τ
. A small value can be interpreted as a
high degree of trust of the investor in the benchmark – in this study, the value is 0.1 (=
T
1,
T
is the sample length in years, Rachev et al. [2008]). The variance covariance matrix express-
ing the confidence in the views can be shown as follows (Idzorek [2005]):
(
)
( )
.
00
00
00
11
⋅
′
⋅Σ⋅
⋅
′
⋅Σ⋅
=Ω
τ
τ
kk
pp
pp
O (20)
The variance covariance matrix expressing the confidence for this study can be depicted as
follows:
.
0.00427000000
00.0096800000
0000510.00000
00000140.0000
0000 0.0004500
000000.000090
0000000.00315
=Ω
15
The Black–Litterman return estimators are a confidence-weighted linear combination of mar-
ket equilibria and the expected return implied by the investor’s views. The Black–Litterman
expected return can be written in the following form (Fabozzi et al. [2006]):
).()()(
implizit
j
j j
implizit
ij
views
j
views
ij
BL
i
rEwrEwrE ⋅+⋅=
∑
∑
(21)
The two weighting matrices are given by (Fabozzi et al. [2006]):
PPPPw
views
ij 1111
])[(
−−−−
Ω
′
Ω
′
+Σ=
τ
(22)
,)(])[(
1111 −−−−
ΣΩ
′
+Σ=
ττ
PPw
implizit
ij (23)
where
.1=+
implizit
ij
views
ij
ww
If the investor has full trust in his subjective “Views,” thus
1=
views
ij
w
and
0=
implizit
ij
w
, the ex-
pected returns of Black–Litterman correspond exactly to the views. If the investor has no trust
in his subjective “Views,”
0=
views
ij
w
and
1=
implizit
ij
w
, one receives exactly the expected returns
of the CAPM for the expected returns with the Black–Litterman estimators.
For this empirical study, the following values for views
ij
w
and implizit
ij
w
can be calculated:
=
0.42890.05130.04360.07390.30630.0341-0.0578
0.11000.29510.22200.1000-0.66210.0951-0.3147
0.04710.10430.35510.0031-0.12390.28770.2248
0.02290.0147-0.0002-0.48530.10270.15270.0252-
0.03390.03190.01000.03270.37950.00610.0331
0.0010-0.0008-0.00490.00990.0001-0.49360.0029
0.04300.09730.12420.0559-0.24730.09080.3196
views
ij
w
16
=
0.57110.0513-0.0436-0.0739-0.3063-0.03410.0578-
0.1100-0.70490.2220-0.10000.6621-0.09510.3147-
0.0471-0.1043-0.64490.00310.1239-0.2877-0.2248-
0.0229-0.01470.00020.51470.1027-0.1527-0.0252
0.0339-0.0319-0.0100-0.0327-0.62050.0061-0.0331-
0.00100.00080.0049-0.0099-0.00010.50640.0029-
0.0430-0.0973-0.1242-0.05590.2473-0.0908-0.6804
implizit
ij
w.
Table 7 shows the implied expected returns and the expected Black–Litterman returns for all
the asset classes. Concerning the current situation on the international markets, especially the
dramatic situations on the international stock market during the last months, the values of all
views are assumed to be 2.79% p.a. The calculations were performed on the basis of a 90%
trust of the investor in his views.
All the described alternative estimators for the expected returns can be used in the asset allo-
cation and compared with the historical estimators. Therefore, Table 8 summarizes the ex-
pected returns for all four methods of estimates. Expected returns, which were calculated with
the Bayes–Stein estimators, are similar to the expected returns with historical estimators. This
results from the MVP (15), which are included in the Bayes–Stein estimators. Also, with
CAPM and Black–Litterman estimators, the connection is clearly recognizable.
The purpose of this study is also to indicate the influence of the higher moments in connection
with different risk measures and various estimators for the expected returns on the asset allo-
cation. In addition, we raise the question of which asset class in which circumstance is con-
tained in an efficient global portfolio with different risk measures and with various estimators
for the expected returns.
Asset Allocation
For this empirical study, the following investment restrictions are constituted of the asset allo-
cation for each asset class, which are entered as additional constraints directly into the optimi-
zation models. The reason is that investors with a low risk appetite invest in a more security-
oriented manner than investors with a higher risk appetite. Therefore, investors’ preferences
can be taken into consideration with an efficient asset allocation.
17
Traditional Investments: Alternative Investments:
stocks max. 40% hedge funds max. 25%
bonds unlimited managed futures max. 25%
real estate max. 40%
private equities max. 40%
commodities max. 40%
To investigate the influence of the aforementioned parameters (different estimators for the
expected returns, different risk measures and higher moments) on an asset allocation, it is
important to compare global portfolios with each other. Tables 9 and 10 give an overview of
the calculated portfolios (MVPs and MRPPs). For the different estimators for the expected
returns (historical estimators, Bayes–Stein estimators, CAPM estimators and Black–Litterman
estimators) and for the different risk measures (Volatility, (m)VaR and (m)CVaR), the ex-
pected returns and the analytical risk measures of the optimal portfolios as well as the portfo-
lio weights can be calculated. These tables therefore combine the results for different return
estimators with the different risk measures, which are used as an objective function.
Analysis of the Minimum Risk Portfolios
The impact of return estimators
Volatility as the risk measure:
Let us first take a look at the results in Table 9 obtained for the classical model MVP of
Markowitz: since this optimization problem in equation (3) is independent of the expected
returns for the asset classes, there is no impact of the return estimators and all the optimal
portfolio weights are the same for the volatility as for the risk measure. Due to the small val-
ues of their volatilities and correlations, the MVP consists of more than 84% bonds, almost
15% hedge funds and almost 1% managed futures. All the other asset classes have an optimal
share of zero. The volatility of the MVP is 2.78% p.a. and the expected return of the MVP
depends on the return estimates.
VaR as the risk measure:
If we choose the Value at Risk as the risk measure for the MRP, the optimal portfolio weights
depend on the return estimators (see equation (7)). Due to the small values of the VaR in Ta-
18
ble 4, for all 4 kinds of return estimators the MRPs still consist only of bonds, hedge funds
and managed futures, but now their optimal weights are different from the results of the case
where we take the volatility as the risk measure. For the historical return estimator then, the
optimal weight of bonds declines to almost 58%, the hedge funds’ share rises to its upper
limit of 25% and the share of managed futures increases to more than 17%. The results for the
optimal portfolio weights that are obtained when we use the Bayes–Stein return estimators
also show a decrease in bonds to approximately 70%, an increase of hedge funds to the upper
limit of 25% and also an increase of managed futures to almost 5%. If the returns are esti-
mated with the CAPM or with the Black–Litterman approach, the results are very similar to
each other and very similar to the results of the MVP: again, the MRPs consist only of the
same 3 asset classes with a large share of bonds (more than 80%), a medium share of hedge
funds (approx. 18% and 15%, respectively) and a small share of managed futures.
CVaR as the risk measure:
Also, in this case, the optimal portfolio weights depend on the return estimators (see equation
(8)). Due to the small CVaR value of bonds in Table 4, this asset class always has the largest
share of up to 98% of the MRPs for all 4 kinds of return estimators. The shares of the other
asset classes depend on the return estimators and may be very small. Again, real estate as well
as commodities never appear in the MRPs. For the historical return estimators, the Bond share
is almost 98% and the other asset classes in the MRP are stocks and private equities. For the
Bayes–Stein return estimators, the Bond share is almost 95% and the other asset classes in the
MRP are hedge funds and stocks. If the returns are estimated with the CAPM or with the
Black–Litterman approach, the results are again very similar to each other and very similar to
the results of the MVP: again, the MRPs consist only of the same 3 asset classes with a large
share of bonds (more than 85%), a medium share of hedge funds (approx. 12% and 14%, re-
spectively) and a small share of managed futures.
The impact of risk measures
Historical return estimators:
Let us again start with the results of the optimal weights for the MVP: the bonds’ share was
more than 84%, the hedge funds’ share was almost 15% and the managed futures’ share al-
most 1%. If we now change the risk measure from the volatility to the VaR or CVaR and use
the historical return estimators, the optimal weights look quite different from each other and
19
from the results of the MVP. For the VaR, due to the small VaR values in Table 4, the bonds’
share decreases to less than 60%, the hedge funds’ share increases to its upper limit of 25%
and the managed futures’ share increases to more than 17%. Again, the optimal portfolio con-
sists only of these 3 asset classes. For the CVaR, only bonds have a small CVaR value in Ta-
ble 4 and therefore the MRP mainly consists of bonds (almost 98%) and small values of
stocks and private equities.
Bayes–Stein return estimators:
Similar results are obtained if we look at the optimal weights for Bayes–Stein return estima-
tors for the 3 different risk measures: for the VaR, the bonds’ share decreases to approx. 70%,
hedge funds increase to their upper limit of 25% and managed futures increase to almost 5%.
For the CVaR, the bonds’ share increases to more than 94%, the hedge funds’ share decreases
to approx. 4% and the stocks have close to 1%.
CAPM and Black–Litterman return estimators:
If we investigate the results for CAPM and Black–Litterman return estimators for the three
different risk measures, the optimal portfolio weights now do not change as dramatically as in
the two cases above (historical and Bayes–Stein return estimators). For both kinds of return
estimators and for all three kinds of risk measures, the MRPs look very similar and consist of
a large proportion of bonds, a medium share of hedge funds and a small share of managed
futures. All the other asset classes do not appear in the portfolios.
The impact of non-normality of returns
From the results in Table 9, calculated for the historical return estimators, we can also analyze
the impact of normally distributed or non-normally distributed returns of the asset classes or,
in other words, the impact of including the skewness and excess kurtosis in the analysis of the
MRPs. With the comparison of the optimal weights for the risk measures VaR and mVaR, the
influence of the higher moments is very easily recognizable. Due to the different values for
the modified VaR as compared with the standard VaR in Table 4, the portfolio weights show
severe differences. The share of bonds increases from almost 58% to almost 98%. This is
caused by the dramatic change of the VaR value of bonds when it is modified for the skew-
ness and excess kurtosis. The share of hedge funds decreases from 25% to 0 and the share of
managed futures decreases from 17% to almost 6%. On the other hand, real estate appear in
20
the portfolio with 2.7%. If we compare the optimal weights for the risk measures CVaR and
mCVaR, we see that the results are almost identical to each other.
Analysis of the Maximum Relative Performance Portfolios
The impact of return estimators
Volatility as the risk measure:
Let us next take a look at the results in Table 10 obtained for the classical Tangency Portfolio
(MRPP) of Tobin: since the objective function of this optimization problem in equation (4)
now depends on the expected returns of the portfolio, here there is an impact of the return
estimators and all the optimal portfolios are different for all 4 kinds of risk measure. If we
choose the volatility as a risk measure and the historical return estimators, the optimal portfo-
lio for the historical estimators consists of bonds (approx. 50%), hedge funds and managed
futures (with their upper limits of 25%) (see Figure 8). Stocks, real estate and private equities
do not appear. These optimal weights result from the reward-to-variability ratios (see Table 4)
and the correlations of the asset classes. The results for the optimal portfolio that are obtained
when we use the Bayes–Stein return estimators also consist of 3 asset classes (bonds, hedge
funds and managed futures), but in this case the share of bonds is much higher (66%) and the
share of managed futures is much lower (approx. 9%). The upper limit of 25% exists only for
hedge funds. The results calculated with the CAPM return estimators attract our attention
next. Compared with the optimal portfolio weights of the other alternative return estimators,
there are no similarities. Stocks are included with their limit of 40%; bonds have a share of
44%. Therefore, the traditional investments represent 84% of the optimal portfolio weights.
The remaining portfolio consists of small shares for hedge funds, real estate and private equi-
ties and almost 10% commodities. In this portfolio, managed futures are not included. The
reason for this strong deviation of the optimal portfolio for CAPM return estimators from the
solutions for all the other return estimators is that, without upper limits for our 7 asset classes,
the optimal portfolio weights would be identical to the market caps in % in Table 6. If the
returns are estimated with the Black–Litterman approach, the share of bonds is almost 82%
and the other asset classes in the MRPP are stocks (approx. 2%), hedge funds (approx. 12%),
managed futures (approx. 3%) and 0.5% commodities (see Figure 9). This portfolio is more
diversified since it consists of 5 of 7 asset classes.
21
VaR as the risk measure:
If we choose the VaR as the risk measure, we obtain approximately identical portfolio
weights as in the case with the volatility as the risk measure for the corresponding return es-
timates.
CVaR as the risk measure:
Similar results are obtained when the CVaR is used as a risk measure. In this case, the optimal
portfolio weights also depend on the return estimators. If the returns are estimated with the
historical or Bayes–Stein approach, the MRPP consists of bonds, hedge funds and managed
futures. Stocks, real estate and private equities never appear. The results of the optimal portfo-
lio for CAPM return estimators show stocks with a share of 40% and bonds with almost 44%.
If the returns are estimated with the Black–Litterman approach, the MRPP consists of a large
share of bonds (more than 82%), a medium share of hedge funds (approx. 11%) and a small
share of managed futures (approx. 4%).
The impact of risk measures
Historical return estimators:
Let us again start with the result of the optimal portfolio weights for the MRPP with the vola-
tility as the risk measure. This portfolio consists of bonds (approx. 49%), and hedge funds and
managed futures have their upper limits with 25%. commodities have a very small share and
stocks, real estate and private equities do not appear. If we change the risk measure from the
volatility to the VaR or CVaR and still use the historical estimators, the optimal portfolio
weights look almost identical to the results that are calculated with the volatility. In this case,
the use of an alternative risk measure has no influence on the portfolio weights.
Bayes–Stein return estimators:
The same result is obtained if we look at the optimal weights for Bayes–Stein return estima-
tors. For all three different risk measures (Volatility, VaR or CVaR), the MRPP consists of
three asset classes (bonds, hedge funds and managed futures) with shares almost independent
of the risk measure. Stocks, real estate, private equities and commodities have no share in the
optimal portfolio.
22
CAPM return estimators:
If we investigate the results for CAPM return estimators, the optimal portfolio weights are
again almost independent of the chosen risk measure: all the solutions are near their market
caps in %.
Black–Litterman return estimators:
If we use the Black–Litterman return estimators, the optimal portfolio weights for all three
risk measures are very similar. Bonds have the largest share in the MRPP, followed by hedge
funds, managed futures and stocks. Real estate and private equities do not appear.
The impact of non-normality of returns
From Table 10, we can also analyze the impact of normally distributed or non-normally dis-
tributed returns of the asset classes for the historical return estimators. Comparing the optimal
weights for the risk measures VaR and mVaR, the influence of the higher moments (skewness
and excess kurtosis) is visible. If the VaR is used as the risk measure, hedge funds and man-
aged futures are included with the upper limit of 25% and bonds with approximately 50%.
Due to considerations of the higher moments (mVaR), the shares of hedge funds and managed
futures are lower and the share of the bonds increases from approximately 50% to 73%. This
effect results from the VaR values of the asset classes (see Table 4) when they are modified
for the skewness and excess kurtosis. On the other hand, real estate appears in the portfolio
with a small share and commodities are excluded. The same results can be shown if mCVaR
instead of CVaR is used for the calculation. Also in this case, hedge funds and managed fu-
tures are limited to 25% without consideration of the higher moments. With higher moments,
the share of bonds increases to 75%, followed by hedge funds and managed futures. There-
fore, in sum, it can be said that higher moments play an important role in the determination of
optimal global portfolios.
Efficient Frontiers
All the efficient frontiers are shown in Figure 7 for the volatility as a risk measure with our
four different return estimators. As a riskless interest rate, the 1-month USD-LIBOR of
April
1, 2009 with the amount of 0.49% p.a. is used. Except for the CAPM estimators, the capital
market lines have a very steep slope, caused by the extremely low riskless interest rate. There-
fore, the MRP and the MRPP are very close.
23
Figures 10 and 11 show the optimal weights of all the asset classes for the efficient portfolios
on the efficient frontier. The horizontal axis shows the amount of the expected portfolio re-
turns with the percentages of the different asset allocations on the vertical axis from the MRP
to the portfolio with the maximum expected returns. From these figures, it is very easily rec-
ognizable how the asset allocation changes with increasing expected portfolio returns.
Final Remarks
We have seen that the international stock markets have suffered recently from several crises.
The comparison of the international stock markets with the MRPPs is given in Figure 12 in
the form of an ex-post analysis. For the period from April 1999 to April 2009, an asset alloca-
tion with traditional and alternative investments had a fundamentally higher performance than
a single investment in stocks, independently of which estimators for the expected returns are
used. If an investor had put 100 US dollars into a mixed portfolio, this investment would al-
ready be worth 155 US dollars after 10 years. In comparison with that, these 100 US dollars
would be worth 81.54 US dollars after an investment in only international stocks, thus about
52% less than with a mixed global portfolio. From this, it can be derived that above all alter-
native investments own high diversification potential. This is caused by the fact that above all
alternative investments show low correlations with traditional investments (see Table 2).
Therefore, they develop over time independently of traditional investments and are more neu-
tral to the international stock markets.
This knowledge also counts in periods of crisis. Just in the times in which international stock
markets have to register enormous losses, e.g., in the dot.com crisis or in the current subprime
crisis, mixed global portfolios turn out to be more resistant to the loss in value and therefore
more resistant to crisis. With a comparison of the performance during and after the dot.com
crisis with stocks and a mixed global portfolio (Figure 13), it becomes particularly clear that a
disposition in a mixed global portfolio is substantially more resistant to the loss in value than
a single asset class. Besides, the calculation method used for the expected returns plays no
role. Historical estimators as well as alternative estimators for the expected returns show an
essentially better performance. Figure 13 shows this effect. Especially during the time of the
dot.com crisis, the difference between a mixed global portfolio and the international stock
markets arises clearly. While stock markets suffered substantial losses (the value depreciated
24
by approximately 50% from May 2000 to October 2002), global portfolios of a good mixture
of traditional and alternative investments remained nearly unaffected. The value of the portfo-
lios from May 2000 to October 2002 has changed marginally. One receives the same result in
the case of the consideration of the latest subprime crisis.
In Figure 14, it is shown that a dis-
position in a well-diversified portfolio is substantially better than a disposition in only one
single asset class. For the time from November 2007 to April 2009, Figure 14 shows that from
the latest financial crisis international stock markets had to suffer strong losses. Since No-
vember 2007, stocks have lost more than 48% of their value. In a globally mixed portfolio,
however, there were practically no losses. It can be noted again that in all periods of crisis
globally diversified portfolios are more resistant and are only marginally influenced by the
depreciations in the international stock markets.
Summary
In this empirical study, we expand a traditional portfolio of stocks and bonds with the most
important alternative investments (hedge funds, managed futures, real estate, private equities
and commodities) and investigate the resulting optimal global asset allocation. For this analy-
sis, we combine various estimators for the expected returns (historical, Bayes–Stein, CAPM
and Black–Litterman) with different frequently used risk measures (Volatility, (m)VaR and
(m)CVaR): the global asset allocation is examined on the basis of global index data for the
period from April 1999 to April 2009. All calculations are from a US-dollar investor’s view.
For each combination of return estimator and risk measure, we analyze two characteristic
portfolios in detail – the portfolio with the lowest risk (MRP) and the portfolio with the high-
est (modified) Sharpe Ratio (MRPP) – and compare the resulting optimal shares of the seven
asset classes. This enables us to analyze the impacts of the chosen return estimator and the
chosen risk measure on the optimal structure of the global portfolio. Moreover, we augment
our study by taking the skewness and the excess kurtosis of the asset’s returns into account.
Furthermore, we investigate the shapes of the efficient frontiers resulting for the volatility as
the risk measure and for all four return estimators.
Finally, we take a look at the two deep declines of global stock prices in the last decade (the
dot.com crisis and subprime crisis) and compare these global stock market shocks with the
performances of the optimal global portfolios. The results of this empirical study unambigu-
25
ously show that investors would be well advised to add some alternative investments to a tra-
ditional global portfolio. This is due to their outperforming reward–risk ratios and to their low
correlations with traditional investments. Therefore, portfolios with the right proportions of
alternative investments are resistant to the developments in traditional markets, particularly in
times of extreme losses in the international stock markets.
26
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29
Figure 1: Performances in times of crisis
-20.71%
0.42% -6.51% 5.63%
-32.33%
-38.82% -29.31%
-45.00%
-40.00%
-35.00%
-30.00%
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
Stocks
Bonds
Hedge Funds
Managed Futures
Real Estate
Private Equities
Commodities
Expected Return in % p.m.
October 2008
30
Figure 2: Performances in times of crisis
-14.60%
-0.12% -0.89% -0.08%
-23.25% -23.17%
-7.16%
-40.00%
-35.00%
-30.00%
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
Stocks
Bonds
Hedge Funds
Managed Futures
Real Estate
Private Equities
Commodities
Expected Return in % p.m.
February 2009
31
Figure 3: Performances in times of crisis
-14.33%
0.31%
-4.23%
1.94%
-25.25%
-38.23%
-15.90%
-40.00%
-35.00%
-30.00%
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
Stocks
Bonds
Hedge Funds
Managed Futures
Real Estate
Private Equities
Commodities
Expected Return in % p.m.
November 2008
32
Figure 4: Ranking according to performance per year
from April until April
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
best
Private Equities Commodities Managed Futures Commodities Private Equities Real Estate Private Equities Real Estate Commodities Managed Futures Hedge Funds
94.03% 23.46% 4.80% 29.33% 40.74% 31.22% 20.19% 35.24% 26.24% 19.69% 3.40%
Commodities Real Estate Hedge Funds Managed Futures Real Estate Private Equities Commodities Private Equities Hedge Funds Bonds Bonds
30.67% 12.43% 4.32% 12.58% 34.14% 20.66% 19.61% 23.56% 11.83% 5.64% -0.20%
Hedge Funds Managed Futures Bonds Bonds Commodities Commodities Real Estate Stocks Managed Futures Hedge Funds Managed Futures
27.13% 9.95% 0.63% 4.12% 29.09% 17.17% 15.32% 18.66% 10.95% -21.16% -8.85%
Stocks Hedge Funds Real Estate Hedge Funds Stocks Stocks Stocks Hedge Funds Stocks Stocks Commodities
25.33% 4.73% -3.89% 3.00% 29.09% 13.52% 10.33% 12.98% 9.14% -51.64% -22.23%
Real Estate Bonds Stocks Real Estate Hedge Funds Hedge Funds Hedge Funds Managed Futures Bonds Commodities Stocks
11.89% -44.60% -18.06% 2.78% 14.36% 9.20% 7.33% 5.51% 0.93% -53.35% -43.99%
Bonds Stocks Commodities Private Equities Managed Futures Managed Futures Managed Futures Commodities Private Equiti es Real Estate Real Estate
-0.67% -13.83% -20.81% -19.52% 10.50% 3.76% 2.41% 3.00% -5.54% -64.86% -96.44%
Managed Futures Private Equities Private Equities Stocks Bonds Bonds Bonds Bonds Real Estate Private Equities Private Equities
worst
-0.81% -44.60% -28.64% -21.74% 3.54% 0.53% -0.85% -1.15% -7.21% -108.00% -99.13%
R an k ing acco rd in g to p e rfo rm a n ce p e r ye a r
33
Figure 5: Performance in US dollars
0
50
100
150
200
250
300
350
400
450
500
550
1
-
May
-
99
1
-
Aug
-
99
1
-
Nov
-
99
1
-
Feb
-
00
1
-
May
-
00
1
-
Aug
-
00
1
-
Nov
-
00
1
-
Feb
-
01
1
-
May
-
01
1
-
Aug
-
01
1
-
Nov
-
01
1
-
Feb
-
02
1
-
May
-
02
1
-
Aug
-
02
1
-
Nov
-
02
1
-
Feb
-
03
1
-
May
-
03
1
-
Aug
-
03
1
-
Nov
-
03
1
-
Feb
-
04
1
-
May
-
04
1
-
Aug
-
04
1
-
Nov
-
04
1
-
Feb
-
05
1
-
May
-
05
1
-
Aug
-
05
1
-
Nov
-
05
1
-
Feb
-
06
1
-
May
-
06
1
-
Aug
-
06
1
-
Nov
-
06
1
-
Feb
-
07
1
-
May
-
07
1
-
Aug
-
07
1
-
Nov
-
07
1
-
Feb
-
08
1
-
May
-
08
1
-
Aug
-
08
1
-
Nov
-
08
1
-
Feb
-
09
1-Apr-99 = 100
Time
Performance in USD
Stocks Bonds Hedge Funds Managed Futures Real Esta te Private Equities Commodities
34
Figure 6: Maximum Drawdown and Time under Water
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0
1000
2000
3000
4000
5000
6000
Nov/98
Mar/99
Jul/99
Nov/99
Mar/00
Jul/00
Nov/00
Mar/01
Jul/01
Nov/01
Mar/02
Jul/02
Nov/02
Mar/03
Jul/03
Nov/03
Mar/04
Jul/04
Nov/04
Mar/05
Jul/05
Nov/05
Mar/06
Jul/06
Nov/06
Mar/07
Jul/07
Nov/07
Mar/08
Jul/08
Nov/08
Stock Price
Time
Time under Water
Maximum
Drawdown
Period of Maximum
Drawdown Period of Recovery
35
Figure 7: Efficient frontier with volatility as a risk measure
MRP-histor.
MRPP-histor.
with historical estimators
with Bayes-Stein estimators
MRPP -BS
with CAPM estimators
MRPP-CAPM
with Black-Litterman
estimators
MRP-CAPM
MRP-BL
MRPP-BL
MRP-BS
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
0.00 2.50 5.00 7.50 10.00 12.50 15.00 17.50 20.00 22.50 25.00
Expected Return in % p.a.
Volatility in % p.a.
With Limits for Portfolio Weights
36
Figure 8: Maximum Relative Performance Portfolio with historical estimators for expected returns
Bonds
49.65%
Hedge Funds
25.00%
Managed
Futures
25.00%
Commodities
0.35%
Expected Return: 4.51% p.a.
Volatility: 3.56% p.a.
37
Figure 9: Maximum Relative Performance Portfolio with Black–Litterman estimators for expected returns
Stocks
2.29%
Bonds
82.08%
Hedge Funds
11.81%
Managed
Futures
3.31%
Commodities
0.50%
Expected Return: 2.07% p.a.
Volatility: 2.88% p.a.
38
Figure 10: Optimal portfolio weights for historical estimators
Bonds
Hedge Funds
Managed Futures
Real Estate
Commodities
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2.59 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 7.62
Portfolio Weights in %
Expected Return in % p.a.
39
Figure 11: Optimal portfolio weights for Black–Litterman estimators
Stocks
Bonds
Hedge Funds
Managed Futures
Real Estate
Private Equities
Commodities
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2.00 3.00 4.00 5.00 5.11
Portfolio Weights in %
Expected Return in % p.a.
40
Figure 12: Review of stocks and Maximum Performance Portfolios – from April 1999 to April 2009
155.51
142.26
109.54
128.11
81.54
60.0
70.0
80.0
90.0
100.0
110.0
120.0
130.0
140.0
150.0
160.0
170.0
1
-
Apr
-
99
1
-
Jul
-
99
1
-
Oct
-
99
1
-
Jan
-
00
1
-
Apr
-
00
1
-
Jul
-
00
1
-
Oct
-
00
1
-
Jan
-
01
1
-
Apr
-
01
1
-
Jul
-
01
1-Oct-01
1
-
Jan
-
02
1-Apr-02
1-Jul-02
1-Oct-02
1-Jan-03
1-Apr-03
1-Jul-03
1-Oct-03
1-Jan-04
1-Apr-04
1-Jul-04
1-Oct-04
1-Jan-05
1-Apr-05
1-Jul-05
1-Oct-05
1-Jan-06
1-Apr-06
1-Jul-06
1-Oct-06
1-Jan-07
1-Apr-07
1-Jul-07
1
-
Oct
-
07
1
-
Jan
-
08
1
-
Apr
-
08
1
-
Jul
-
08
1
-
Oct
-
08
1
-
Jan
-
09
1
-
Apr
-
09
1-April 99 = 100
Stocks & Maximum Relative Performance Portfolios
historical estimators Bayes-Stein estimators CAPM estimators Black-Litterman estimators Stocks
41
Figure 13: Consequences of the dot.com crisis
96.06
99.59
100.42
99.99
54.96
54.0
56.0
58.0
60.0
62.0
64.0
66.0
68.0
70.0
72.0
74.0
76.0
78.0
80.0
82.0
84.0
86.0
88.0
90.0
92.0
94.0
96.0
98.0
100.0
102.0
104.0
106.0
108.0
110.0
1
-
Apr
-
00
1-May-00
1
-
Jun
-
00
1-Jul-00
1-Aug-00
1-Sep-00
1-Oct-00
1
-
Nov
-
00
1-Dec -00
1
-
Jan
-
01
1-Feb-01
1-Mar-01
1-Apr -01
1-May-01
1
-
Jun
-
01
1-Jul-01
1
-
Aug
-
01
1-Sep-01
1-Oct-01
1
-
Nov
-
01
1-Dec -01
1
-
Jan
-
02
1-Feb-02
1
-
Mar
-
02
1-Apr -02
1-May-02
1-Jun-02
1-Jul-02
1
-
Aug
-
02
1-Sep-02
1
-
Oct
-
02
1-April-00 = 100
dot.com crisis
historical estimators Bayes-Stein estimators CAPM estimators Black-Litterman estimators Stocks
42
Figure 14: The subprime crisis
104.00
69.84
101.62
101.39
51.72
45.0
50.0
55.0
60.0
65.0
70.0
75.0
80.0
85.0
90.0
95.0
100.0
105.0
110.0
1
-
Nov
-
07
1-Dec-07
1-Jan-08
1
-
Feb
-
08
1-Mar-08
1
-
Apr
-
08
1-May-08
1-Jun-08
1
-
Jul
-
08
1-Aug-08
1
-
Sep
-
08
1
-
Oct
-
08
1-Nov-08
1
-
Dec
-
08
1-Jan-09
1
-
Feb
-
09
1
-
Mar
-
09
1-Apr-09
1-Nov-07 = 100
subprime crisis
historical estimators Bayes-Stein estimators CAPM estimators Black-Litterman estimators Stocks
43
Table 1: Traditional and alternative investments
Asset Class Database Asset Class Database
Stocks MSCI World Hedge Funds CS/Tremont/Hedge
Funds
Bonds Barclay’s (former
Lehman Brothers) Global
Aggregate Index Managed Futures CISDM CTA
Real Estate FTSE/EPRA Global Real
Estate
Private Equities LPX50
Commodities Jim Rogers Commodity
Index
Investment Universe
Traditional Investments Alternative Investments
44
Table 2: Correlation matrix in USD
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
Stocks
1
Bonds
0.100 1
Hedge Funds
0.599 0.060 1
Managed Futures
-0.102 0.149 0.176 1
Real Estate
0.813 0.158 0.477 -0.022 1
Private Equities
0.868 0.042 0.714 -0.088 0.790 1
Commodities
0.455 0.031 0.550 0.155 0.420 0.521 1
Period from 04/1999 to 04/2009 in USD
45
Table 3: Statistical analysis of the assets
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
Mean in % p.a. -1.65 1.72 6.78 7.73 4.00 -5.54 8.98
Volatility in % p.a. 17.76 2.99 6.73 8.57 22.58 31.12 20.67
Mean in % p.m. -0.14 0.14 0.57 0.64 0.33 -0.46 0.75
Volatility in % p.m. 5.13 0.86 1.94 2.47 6.52 8.98 5.97
Minimum in % p.m. -20.71 -2.11 -6.78 -4.55 -32.33 -38.82 -29.31
Maximum in % p.m. 13.91 6.02 8.18 7.57 18.95 29.34 14.23
Skewness
-0.9184 3.8390 -0.3241 0.3524 -1.9671 -1.2459 -1.4141
Excess Kurtosis
2.3254 22.3765 4.5674 0.2129 7.6928 5.3213 4.7954
T
120 120 120 120 120 120 120
JB Test on normal distribution
43.91 2798.29 106.41 2.71 373.29 172.63 154.97
p-Value*
0.00 0.00 0.00 0.26 0.00 0.00 0.00
normally distributed
No No No Yes No No No
*significance level 95%
Period from 04/1999 to 04/2009 in USD
46
Table 4: Maximum Drawdown, Time under Water, Risk and Performance measures
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
Max. Drawdown in % 80.22 2.63 21.91 9.15 117.70 172.53 94.38
Period in months 16 48 14 5 21 21 8
from 12/1/2007 10/1/2004 12/1/2007 7/1/2003 5/1/2007 7/1/2007 4/1/2008
to 4/1/2009 9/1/2008 1/1/2009 9/1/2004 3/1/2009 3/1/2009 3/1/2009
Under Water - Period in months 69 51 17 19 22 81 19
from 2/1/2000 10/1/2004 12/1/2007 7/1/2003 5/1/2007 2/1/2000 3/1/2001
to 1/1/2006 12/1/2008 4/1/2009 11/1/2005 4/1/2009 12/1/2006 9/1/2002
analyt. VaR -30.86 -3.19 -4.29 -6.36 -33.14 -56.73 -25.02
modified VaR -34.38 2.24 -4.28 -5.44 -40.61 -63.50 -30.55
analyt. CVaR -34.97 -7.88 -20.67 -25.40 -50.57 -58.65 -51.63
modfied CVaR -38.04 -3.78 -20.65 -24.62 -57.10 -64.54 -56.45
SR -0.12 0.41 0.93 0.84 0.16 -0.19 0.41
mSR
analyt.VaR
-0.07
0.33
1.31
1.06
0.10
-0.11
0.33
mSR
mVaR
-0.06
-0.70
1.32
1.22
0.09
-0.09
0.27
mSR
analyt.CVaR
-0.06
0.16
0.30
0.28
0.07
-0.10
0.16
mSR
mCVaR
-0.06 0.32 0.30 0.29 0.06 -0.09 0.15
Period from 04/1999 to 04/2009 in USD
Risk and Performance measures for α = 5 % in % p.a.
47
Table 5: Comparison of the historical and Bayes–Stein return estimators in % p.a.
Expected Return in % p.a.
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
historical estimators
-1.65 1.72 6.78 7.73 4.00 -5.54 8.98
Bayes-Stein estimators
-0.01 2.23 5.59 6.22 3.74 -2.59 7.06
Period from 04/1999 to 04/2009 in USD
48
Table 6: Expected returns with the CAPM estimators
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
Marktkapitalisierung in Mrd. EUR 15.946,62 10.738,13 849,05 132,60 257,58 21,08 3.105,01
Marktkapitalisierung in % 51,36 34,58 2,73 0,43 0,83 0,07 10,00
Beta 1,38 0,50 0,75 0,23 1,03 1,79 0,85
erwartete CAPM-Rendite
in % p.a. 9,35 6,21 6,81 5,19 8,07 10,80 7,41
Zeitraum von 11/1998 bis 10/2008 in EUR
49
Table 7: Expected returns with Black–Litterman estimators on the base to 90% trust of the investor in “Views”
Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
Views in % p.a.
2.79 2.79 2.79 2.79 2.79 2.79 2.79
Implicit Expected Return in % p.a.
5.23 0.66 1.70 0.33 5.52 7.95 3.95
Black-Litterman Return in % p.a. 4.31 1.97 2.16 2.46 4.71 5.91 3.83
Period from 04/1999 to 04/2009 in USD
50
Table 8: Comparison of the expected returns in % p.a.
Expected Return in % p.a. Stocks Bonds Hedge
Funds Managed
Futures Real Estate Private
Equities Commodities
historical estimators
-1.65 1.72 6.78 7.73 4.00 -5.54 8.98
Bayes-Stein estimators
-0.01 2.23 5.59 6.22 3.74 -2.59 7.06
CAPM estimators
5.23 0.66 1.70 0.33 5.52 7.95 3.95
Black-Litterman estimators 4.31 1.97 2.16 2.46 4.71 5.91 3.83
Period from 04/1999 to 04/2009 in USD
51
Table 9: Comparison of Minimum Risk Portfolios
Volatility VaR CVaR mVaR mCVaR
Volatility 2.52 2.78 -2.06 -8.27 0.96 -5.82 0.00 84.35 14.77 0.88 0.00 0.00 0.00
VaR 4.01 3.20 -1.25 -10.60 0.00 -9.55 0.00 57.98 25.00 17.02 0.00 0.00 0.00
CVaR 1.61 3.00 -3.33 -7.80 0.98 -4.41 1.09 97.84 0.00 0.00 0.00 1.07 0.00
mVaR
2.12
3.03
-2.86
-8.36
1.63
-4.85
0.00
91.75
0.00
5.55
2.70
0.00
0.00
mCVaR 1.63 2.99 -3.29 -7.80 1.16 -4.32 2.71 97.29 0.00 0.00 0.00 0.00 0.00
Volatility
2.77
2.78
-1.81
-8.50
0.00
84.35
14.77
0.88
0.00
0.00
0.00
VaR 3.27 2.94 -1.57 -9.32 0.00 70.19 25.00 4.81 0.00 0.00 0.00
CVaR
2.35
2.89
-2.41
-8.32
1.19
94.51
4.31
0.00
0.00
0.00
0.00
Volatility 0.81 2.78 -3.77 -6.54 0.00 84.35 14.77 0.88 0.00 0.00 0.00
VaR 0.85 2.79 -3.75 -6.61 0.00 81.53 18.43 0.04 0.00 0.00 0.00
CVaR 0.77 2.79 -3.81 -6.53 0.00 86.60 11.85 1.55 0.00 0.00 0.00
Volatility
2.00
2.78
-2.57
-7.74
0.00
84.35
14.77
0.88
0.00
0.00
0.00
VaR
2.01
2.78
-2.57
-7.75
0.00
83.37
15.20
1.43
0.00
0.00
0.00
CVaR
2.00
2.78
-2.57
-7.74
0.00
85.13
14.42
0.45
0.00
0.00
0.00
Black-
Litterman
normal
distributed
CAPM
normal
distributed
Managed
Futures
in %
Real Estate
in %
Private
Equities in
%
historical
normal
distributed
non-normal
distributed
Bayes-Stein
normal
distributed
Commodities
in %
normally distributed non-normally distributed
Minimum Risk Portfolio
Traditional Investments Alternative Investments
Estimator Goal Function Expected
Return
in % p.a.
Analytical Risk Measures in % p.a.
Stocks
in %
Bonds
in %
Hedge
Funds
in %
52
Table 10: Comparison of Maximum Relative Performance Portfolio
Volatility VaR CVaR mVaR mCVaR
Volatility 4.51 3.56 -1.34 -11.86 -0.40 -11.06 0.00 49.65 25.00 25.00 0.00 0.00 0.35
VaR 4.51 3.56 -1.34 -11.86 -0.40 -11.06 0.00 49.65 25.00 25.00 0.00 0.00 0.35
CVaR 4.54 3.59 -1.36 -11.95 -0.44 -11.17 0.00 49.25 25.00 25.00 0.00 0.00 0.75
mVaR 3.46 3.01 -1.49 -9.67 0.49 -8.02 0.00 67.33 18.70 12.74 1.22 0.00 0.00
mCVaR 3.08 2.90 -1.69 -9.06 0.95 -6.90 0.00 74.96 13.05 11.32 0.67 0.00 0.00
Volatility 3.43 3.07 -1.63 -9.77 0.00 66.05 25.00 8.95 0.00 0.00 0.00
VaR 3.43 3.07 -1.63 -9.77 0.00 66.05 25.00 8.95 0.00 0.00 0.00
CVaR 3.46 3.11 -1.65 -9.88 0.00 65.21 25.00 9.79 0.00 0.00 0.00
Volatility 3.17 9.75 -12.88 -23.29 40.00 44.01 0.75 0.00 2.12 3.44 9.68
VaR 2.98 9.10 -11.99 -21.76 40.00 43.48 5.32 0.00 0.93 1.32 8.96
CVaR 3.28 10.23 -13.55 -24.39 40.00 40.05 2.84 0.00 2.68 4.04 10.39
Volatility 2.07 2.88 -2.66 -8.01 2.29 82.08 11.81 3.31 0.00 0.00 0.50
VaR 2.07 2.88 -2.67 -8.02 2.34 82.15 11.71 3.37 0.00 0.00 0.43
CVaR 2.09 2.91 -2.70 -8.10 2.86 82.06 10.79 3.70 0.00 0.00 0.58
CAPM
normal
distributed
Black-
Litterman
normal
distributed
normally distributed non-normally distributed
historical
normal
distributed
non-normal
distributed
Bayes-Stein
normal
distributed
Commodities
in %
Maximum Relative Performance Portfolio
Traditional Investments Alternative Investments
Estimator Goal Function Expected
Return
in % p.a.
Analytical Risk Measures in % p.a.
Stocks
in %
Bonds
in %
Hedge
Funds
in %
Managed
Futures
in %
Real Estate
in %
Private
Equities in
%
