Download Optimal Portfolio Selection with a Shortfall Probability Constraint

Optimal Portfolio Selection with a Shortfall
Probability Constraint: Evidence from Alternative
Distribution Functions
Yalcin Akcay
College of Administrative Sciences and Economics, Koc University, Rumeli Feneri Yolu, Istanbul, Turkey, 34450,
[email protected]
Turan G. Bali*
Department of Economics and Finance, Zicklin School of Business, Baruch College, One Bernard Baruch Way, Box 10-225,
New York, NY 10010, turan [email protected]
Atakan Yalcin
College of Administrative Sciences and Economics, Koc University, Rumeli Feneri Yolu, Istanbul, Turkey, 34450,
[email protected]
This paper proposes a new approach to optimal portfolio selection in a downside risk framework that allocates
assets by maximizing expected return of a portfolio subject to a constraint on shortfall probability. The
shortfall constraint reflects the typical desire of a loss-averse investor to limit downside risk by putting a
probabilistic upper bound on the maximum likely loss. The paper compares the empirical performance of the
mean-variance approach with the newly proposed asset allocation model in terms of their power to generate
higher cumulative and risk-adjusted returns. The results indicate that the loss-averse portfolio with normal
density outperforms the mean-variance approach based on the cumulative cash values, geometric mean
returns, and average risk-adjusted returns. The relative performance of the symmetric thin-tailed, symmetric
fat-tailed, and skewed fat-tailed distributions are evaluated in terms of average return, average risk, and
average risk-adjusted return for the loss-averse portfolio with a constraint on the maximum expected loss.
The asset allocation model with a downside risk constraint generates a more profitable trading strategy than
the widely used mean-variance approach. This finding holds for alternative distribution functions (Normal,
Student-t, Skewed-t), different measures of portfolio risk (standard deviation, VaR, Expected Shortfall),
different levels of shortfall return (Rlow = 0%, –1%, –2%), and shortfall probability (θ = 2.5%, 5%, 10%).
Key words : finance, risk analysis, risk management, optimal asset allocation, optimal portfolio selection.
* Corresponding author
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1.
Introduction
The modern theory of portfolio choice determines the optimum asset mix by maximizing (1) the
expected excess return of a portfolio per unit of risk in a mean-variance framework or (2) the
expected value of some utility function approximated by the expected return and variance of the
portfolio. In both cases, market risk of the portfolio is defined in terms of the variance (or standard
deviation) of portfolio returns. Modeling portfolio risk with the traditional volatility measures
implies that investors are concerned only about the average variation (and co-variation) of asset
returns, and they 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.1 Therefore, the set of meanvariance efficient portfolios may produce an inefficient strategy for maximizing expected return of
the portfolio while minimizing its risk.
This paper proposes 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 disaster level of
returns, 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 are several reasons why we consider downside risk and shortfall constraints in optimal
portfolio selection. First, there is an extensive literature on safety-first investors who minimize the
chance of disaster, introduced by Roy (1952), Telser (1955), Baumol (1963), and Levy and Sarnat
(1972). Safety-first investor uses a downside risk measure which is a function of Value-at-Risk
(VaR).2 Roy (1952) indicates that most investors are principally concerned with avoiding a possible
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2
See Longin (2000), McNeil and Frey (2000), Neftci (2000), and Bali (2003).
Value at risk (VaR) is defined as the expected maximum loss over a given time interval at a given confidence level.
For example, if the given period of time is one day and the given probability is 1%, the VaR measure would be an
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disaster and that the principle of safety plays a crucial role in the decision-making process. In other
words, the idea of a disaster exists and a risk averse safety-first investor will seek to reduce the
chance of such a catastrophe occurring as far as possible.3
Second, 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 firstorder approximation they would be mean-variance 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 often impose restrictions that elicit asymmetric treatment
of upside potential and downside risk.
Third, the mean-variance portfolio theory developed by Markowitz (1952a, 1959) critically relies
on two assumptions. Either the investors have a quadratic utility or the asset returns are jointly
normally distributed (see Levy and Markowitz (1979), Chamberlain (1983) and Berk (1997)). Both
assumptions are not required, just one or the other: (i) If an investor has quadratic preferences, she
cares only about the mean and variance of returns; and the skewness and kurtosis of returns have
no effect on expected utility, i.e., she will not care, for example, about extreme losses. (ii) Meanvariance optimization can be justified if the asset returns are jointly normally distributed since the
mean and variance will completely describe the distribution. However, the empirical distribution
of asset returns is typically skewed, peaked around the mode, and has fat-tails, implying that
extreme events occur much more frequently than predicted by the normal distribution. Therefore,
the traditional measures of market risk (e.g., variance or standard deviation) are not appropriate
to approximate the maximum likely loss that a firm can expect to lose, especially under highly
volatile periods.
Fourth, financial institutions hold portfolios of wide variety of assets. Each institution needs to
quantify the amount of risk its portfolio may incur in a certain time period. For example, a bank
needs to assess its potential losses in order to put aside enough capital to cover them. Similarly,
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.
3
There is a wealth of experimental evidence for loss aversion (see Markowitz (1952b) and Kahneman et al. (1990)).
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a company needs to track the value of its assets and any cash flows resulting from losses on its
portfolio. In addition, credit-rating and regulatory agencies must be able to assess likely losses
on portfolios as well, since they need to set capital requirements and issue credit ratings. These
institutions can judge the likelihood and magnitude of potential losses on their portfolios using
value at risk. Regulatory concerns require financial institutions to report a single number, the socalled VaR, which measures the maximum loss on their trading portfolio if the lowest 1% quantile
return would materialize. Capital adequacy is judged on the basis of the size of this expected loss.
Finally, there is ample evidence for investors’ preference for positive skeweness and lower kurtosis.
Arditti (1967), Arditti and Levy (1975) and Kraus and Litzenberger (1976) extend the meanvariance portfolio theory to incorporate the effect of skewness on valuation. Harvey and Siddique
(2000) present an asset-pricing model with conditional co-skewness, where risk-averse investors
prefer positively skewed assets to negatively skewed assets. Dittmar (2002) introduces a fourmoment asset pricing model in which investors with decreasing absolute prudence dislike co-kurtosis
(see Pratt and Zeckhauser (1987) and Kimball (1993)). His findings suggest that investors are
averse to kurtosis, and prefer stocks with lower probability mass in the tails of the distribution to
stocks with higher probability mass in the tails of the distribution. Since the magnitude of downside
risk becomes larger for negatively skewed and thicker-tailed asset distributions, the findings of the
three-moment and four-moment asset-pricing models suggest significant impact of the shortfall
probability (or downside risk) constraint on optimal asset allocation.
Due to a wide variety of motivating points discussed above, we investigate the effects of extreme
returns and shortfall probability constraints on optimal asset allocation problem.4 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 fat-tailed, and normal distribution. These effects concern
4
The interested reader may wish to consult with Basak and Shapiro (2001), Basak et al. (2006), and Basak et al.
(2007) for theoretical work on optimal portfolio selection with a constraint on value-at-risk or shortfall return.
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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, on the one hand, different distributional assumptions made by
the portfolio manager and, on the other hand, the resulting optimal asset allocation decisions and
downside risk measures. We compare the empirical performance of the mean-variance 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 in order to determine the effects of alternative
distributions on optimal asset allocation.5
This paper is organized as follows. Section 2 provides an asset allocation model with a shortfall
probability constraint. Section 3 describes the data and estimation procedure. Section 4 presents
results from the mean-variance and loss-averse portfolio selection models with the normal distribution. Section 5 compares the relative performance of the symmetric fat-tailed, skewed fat-tailed,
and normal distributions in terms of their power to generate higher cumulative and risk-adjusted
returns. Section 6 concludes the paper.
2.
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 Rlow . Formally, the asset allocation problem is given by:
5
While there exists some studies 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.
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maxn
ω∈
subject to
P
n
i=1
n
n
ωi E(Ri )
(1)
i=1
ωi Ri < Rlow
ωi = 1,
≤θ
(2)
ωi ≥ 0
i=1
where ωi and Ri , i = 1, 2, . . . , n, denote the fraction invested in asset class i and the return on
n
asset class i, respectively. E(Rp ) =
ωi E(Ri ) is the expected return of the portfolio, where E(.)
i=1
is the expectation operator with respect to the probability distribution P of the asset returns.
P (Rp < Rlow ) ≤ θ in equation (2) is the constraint on shortfall probability that the portfolio’s
n
return, Rp =
ωi Ri , will not fall below the shortfall return Rlow .
i=1
We can interpret the probabilistic constraint in equation (2) using the more popular concept of
value at risk. In effect, the constraint fixes the permitted VaR for feasible portfolios. Value at risk
is the maximum amount that can be lost with a certain confidence level. In the setting of equation
(2) with Rlow < 0, the VaR per dollar invested is −Rlow 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)-(2). 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 (1992) use the standardized Student-t distribution. The symmetric standardized t density with 2 < v < ∞ is given by:
−1
−(v+1)/2
v+1
v
(Rp − µ)
2 −1/2
Γ
(v − 2)σ
1+
f (Rp ; µ, σ, v) = Γ
2
2
(v − 2)σ 2
(3)
where µ and σ are, respectively, the mean and standard deviation of portfolio returns Rp , v is the
degrees of freedom parameter governing the tail-thickness of the return distribution, and Γ(a) =
∞ a−1 −x
x e dx is the gamma function. It is well known that for 1/v → 0 the t distribution approaches
0
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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:
⎧ 2 − v+1
2
⎪
⎪
1
bz+a
⎪
if z < −a/b
⎨ bc 1 + v−2 1−λ
f (z; µ, σ, v, λ) =
2 − v+1
2
⎪
⎪
1
bz+a
⎪
⎩ bc 1 + v−2 1+λ
if z ≥ −a/b
where z =
Rp −µ
σ
(4)
is the standardized return, and the constants a, b, and c are given by
v−2
a = 4λc
,
v−1
2
2
2
b = 1 + 3λ − a ,
Γ
v+1 2
c= π(v − 2)Γ v2
where µ and σ are, respectively, the mean and standard deviation of portfolio returns Rp , v is the
the tail-thickness parameter, and λ is the skewness parameter.
Hansen (1994) shows that the skewed t density is defined for 2 < 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:
1
f (Rp ; µ, σ) = √
e
2πσ 2
−1
2
R −µ 2 p
σ
,
(5)
where µ and σ are the mean and standard deviation parameter of the normal density function,
respectively.
We first model the shortfall probability constraint in equation (2) using the normal distribution.
That is, P (Rp < Rlow ) ≤ θ is assumed to follow the normal probability density function given in
equation (5). Then, we use the symmetric t and skewed t distributions in equations (3) and (4)
when modeling the tail-thickness and asymmetry of the portfolio return distribution in order to
investigate the effects of kurtosis and skewness on optimal asset allocation.
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3.
Data and Estimation
We use daily and monthly data on the 3-month Treasury bills, and the Dow Jones Industrial
Average (DJIA) , S&P500 composite and NASDAQ composite indices. Daily 3-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
Financial 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
2006, yielding a total of 430 monthly observations.
Table 1 provides descriptive statistics for the four asset classes. The average monthly return is
about 0.50%, 0.71%, 0.72%, and 0.89% for the 3-month T-bill, DJIA, S&P500, and NASDAQ,
respectively. The standard deviations of monthly returns are about 0.24%, 4.35%, 4.28%, and 6.32%
for the 3-month T-bill, DJIA, S&P500, and NASDAQ, respectively. It is clear that the stocks that
form the NASDAQ index have a higher average return accompanied by a higher volatility and the
3-month T-bill has the highest average risk-adjusted return measured by the ratio of average return
to sample standard deviation (2.04=0.50%÷0.24%). The empirical return distribution is negatively
skewed for the stock market indices (DJIA, S&P500, NASDAQ) and the skewness statistic for
NASDAQ is highly significant, both economically and statistically. 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.6 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.7
In this section, we test whether the symmetric thin-tailed, the symmetric fat-tailed, or the
skewed fat-tailed density provides an accurate characterization of the empirical return distribution
for the 3-month T-bill, DJIA, S&P500, and NASDAQ series. First, we use maximum likelihood
6
Standard errors of the skewness and kurtosis statistics are calculated as
the number of observations.
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6/n and
24/n, respectively, where n is
Jarque-Bera, JB = n[(S 2 /6) + (K–3)2 /24], 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.
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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 Rt with respect to the mean, standard deviation, tail-thickness, and skewness parameters (µ, σ,
v, λ):
n
v v+1 v+1
d2t
n
−n ln σ −
LogLSkew−t = n ln b+n ln Γ
ln 1 +
− ln π −n ln Γ(v −2)−n ln Γ
2
2
2
2
(v − 2)
t=1
where dt =
zt =
Rt −µ
σ
bzt +a
,
(1−λs)
and s is a sign dummy taking the value of 1 if bzt + a < 0 and s = –1 otherwise.
is the standardized return, where Rt is the monthly raw return on the 3-month T-bill,
DJIA, S&P500, and NASDAQ.
Table 2 presents the parameter estimates, the t-statistics in parentheses, the maximized loglikelihood values, and the likelihood ratio (LR) test statistics in the last column.8 As shown in Panel
A, the tail-thickness parameter (v) of the skewed t density for the 3-month T-bill is about 5.57 with
a t-statistic of 2.14. The skewness parameter (λ) of the skewed t density for the 3-month Treasury
is about 0.24 with a t-statistic of 2.73. These results indicate that the empirical return distribution
is skewed to the right and thick-tailed for the 3-month Treasury bill. As presented in Table 2,
the tail-thickness parameter of the skewed t density is in the range of 4.79 to 6.56 and highly
significant for the DJIA, S&P500, and NASDAQ indices. The skewness parameter is negative for
all stock market indices considered in the paper, but it is significant only for the NASDAQ index.
Specifically, the estimated value of λ for the DJIA, S&P500, and NASDAQ is –0.0623, –0.0986,
and –0.1347 with the t-statistics of –0.88, –1.42, and –1.99, respectively. Overall, the parameter
estimates indicate that the return distribution has thicker tails than the normal distribution for
all asset classes considered in the paper. With regard to skewness, the return distribution of the
3-month T-bill (NASDAQ index) is positively (negatively) skewed with statistically significant
skewness parameter. Although the distribution of DJIA and S&P500 is also skewed to the left, the
8
The likelihood ratio (LR) test statistic is calculated as LR = –2(LogL* – LogL), where LogL* is the value of the
log likelihood under the null hypothesis, 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 χ2(1,0.01) = 6.63 and χ2(2,0.01) = 9.21, respectively.
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skewness parameter is not statistically significant for the DJIA and S&P500 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,
i.e., the joint hypothesis of λ = 0 and v = ∞ is rejected in favor of the skewed-t for all asset classes
considered in the paper.
The parameters of the symmetric t density are estimated by maximizing the log-likelihood function of Rt with respect to the mean, standard deviation, and tail-thickness parameters (µ, σ, v).
The standardized-t density in equation (8) gives the following log-likelihood function:
n
v n v+1
v+1 ε2t
2
− ln (v − 2) σ −
− n ln Γ
LogLStudent−t = n ln Γ
ln 1 +
.
2
2
2
2
(v − 2)σ 2
t=1
where εt = Rt − µ is the monthly raw return deviated from the mean (µ), and σ is the standard
deviation of asset returns.
Table 2 reports the parameter estimates, the t-statistics in parentheses, the maximized loglikelihood values, and the LR statistics for the symmetric t distribution. The tail-thickness parameter (v) of the standardized t density for the 3-month T-bill is about 4.31 with a t-statistic of 3.56.
The tail-thickness parameter is about 5.61, 6.22, and 4.60 with the corresponding t-statistics of
3.70, 3.25, and 4.05 for the DJIA, S&P500, 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, i.e., the null hypothesis of v= ∞ is rejected in favor of the Student-t
for all asset classes considered in the paper.
The parameters of the normal density are estimated by maximizing the log-likelihood function
of Rt with respect to the mean and standard deviation parameters (µ, σ). The normal density in
equation (10) yields the following log-likelihood function:
2
n n
n
1 Rt − µ
2
LogLN ormal = − ln (2π) − ln σ −
.
2
2
2 t=1
σ
Table 2 reports the parameter estimates, the t-statistics in parentheses, and the maximized loglikelihood values for the normal density. The mean and standard deviation parameters are very
similar to the sample estimates µ and σ for all asset classes considered in the paper. The likelihood
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ratio test statistics reject the normal density in favor of the symmetric t and skewed t density
functions.
The most important ingredient in asset allocation problems is the assumed probability distribution of future returns on risky assets in the portfolio. In most theoretical and empirical work, a
normal or lognormal distribution is assumed because the normal distribution has several attractive
properties. First, it is easy to use, and it produces tractable results in many analytical exercises.
Second, all moments of positive order exist. Finally, the normal distribution is completely characterized by its first two moments, thus establishing the link with mean-variance portfolio theory.
An important drawback of the normal distribution in the context of investment decisions is that
the tails of the normal density decay exponentially toward zero. Hence, large realizations of asset
returns are very unlikely.
As presented in the paper, the normality assumption contradicts with empirical findings on asset
returns, which state that these returns generally exhibit leptokurtic behavior, i.e., have fatter tails
than the normal distribution. This means that extreme returns of either sign occur far more often
in practice than by predicted by the normal model.9 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.
4.
Results: Variance vs. Shortfall Constraint with the Normal
Distribution
This section describes the dynamic asset allocation strategy of two classes of investors: meanvariance (MV) versus loss-averse (LA) investor. We should note that both types of investors go
through the same input estimation and same re-balancing process from March 1971 through December 2006. Although they use the same inputs (i.e., expected returns, variances, and covariances),
the optimal portfolio selection of the MV and LA investors are different 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
9
See Bali (2007) for interest rate markets, Hsieh (1989a,b) for currency markets, and Bollerslev (1987), Nelson (1991),
and Bali and Weinbaum (2007) for stock markets.
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the portfolio. Whereas, the LA investors allocate financial assets by maximizing expected return
subject to a constraint on shortfall probability.
This paper does 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 one-month 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 4 asset classes considered in the paper. For example, daily returns on the 3-month T-bill, DJIA,
S&P500, 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 1971. This optimal portfolio selection process is repeated for
each month until December 2006, yielding a total of 429 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.
When the univariate asset distributions are Student-t and skewed-t, the implied portfolio distributions do not have to follow a multivariate Student-t and skewed-t, respectively. At an earlier
stage of the study, we generate hypothetical portfolios of 3-month T-bill, DJIA, S&P500, and
NASDAQ, and estimate the skewness (λ) and tail-thickness (v) parameters of the skewed-t and
Student-t density using the hypothetical return distributions. The maximum likelihood estimates
of v turn out to be in the range of 3 to 4, and the estimates of λ are in the range of –0.08 to
–0.12. 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 = –0.10. 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.
To determine the optimum asset mix in a mean-variance framework in month t, the MV
investor maximizes the expected return per unit of risk: E(Rp,t+1|t ) σp,t+1|t , where E(Rp,t+1|t ) =
13
n
ωi,t · E(Ri,t+1|t ) is the expected return of the portfolio for month t+1 given the information
n
n
n
2
2
set in month t, and σp,t+1|t =
ωi,t
· E(σi,t+1|t
)+
ωi,t · ωj,t · E(σij,t+1|t ) is the expected
i=1
i=1
i=1 j=1,i=j
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
2
2
the current month’s realized values, i.e., E(Ri,t+1|t ) ≡ Ri,t , E(σi,t+1|t
) = σi,t
, and E(σij,t+1|t ) ≡ σij,t
for all i, j, and t.
To find the optimal portfolio weights in month t, the LA investor maximizes the one-month
n
ahead expected return of the portfolio, E(Rp,t+1|t ) =
ωi,t · E(Ri,t+1|t ), subject to a constraint on
i=1
shortfall probability, P (Rp,t < R
low
) ≤ θ, that the portfolio’s return Rp,t =
n
ωi,t Ri,t will not fall
i=1
below the shortfall return Rlow . As in the MV framework, one-month lagged returns, variances and
covariances are used in the expected return and shortfall constraint of the LA investor, i.e., the
2
2
same inputs are used in the optimization: E(Ri,t+1|t ) ≡ Ri,t , E(σi,t+1|t
) = σi,t
, and E(σij,t+1|t ) ≡ σij,t
for all i, j, and t.
Since the mean-variance framework implicitly assumes that the portfolio’s 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 = 2.5%, θ = 5%, and θ =
10%. We also report the results for three values of the shortfall return, Rlow = 0%, Rlow = –1%,
and Rlow = –2%.
In order to evaluate the relative performance of the mean-variance and loss-averse portfolio selection models, we present the cumulative cash values of $100 investment, annualized geometric mean
return of the optimal portfolio, and the average risk-adjusted return. According to the cumulative
cash values displayed in Figure 1, if one had invested $100 on the mean-variance portfolio in March
1971, she would have received $1,614.47 in December 2006. During the same period, if she had
invested $100 on the loss-averse portfolio with Rlow = –2%, she would have received $3,391.96 for
θ = 2.5%, $3,655.22 for θ = 5%, and $4,028.80 for θ = 10%. Figure 1 clearly demonstrates that
for all months from April 1971 to December 2006, the loss-averse framework performs better than
the mean-variance model in terms of the cumulative cash values and the compounded annualized
14
return (or the annualized geometric mean return).
Figure 2 plots the cumulative cash values of $100 investment on the loss-averse portfolio with a
downside risk constraint of -1% per month. From March 1971 to December 2006, $100 investment
on the LA portfolio with Rlow = -1% grows to $3,174.14 for θ = 2.5%, $3,999.00 for θ = 5%, and
$3,918.52 for θ = 10%. Figure 3 presents the cumulative cash values of $100 investment on the
loss-averse portfolio with Rlow = 0%. Over the sample period of March 1971–December 2006, $100
investment on the LA portfolio with Rlow = 0% grows to $2,310.18 for θ = 2.5%, $2,812.41 for θ
= 5%, and $4,389.18 for θ = 10%.
These cumulative values are summarized in Panel A of Table 3 that also compares the relative
performance of the MV and LA asset allocation models based on the annualized geometric mean
return and the average risk-adjusted return of the optimal portfolio. Panel B of Table 3 shows that
for each value of the shortfall return (Rlow = -2%, -1%, 0%), the annualized geometric mean return
of the optimal LA portfolio increases as the shortfall probability rises from θ = 2.5% to θ = 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 3 presents the average risk-adjusted return of
the optimal MV and LA portfolios. For each level of the shortfall probability (θ = 2.5%, 5%, 10%),
risk-adjusted return of the optimal LA portfolio increases as the shortfall return increases from Rlow
= –2% to Rlow = –1% to Rlow = 0%. Specifically, the average return-to-standard deviation ratio
is in the range of 0.271 to 0.366, which is above the average risk-adjusted return of the optimal
mean-variance portfolio, 0.241.
To determine whether the loss-averse portfolio performs better or worse than the mean-variance
portfolio in terms of risk-adjusted return, we calculate the average return of the optimal portfolio,
average risk of the optimal portfolio, and average risk-adjusted return (i.e., average return-torisk ratio) over the sample period of April 1971–December 2006. Risk of the optimal portfolio is
measured in terms of standard deviation, 1% VaR, and 1% Expected Shortfall (ES).10 Panel A of
10
VaR as a risk measure is criticized for not being sub-additive. 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
15
Table 4 shows that the average return of the optimal mean-variance portfolio is 0.69% per month,
which is below the average return of the optimal loss-averse portfolio for different values of shortfall
probability. As reported in Panel A, the average return of the optimal portfolio with Rlow = –1% is
0.84%, 0.90%, and 0.91% per month for θ = 2.5%, θ = 5%, and θ = 10%, respectively. The average
standard deviation of the optimal mean-variance portfolio is 2.88% per month, whereas the average
standard deviation of the optimal loss-averse portfolio with Rlow = –1% is 2.48%, 2.79%, and 3.07%
per month for θ = 2.5%, θ = 5%, and θ = 10%, respectively.11 In terms of risk-adjusted return,
the last column of Panel A shows superior performance of the loss-averse asset allocation model
over the mean-variance approach. The average return-to-risk ratio of the mean-variance portfolio
is 0.24, which is less than the average risk-adjusted return of the loss-averse portfolio with Rlow =
–1%: 0.34, 0.32, and 0.29 for θ = 2.5%, θ = 5%, and θ = 10%, respectively.
Panel B of Table 4 shows that the 1% VaR of the optimal mean-variance portfolio is on average
8.32% per month, which is significantly greater than the 1% VaR of the optimal loss-averse portfolio
with Rlow = –1%: 4.41%, 5.16%, and 5.83% per month for θ = 2.5%, θ = 5%, and θ = 10%,
respectively. 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 loss-averse portfolio with Rlow = –1%
is 0.19, 0.18, and 0.16 for θ = 2.5%, θ = 5%, and θ = 10%, respectively. Whereas, the average
return-to-VaR ratio of the optimal mean-variance portfolio is 0.08, indicating that the risk-adjusted
return of the optimal loss-averse portfolio is at least two times bigger than the risk-adjusted return
of the optimal mean-variance portfolio.
Similar results are obtained when the portfolio risk is measured with the 1% Expected Shortfall
(ES). Panel C of Table 4 reports that the 1% ES of the optimal mean-variance portfolio is on
average 13.65% per month, which is significantly greater than the 1% ES of the optimal loss-averse
portfolio with Rlow = –1%: 7.23%, 8.22%, and 8.83% per month for θ = 2.5%, θ = 5%, and θ =
10%, respectively. The last column of Panel C shows that the average return-to-ES ratio of the
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.
11
Since value at risk is computed based on the negative tail of the return distribution, the original VaR measures are
negative. However, to be consistent with the standard measures of portfolio risk, we multiply the original VaRs by
–1 so that we can report positive average risk and average risk-adjusted returns in our tables.
16
optimal loss-averse portfolio with Rlow = –1% is in the range of 0.10 to 0.12, whereas the average
return-to-ES ratio of the optimal mean-variance portfolio is 0.05. Similar to our findings from the
VaR, the risk-adjusted return of the optimal loss-averse portfolio is at least two times larger than
the risk-adjusted return of the optimal mean-variance portfolio.
These results indicate that the optimal portfolio selection model with shortfall probability constraint outperforms the standard mean-variance 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.
5.
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 a
loss-averse investor.
Table 5 compares the relative performance of the thin-tailed normal, the symmetric fat-tailed
t, and the skewed fat-tailed t distributions in terms of average return, average risk, and average
risk-adjusted return for the loss-averse portfolio with Rlow = –1% and θ = 2.5%. Risk of the optimal
portfolio is measured in terms of standard deviation, 1% VaR, and 1% Expected Shortfall (ES).
Panel A of Table 5 shows that the average return of the optimal loss-averse portfolio with the
Normal density is 0.84% per month, which performs in between the optimal loss-averse 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 average risk, the Studentt density outperforms both the Skewed-t and Normal density. Specifically, the average standard
deviation of the optimal portfolio is 2.48% per month for the Normal density, whereas the average
standard deviation of the optimal portfolio is in the range of 1.39% to 1.70% per month for the
17
Student-t, and 2.58% to 2.73% 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 5 shows that the 1% VaR of the optimal portfolio with the Normal density is
on average 4.41% per month, whereas the 1% VaR of the optimal portfolio is in the range of 2.67%
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-VaR ratio of the optimal portfolio with the Normal density is 0.19, whereas
the average return-to-VaR ratio of the optimal portfolio with the Skewed-t density is 0.17. In terms
of risk-adjusted return, the Student-t density with the average return-to-VaR ratio of 0.21 to 0.27
outperforms the Skewed-t and Normal distributions.
Qualitative results turn out to be similar when the portfolio risk is measured with the 1%
Expected Shortfall (ES). Panel C of Table 5 reports that the 1% ES of the optimal portfolio with
the Normal density is on average 7.23% 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 0.12 for the Normal density, 0.11 for the Skewed-t density, and between 0.15
and 0.21 for the Student-t density. Similar to our findings from the standard deviation and valueat-risk, the Student-t distribution performs better than the Skewed-t and Normal distributions
when the risk-adjusted return of the loss-averse portfolio is defined in terms of expected shortfall.
As a robustness check, Tables 6 and 7 evaluate the relative performance of the Normal, Student-t,
and Skewed-t distributions in terms of average return, average risk, and average risk-adjusted return
for the loss-averse portfolio with Rlow = –1% and θ = 5% and θ = 10%. The results clearly indicate
superior performance of the Student-t distribution over the Skewed-t and Normal distributions in
term of average risk-adjusted returns.
Overall, we can conclude that the optimal asset allocation model with a shortfall probability
18
constraint provides a more profitable (in terms of cumulative and risk-adjusted returns) trading
strategy than the widely used mean-variance approach. This result holds for alternative distribution
functions (Normal, Student-t, Skewed-t), different measures of portfolio risk (standard deviation,
VaR, Expected Shortfall), different levels of shortfall return (Rlow = 0%, –1%, –2%), and shortfall
probability (θ = 2.5%, 5%, 10%). While investigating the effects of skewness and kurtosis on the
loss-averse portfolio, we observe that incorporating skewness into the asset allocation problem
improves the performance in terms of average return, but it increases risk of the optimal portfolio as
well. Hence, taking skewness of the return distribution into account does not augment profitability
of the loss-averse 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.
6.
Conclusions
This paper introduces a new approach to optimal portfolio selection in a downside risk framework
that allocates assets by maximizing expected return of a portfolio subject to a constraint on
shortfall probability. The shortfall constraint reflects the typical desire of a loss-averse investor
to limit downside risk by putting a probabilistic upper bound on the maximum likely loss. The
shortfall probability constraint is first modeled with the normal distribution. Then, the symmetric
fat-tailed and skewed fat-tailed distributions are used to estimate the tails of the portfolio return
distribution and to determine the effects of skewness and kurtosis on optimal asset allocation.
The paper compares the empirical performance of the mean-variance approach with the newly
proposed asset allocation model in terms of their power to generate higher cumulative and riskadjusted returns. The results indicate that the loss-averse portfolio with normal density outperforms
the mean-variance approach based on the cumulative cash values, geometric mean returns, and
average risk-adjusted returns. The key findings turn out to be robust across different measures of
downside risk, different levels of shortfall return and shortfall probability.
The relative performance of the symmetric thin-tailed, symmetric fat-tailed, and skewed fattailed distributions are evaluated in terms of average return, average risk, and average risk-adjusted
return for the loss-averse portfolio with a constraint on the maximum expected loss. The asset allo-
19
cation model with a downside risk constraint generates a more profitable trading strategy than the
widely used mean-variance approach. This finding holds for alternative distribution functions (Normal, Student-t, Skewed-t), different measures of portfolio risk (standard deviation, VaR, Expected
Shortfall), different levels of shortfall return (Rlow = 0%, –1%, –2%), and shortfall probability (θ
= 2.5%, 5%, 10%).
The paper investigates the effects of skewness and kurtosis of the portfolio return distribution
on the asset allocation decision of a loss-averse investor. The results show that accounting for tailthickness of the return distribution (or excess kurtosis) enhances both the cumulative return and
the average risk-adjusted return of the dynamic asset allocation model with a shortfall probability
constraint. However, incorporating skewness into the portfolio selection problem does not augment
profitability of the loss-averse investment strategy in terms of risk-adjusted return.
20
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23
Table 1
Descriptive Statistics
This table presents descriptive statistics of the monthly percentage returns on the 3-month Treasury bill,
Dow Jones Industrial Average (DJIA), S&P500, and NASDAQ stock market indices over the sample period
of March 1971 to December 2006. Standard errors of the skewness and kurtosis statistics are calculated as
6/n and 24/n, respectively, where n denotes the number of observations (n= 430). Jarque-Bera, JB =
n[(S 2 /6) + (K–3)2 /24], 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.
∗
denotes statistical significance (at least) at the 1% level.
Statistics 3-month T-bill DJIA S&P500 NASDAQ
Mean
0.4969
0.7143
0.7159
0.8860
Median
0.4527
0.8851
0.9618
1.3506
Std. Dev.
0.2435
4.3460
4.2817
6.3196
Maximum
1.3587
13.592
12.501
20.105
Minimum
0.0736
–21.812 –20.662
–29.435
–0.5260∗ –0.5035∗
–0.7159∗
Skewness
0.8925∗
∗
∗
∗
Kurtosis
4.2502
5.3310
4.8299
5.5803∗
∗
∗
∗
Jarque-Bera
85.10
117.19
78.16
156.02∗
24
Table 2
Maximum Likelihood Estimates of Alternative Distribution Functions
This table presents the maximum likelihood parameter estimates of the Skewed-t, Student-t, and Normal
distributions. The results are based on the monthly returns on the 3-month T-bill, DJIA, S&P500, and NASDAQ stock market indices spanning the period March 1971–December 2006 (430 observations). Asymptotic
t−statistics are given in parentheses. LogL is the maximized log-likelihood value. The likelihood ratio (LR)
test statistic is calculated as LR = –2(LogL∗ – LogL), where LogL∗ is the value of the log likelihood under
the null hypothesis, 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 χ2(1,0.01) = 6.63 and χ2(2,0.01) = 9.21. LRnormal
tests whether the empirical return distribution follows the normal distribution as opposed to the Student-t
or Skewed-t density.
Panel A. 3-month Treasury bill
Distribution
µ
Normal
0.4974
(36.83)
Student-t 0.4723
(43.09)
Skewed-t
0.3234
(5.66)
σ
0.2433
(32.46)
0.2529
(11.43)
0.2450
(11.32)
v
4.3140
(3.56)
5.5686
(2.14)
λ
0.2434
(2.73)
LogL
–2.36
LRnormal
———
12.50
29.72
19.75
44.22
Panel B. Dow Jones Industrial Average
Distribution
µ
σ
v
λ
LogL
LRnormal
Normal
0.7093 4.3447
–1238.91 ———
(3.27) (41.68)
Student-t 0.8210 4.3592 5.6051
–1225.08
27.66
(4.24) (18.34) (3.70)
Skewed-t
1.5580 4.3491 5.7992 –0.0623 –1224.66
28.50
(1.82) (18.26) (3.40) (–0.88)
Panel C. S&P 500 Composite Index
Distribution
µ
σ
v
λ
LogL
LRnormal
Normal
0.7091 4.2793
–1232.40 ———
(3.32) (39.17)
Student-t 0.8349 4.2938 6.2161
–1221.74
21.32
(4.30) (19.30) (3.25)
Skewed-t
2.0063 4.2840 6.5635 –0.0986 –1221.01
22.78
(2.37) (19.35) (2.88) (–1.42)
Panel D. NASDAQ Composite Index
Distribution
µ
σ
v
λ
LogL
LRnormal
Normal
0.8776 6.3172
–1399.49 ———
(2.71) (41.76)
Student-t 1.1752 6.3978 4.5978
–1380.22
38.54
(4.32) (14.96) (4.05)
Skewed-t
3.4129 6.3817 4.7928 –0.1347 –1378.15
42.68
(2.97) (14.63) (3.58) (–1.99)
25
Table 3
Mean-Variance vs. Shortfall Constraint with the Normal Distribution: Growth, Annualized Geometric
Mean, Risk-adjusted Return
Panel A presents the cumulative cash values of $100 investment on the optimal loss-averse portfolio with
Rlow = -2%, -1%, 0%, and θ = 2.5%, 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 2006. Risk of the optimal portfolio is defined
in terms of standard deviation.
Panel A. Cumulative Cash Values of $100 Investment
θ = 2.5%
θ = 5%
θ = 10%
Rlow = -2% Rlow = -1%
$3,391.96
$3,174.14
$3,655.22
$3,999.00
$4,028.80
$3,918.52
Rlow = 0%
$2,310.18
$2,812.41
$4,389.18
Cumulative cash value of $100 investment on the optimal
mean-variance portfolio = $1,614.47
Panel B. Annualized Geometric Mean Return
of the Optimal Portfolio
θ = 2.5%
θ = 5%
θ = 10%
Rlow = -2% Rlow = -1%
10.36%
10.16%
10.59%
10.84%
10.89%
10.87%
Rlow = 0%
9.18%
9.78%
11.16%
Annualized geometric mean return of the optimal portfolio
= 8.09%
Panel C. Risk-adjusted Return of the Optimal Portfolio
θ = 2.5%
θ = 5%
θ = 10%
Rlow = -2% Rlow = -1%
0.306
0.339
0.290
0.323
0.271
0.294
Rlow = 0%
0.366
0.340
0.343
Risk-adjusted return of the optimal mean-variance portfolio
= 0.241
26
4500
4000
Portfolio Value ($)
3500
3000
2500
2000
1500
LA (Rlow=-2%,θ=2.5%)
LA (Rlow=-2%,θ=5%)
LA (Rlow=-2%,θ=10%)
MV
1000
500
0
Mar-71
Figure 1
Sep-74
Apr-78
Nov-81
Jun-85
Jan-89
Aug-92
Mar-96
Oct-99
May-03
Dec-06
$100 Investment on the MV and LA Portfolio with Rlow = -2%
4500
4000
Portfolio Value ($)
3500
3000
2500
2000
1500
LA (Rlow=-1%,θ=2.5%)
LA (Rlow=-1%,θ=5%)
LA (Rlow=-1%,θ=10%)
MV
1000
500
0
Mar-71
Figure 2
Sep-74
Apr-78
Nov-81
Jun-85
Jan-89
Aug-92
Mar-96
Oct-99
May-03
Dec-06
$100 Investment on the MV and LA Portfolio with Rlow = -1%
4500
4000
Portfolio Value ($)
3500
3000
2500
2000
1500
LA (Rlow=0%,θ=2.5%)
LA (Rlow=0%,θ=5%)
LA (Rlow=0%,θ=10%)
MV
1000
500
0
Mar-71
Figure 3
Sep-74
Apr-78
Nov-81
Jun-85
Jan-89
Aug-92
Mar-96
Oct-99
May-03
Dec-06
$100 Investment on the MV and LA Portfolio with Rlow = 0%
27
Table 4
Mean-Variance vs. Shortfall Constraint with the Normal Distribution – Portfolio Risk: Standard
Deviation, VaR, Expected Shortfall
This table presents the average return of the optimal portfolio, average risk of the optimal portfolio, and
average risk-adjusted return (i.e., average return-to-risk ratio) over the sample period of April 1971–December
2006. Panel A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% valueat-risk (VaR), and 1% expected shortfall (ES), respectively. The results are presented for the mean-variance
portfolio and the loss-averse portfolio with Rlow = -1%, and θ =2.5%, 5%, and 10%.
Panel A. Return, standard deviation, and risk-adjusted return
LA (Rlow = -1%, θ =2.5%)
LA (Rlow = -1%, θ = 5%)
LA (Rlow = -1%, θ = 10%)
Mean-Variance
Average return Standard deviation Risk-adjusted return
0.84%
2.48%
0.34
0.90%
2.79%
0.32
0.91%
3.07%
0.29
0.69%
2.88%
0.24
Panel B. Return, 1% VaR, and risk-adjusted return
LA (Rlow = -1%, θ =2.5%)
LA (Rlow = -1%, θ = 5%)
LA (Rlow = -1%, θ = 10%)
Mean-Variance
Average return
0.84%
0.90%
0.91%
0.69%
1% VaR
4.41%
5.16%
5.83%
8.32%
Risk-adjusted return
0.19
0.18
0.16
0.08
Panel C. Return, 1% expected shortfall, and risk-adjusted return
LA (Rlow = -1%, θ =2.5%)
LA (Rlow = -1%, θ = 5%)
LA (Rlow = -1%, θ = 10%)
Mean-Variance
Average return
0.84%
0.90%
0.91%
0.69%
1% ES
7.23%
8.22%
8.83%
13.65%
Risk-adjusted return
0.12
0.11
0.10
0.05
28
Table 5
Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with
Rlow = −1%, θ = 2.5%
This table presents the average return, average risk, and average risk-adjusted return of the optimal lossaverse portfolio with Rlow = –1% and θ = 2.5% over the sample period of April 1971–December 2006. Panel
A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% value-at-risk (VaR),
and 1% expected shortfall (ES), respectively. The Student-t density is assumed to have a tail-thickness
parameter of v = 3, 4. The Skewed-t density is assumed to have a tail-thickness parameter of v = 3, 4, and
skewness parameter of λ = -0.1.
Panel A. Return, standard deviation, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return Standard deviation Risk-adjusted return
0.84%
2.48%
0.34
0.71%
1.39%
0.51
0.73%
1.70%
0.43
0.89%
2.73%
0.33
0.85%
2.58%
0.33
Panel B. Return, 1% VaR, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.84%
0.71%
0.73%
0.89%
0.85%
1% VaR
4.41%
2.67%
3.45%
5.16%
5.09%
Risk-adjusted return
0.19
0.27
0.21
0.17
0.17
Panel C. Return, 1% expected shortfall, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.84%
0.71%
0.73%
0.89%
0.85%
1% ES
7.23%
3.44%
4.74%
8.19%
7.89%
Risk-adjusted return
0.12
0.21
0.15
0.11
0.11
29
Table 6
Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with
Rlow = −1%, θ = 5%
This table presents the average return, average risk, and average risk-adjusted return of the optimal lossaverse portfolio with Rlow = –1% and θ = 5% over the sample period of April 1971–December 2006. Panel A,
B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% value-at-risk (VaR),
and 1% expected shortfall (ES), respectively. The Student-t density is assumed to have a tail-thickness
parameter of v = 3, 4. The Skewed-t density is assumed to have a tail-thickness parameter of v = 3, 4, and
skewness parameter of λ = -0.1.
Panel A. Return, standard deviation, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return Standard deviation Risk-adjusted return
0.90%
2.79%
0.32
0.77%
2.06%
0.37
0.81%
2.31%
0.35
0.91%
2.94%
0.31
0.91%
2.84%
0.32
Panel B. Return, 1% VaR, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.90%
0.77%
0.81%
0.91%
0.91%
1% VaR
5.16%
3.55%
3.63%
5.16%
5.16%
Risk-adjusted return
0.17
0.22
0.22
0.18
0.18
Panel C. Return, 1% expected shortfall, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.90%
0.77%
0.81%
0.91%
0.91%
1% ES
8.22%
6.08%
6.45%
8.23%
8.23%
Risk-adjusted return
0.11
0.13
0.13
0.11
0.11
30
Table 7
Performance of the Normal, Student-t, and Skewed-t Distributions: Loss-Averse Portfolio with
Rlow = −1%, θ = 10%
This table presents the average return, average risk, and average risk-adjusted return of the optimal lossaverse portfolio with Rlow = –1% and θ = 10% over the sample period of April 1971–December 2006. Panel
A, B, and C measure the risk of the optimal portfolio in terms of standard deviation, 1% value-at-risk (VaR),
and 1% expected shortfall (ES), respectively. The Student-t density is assumed to have a tail-thickness
parameter of v = 3, 4. The Skewed-t density is assumed to have a tail-thickness parameter of v = 3, 4, and
skewness parameter of λ = -0.1.
Panel A. Return, standard deviation, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return Standard deviation Risk-adjusted return
0.91%
3.07%
0.30
0.90%
2.80%
0.32
0.91%
2.87%
0.32
0.91%
3.44%
0.26
0.93%
3.35%
0.28
Panel B. Return, 1% VaR, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.91%
0.90%
0.91%
0.91%
0.93%
1% VaR
5.83%
5.16%
5.16%
8.43%
7.63%
Risk-adjusted return
0.16
0.17
0.18
0.11
0.12
Panel C. Return, 1% expected shortfall, and risk-adjusted return
Normal
Student-t (v = 3)
Student-t (v = 4)
Skewed-t (v = 3, λ = -0.1)
Skewed-t (v = 4, λ = -0.1)
Average return
0.91%
0.90%
0.91%
0.91%
0.93%
1% ES
8.83%
8.22%
8.23%
9.79%
9.69%
Risk-adjusted return
0.10
0.11
0.11
0.09
0.10