Downside Loss Aversion and Portfolio Management

Transcription

1 Downside Loss Aversion and Portfolio Management Robert Jarrow a and Feng Zhao b September 2005 Abstract: Downside loss averse preferences have seen a resurgence in the portfolio management literature. This is due to the increasing usage of derivatives in managing equity portfolios, and the increased usage of quantitative techniques for bond portfolio management. We employ the lower partial moment as a risk measure for downside loss aversion, and compare mean-variance (M-V) and mean-lower partial moment (M-LPM) optimal portfolios under non-normal asset return distributions. When asset returns are nearly normally distributed, there is little difference between the optimal M-V and M-LPM portfolios. When asset returns are non-normal with large left tails, we document significant differences in M-V and M-LPM optimal portfolios. This observation is consistent with industry usage of M-V theory for equity portfolios, but not for fixed income portfolios. a Jarrow is from Johnson Graduate School of Management, Cornell University, Ithaca, NY (raj15@cornell.edu). b Zhao is from Rutgers Business School, Rutgers University, Newark, NJ (FengZhao@rbsmail.rutgers.edu). We thank David Hsieh (the editor), the associate editor and an anonymous referee for the helpful suggestions. We are responsible for any remaining errors.

2 1 Introduction The study of downside risk measures in portfolio management and evaluation, after two decades of silence, has been rekindled recently for five related reasons. One, in the financial community, the determination of capital is a topic of current debate due to numerous financial catastrophes and the Basel I, II accords. Downside risk measures such as the Value at Risk (VaR) are crucial to this debate (see also the literature related to coherent risk measures, for example, Artzner et al. (1999) and Jarrow (2002)). Two, foreign currency and equity derivatives are becoming more popular in managing equity portfolios. As such, they can potentially change the equity portfolio s distribution from symmetric to non-symmetric. This arguably invalidates standard mean-variance analysis (see Leland (1999) and Pedersen (2002)). Three, behavioral finance has flourished and the literature has documented many investor characteristics, including downside loss aversion (see Kahneman and Tversky (1979)). Fourth, with the downturn in equity markets and the development of new tools to evaluate credit risk (see Bielecki and Rutkowski (2001)), fixed income portfolio management has become more conducive to quantitative analysis. Due to heavy left tails for bonds distributions, mean-variance (M-V) portfolio analysis is less useful in this context. Last, recent studies on event risks (see Liu, Longstaff and Pan(2003)) show that downside risk exists, even for stocks. The purpose of this paper is to study downside loss averse portfolio theory. We first motivate the use of the Lower Partial Moment (LPM) as an appropriate risk measure for downside loss-averse preferences. Second, we compare 2

3 optimal M-V portfolios with optimal M-LPM portfolios. The optimal portfolios are compared under two asset return scenarios, one for high-yield bonds and the other for stocks with event risks. The analysis generates two insights useful for portfolio management. First, the two types of preferences lead to similar optimal portfolio choices, if the portfolio s return distribution is normal or log-normal. This is because for a fixed expected return, LPM is strictly increasing in variance, and therefore M-V and M-LPM optimal portfolios are similar. 1 This appears to be the case for the traditional equity portfolio analysis (see Grootveld and Hallerbach (1999)). However, for portfolios that consist of fixed income securities, derivatives, or stocks with event risks, the M-V optimal portfolio can differ significantly from the M-LPM optimal portfolio. These observations explain why quantitative equity portfolio management (in the absence of derivatives) is almost exclusively concerned with M-V analysis, despite the evidence supporting the usage of downside loss portfolio theory as documented above. These observations also clarify why M-V analysis is inappropriate for use in fixed income portfolio management. Our paper is related to other studies of downside loss aversion in the portfolio management and asset allocation literature. Bawa (1978) and Fishburn (1977) introduced the lower partial moment (LPM) as an alternative risk measure to variance. Using insights from Kahneman and Tversky (1979) 2,weshow 1 We can extend this argument to some non-gaussian distributions. In general, as long as the downside risk measure, defined over the portfolio s return distribution, is increasing in variance for all expected return levels, M-V portfolios are also optimal for downside loss averse preferences. 2 We use the kink-shaped utility function of prospect theory, but do not consider the probability transformation used in the Cumulative Prospect Theory of Tversky and Kahneman 3

4 that the LPM can be properly used to measure downside loss aversion. Other down-side risk measures have also been proposed in the literature, for example, VaR, conditional VaR, or expected shortfall (see Basak and Shapiro (2001) and Rockafellar and Uryasev (2002)). Usually, these measures are not generated from investor preferences, but enter as constraints in the utility maximization problem. In portfolio management, conditional VaR has been studied by Krokhmal, Palmquist and Uryasev (2001). With respect to dynamic asset allocation, Basak and Shapiro (2001) and Cuoco, He and Issaenko (2001) study the dynamic utility maximization problem with a downside risk constraint under Gaussian distributions. Liu, Longstaff and Pan (2003) study event risk, but they do not consider loss aversion. An outline for this paper is as follows. Section 2 reviews downside risk portfolio theory for its use in comparing portfolio performance. Section 3 provides a comparison of M-V and M-LPM optimal portfolios under two return distribution scenarios which are non-gaussian. Lastly, section 4 concludes the paper. 2 Downside Risk and Portfolio Theory This section reviews and extends downside loss portfolio theory. We consider a single period economy in which agents invest in period 0 and the investment outcomes are realized in period 1. (1992). The effect of the probability transformation is to place more weight on the tails of the return distribution. In this study, we consider non-gaussian distributions directly, without imposing the probability transformation. 4

5 2.1 DownsideLossAverseUtilityFunctions An agent s downside loss averse utility is defined over the portfolio s return: ½ f(x) u(x) = f(x)+g(x) for x a for x<a (1) where f and g are increasing and f is concave. 3 The function g embodies the investor s aversion toward downside losses (x <a), while the function f represents a standard risk averse utility function. The constant a is called the reference level. If the utility function is continuous at the kink, we need to have g(a) =0. When g 0, investors do not exhibit downside loss aversion, yielding the structure used in standard portfolio analysis. One can obtain mean-lower partial moment (M-LPM) utility functions by assuming that the function f is linear and the function g is a power function, i.e. f(x) =c 1 + c 2 x g(x) = c 3 (a x) n (2) where c i 0 for i =1, 2, 3. The equivalence between M-LPM and the utility function characterized by (2) is shown in Fishburn (1977). The utility function postulated in (1) is also a modest generalization of the 3 In contrast, prospect theory states that the value function is convex below the reference point, i.e. f 00 + g

6 downside risk averse utility function given in Kahneman and Tversky (1979) 4 : ½ x a for x a u(x) = (1 + b) (x a) for x<a (3) and b>0. In this special case, f and g are linear (risk neutral), and downside aversion is exhibited by the kink in the utility function at the return level a. Notably, asymmetric loss functions have long been used in the statistical decision theory literature (e.g., see Granger (1969), Varian (1974), Zellner (1986) and Christoffersen and Diebold (1997)). Two commonly used loss functions are the so-called LINLIN and LINEX Expected Utility Given the downside loss utility function (1), letting the portfolio return X follow the distribution F X, we can decompose expected utility into three parts: E[u(X)] = f (E[X]) + {E[f(X)] f (E[X])} + Z a g(x)df X (x). (5) The first part f (E[X]) is the transformed expected return, the second part {E[f(X)] f (E[X])} incorporates the (standard) risk due to the concavity 4 In prospect theory, three characteristics were present in the utility function: (1) a reference level, (2) concavity in gains and convexity in losses, and (3) a steeper slope for losses. 5 The LINLIN loss function is piecewise linear, which is identical to (3). The LINEX loss function is obtained by setting f(x) =c 1 + c 2 x (4) g(x) =c 3 [exp (c 4 x) c 4 x 1] with c 1,c 2 0, c 3 > 0, c 4 6=0. 6

7 of f, and the third part is named the Downside Risk Measure and DRM X R a g(x)df X(x). An important class of expected functions E[f(X)] are those for which there exists a convex, non-decreasing function f such that f (E[X]) E[f(X)] = f(sd(x)) where SD(X) p E[X 2 ] E[X] 2. For example, such an f exists (i) for arbitrary return distributions if f is a quadratic function, or (ii) for arbitrary functions f if returns have an elliptical distribution (see Meyer (1987)). We will call such functions f mean-variance (M-V) preferences. 6 Given mean-variance preferences f, using the decomposition in expression (5), we can rewrite expected utility as E[u(X)] = f (E[X]) f(sd(x)) DRM X. (6) Under these hypotheses, expected utility is seen to be non-decreasing in E[X] and non-increasing in both SD(X) and DRM X. When the utility function assumes the M-LPM form as in expression (2), to be consistent with the existing literature, we rewrite the downside risk compo- 6 This characterization can also be generalized to higher order moments such as skewness and kurtosis. 7

8 nent as DRM X c 3 LPM n (a; F X ) where LPM n (a; F X ) Z a (a x) n df X (x). (7) Here, LPM n (a; F X ) represents the n-th order LPM of the probability distribution F X with respect to the reference level a. In this case, expression (6) can be written as 7 E[u(X)] = c 1 + c 2 E(X) c 3 LPM n (a; F X ). (8) This decomposition will prove useful in Section 2.3. The reference level a can also depend on the distribution of the portfolio s return. For instance, if a =VaR α (X), theα-quantile of X, we can relate the first order lower partial moment to Expected Shortfall (ES), also called conditional VaR, which is defined to be ES α (X) = 1 α E X 1 {X VaR α (X)}. Using the definition for LPM, we have LPM 1 (VaR α (X); F X )=αes α (X)+αVaR α (X). The recent risk management literature advocates the use of ES in portfolio 7 In this case, f(sd(x) =0. 8

9 management because VaR violates the subadditivity axiom for a coherent risk measure (see Artzner et al. (1999), Frey and McNeil (2002) and Tasche (2002)). 2.3 The Portfolio Optimization Problem Let the economy consist of N risky assets and a risk-free asset with (one plus) returns denoted by {R k } N k=1 and R 0, respectively. Let w n denote the portfolio weight invested in asset n. The set of possible returns is given by ( ) NX NX Z = X = w k R k : w k =1. (9) k=0 k=0 Also, let Z 0 = {X Z : w 0 =0} be the set of risky asset only portfolios. The investor s portfolio problem is to maximize expected utility subject to the budget constraint, i.e. sup E[u(X)]. (10) X Z As is well known, the solution to this problem may not exist if there is arbitrage opportunity in Z, or if, heuristically speaking, returns overshadow risk when taking an infinite position. This stems from the fact that Z is not a compact set. We assume, therefore, that there is no arbitrage in this general sense, and that the solution to (10) is finite. Under this maintained assumption, the following proposition characterizes the solution to (10) in a form that is more convenient for computation. Proposition 1 Given mean-variance preferences f and concave g, letx be 9

10 the solution to (10), then it solves min X Z DRM X s.t. E[X] µ (11) SD(X) s for (µ, s) =(E[X ],SD(X )). Conversely, any solution to (11) solves (10) for a utility function u( ) that admits the representation in expression (1). Proof. Suppose X solves (10) but does not solve (11). Then, there exists X 0 Z such that E[X 0 ] E[X ],SD(X 0 ) SD(X ) and DRM X 0 <DRM X. From (6), E[u(X 0 )] = f (E[X 0 ]) f(sd(x)) DRM X 0. Since f is increasing and f is non-decreasing, we have E[u(X 0 )] >E[u(X )]. This contradicts that X solves (10). Conversely, the efficient frontier (E[X ], SD(X ), DRM X ) is a 2-dimensional manifold in R 3. We transform this frontier to f(e[x ]), f(sd(x )), DRM X for convenience. Denote d (µ, s) as the solution to (11) for some µ, s. First, we needtoshowtheconvexityofthefeasibleset f(µ), f(s),d µ, s > 0,d>d (µ, s) ª. But, this follows directly from the concavity of the functions f, f and g. Next, note that the indifference curve generated by the expected utility function has 10

11 the form c 1 f(µ)+c 2 f(s)+c 3 d = U 0 for some constant U 0. Since every point on the frontier is a tangent point to this 2-dimensional plane, this proves our result. This proposition shows that the solution to the portfolio optimization problem can be viewed as that portfolio which minimizes downside risk, subject to an expected return target µ andanupperbounds on the standard deviation. From the proof, we see that this optimization problem can alternatively be written with the standard deviation as the objective function, and the expected return and the DRM as the constraints. We use this alternative formulation in our simulation study to find the LPM-constrained M-V optimal portfolios. Unfortunately, computing the solution to (11) is not as straightforward as in the M-V quadratic programming case. Instead, a more general convex programming algorithm needs to be applied (see Steinbach (2001) for related discussion). Under the M-LPM utility function (2), the solution to (10) simplifies further to generate the following corollary. Corollary 2 Given M-LPM preferences (2), the solution X to (10) solves min LPM n(a; F X ) X Z s.t. E(X) µ. (12) Parallel to M-V analysis, M-LPM analysis can be studied by considering a 11

12 two-parameter efficient frontier (see Bawa (1975), Bawa and Lindenberg (1977), and Harlow and Rao (1989)). The M-LPM n efficient frontier is the solution to expression (12) for different (a, µ). If in (12), instead of X Z we have X Z 0, the efficient frontier is generated by only the risky assets. Different from the M-V frontier, the M-LPM n efficient frontier changes for different values of the pair (a, µ). For easy reference, the properties of the M-LPM efficient frontier are collected in an appendix to this paper. Interestingly, M-LPM analysis can be used to understand the solution to the portfolio problem under the Kahneman and Tversky utility function in expression (3). Corollary 3 Under expression (3), for a fixed a, the solution to (10) is on the M-LPM 1 efficient frontier for some (µ,lpm 1 (a; F X )). Conversely, for any (µ,lpm 1 (a; F X )), there is a constant b such that this pair solves (10). Proof. For the first part, we note from (3) that f(x) =x a, g(x) =b(a x) + and thus f(e[x]) = E[X] a, f(sd(x)) = 0, DRM X = b LPM 1 (a; F X ). From Proposition 1, any solution to (10) with the utility function (3) solves (11), which degenerates to the M-LPM 1 optimization problem. Conversely, we show that the tangent point between the M-LPM 1 frontier and the indifference curve generated by the utility function gives the highest value of expected utility along the frontier. Note that the indifference curve generated by 12

13 the utility function (3) has the following form E[X] b LPM 1 (a; F X )=U 0 for some constant U 0. This is a straight line with intercept U 0 and slope b. From Proposition 10 in the appendix, the M-LPM 1 frontier is convex in the mean. Therefore, for any point (E[X ], LPM 1 (a; F X )) on the M-LPM 1 frontier there is a tangent indifference curve for some U 0 and b. Specifically, if the point is not a kink on the frontier, the gradient dlpm1(a;fx) de[x] X=X = 1 b. 3 Numerical Implementation This section uses simulation to compare M-V and M-LPM optimal portfolios under two different portfolio return distributions. We first describe the computation procedure and then present the results. 3.1 The Computational Procedure To compute the solution to expression (12) in the general case, analytic expressions for LPM n are unavailable, and integration has to be done numerically. Given the large dimension of the integral - equal to the number of risky assets - we use Monte-Carlo simulation. We investigate portfolios consisting of 5 assets in order to apply deterministic optimization techniques on the objective function estimated from Monte-Carlo simulations. Alternatively, optimization of larger asset portfolios could use the newly-developed simulated optimization 13

14 techniques (see Fu (2002)). Denoting R =(R 1,...,R N ) 0 as the vector of asset returns, the computation of DRM involves the simulation of l copies of nr (j)o l j=1. Given the portfolio weights {w i } N i=1, we can compute l copies of the portfolio returns X (j)ª l j=1. To evaluate the expectation of any function of X, say y(x), we can simply take the average of the l copies of y(x (j) ). These l copies are i.i.d. random variables. If they are of finite variance, by the central limit theorem, l 1 l lx y(x (j) ) Ey(X) = d Normal 0,σ 2 (y(x)). j=1 The accuracy of the estimate can be controlled by choosing the appropriate simulation length l. In our case, we let l = The resulting accuracy for our simulated objective function with a unit variance is In most cases, the standard deviation of the simulated objective function, σ (y(x)), is less than 0.20, which implies that the accuracy of the estimated objective function is of the order We also pick the optimization tolerance on the objective function to be 10 6 in order to ensure that the portfolio weights from the optimization are reasonably accurate. For comparison, we use the optimal weights from the M-V analysis as the initial point for the M-LPM optimization. The optimization algorithm applied is sequential quadratic programming. To compare portfolios, we examine the portfolio weights themselves, instead of their variances or LPMs. We first compute a vector of normalized differences 14

15 of the portfolio weights: 4w 0 = w MV w MLPM /w 0 where the normalization represents an equally weighted portfolio, w 0 = 1 N.We construct three measures to quantify the differences in the optimal portfolio weights: sup 4w 0, mean 4w 0 and median 4w 0. We analyze these differences for a representative set of mean returns. As shown in section 2.3, each point on the efficient frontier is the optimal choice for a specific utility function (3) with different b values. The reference return level a is set equal to 0.5. This implies that investors exhibit loss aversion when losing 50% or more on their investments. In previous studies using M-LPM analysis, the most commonly used reference level was based on the risk free rate. We choose the lower level because, consistent with the behavioral literature, investors only appear to be loss averse on the downside tail of the distribution. 3.2 M-V versus M-LPM Comparison We study portfolio optimization under two scenarios. First, when the asset return distribution is assumed to be lognormal with a point mass at zero. This provides a reasonable approximation to the return distribution of a high-yield bond. Second, we assume that the log(asset) price follows a jump-diffusion process, which is widely used for modelling a stock price with downside event 15

16 risk. Under each scenario, we compute the optimal M-V and M-LPM portfolios, and study their differences. We also determine the optimal M-V portfolio subject to its LPM being below a given level. This corresponds to the solution to problem (12) which contains preferences exhibiting both risk and loss aversion (as in expression (6)). The optimal solution to (12) lies between the M-V and M-LPM optimal portfolios. To illustrate the computations, we use LP M 1 as the downside risk measure. Although not reported here, we repeated our computations using LP M 2, with qualitatively similar results. In general, the difference is that higher order LP Ms impose higher penalties on portfolios with larger losses in the tail of the distribution High Yield Bond Return Distributions We first consider portfolios consisting of high-yield bonds. High yield bonds pay a promised return unless default occurs. As such, (one plus) returns of high-yield bonds R are distributed as follows: R =(1 N Θ) A + Θ 0 N where the non-default return distribution, A are log-normal with mean µ A and variance Σ A, and the Bernoulli random vector Θ with parameter q represents the occurrence of a default for each asset. When default occurs, the (one plus) return is zero, which means that all the investment is lost. Since our goal is to study the difference in the portfolio weights between M-V and M-LPM portfo- 16

17 lios, we assume that the asset returns are independent. Thus, the correlation among assets can be represented by Σ A, a diagonal matrix, and Θ, a vector of independent Bernoulli random variables 8. Since we are studying the static portfolio problem, it is realistic to set the time frame to be one year. The parameters used in the simulation are calibrated to historical observations. The volatility of the non-default component A is set to be 10% for all five bonds The default rates are 0%, 0.3%, 1%, 3%, 7%. We pick these values based on the historical default rates of corporate bonds with decreasing Moody s credit ratings 9 (see Carty and Lieberman (1997)). The mean of the non-default component A is set between 3% and 23% with a default risk premium determined by the ratio of the asset s default probability to the maximum probability of 7%. The simulation parameters are reported in Table 1.A. Note that the riskier assets have a higher variance and LPM. The computed measures are reported in Table 1.B. We report our results at four representative (one plus) mean return levels for the optimal M-V and M-LPM portfolios, 1.05, 1.07, 1.09 and The difference in portfolio weights is evidenced. The largest difference occurs at a mean return level of 1.05, where the maximum portfolio weight difference is more than twice the benchmark weight, and the mean difference is equal to 116% of the benchmark weight. The 8 This can be generalized to the Bernoulli mixture. Let Ψ be a M 1 vector with M N and functions T i : R M [0, 1] for i =1,..., N. The Bernoulli mixture Θ is such that P (Θ i =1 Ψ) =T i (Ψ) for i =1,..., N. 9 To be precise, 0% is for Aaa rated bonds, 0.3% for Baa to Ba, 1% for Ba, 3% for Ba3-B1, 7% for B2. 17

18 smallest difference occurs at the mean return of 1.11 where the maximum is 78% and the average is 35%. Figures 1.A provides a visual comparison of the portfolio weights. At the relatively low mean return levels of 1.05 and 1.07, the M-V portfolio spreads the weights more equally across the assets, while the M-LPM portfolio puts more weight on the safest asset. Intuitively, M-V investors choose a portfolio with low variance but high LPM (negatively skewed) and M-LPM investors do the opposite. At a relatively high mean return level of 1.09, them-vandm- LPM portfolios exhibit distinctly different patterns. At the highest mean return level of 1.11, both preferences place more weight on the assets with the highest returns, but they differ in the remaining asset weights. This indicates that both investors pick assets with higher default probabilities, which lead to both larger variance and LPM and therefore there is less divergence in the portfolio weights. Finally, we graph in M-V space the effect of an LPM constraint on the M-V optimization problem. We fix the LPM constraint to be a constant, implying that the investor s downside risk does not exceed this upper bound. There are two issues in choosing a constant. A small constant makes the constraint infeasible for very high and low mean return levels. In contrast, a large constant would make the constraint non-binding for the middle mean return levels. Given that the LPM values range from less than 10 8 to above 10 2 along the M-V frontiers, we set the constant equal to This value makes the constraint feasible for a wide range of mean return levels, yet binding for many levels within this range. For mean return levels outside the range of feasibility (for 18

19 the constant 10 5 ), we changed the constraint to equal the LPM values of the M-LPM portfolios. As shown in Figure 2.A, when the mean return is near the global minimum variance level, the LPM constraint is not binding, and therefore the two frontiers coincide. At higher mean returns, the constraint becomes binding and the constrained M-V frontier deviates from the unconstrained one. When the mean return is even higher, no portfolios are feasible. However, if we switch to the variable constraint, the deviation continues. 10 When the mean return is below the global minimum variance level, the deviation continues but to a lesser extent. This graph illustrates the fact that loss averse investors choose optimal portfolios off the traditional M-V frontier Jump-Diffusion Processes for Stock Prices This section studies portfolios of stocks whose prices follow jump-diffusion processes. Letting S denote the stock price, the incremental log stock price over the interval 4t evolves according to the following equation. M(λ4t) p X ln S t+4t ln S t = µ 4 t + σε t+4t 4t + m=1 J m where µ and σ are constants, and ε t+4t is an Normal(0, 1) random variable. The number of jumps M that occur during the interval 4t is a Poisson random 10 Note that the M-V frontier is concave in the mean return. The locus of the M-LPM portfolios is not necessarily concave in M-V space, although it is in M-LPM space. When the LPM constraint is constant, the constrained M-V frontier is concave, but not when the constraint varies across the mean return levels. 19

20 variable with parameter λ 4 t. The jump size is given by J m. We assume that J m is a normal random variable. As is well known, the jump term makes the distribution deviate from the diffusion case by generating fatter tails. Since we are studying the static portfolio problem, only the unconditional distribution of the stock price is needed. Therefore, adding stochastic volatility will not alter our conclusions since stochastic volatility has the same effect of fattening the tails of the unconditional distribution. Our simulation parameters are close to those in Liu, Longstaff and Pan (2003). We let the investment horizon be 1 year, µ =5%per annum, and σ = 15% per annum. We set the average jump frequency to be one jump every year 11. We let the five stocks have Gaussian jump sizes with mean -0.9, -0.5, 0, 0.5, 0.7, respectively 12 and a standard deviation equal to one third the jump size. The summary statistics of the simulated sample are reported in Table 2.A. In contrast to high-yield bonds where bonds with higher variance also have higher LPM (at the same mean return level), stocks with positive jumps have higher variance but thinner downside tails than those with negative jumps. So, we expect that M-V and M-LPM investors will make quite different choices for high yield bonds. Table 2.B. reports the three distance measures for the portfolio weights. The 11 We also set the average jump frequency to be once every two years and once every ten years, with similar results. When the jumps are less frequent, i.e. less skewed return distributions, at low return levels the optimal portfolios have distributions very close to lognormal, and therefore the difference between the M-V and M-LPM optimal portfolios are small or zero if the M-V portfolios have zero LPM. 12 Because the exponential function skews returns to the right, we make the magnitude of the largest positive jump smaller than the largest negative jump. In contrast, Liu, Longstaff and Pan (2003) used symmetric jump sizes. 20

21 four representative (one plus) mean return levels for the optimal portfolios are 1.07, 1.17, 1.27, and The differences are greater for the high mean return levels and smaller for the lower mean return levels. Figure 1.B plots the portfolio weights. We see that at lower mean return levels, the stock without jumps (log-normally distributed) is the major component of the portfolio, and therefore the portfolio s return distribution is close to a log-normal, where the difference between M-V and M-LPM is small. At higher return levels, M-LPM investors prefer the stocks with positive jumps, while M-V investors prefer the opposite. Figures2.Bshowsthedifference between the M-V portfolios with and without the LPM constraints in M-V space. For the reasons discussed earlier, we set the constraint equal to Similar to the high-yield bonds scenario, the unconstrained M-V and constrained M-V portfolios coincide when the LPM constraint is non-binding and they deviate otherwise. More portfolios are feasible given the LPM constraint when the mean return is above the global minimum variance level because stocks with positive jumps have relatively low LPM and high variance, thereby entering the optimal LPM-constrained portfolios. As before, investors with downside loss aversion would choose portfolios off the traditional M-V frontier. 21

22 4 Conclusion This paper studies portfolio management of high yield bonds or stocks with event risk for downside loss averse investors. We motivate the use of the wellstudied Lower Partial Moment (LPM) as an appropriate risk measure for loss aversion. We show that the portfolio weights contained in Mean-LPM (M-LPM) and Mean-Variance (M-V) optimal portfolios are quite different for portfolios of high-yield bonds or portfolios of stocks with event risks. The closer the portfolio s returns are to lognormality, the less difference there is in the optimal portfolio weights for M-LPM and M-V portfolios. Because portfolios consisting of only equities with small event risks are arguably more normally distributed than are fixed income portfolios, this analysis clarifies why the existing practice of using M-V analysis to manage equity portfolios is reasonable, despite the mounting empirical evidence supporting the usage of downside loss portfolio theory in investment management. References [1] Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999), Coherent Measures of Risk, Mathematical Finance, v9, n3 (July), [2] Basak, S. and Shapiro, A. (2001), Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices, Review of Financial Studies, Vol. 14, No.2,

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26 Table 1: Simulation Study for High-Yield Bonds This table reports the statistics of the simulated sample in Panel A. There are five bonds being simulated. The length of the simulation is 4 million. The reference level is Panel B reports the three distance measures defined over the portfolio weights. We compare the normalized differences of the vector of portfolio weights, w 0 = w M-V w M-LPM /w 0, where the normalization corresponds to an equally weighted portfolio with w 0 = 1/5 in this table. The comparison is provided at four mean return levels for the optimal portfolios, 1.05, 1.07, 1.09 and Panel A: Simulation Parameters and Bonds Statistics Default Prob. Bond % 0.3% 1% 3% 7% Mean Std. Dev % Quantile LPM 1 (0.5) Panel B: Distance Measures between the M-V and M-LPM Optimal Portfolios Mean Return Level Sup( w 0 ) Mean( w 0 ) Median( w 0 )

27 Table 2: Simulation Study for Stocks with Gaussian Jumps This table reports the statistics of the simulated sample in Panel A. There are five stocks being simulated. The length of the simulation is 4 million. The reference level is The mean jump sizes for stocks, from 1 to 5, are -0.9, 0.5, 0, 0.5, and 0.7. The standard deviations of the jump sizes are 1/3 of the mean jump sizes. Panel B reports the three distance measures defined over the portfolio weights. We compare the normalized differences of the vector of portfolio weights, w 0 = w M-V w M-LPM /w 0, where the normalization corresponds to an equally weighted portfolio with w 0 = 1/5 in this table. The comparison is provided at four mean return levels for the optimal portfolios, 1.07, 1.17, 1.27 and Panel A: Simulation Parameters and Stocks Statistics Stock Mean Std. Dev % Quantile LPM 1 (0.5) Panel B: Distance Measures Between the M-V and M-LPM Optimal Portfolios Mean Return Level Sup( w 0 ) Mean( w 0 ) Median( w 0 )

28 Panel A: High-Yield Bonds (1) Mean Return 1.05 (2) Mean Return (3) Mean Return 1.09 (4) Mean Return Panel B: Stocks with Gaussian Jumps (1) Mean Return 1.07 (2) Mean Return (3) Mean Return 1.27 (4) Mean Return Figure 1: Comparison of Optimal Portfolio Weights (Black: M-V; White: M-LPM 1 ) This figure plots the optimal portfolios weights on the fives assets at representative mean return levels. In Panel A, the default probabilities are, from bond 1 to 5, 0%, 0.3%, 1%, 3% and 7%. In Panel B and C, the mean jump sizes are, from stock 1 to 5, -0.9, 0.5, 0, 0.5, and 0.7.

29 Panel A: High-Yield Bonds Mean Standard Deviation 1.4 Panel B: Stocks with Gaussian Jumps Mean Standard Deviation Figure2: M-V Frontiers from the Simulation Studies with LPM 1 Constraints (Solid: M-V portfolios; Dash: M-V portfolios with LPM 1 constraints) This figure plots the M-V frontier and the LPM 1 -constrained M-V frontiers. The thick dashed line represents a constant LPM 1 constraint set at The thin dashed line represents a variable LPM 1 constraint set at the minimum LPM 1 at each mean return level. When the constant constraint is set smaller, the thick dashed line contracts and the deviations occur closer to the global variance minimum portfolio. The opposite is true when the constraint is set higher.