Optimal Portfolios with Traditional and Alternative Investments: An Empirical Investigation. Edwin O. Fischer* Susanne Lind-Braucher** August 2009

Transcription

1 Optimal Portfolios with Traditional and Alternative Investments: An Empirical Investigation Edwin O. Fischer* Susanne Lind-Braucher** August 29 * Professor at the Institute for Finance at the Karl-Franzens-University Graz, Universitätsstrasse 15/G2, A-81 Graz, Austria, edwin.fischer@uni-graz.at ** Research Assistant at the Institute for Finance at the Karl-Franzens-University Graz, Universitätsstrasse 15/G2, A-81 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 introducing 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 obective function for the portfolio optimization and different estimates for the expected return (historical estimates, robust Bayes Stein estimates, CAPM estimates and Black Litterman estimates). Furthermore, the alternative risk measures are additionally modified for the skewness and the kurtosis ((modified) VaR and (modified) CVaR). In this manner, the influences of the higher moments 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 suffered 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 already increased by over 13 percent on average in Europe (JP Morgan Asset Management [27]). 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 28 with -2.71% 2

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 27. Since the middle of 28, 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 28 (see Figure 3). The stock market registered a loss of %, private equities even %, real estate % and commodities -15.9%. The world economy was therefore in an unstable constitution, which continued until February 29. The situation on the international capital markets was alarming, as shown by the negative performances of all investments in February 29 (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. [21, 22]); 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 26, but since 27, 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 performance than stocks, especially in times of crisis. Hitherto published empirical studies, e.g. Kaiser et al. [28]; Kat [27]; Lee and Stevenson [25]; Maurer and Reiner [22]; Schweitzer [28]) show positive effects of portfolio diversification if alternative investments, e.g. hedge funds or real estate, are added to a traditional portfolio of stocks and bonds. In summer 22, Schneeweis et al. published an empirical 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 [23, 24]). In that case, it is necessary for the portfolio optimization to take the skewness and the excess kurtosis into consideration (see also Kat [25]). 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 [22]). The modified VaR allows us to measure 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- 3

4 pared with the Value at Risk (Acerbi and Tasche 22; 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, 199]; Kan and Zhou [27]; Litterman and the Quantitative Resources Group [23]) show that in the portfolio analysis the results are never optimal with parameters uncertainty. However, alternatives 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 return 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 estimator 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 return 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- 4

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, 29. This study is from the view of an investor investing in USD. An extensive analysis in EUR is presented by Fischer and Lind-Braucher [29]. 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 behave towards each other. This is expressed by a correlation coefficient ρ i, 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 dependency exists; in that case, the development is accurately contrary. The correlation coefficients of this study (see Table 2) show a clear result: e.g. stocks and managed futures with a correlation coefficient of.12 show rather independent values. That is also valid for managed futures and private equities (-.88). Also, in that case, the correlation coefficients are negative. Stocks and private equities show a very high positive correlation of.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. Table 3 shows an overview of the returns and volatilities of the examined asset classes. The average returns and volatilities per year and month are calculated for all the alternatives of investment 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 (2.67%) range respectively in the most risky investment classes with annual volatility. Furthermore, higher 5

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. ( P The expected portfolio return E r ) can be shown as a weighted average of the expected returns of the asset classes E ( r ), whereas x depicts the weight proportions N as the number of assets: N E( r ) = E( r ) x. (1) P = 1 The risk of a portfolio conforms to the standard deviation with the covariance σ = ρ σ σ : i i i N N σ ( r ) = σ x x. (2) P = 1 i= 1 i i 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- 6

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 calculated by minimizing the risk without restricting the expected return (Elton et al. [23]): Minimum Variance Portfolio: (3) σ ( r p ) N N = i= 1 = 1 x x σ i i min! s.t. N = 1 x x. = 1 The Portfolio Theory of Markowitz was extended by Tobin [1958] by additionally viewing a riskless investment. The Maximum Relative Performance Portfolio MRPP (Tangency Portfolio) can be calculated by maximizing the Sharpe Ratio (Sharpe [1994]): Maximum Relative Performance Portfolio: (4) SR = s.t. E( r P σ ( r ) r P ) max! E( r P ) = N = 1 E( r ) x σ ( r N = 1 x p x ) = = 1, N N i= 1 = 1 x x σ i i 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 7

8 measure. If alternative risk measures (VaR und CVaR) are used instead of the volatility, only the obective 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 is defined as follows (Favre and Galeano [22]): 1 α = 95%). Formally, the VaR VaR r = F 1 ( ) ( )( α) = z α (5) r 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 [22]): CVaR r ) = E( r r < VaR( r )). (6) ( In the case of normally distributed returns, the VaR and the CVaR are calculated as follows (Favre and Galleano [22]): with Value at Risk: VaR r ) = E( r ) + z σ ( r ) (7) ( α z α = quantile of the standard normal distribution with Conditional Value at Risk: CVaR ( z ) ) = E( r ) + ϕ α σ ( r ) (8) α ( r (.) ϕ = density function of the standard normal distribution. 8

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 [22]): msr msr VaR CVaR E( rp ) r = (9) VaR + r E( rp ) r =. (1) CVaR Two further measures, which are of great interest in risk analysis, are Maximum Drawdown and Time under Water (López de Prado and Peian [24]). Maximum Drawdown is the maximum value deficit that occurs by assessing an investment. By using the term Time under 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. Regarding 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 nonnormally 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 [23]). These risk measures can be calculated by (Favre and Galeano [22]): modified VaR: mvar r ) = E( r ) + z σ ( r ) (11) ( CF with modified CVaR: ϕ (.) N (.) ϕ( z CF ) mcvar( r ) = E( r) + σ ( r ) N( zcf ) = density function of the standard normal distribution = distribution function of the standard normal distribution (12) 9

10 and with the Cornish Fisher expansion: ( z 1) S + ( z 3z ) K ( 2 z 5z ) S. z z 1 2 CF = α α α α α α Additionally, modified performance measures can be defined on the basis of the Cornish Fisher expansion (Gregoriou and Gueyie [23]): msr msr mvar mcvar E( rp ) r = (13) mvar + r E( rp ) r =. (14) mcvar 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 α = 5%. As a riskless interest rate, the 1-month USD-LIBOR of April 1, 29 with the amount of.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 theories, 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 historical 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 optimization. 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 1

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]): E( r ) BS. hist = (1 w) E( r ) + w E( r ). (15) MVP r ( MVP hist E ( ) stands for the historical return and E r ) for the return of the MVP without shortsale 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 calculated, 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 [199]): 1 Σ is the inverse of the variance covariance matrix (Jorion w = N + 2 ( ) ( ) 1 N E( r ) E( r )1 TΣ ( E( r ) E( r )1) MVP MVP (16) where Σ = σ i. Since the variance covariance matrix Σ is not known in practice, it is replaced by (Kan and Zhou [27]): ˆ T 1 Σ = σ i. (17) T N 2 For the MVP and for the Shrinkage Factor, the following values are calculated: E( r MVP ) = w = 3.25 %.33. p. a. 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 11

12 factor w has on the calculation of the expected returns: high historical returns, e.g. commodities (8.98%), managed futures (7.73%), hedge funds (6.78%) and real estate (4.%), 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]): [ E( r 4 M r 2 ] 4 CAPM E( r ) = r + ) β 4 3 (18) 1 risk premium with E ( r M ) = expected capital market return E( r M ) r = market risk premium σ β = Beta Factor = ρ M. σ M For the calculation of expected returns according to the CAPM, the expected return E r ) and the risk σ M 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, 29 and are based on Thomson Financial Datastream and on the homepage of the used hedge funds index (see The value of commodities (1%) was calibrated following Idzorek [26]. 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 expected capital market returns, we use the following value: ( M E( r M ) = 2.79% p. a. 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 [26]). 12

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 portfolios. The idea of these estimators is that the expected returns of the market equilibrium are connected with individual subective 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 actual 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 implizit the market equilibrium E ( ) serve as neutral reference returns, which are combined with r subective return views for the Black Litterman estimators. The advantage of the Black Litterman estimators is that neutral return expectations are combined with subective return forecasts. Thereby, a consistent adaption of the return expectation to subective market evaluations 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 [25]): p P = M p 1,1 k,1 L O L p 1, n p M k, n. (19) For our study with seven asset classes, the forecasts matrix can be depicted as follows: 13

14 = P Next, it can be considered how much trust an investor gives to his subective returns. This is achieved with the variance covariance matrix Ω expressing the confidence in the views. In this study, it is assumed to be 9%. Simplifying, it is assumed in the Black Litterman estimators 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 returns 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.1 (= T 1, T is the sample length in years, Rachev et al. [28]). The variance covariance matrix expressing the confidence in the views can be shown as follows (Idzorek [25]): ( ) ( ). 1 1 Σ Σ Ω = τ τ p k p k p p O (2) The variance covariance matrix expressing the confidence for this study can be depicted as follows: = Ω

15 The Black Litterman return estimators are a confidence-weighted linear combination of market 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. [26]): E( r BL i views views implizit implizit ) = w E( r ) + w E( r ). (21) i i The two weighting matrices are given by (Fabozzi et al. [26]): w views i = [( τσ) 1 + P Ω 1 P] 1 P Ω 1 P (22) i = [( τσ) + P Ω P] ( τσ), (23) w implizit where views + implizit i w i w =1. views implizit If the investor has full trust in his subective Views, thus w = 1 and w =, the expected returns of Black Litterman correspond exactly to the views. If the investor has no trust views implizit in his subective Views, w = and w = 1, one receives exactly the expected returns i of the CAPM for the expected returns with the Black Litterman estimators. i i i For this empirical study, the following values for views w i and implizit w i can be calculated: views w i =

16 implizit w i = 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 9% trust of the investor in his views. All the described alternative estimators for the expected returns can be used in the asset allocation and compared with the historical estimators. Therefore, Table 8 summarizes the expected 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 allocation. In addition, we raise the question of which asset class in which circumstance is contained 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 allocation for each asset class, which are entered as additional constraints directly into the optimization models. The reason is that investors with a low risk appetite invest in a more securityoriented manner than investors with a higher risk appetite. Therefore, investors preferences can be taken into consideration with an efficient asset allocation. 16

17 Traditional Investments: Alternative Investments: stocks max. 4% hedge funds max. 25% bonds unlimited managed futures max. 25% real estate max. 4% private equities max. 4% commodities max. 4% 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 1 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 expected returns and the analytical risk measures of the optimal portfolios as well as the portfolio weights can be calculated. These tables therefore combine the results for different return estimators with the different risk measures, which are used as an obective 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 values 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- 17

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 7%, 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 estimated 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 8%), 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%, respectively) 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 almost 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 18

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 6%, 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 consists only of these 3 asset classes. For the CVaR, only bonds have a small CVaR value in Table 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 estimators for the 3 different risk measures: for the VaR, the bonds share decreases to approx. 7%, 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 skewness and excess kurtosis. The share of hedge funds decreases from 25% to and the share of managed futures decreases from 17% to almost 6%. On the other hand, real estate appear in 19

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 1 obtained for the classical Tangency Portfolio (MRPP) of Tobin: since the obective 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 portfolio for the historical estimators consists of bonds (approx. 5%), 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 4%; 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 equities and almost 1% 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.5% commodities (see Figure 9). This portfolio is more diversified since it consists of 5 of 7 asset classes. 2

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 estimates. 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 portfolio for CAPM return estimators show stocks with a share of 4% 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 volatility 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 estimators. 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. 21

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 1, we can also analyze the impact of normally distributed or non-normally distributed 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 managed futures are included with the upper limit of 25% and bonds with approximately 5%. 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 5% 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 futures 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. Therefore, 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, 29 with the amount of.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. Therefore, the MRP and the MRPP are very close. 22

23 Figures 1 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 returns 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 recognizable 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 29, an asset allocation 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 1 US dollars into a mixed portfolio, this investment would already be worth 155 US dollars after 1 years. In comparison with that, these 1 US dollars would be worth 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 alternative 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 neutral 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 23

24 by approximately 5% from May 2 to October 22), global portfolios of a good mixture of traditional and alternative investments remained nearly unaffected. The value of the portfolios from May 2 to October 22 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 disposition in a well-diversified portfolio is substantially better than a disposition in only one single asset class. For the time from November 27 to April 29, Figure 14 shows that from the latest financial crisis international stock markets had to suffer strong losses. Since November 27, 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 analysis, 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 29. 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 highest (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- 24

25 ously show that investors would be well advised to add some alternative investments to a traditional 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. 25

26 References Acerbi, C., and D. Tasche. Expected shortfall: a natural coherent alternative to Value at Risk. Economic Notes, Vol. 31, No. 2 (22), pp Artzner, P., F. Delbaen, J-M. Eber, and David Heath. Coherent measures of risk. Mathematical Finance, Vol. 9, No. 3 (1999), pp Bera, A.K., and C.M. Jarque. A test for normality of observations and regression residuals. International Statistical Review, 55 (1987), pp Black, F., and R. Litterman. Global portfolio optimization. Financial Analysts Journal, September/Oktober (1992), pp Cornish, E.A., and R.A. Fisher. Moments and cumulates in the specification of distribution. Revue of the International Statistical Institute, Vol. 5, No. 4 (1938), pp Damodaran, A. Damodaran on Valuation: Security Analysis for Investment and Corporate Finance. 2 nd edition, Hoboken: John Wiley & Sons, 26. Fabozzi, FJ., S.M. Focardi, and P.N. Kolm. Financial Modelling of the Equity Market, From CAPM to Cointegration. Hoboken: John Wiley & Sons, 26. Elton, E.J., and Martin J. Gruber. Modern portfolio theory and investment analysis. 6 th edition, New York: John Wiley & Son, 23. Favre, L, and J-A. Galeano. Mean-modified value-at-risk optimization with hedge funds. Journal of Alternative Investments (22), pp Fischer, E.O., and S. Lind-Braucher, Effiziente Asset Allocation im Globalen Portfolio Management. Finanz Betrieb 9 (29), pp Gregoriou, G.N., and J-P. Gueyie. Risk-adusted performance of funds of hedge funds using a modified Sharpe Ratio. The Journal of Alternative Investments, Vol. 6, No. 3 (23), pp He, G., and R. Litterman. The Intuition behind Black-Litterman Model Portfolios. Investment Management Research, Goldman Sachs Quantitative Resources Group working paper, Idzorek, T.M. A Step-by-Step Guide to the Black-Litterman-Model: Incorporation User- Specified Confidence Level. Zephyr Associates, Zephyr Cove, Working paper (24). -. Strategic Asset Allocation and commodities. Ibbotson Association, Chicago (26). Jorion, P. Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21, No. 3 (1986), pp

27 -. Bayesian and CAPM estimators of the means: implications for portfolio selection. Journal of Banking and Finance, 15 (199), pp Portfolio optimization in practice. Financial Analysts Journal (1992), pp JPMorgan Asset Management: The Alternative Asset Survey 27, Are Alternatives Mainstream? London, 27. Kaiser, D.G., D. Schweizer, and L. Wu. Strategic hedge fund Portfolio Construction that Incorporates Higher Moments. Unpublished Work (28). Kan, R, and G. Zhou, Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, Vol. 42, No. 3 (27), pp Kat, H.M. 1 things investors should know about hedge funds. Journals of Private Wealth Management, Spring (23), pp Managed futures and hedge funds: a match made in heaven. Journal of Investment Management, Vol. 2, No. 1 (24), pp Integrating hedge funds into the traditional portfolio. Journals of Private Wealth Management, Spring (25), pp How to evaluate a new diversifier with 1 simple questions. Journals of Private Wealth Management, Spring (27), pp Lee, St., and S. Stevenson, The case for REITs in the mixed-asset-portfolio in the short and long-run. Journal of Real Estate Portfolio Management, Vol. 11, No. 1 (25), pp Litterman, R. and the Quantitative Resources Group, Goldman Sachs Asset Management: Modern Investment Management. An Equilibrium Approach. New Jersey: John Wiley & Sons, 23. López de Prado, M.M., and A. Peia. Measuring loss potential of hedge fund strategies. Journal of Alternative Investments, (24), pp Markowitz, H.M. Portfolio selection. Journal of Finance, Vol. 7, 1 (1952), pp Maurer, R., and F. Reiner. International asset allocation with real estate securities in a shortfall risk framework: The viewpoint of German and U.S. investors. Journal of Real Estate Portfolio Management, Vol. 8, No. 1 (22), pp Rachev, S.T., J.S.J. Hsu, B.S. Bagasheva, and F.J. Fabozzi. Bayesian Methods in Finance. Hoboken: John Wiley, 28. Rockafellar, R.T., and St. Uryasev. Optimization of conditional value-at-risk. Journal of Risk (1999), pp