OPTIMAL PORTFOLIO SELECTION WITH A SHORTFALL PROBABILITY CONSTRAINT: EVIDENCE FROM ALTERNATIVE DISTRIBUTION FUNCTIONS. Abstract

Transcription

1 The Journal of Financial Research Vol. XXXIII, No. 1 Pages Spring 010 OPTIMAL PORTFOLIO SELECTION WITH A SHORTFALL PROBABILITY CONSTRAINT: EVIDENCE FROM ALTERNATIVE DISTRIBUTION FUNCTIONS Yalcin Akcay and Atakan Yalcin Koc University Abstract We propose a new approach to optimal portfolio selection in a downside risk framework that allocates assets by maximizing expected return subject to a shortfall probability constraint, reflecting the typical desire of a risk-averse investor to limit the maximum likely loss. Our empirical results indicate that the lossaverse portfolio outperforms the widely used mean-variance approach based on the cumulative cash values, geometric mean returns, and average risk-adjusted returns. We also evaluate the relative performance of the loss-averse portfolio with normal, symmetric thin-tailed, symmetric fat-tailed, and skewed fat-tailed return distributions in terms of average return, risk, and average risk-adjusted return. JEL Classification: C13, C, G1 I. Introduction The optimum asset mix in modern portfolio theory is determined by maximizing either the expected excess return of a portfolio per unit of risk in a mean-variance (MV) framework or the expected value of some utility function approximated by the expected return and variance of the portfolio. In both cases, portfolio risk is defined as the variance (or standard deviation) of portfolio returns. Modeling portfolio risk with these traditional volatility measures implies that investors are not allowed to treat the negative and positive tails of the return distribution separately. The standard risk measures determine the volatility of unexpected outcomes under normal market conditions, which corresponds to the normal functioning of financial markets during ordinary periods. However, neither the variance nor the standard deviation can yield an accurate characterization of actual portfolio risk during highly volatile periods (see Bali, Demirtas, and Levy 009). Therefore, the set of MV efficient portfolios We are grateful for valuable suggestions from the editor, Gerald D. Gay, and an anonymous referee that greatly improved the quality of this article. We also thank Turan G. Bali for valuable comments on an earlier version of this article. 77

2 78 The Journal of Financial Research may produce an inefficient strategy for maximizing expected return of the portfolio while minimizing its risk. In this article we propose a new approach to optimal asset allocation in a downside risk framework with a constraint on shortfall probability. We assume that investors have in mind some disaster level of returns and that they behave so as to minimize the probability of disaster. While minimizing the shortfall probability that the return on the portfolio will not fall below some given level, investors are also assumed to maximize expected return of the portfolio. Hence, the new approach allocates financial assets by maximizing expected return of the portfolio subject to a constraint on shortfall probability. There is a significant amount of prior work on safety-first investors minimizing the chance of a disaster, introduced by Roy (195) and Levy and Sarnat (197). A safety-first investor uses a downside risk measure that is a function of value-at-risk (VaR). VaR is defined as the expected maximum loss over a given interval at a given confidence level. For example, if the given period is one day and the given probability is 1%, the VaR measure would be an estimate of the decline in the portfolio value that could occur with a 1% probability over the next trading day. In other words, if the VaR measure is accurate, losses greater than the VaR measure should occur less than 1% of the time. Roy points out that safety-first investors try to reduce the probability and magnitude of extreme losses in their portfolios. 1 In addition, we believe optimal portfolio selection under limited downside risk to be a practical problem. Even if agents are endowed with standard concave utility functions such that to a first-order approximation they would be MV optimizers, practical circumstances such as short-sale and liquidity constraints as well as some loss constraints such as maximum drawdown commonly used by portfolio managers impose restrictions that elicit asymmetric treatment of upside potential and downside risk. Moreover, the MV optimization holds if the multivariate distribution of risky assets follows a normal density. However, as discussed by Bali, Demirtas, and Levy (009), the physical distribution of asset returns is generally skewed and has thicker tails than the normal distribution. Hence, the standard measures of portfolio risk are not appropriate to use in dynamic asset allocation, especially during large market moves. Finally, earlier studies indicate that investors have preference for positive skewness and lower kurtosis. For example, Arditti and Levy (1975) and Harvey and Siddique (000) provide theoretical and empirical evidence that investors prefer to hold stocks with positively skewed return distributions and they are averse to stocks with high kurtosis. Because the shortfall probability is a function of higher order 1 There is a wealth of experimental evidence for loss aversion (e.g., Markowitz 195; Kahneman, Knetsch, and Thaler 1990).

3 Optimal Portfolio Selection 79 moments, the extended asset pricing models introduced by the aforementioned studies suggest significant impact of the shortfall probability (or downside risk) constraint on optimal asset allocation. Because of a wide variety of motivating points discussed previously, we investigate the effects of extreme returns and shortfall probability constraints on optimal asset allocation problem (we refer the reader to Basak and Shapiro 001; Basak, Shapiro, and Tepla 006 for theoretical work on optimal portfolio selection with a constraint on VaR or shortfall return). The shortfall constraint reflects the typical desire of an investor to limit downside risk by putting a probabilistic upper bound on the maximum loss. Put differently, the investor wants to determine an optimal asset allocation for a given level of maximum likely loss (or VaR). Our analysis reveals that the degree of the shortfall probability plays a crucial role in determining the effects of the choice among a symmetric fat-tailed, skewed fattailed, and normal distribution. These effects concern the composition of optimal asset allocations as well as the consequences of misspecification of the degree of skewness and kurtosis for the downside risk measure. We consider VaR bounds and optimal portfolio selection in a unified framework. We concentrate on the interaction between different distributional assumptions made by the portfolio manager and the resulting optimal asset allocation decisions and downside risk measures. We compare the empirical performance of the MV approach with the newly proposed asset allocation model in terms of their power to generate higher cumulative and risk-adjusted returns. We first model the shortfall probability constraint using the normal distribution. Then, we use the symmetric fat-tailed and skewed fat-tailed distributions in modeling the tails of the portfolio return distribution to determine the effects of alternative distributions on optimal asset allocation. II. Asset Allocation Model with Shortfall Constraint We consider a one-period model with n asset classes. At the beginning of the period, the portfolio manager can invest the money available in any of the n asset classes. The manager is not allowed to hold short positions. The objective of the portfolio manager is to maximize the expected return on the portfolio subject to a shortfall constraint. This shortfall constraint states that with a sufficiently high probability 1 θ (with θ being a small number), the return on the portfolio will not fall below the shortfall return R low. Formally, the asset allocation problem is given by: Although some studies exist on the mean-var approach mentioned earlier, our contribution consists of not only proposing a new asset allocation model but also providing an empirical comparison using several different density functions to compute shortfall probabilities.

4 80 The Journal of Financial Research subject to max ω R n n ω i E(R i ) (1) i=1 ( ) n P ω i R i <R low θ () i=1 n ω i = 1, ω i 0, (3) i=1 where ω i and R i,i = 1,,..., n, denote the fraction invested in asset class i and the return on asset class i, respectively. E(R p ) = n i=1 ω i E(R i ) is the expected return of the portfolio, where E(.) is the expectation operator with respect to the probability distribution P of the asset returns. P(R p <R low ) θ in equation () is the constraint on shortfall probability that the portfolio s return, R p = n i=1 ω i R i, will not fall below the shortfall return R low. We can interpret the probabilistic constraint in equation () using the more popular concept of VaR. In effect, the constraint fixes the permitted VaR for feasible portfolios. VaR is the maximum amount that can be lost with a certain confidence level. In the setting of equation () with R low < 0, the VaR per dollar invested is R low with a confidence level of 1 θ. Frequently, the probability distribution P of the asset returns is assumed to be normal or lognormal. Yet the normal distribution is inadequate for describing the probability of extreme returns as usually encountered in practice. Our goal is to investigate the effect of extreme returns (or skewness and fat tails) on the solution of the asset allocation problem in equations (1) and (). Therefore, we need to introduce a class of probability distributions that allow for skewness and fat tails while simultaneously yielding tractable solutions. The class of Student-t distributions meets these requirements. Bollerslev (1987) and Bollerslev and Wooldridge (199) use the standardized Student-t distribution. The symmetric standardized t density with < v < is given by: ( ) v + 1 ( v 1 f (R p ;μ, σ, v) = Ɣ Ɣ [(v )σ ) ] [1 1/ + (R p μ) ] (v+1)/, (v )σ (4) where μ and σ are, respectively, the mean and standard deviation of portfolio returns R p,v is the degrees of freedom parameter governing the tail thickness of the return distribution, and Ɣ(a) = 0 x a 1 e x dx is the gamma function. It is well known

5 Optimal Portfolio Selection 81 that for 1/v 0 the t distribution approaches a normal distribution, but for 1/v >0 the t distribution has fatter tails than the corresponding normal distribution. Hansen (1994) is the first to introduce a generalization of the Student-t distribution where asymmetries may occur, while maintaining the assumption of a zero mean and unit variance. Hansen s skewed-t density is defined as: bc f (z;μ, σ, v, λ) = ( bc ( v v ( ) ) bz + a v+1 1 λ ( ) ) bz + a v λ if z< a/b if z a/b, where z = R p μ is the standardized return, and the constants a, b, and c are given σ by a = 4λc ( ) v, b = 1 + 3λ a, c = v 1 (5) ( ) v + 1 Ɣ ( v (6) π(v )Ɣ ), where μ and σ are, respectively, the mean and standard deviation of portfolio returns R p,v is the the tail thickness parameter, and λ is the skewness parameter. Hansen (1994) shows that the skewed-t density is defined for <v< and 1 < λ < 1. This density has a single mode at a/b, which is of opposite sign with the parameter λ. Thus, if λ > 0, the mode of the density is to the left of zero and the variable is skewed to the right, and vice versa when λ < 0. Furthermore, if λ = 0, Hansen s distribution reduces to the traditional standardized t distribution. If λ = 0 and v =, it reduces to a normal density: f (R p ;μ, σ ) = [ 1 e 1 πσ ( ) ] Rp μ σ (7) where μ and σ are the mean and standard deviation of the normal density function, respectively. We first model the shortfall probability constraint in equation () using the normal distribution. That is, P(R p <R low ) θ is assumed to follow the normal probability density function given in equation (7). Then, we use the symmetric-t and skewed-t distributions in equations (4) and (5) when modeling the tail thickness and asymmetry of the portfolio return distribution to investigate the effects of kurtosis and skewness on optimal asset allocation.

6 8 The Journal of Financial Research III. Data and Estimation We use daily and monthly data on the three-month Treasury bills (T-bills), and the Dow Jones Industrial Average (DJIA), and S&P 500 composite and NASDAQ composite indices in our main empirical analysis. In the Robustness section, we repeat the analysis for a sample of stocks included in the DJIA index. Daily three-month Treasury data as reported in the Federal Reserve H.15 database are obtained from the Wharton Research Data Services (WRDS). Daily returns on stock market indices are obtained from the Thomson Reuters Financial database, whereas the returns on individual stocks are from the Center for Research in Security Prices (CRSP) database. We compute monthly returns by compounding daily returns within a month. Monthly standard deviations and covariances are also computed based on daily returns within a month. In our empirical analysis, we use the common sample period from March 1971 to December 006, yielding a total of 430 monthly observations. Table 1 provides descriptive statistics for the four asset classes. The average monthly returns are about 0.50%, 0.71%, 0.7%, and 0.89% for the three-month T-bill, and DJIA, S&P 500, and NASDAQ indices, respectively, whereas the standard deviations of monthly returns are about 0.4%, 4.35%, 4.8%, and 6.3%. It is clear that the stocks that form the NASDAQ index have a higher average return accompanied by a higher volatility and the three-month T-bill has the highest average risk-adjusted return measured by the ratio of average return to sample standard deviation (.04 = 0.50% 0.4%). The empirical return distribution is significantly TABLE 1. Descriptive Statistics. Statistics 3-Month T-bill DJIA S&P 500 NASDAQ Mean Median St. dev Maximum Minimum Skewness Kurtosis Jarque-Bera Note: Descriptive statistics of the monthly percentage returns on the three-month Treasury bill, Dow Jones Industrial Average (DJIA), S&P 500, and NASDAQ stock market indices over the sample period of March 1971 to December 006 are listed. Standard errors of the skewness and kurtosis statistics are calculated as 6/n and 4/n, respectively, where n denotes the number of observations (n = 430). Jarque-Bera, JB = n[(s /6) + (K 3) /4, is a formal test statistic for testing whether the returns are normally distributed, where S is skewness and K is kurtosis. The JB statistic is distributed as the chi-square with two degrees of freedom. Significant at the 1% level.

7 Optimal Portfolio Selection 83 negatively skewed for any of the stock market indices (DJIA, S&P 500, NASDAQ). The (excess) kurtosis statistics are greater than (zero) three and statistically significant. Both the skewness and the excess kurtosis measures indicate that the return distributions are far from being normal. Note that standard errors of the skewness and kurtosis statistics are calculated as 6/n and 4/n, respectively, where n is the number of observations. The Jarque-Bera statistics are also very large and statistically significant, rejecting the assumption of normality and implying that the distribution of returns has much thicker tails than the normal distribution. 3 In the remainder of this section, we test whether the symmetric fat-tailed or the skewed fat-tailed density provides an accurate characterization of the empirical return distribution for the three-month T-bill, DJIA, S&P 500, and NASDAQ series. First, we use maximum likelihood methodology to estimate the parameters of the normal, symmetric-t, and skewed-t density functions. Then, based on the statistical significance of the skewness and tail thickness parameters as well as the likelihood ratio tests, we evaluate the relative performance of alternative distribution functions in terms of their power to capture the actual distribution of realized returns. The parameters of the skewed-t density are estimated by maximizing the log-likelihood function of R t with respect to the mean, standard deviation, tail thickness, and skewness parameters (μ, σ,v,λ): ( v + 1 LogL Skew-t = n ln b + n ln Ɣ ( v n ln Ɣ n ln σ ) ) n ln π n ln(v ) ( ) v + 1 n t=1 ( ) ln 1 + d t, (v ) where d t = bz t +a (1 λs), and s is a sign dummy taking the value of 1 if bz t + a<0, and 1 otherwise. z t = R t μ is the standardized return, where R σ t is the monthly raw return on the three-month T-bill, DJIA, S&P 500, and NASDAQ. Our results show that the tail thickness parameter (v) of the skewed-t density for the three-month T-bill is about 5.63 with a t-statistic of.87. The skewness parameter (λ) of the skewed-t density for the three-month T-bill is about 0.5 with a t-statistic of 4.1. These results indicate that the empirical return distribution is skewed to the right and thick tailed for the three-month T-bill. On the other hand, the tail thickness parameter of the skewed-t density is in the range of 4.79 to 6.57 and highly significant for the stock indices. Similar to the sample estimates in Table 1, the skewness parameter is negative for all of the stock market indices considered in the article but is significant only for the NASDAQ index. Overall, (8) 3 Jarque-Bera, JB = n[(s /6) + (K 3) /4], is a formal test statistic for testing whether the returns are normally distributed, where n denotes the number of observations, S is skewness, and K is kurtosis. The JB statistic is distributed as the chi-square with two degrees of freedom.

8 84 The Journal of Financial Research the parameter estimates indicate that the return distribution has thicker tails than the normal distribution for all asset classes considered in this article. With regard to skewness, the return distribution of the three-month T-bill (NASDAQ index) is positively (negatively) skewed with statistically significant skewness parameter. Although the distribution of DJIA or S&P 500 is also skewed to the left, the skewness parameter is not statistically significant for either of these two indices. Comparing the maximized log-likelihood values of the skewed-t and normal density functions provide strong evidence based on the likelihood ratio test that the empirical return distribution is not normal; that is, the joint hypothesis of λ = 0 and v = is rejected in favor of the skewed-t for all asset classes considered in the article. 4 We estimate the parameters of the symmetric-t density by maximizing the log-likelihood function of R t with respect to the mean, standard deviation, and tail thickness parameters (μ,σ, v). The standardized-t density in equation (4) gives the following log-likelihood function: ( v + 1 LogL Student-t = n ln Ɣ ( ) v + 1 n ) ( v n ln Ɣ ) n ln[(v )σ ] t=1 ( ln 1 + εt ), (v )σ where ε t = R t μ is the monthly raw return deviated from the mean (μ), and σ is the standard deviation of asset returns. The tail thickness parameter (v) of the standardized-t density for the threemonth T-bill is about 4.33 with a t-statistic of The tail thickness parameters are about 5.59, 6., and 4.60 with the corresponding t-statistics of 3.79, 3.46, and 4.53 for the DJIA, S&P 500, and NASDAQ indices, respectively. Comparing the maximized log-likelihood values of the symmetric-t and normal density functions provide strong evidence based on the likelihood ratio test that the empirical return distribution has thicker tails than the normal distribution; that is, the null hypothesis of v = is rejected in favor of the Student-t for all asset classes considered in the article. The evidence so far shows that the normality assumption contradicts empirical findings on asset returns, which suggests that these return series generally exhibit leptokurtic behavior, that is, have fatter tails than the normal distribution. This means that extreme returns of either sign occur far more often in practice than (9) 4 The likelihood ratio (LR) test statistic ranges from 17 to 44.5 across asset classes. This test statistic is calculated as LR = (LogL LogL), where LogL is the value of the log likelihood under the null hypothesis (i.e., normal), and LogL is the log likelihood under the alternative. The critical values with one and two degrees of freedom at the 1% level of significance are χ (1,0.01) = 6.63 and χ (,0.01) = 9.1, respectively.

9 Optimal Portfolio Selection 85 by predicted by the normal model. 5 Fat tails need not be important if one is interested only in, for example, expected returns. If one is also interested in measures that describe the uncertainty or the risk associated with a given asset allocation strategy, however, the precise shape of the tails of the distribution may matter a great deal. IV. Results: Variance versus Shortfall Constraint with the Normal Distribution This section describes the dynamic asset allocation strategy of two classes of investors: MV versus loss-averse (LA) investor. We should note that both types of investors go through the same input estimation and same rebalancing process from March 1971 through December 006. Although they use the same inputs (i.e., expected returns, variances, and covariances), the optimal portfolio selection of the MV and LA investors differ because the inputs are processed in a different asset allocation framework. For the MV investors, the optimal portfolio weights are determined by maximizing expected return subject to a constraint on the variance of the portfolio. In contrast, the LA investors allocate financial assets by maximizing expected return subject to a constraint on shortfall probability. In this article, we do not allow the MV and LA investors to use any structural model or econometric forecasting model to predict the one-month-ahead conditional mean and variance covariance matrix. The returns, variances, and covariances are assumed to follow a first-order Markov process. Hence, the onemonth-ahead expected return, expected variance, and covariance are proxied by the current month s return and variance covariance matrix. The MV and LA investors first use daily returns within a month and calculate the monthly returns, monthly variances, and covariances for the asset classes considered. For example, daily returns on the three-month T-bill, DJIA, S&P 500, and NASDAQ from March 1, 1971 to March 31, 1971 are used to compute the monthly returns, monthly variances, and covariances. Then, these estimates are used as input to find the optimal portfolio weights for April This optimal portfolio selection process is repeated for each month until December 006, yielding a total of 49 observations for the optimal portfolios realized return and risk. Finally, the relative performance of the MV and LA investors is assessed in terms of the optimal portfolios cumulative and risk-adjusted returns. We display in Figure I the time series of monthly realized raw returns, standard deviation, skewness, and kurtosis for the DJIA index calculated from within-month daily data. Notice that there is significant time-series variation in the 5 As examples, refer to Bali (007) for interest rate markets and Nelson (1991) for stock markets.

10 86 The Journal of Financial Research 0.75% Return 7.00% Standard Deviation 0.50% 6.00% 0.5% 5.00% 0.00% -0.5% -0.50% -0.75% 4.00% 3.00%.00% -1.00% 1.00% -1.5% Mar-71 Jun-74 Sep-77 Dec-80 Mar-84 Jun-87 Sep-90 Dec-93 Mar-97 Jun-00 Sep-03 Dec Skewness -3 Mar-71 Jun-74 Sep-77 Dec-80 Mar-84 Jun-87 Sep-90 Dec-93 Mar-97 Jun-00 Sep-03 Dec % Apr Jul-74 Oct-77 Jan-81 Apr-84 Jul-87 Oct-90 Jan-94 Apr-97 Jul-00 Oct-03 Kurtosis 0 Mar-71 Jun-74 Sep-77 Dec-80 Mar-84 Jun-87 Sep-90 Dec-93 Mar-97 Jun-00 Sep-03 Dec-06 Figure I. Monthly Realized Measures of Volatility, Skewness, and Kurtosis for the Dow Jones Industrial Average Index. higher order moments of the realized return distribution, justifying monthly updates of optimal portfolio weights. To determine the optimum asset mix in a MV framework in month t, the MV investor maximizes the expected return per unit of risk: E(R p,t+1 t)/σ p,t+1 t, where E(R p,t+1 t) = n i=1 ω i,t E(R i,t+1 t) is the expected return of the portfolio for month t+1 given the information set in month t, and σ p,t+1 t = n ωi,t E( ) n σi,t+1 t + i=1 i=1 n j=1,i j ω i,t ω j,t E(σ ij,t+1 t) (10) is the expected standard deviation of the portfolio for month t+1 given the information set in month t. As mentioned earlier, the one-month-ahead expected returns, variances, and covariances are proxied by the current month s realized values, that is, E(R i,t+1 t) R i,t,e(σi,t+1 t ) = σ i,t, and E(σ ij,t+1 t) σ ij,t for all i, j, and t. To find the optimal portfolio weights in month t, the LA investor maximizes the one-month-ahead expected return of the portfolio, E(R p,t+1 t) = n i=1 ω i,t E(R i,t+1 t), subject to a constraint on shortfall probability, P(R p,t+1 <R low ) θ, that the portfolio s return will not fall below the shortfall return R low.asinthe MV framework, one-month lagged returns, variances, and covariances are used in the expected return and shortfall constraint of the LA investor; that is, the same inputs are used in the optimization: E(R i,t+1 t) R i,t,e(σi,t+1 t ) = σ i,t, and E(σ ij,t+1 t) σ ij,t for all i, j, and t.

11 Optimal Portfolio Selection Portfolio Value ($) LA (R low =-%, θ=.5%) LA (R low =-%, θ=5%) LA (R low =-%, θ=10%) MV Mar-71 Sep-74 Apr-78 Nov-81 Jun-85 Jan-89 Aug-9 Mar-96 Oct-99 May-03 Dec-06 Figure II. $100 Investment on the Mean-Variance (MV) and Loss-Averse (LA) Portfolio with R low = %. Because the MV framework implicitly assumes that the portfolio return follows a normal distribution, the shortfall probability constraint in the LA framework is modeled with the normal density as well. We consider three values for the shortfall probability: θ =.5%, θ = 5%, and θ = 10%. We also report the results for three values of the shortfall return: R low = 0%, R low = 1%, and R low = %. To evaluate the relative performance of the MV and LA portfolio selection models, we present the cumulative cash values of $100 initial investment, annualized geometric mean return of the optimal portfolio, and the average riskadjusted return. According to the cumulative cash values displayed in Figure II, if one had invested $100 on the MV portfolio in March 1971, she would have received $1, in December 006. During the same period, if she had invested $100 on the LA portfolio with R low = %, she would have received $3, for θ =.5%, $3,655. for θ = 5%, and $4,08.80 for θ = 10%. Figure II clearly demonstrates that for all months from April 1971 to December 006, the LA framework performs better than the MV model in terms of the cumulative cash values and the compounded annualized return (or the annualized geometric mean return). Figure III plots the cumulative cash values of $100 investment on the LA portfolio with a downside risk constraint of 1% per month. From March 1971 to December 006, $100 investment on the LA portfolio with R low = 1% grows to $3, for θ =.5%, $3, for θ = 5%, and $3,918.5 for θ = 10%. Figure IV presents the cumulative cash values of $100 investment on the LA portfolio with R low = 0%. Over the sample period of March 1971 to December

12 88 The Journal of Financial Research Portfolio Value ($) LA (R low =-1%, θ=.5%) LA (R low =-1%, θ=5%) LA (R low =-1%, θ=10%) MV Mar-71 Sep-74 Apr-78 Nov-81 Jun-85 Jan-89 Aug-9 Mar-96 Oct-99 May-03 Dec-06 Figure III. $100 Investment on the Mean-Variance (MV) and Loss-Averse (LA) Portfolio with R low = 1% Portfolio Value ($) LA (R low =0%, θ=.5%) LA (R low =0%, θ=5%) LA (R low =0%, θ=10%) MV Mar-71 Sep-74 Apr-78 Nov-81 Jun-85 Jan-89 Aug-9 Mar-96 Oct-99 May-03 Dec-06 Figure IV. $100 Investment on the Mean-Variance (MV) and Loss-Averse (LA) Portfolio with R low = 0%. 006, $100 investment on the LA portfolio with R low = 0% grows to $, for θ =.5%, $,81.41 for θ = 5%, and $4, for θ = 10%. These cumulative values are summarized in Panel A of Table, which also compares the relative performance of the MV and LA asset allocation models

13 Optimal Portfolio Selection 89 TABLE. Mean-Variance versus Shortfall Constraint with the Normal Distribution: Growth, Annualized Geometric Mean, and Risk-Adjusted Return. Panel A. Cumulative Cash Values of $100 Investment a R low = % R low = 1% R low = 0% θ =.5% $3, $3, $, θ = 5% $3,655. $3, $,81.41 θ = 10% $4,08.80 $3,918.5 $4, Panel B. Annualized Geometric Mean Return of the Optimal Portfolio b θ =.5% 10.36% 10.15% 9.18% θ = 5% 10.59% 10.87% 9.78% θ = 10% 10.89% 10.81% 11.16% Panel C. Risk-Adjusted Return of the Optimal Portfolio c θ =.5% θ = 5% θ = 10% Note: Panel A presents the cumulative cash values of $100 investment on the optimal loss-averse portfolio with R low = %, 1%, 0%, and θ =.5%, 5%, 10%. Panel B shows the annualized geometric mean return of the optimal loss-averse portfolio. Panel C reports the average risk-adjusted return of the optimal loss-averse portfolio. The sample period is from April 1971 to December 006. Risk of the optimal portfolio is defined in terms of standard deviation. a Cumulative cash value of $100 investment of the optimal mean-variance portfolio = $1, b Annualized geometric mean return of the optimal mean-variance portfolio = 8.09%. c Risk-adjusted return of the optimal mean-variance portfolio = based on the annualized geometric mean return and the average risk-adjusted return of the optimal portfolio. Panel B shows that for each value of the shortfall return (R low = %, 1%, 0%), the annualized geometric mean return of the optimal LA portfolio almost monotonically increases as the shortfall probability rises from θ =.5 toθ = 5% to θ = 10%. The annualized geometric mean return of the optimal LA portfolio is in the range of 9.18% to 11.16% per annum, which is above the annualized geometric mean return of the optimal MV portfolio, 8.09% per annum. Panel C of Table presents the average risk-adjusted return of the optimal MV and LA portfolios. For each level of the shortfall probability (θ =.5%, 5%, 10%), the risk-adjusted return of the optimal LA portfolio increases as the shortfall return increases from R low = % to R low = 1% to R low = 0%. Specifically, the average return-to-standard-deviation ratio is in the range of 0.71 to 0.366, which is above the average risk-adjusted return of the optimal MV portfolio, To determine whether the LA portfolio performs better or worse than the MV portfolio in terms of risk-adjusted return, we calculate the average return, risk, and average risk-adjusted return (i.e., average return-to-risk ratio) of the optimal portfolio over the sample period of April 1971 to December 006. Risk of the

14 90 The Journal of Financial Research TABLE 3. Mean-Variance versus Shortfall Constraint with the Normal Distribution Portfolio Risk: Standard Deviation, Value-at-Risk (VaR), and Expected Shortfall (ES). Panel A. Return, Standard Deviation, and Risk-Adjusted Return Average Return Standard Deviation Risk-Adjusted Return LA (R low = 1%, θ =.5%) 0.84%.48% 0.34 LA (R low = 1%, θ = 5%) 0.90%.79% 0.3 LA (R low = 1%, θ = 10%) 0.91% 3.07% 0.9 Mean-variance 0.69%.88% 0.4 Panel B. Return, 1% VaR, and Risk-Adjusted Return Average Return 1% VaR Risk-Adjusted Return LA (R low = 1%, θ =.5%) 0.84% 4.41% 0.19 LA (R low = 1%, θ = 5%) 0.90% 5.16% 0.18 LA (R low = 1%, θ = 10%) 0.91% 5.83% 0.16 Mean-Variance 0.69% 8.3% 0.08 Panel C. Return, 1% ES, and Risk-Adjusted Return Average Return 1% ES Risk-Adjusted Return LA (R low = 1%, θ =.5%) 0.84% 7.3% 0.1 LA (R low = 1%, θ = 5%) 0.90% 8.% 0.11 LA (R low = 1%, θ = 10%) 0.91% 8.83% 0.10 Mean-variance 0.69% 13.65% 0.05 Note: This table presents the average return, average risk, and average risk-adjusted return (i.e., average return-to-risk ratio) of the optimal portfolio. The results are based on monthly returns over the sample period of April 1971 to December 006. Panels A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% VaR, and 1% ES, respectively. The results are presented for the mean-variance portfolio and the loss-averse portfolio with R low = 1%, and θ =.5%, 5%, and 10%. optimal portfolio is measured in terms of standard deviation, 1% VaR, and 1% expected shortfall (ES). Note that VaR as a risk measure is sometimes criticized for not being subadditive. Because of this, the risk of a portfolio can be larger than the sum of the stand-alone risks of its components when measured by VaR. Moreover, VaR does not take into account the severity of an incurred damage event. To alleviate these deficiencies, Artzner et al. (1999) introduce an alternative downside risk measure called expected shortfall, defined as the conditional expectation of loss (or average loss) given that the loss is beyond the VaR level. Panel A of Table 3 shows that the average return of the optimal MV portfolio is 0.69% per month, which is below the average return of the optimal LA portfolio for different values of shortfall probability. As reported in Panel A, the average return of the optimal portfolio with R low = 1% is 0.84%, 0.90%, and 0.91% per month for θ =.5%, θ = 5%, and θ = 10%, respectively. The standard deviation of the optimal MV portfolio return is.88% per month, whereas the standard deviation of the optimal LA portfolio with R low = 1% is in the range of.48% to 3.07% per month. In terms of

15 Optimal Portfolio Selection 91 risk-adjusted return, the last column of Panel A shows superior performance of the LA asset allocation model over the MV approach. The average return-to-risk ratio of the MV portfolio is 0.4, which is less than the average risk-adjusted return of the LA portfolio, ranging from 0.9 to Panel B of Table 3 shows that the 1% VaR of the optimal MV portfolio is on average 8.3% per month, which is significantly greater than the 1% VaR of the optimal LA portfolio with R low = 1% : 4.41%, 5.16%, and 5.83% per month for θ =.5%, θ = 5%, and θ = 10%, respectively. To be consistent with the standard measures of portfolio risk, we multiply the original VaRs by 1 so that we can report positive risk and risk-adjusted returns in our tables (because VaR is computed based on the negative tail of the return distribution, the original VaR measures are negative). As presented in the last column of Panel B, when the portfolio risk is measured with 1% VaR, the average return-to-var ratio of the optimal LA portfolio with R low = 1% is in the range of 0.16 to On the other hand, the average return-to-var ratio of the optimal MV portfolio is 0.08, indicating that the riskadjusted return of the optimal loss-averse portfolio is at least two times larger than the risk-adjusted return of the optimal MV portfolio. Similar results are obtained when the portfolio risk is measured with the 1% ES. Panel C of Table 3 reports that the 1% ES of the optimal MV portfolio is on average 13.65% per month, which is significantly greater than the 1% ES of the optimal LA portfolio, ranging from 7.3% to 8.83% per month. The last column of Panel C shows that the average return-to-es ratio of the optimal LA portfolio with R low = 1% is in the range of 0.10 to 0.1, whereas the average return-to-es ratio of the optimal MV portfolio is Similar to our findings in Panel B, the risk-adjusted return of the optimal LA portfolio is at least two times larger than the risk-adjusted return of the optimal MV portfolio. These results indicate that the optimal portfolio selection model with shortfall probability constraint outperforms the standard MV approach based on the cumulative cash values, geometric mean returns, and average risk-adjusted returns. The key findings are also robust across different measures of portfolio risk and shortfall probability. V. Results: Shortfall Constraint with the Normal, Student-t, and Skewed-t Distributions We have so far modeled the shortfall probability constraint using the normal distribution. We now use the symmetric-t and skewed-t density in characterizing the tails of the portfolio return distribution to examine the effect of higher order moments on the optimal portfolio selection of an LA investor. When the univariate asset distributions are Student-t and skewed-t, the implied portfolio distributions do not have to follow the multivariate counterparts.

16 9 The Journal of Financial Research TABLE 4. Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with R low = 1%, θ =.5%. Panel A. Return, Standard Deviation, and Risk-Adjusted Return Average Return Standard Deviation Risk-Adjusted Return Normal 0.84%.48% 0.34 Student-t (v = 3) 0.71% 1.39% 0.51 Student-t (v = 4) 0.73% 1.70% 0.43 Skewed-t (v = 3, λ = 0.1) 0.89%.73% 0.33 Skewed-t (v = 4, λ = 0.1) 0.85%.58% 0.33 Panel B. Return, 1% Value-at-Risk (VaR), and Risk-Adjusted Return Average Return 1% VaR Risk-Adjusted Return Normal 0.84% 4.41% 0.19 Student-t (v = 3) 0.71%.67% 0.7 Student-t (v = 4) 0.73% 3.45% 0.1 Skewed-t (v = 3, λ = 0.1) 0.89% 5.16% 0.17 Skewed-t (v = 4, λ = 0.1) 0.85% 5.09% 0.17 Panel C. Return, 1% Expected Shortfall (ES), and Risk-Adjusted Return Average Return 1% ES Risk-Adjusted Return Normal 0.84% 7.3% 0.1 Student-t (v = 3) 0.71% 3.44% 0.1 Student-t (v = 4) 0.73% 4.74% 0.15 Skewed-t (v = 3, λ = 0.1) 0.89% 8.19% 0.11 Skewed-t (v = 4, λ = 0.1) 0.85% 7.89% 0.11 Note: This table presents the average return, average risk, and average risk-adjusted return of the optimal loss-averse portfolio with R low = 1% and θ =.5%. The results are based on monthly returns over the sample period of April 1971 to December 006. Panels A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% VaR, and 1% ES, respectively. The Student-t density is assumed to have a tail thickness parameter of v = 3 or 4. The Skewed-t density is assumed to have a tail thickness parameter of v = 3 or 4, and skewness parameter of λ = 0.1. At an earlier stage of the study, we generated hypothetical portfolios of three-month T-bill, DJIA, S&P 500, and NASDAQ indices, and estimated skewness (λ) and tail thickness (v) parameters of the skewed-t and Student-t density functions using the hypothetical return distributions. The maximum likelihood estimates of v turned out to be in the range of 3 to 4, and the estimates of λ were in the range of 0.08 to 0.1. Hence, in our asset allocation models with the skewed-t and Student-t density, we use the estimated values of v = 3, 4 and λ = In other words, we assume that the portfolio returns follow a Normal, Student-t, or skewed-t distribution instead of assuming that each asset class is distributed as Normal, Student-t, or skewed-t, which may or may not imply the corresponding multivariate distribution. Table 4 compares the relative performance of the thin-tailed Normal, the symmetric fat-tailed t (Student-t), and the skewed fat-tailed t distributions in terms

17 Optimal Portfolio Selection 93 of average return, risk, and average risk-adjusted return for the LA portfolio with R low = 1% and θ =.5%. Risk of the optimal portfolio is measured in terms of standard deviation, 1% VaR, and 1% ES. Panel A of Table 4 shows that the average return of the optimal LA portfolio with the Normal density is 0.84% per month, which is between the performance of the optimal LA portfolios with the Student-t and skewed-t density functions. As reported in Panel A, the average return of the optimal portfolio with the Student-t is in the range of 0.71% to 0.73% per month, whereas the average return of the optimal portfolio with the skewed-t is in the range of 0.85% to 0.89% per month. Thus, the skewed-t density performs better than the Student-t and Normal density in terms of average return of the optimal portfolio. However, in terms of risk, the Student-t density outperforms both the skewed-t and Normal density. Specifically, the standard deviation of the optimal portfolio return is.48% per month for the Normal density, whereas the standard deviation of the optimal portfolio is in the range of 1.39% to 1.70% per month for the Student-t density and.58% to.73% per month for the skewed-t density. In terms of risk-adjusted return, the last column of Panel A shows the superior performance of the Student-t over the skewed-t and Normal distributions. The average return-to-risk ratio ranges from 0.43 to 0.51 for the Student-t density, whereas the average risk-adjusted return is 0.34 for the Normal density and 0.33 for the skewed-t density. Panel B of Table 4 shows that the 1% VaR of the optimal portfolio with the Normal density is 4.41% per month, whereas the 1% VaR of the optimal portfolio is in the range of.67% to 3.45% for the Student-t density and 5.09% to 5.16% per month for the skewed-t density. As displayed in the last column of Panel B, when the portfolio risk is measured in terms of 1% VaR, the average return-to-risk ratio of the optimal portfolio with the Normal density is 0.19, whereas the average return-to-risk ratio of the optimal portfolio with the skewed-t density is In terms of risk-adjusted return, the Student-t density with the average return-to-risk ratio of 0.1 to 0.7 outperforms both the skewed-t and Normal distributions. Qualitative results turn out to be similar when the portfolio risk is measured with the 1% ES. Panel C of Table 4 reports that the 1% ES of the optimal portfolio with the Normal density is 7.3% per month, whereas the 1% ES of the optimal portfolio is in the range of 3.44% to 4.74% per month for the Student-t density and 7.89% to 8.19% per month for the skewed-t density. The last column of Panel C shows that the average return-to-es ratio of the optimal portfolio is 0.1 for the Normal density, 0.11 for the skewed-t density, and between 0.15 and 0.1 for the Student-t density. Similar to our findings from the standard deviation and VaR, the Student-t distribution performs better than the skewed-t and Normal distributions when the risk-adjusted return of the LA portfolio is defined in terms of expected shortfall. As a robustness check, Tables 5 and 6 evaluate the relative performance of the Normal, Student-t, and skewed-t distributions in terms of average return, risk,

18 94 The Journal of Financial Research TABLE 5. Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with R low = 1%, θ = 5%. Panel A. Return, Standard Deviation, and Risk-Adjusted Return Average Return Standard Deviation Risk-Adjusted Return Normal 0.90%.79% 0.3 Student-t (v = 3) 0.77%.06% 0.37 Student-t (v = 4) 0.81%.31% 0.35 Skewed-t (v = 3, λ = 0.1) 0.91%.94% 0.31 Skewed-t (v = 4, λ = 0.1) 0.91%.84% 0.3 Panel B. Return, 1% Value-at-Risk (VaR), and Risk-Adjusted Return Average Return 1% VaR Risk-Adjusted Return Normal 0.90% 5.16% 0.18 Student-t (v = 3) 0.77% 3.55% 0. Student-t (v = 4) 0.81% 3.63% 0. Skewed-t (v = 3, λ = 0.1) 0.91% 5.16% 0.18 Skewed-t (v = 4, λ = 0.1) 0.91% 5.16% 0.18 Panel C. Return, 1% Expected Shortfall (ES), and Risk-Adjusted Return Average Return 1% ES Risk-Adjusted Return Normal 0.90% 8.% 0.11 Student-t (v = 3) 0.77% 6.08% 0.13 Student-t (v = 4) 0.81% 6.45% 0.13 Skewed-t (v = 3, λ = 0.1) 0.91% 8.3% 0.11 Skewed-t (v = 4, λ = 0.1) 0.91% 8.3% 0.11 Note: This table presents the average return, average risk, and average risk-adjusted return of the optimal loss-averse portfolio with R low = 1% and θ = 5%. The results are based on monthly returns over the sample period of April 1971 to December 006. Panels A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% VaR, and 1% ES, respectively. The Student-t density is assumed to have a tail thickness parameter of v = 3 or 4. The Skewed-t density is assumed to have a tail thickness parameter of v = 3 or 4, and skewness parameter of λ = 0.1. and average risk-adjusted return for the LA portfolio with R low = 1%, and θ = 5% or θ = 10%. The results clearly indicate superior performance of the Student-t distribution over the skewed-t and Normal distributions in term of average riskadjusted returns. Overall, we can conclude that the optimal asset allocation model with a shortfall probability constraint provides a more profitable (in terms of cumulative and risk-adjusted returns) trading strategy than the widely used MV approach. This result holds for alternative distribution functions (Normal, Student-t, skewed-t), different measures of portfolio risk (standard deviation, VaR, ES), different levels of shortfall return (R low = 0%, 1%, %), and shortfall probability (θ =.5%, 5%, 10%). While investigating the effects of skewness and kurtosis on the LA portfolio, we observe that incorporating skewness into the asset allocation problem

19 Optimal Portfolio Selection 95 TABLE 6. Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with R low = 1%, θ = 10%. Panel A. Return, Standard Deviation, and Risk-Adjusted Return Average Return Standard Deviation Risk-Adjusted Return Normal 0.91% 3.07% 0.9 Student-t (v = 3) 0.90%.80% 0.3 Student-t (v = 4) 0.91%.87% 0.3 Skewed-t (v = 3, λ = 0.1) 0.91% 3.44% 0.6 Skewed-t (v = 4, λ = 0.1) 0.93% 3.35% 0.8 Panel B. Return, 1% Value-at-Risk (VaR), and Risk-Adjusted Return Average Return 1% VaR Risk-Adjusted Return Normal 0.91% 5.83% 0.16 Student-t (v = 3) 0.90% 5.16% 0.17 Student-t (v = 4) 0.91% 5.16% 0.18 Skewed-t (v = 3, λ = 0.1) 0.91% 8.43% 0.11 Skewed-t (v = 4, λ = 0.1) 0.93% 7.63% 0.1 Panel C. Return, 1% Expected Shortfall (ES), and Risk-Adjusted Return Average Return 1% ES Risk-Adjusted Return Normal 0.91% 8.83% 0.10 Student-t (v = 3) 0.90% 8.% 0.11 Student-t (v = 4) 0.91% 8.3% 0.11 Skewed-t (v = 3, λ = 0.1) 0.91% 9.79% 0.09 Skewed-t (v = 4, λ = 0.1) 0.93% 9.69% 0.10 Note: This table presents the average return, average risk, and average risk-adjusted return of the optimal LA portfolio with R low = 1% and θ = 10%. The results based on monthly returns over the sample period of April 1971 to December 006. Panels A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% VaR, and 1% ES, respectively. The Student-t density is assumed to have a tail thickness parameter of v = 3 or 4. The Skewed-t density is assumed to have a tail thickness parameter of v = 3 or 4, and skewness parameter of λ = 0.1. improves the performance in terms of average return, but it increases the risk of the optimal portfolio as well. Hence, taking skewness of the return distribution into account does not augment profitability of the LA investment strategy. In terms of risk-adjusted return, accounting for tail thickness of the return distribution (or excess kurtosis) enhances performance of the dynamic asset allocation model with a shortfall probability constraint. VI. Robustness Analysis The period covering March 1971 to December 006 is a long interval with large price fluctuations in the stock market and high inflationary cycles. The performance

20 96 The Journal of Financial Research TABLE 7. Mean-Variance versus Shortfall Constraint: Subsample Analysis. MV LA (R low = 1%, θ = 10%) Cash val. of $100 inv. $64.38 $8.93 $56.70 $ $38.5 $30.81 Ann. geo. mean return 8.63% 7.15% 8.17% 1.0% 10.4% 9.67% Avg. return 0.7% 0.63% 0.73% 1.00% 0.88% 0.84% Std. dev. of return.19%.38% 3.80%.68%.84% 3.63% Risk-adjusted return % VaR 4.9% 7.30% 9.37% 4.10% 5.33% 8.5% Risk-adjusted return % ES 9.63% 8.85% 17.7% 4.81% 7.34% 10.88% Risk-adjusted return Note: This table presents the cumulative cash values of $100 initial investment, annualized geometric mean return, average return, risk, and risk-adjusted return (i.e., return-to-risk ratio) of the optimal portfolio. We measure the risk of the optimal portfolio in terms of standard deviation, 1% value-at-risk (VaR) or 1% expected shortfall (ES). The results are based on monthly returns over three different sample periods of roughly equal length covering April 1971 December 198, January 1983 December 1994, and January 1995 December 006. We present the results for the mean-variance (MV) portfolio and the loss-averse (LA) portfolio with R low = 1% and θ = 10%. The shortfall probability constraint in the LA framework is modeled with the normal density. of the LA portfolio selection model over periods with different regimes remains an open question. In addition, our empirical analyses are based on the DJIA, S&P 500 composite and NASDAQ composite indices. Although these are standard asset classes used in optimal portfolio selection models, it would be interesting to implement the newly proposed model using an alternative set of risky assets such as individual stocks. Subsample Analysis As a robustness check, we first divide our original data set into three subsamples roughly of equal length and repeat the empirical analyses of Section IV in each. 6 Table 7 presents results comparing the relative performance of the two asset allocation models based on the cumulative cash values of $100 initial investment, annualized geometric mean return, and alternative measures of risk-adjusted return over periods covering April 1971 December 198, January 1983 December 1994, and January 1995 December 006. The results clearly demonstrate that, 6 Because our earlier findings suggest that using different distribution functions does not make much of a difference in terms of performance evaluation, we only report the results where the shortfall probability constraint in the LA framework is modeled with the normal density. We consider θ = 10% for the shortfall probability and R low = 1% for the shortfall return.