Downside Loss Aversion and Portfolio Management

Robert Jarrowa and Feng Zhaob

September 2005

Abstract: Downside loss averse preferences have seen a resurgence in the port-

folio 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 as-

set 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 dif-

ferences 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.

aJarrow is from Johnson Graduate School of Management, Cornell University, Ithaca,

NY 14853 (raj15@cornell.edu). bZhao is from Rutgers Business School, Rutgers University,

Newark, NJ 07102 (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.

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 cur-

rent 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 flour-

ished 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 appro-

priate risk measure for downside loss-averse preferences. Second, we compare

2

optimal M-V portfolios with optimal M-LPM portfolios. The optimal portfo-

lios 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 nor-

mal 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 ob-

servations 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 port-

folio management and asset allocation literature. Bawa (1978) and Fishburn

(1977) introduced the lower partial moment (LPM) as an alternative risk mea-

sure to variance. Using insights from Kahneman and Tversky (1979)2, we show

1We can extend this argument to some non-Gaussian distributions. In general, as longas the downside risk measure, defined over the portfolio’s return distribution, is increasingin variance for all expected return levels, M-V portfolios are also optimal for downside lossaverse preferences.

2We use the kink-shaped utility function of prospect theory, but do not consider the prob-ability transformation used in the Cumulative Prospect Theory of Tversky and Kahneman

3

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 exam-

ple, VaR, conditional VaR, or expected shortfall (see Basak and Shapiro (2001)

and Rockafellar and Uryasev (2002)). Usually, these measures are not gener-

ated from investor preferences, but enter as constraints in the utility maximiza-

tion problem. In portfolio management, conditional VaR has been studied by

Krokhmal, Palmquist and Uryasev (2001). With respect to dynamic asset allo-

cation, 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 pro-

vides 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 thereturn distribution. In this study, we consider non-Gaussian distributions directly, withoutimposing the probability transformation.

4

2.1 Downside Loss Averse Utility Functions

An agent’s downside loss averse utility is defined over the portfolio’s return:

u(x) =

½f(x) for x ≥ a

f(x) + g(x) 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) = c1 + c2x

g(x) = −c3(a− x)n(2)

where ci ≥ 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 referencepoint, i.e. f 00 + g00 ≤ 0.

5

downside risk averse utility function given in Kahneman and Tversky (1979)4:

u(x) =

½x− a for x ≥ a

(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.” 5

2.2 Expected Utility

Given the downside loss utility function (1), letting the portfolio return X follow

the distribution FX , we can decompose expected utility into three parts:

E[u(X)] = f (E[X]) + {E[f(X)]− f (E[X])}

+

Z a

−∞g(x)dFX(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 referencelevel, (2) concavity in gains and convexity in losses, and (3) a steeper slope for losses.

5The “LINLIN” loss function is piecewise linear, which is identical to (3). The “LINEX”loss function is obtained by setting

f(x) = c1 + c2x

g(x) = c3 [exp (c4x)− c4x− 1](4)

with c1, c2 ≥ 0, c3 > 0, c4 6= 0.

6

of f , and the third part is named the Downside Risk Measure and DRMX ≡

− R a−∞ g(x)dFX(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) ≡pE[X2]−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))−DRMX . (6)

Under these hypotheses, expected utility is seen to be non-decreasing in E[X]

and non-increasing in both SD(X) and DRMX .

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-

6This characterization can also be generalized to higher order moments such as skewnessand kurtosis.

7

nent as

DRMX ≡ c3 · LPMn(a;FX) where LPMn(a;FX) ≡Z a

−∞(a− x)ndFX(x). (7)

Here, LPMn(a;FX) represents the n-th order LPM of the probability distribu-

tion FX with respect to the reference level a. In this case, expression (6) can

be written as7

E[u(X)] = c1 + c2E(X)− c3 · LPMn(a;FX). (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 =VaRα(X), the α-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

ESα(X) = − 1αE£X · 1{X≤ VaRα(X)}

¤.

Using the definition for LPM, we have

LPM1(VaRα(X);FX) = αESα(X) + αVaRα(X).

The recent risk management literature advocates the use of ES in portfolio

7 In this case, f(SD(X) = 0.

8

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 {Rk}Nk=1 and R0, respectively. Let wn denote the portfolio

weight invested in asset n. The set of possible returns is given by

Z =

(X =

NXk=0

wkRk :NXk=0

wk = 1

). (9)

Also, let Z0 = {X ∈ Z : w0 = 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.

supX∈Z

E[u(X)]. (10)

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, let X∗ be

9

the solution to (10), then it solves

minX∈Z

DRMX

s.t. E[X] ≥ µ

SD(X) ≤ s

(11)

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 DRMX0 <DRMX∗ .

From (6), E[u(X 0)] = f (E[X 0]) − f(SD(X))−DRMX0 . 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∗),DRMX∗) is a 2-dimensional

manifold in R3. We transform this frontier to¡f(E[X∗]), f(SD(X∗)),DRMX∗

¢for convenience. Denote d∗(µ, s) as the solution to (11) for some µ, s. First, we

need to show the convexity of the feasible set

©¡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

the form

c1 · f(µ) + c2 · f(s) + c3 · d = U0 for some constant U0.

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 prob-

lem can be viewed as that portfolio which minimizes downside risk, subject to an

expected return target µ and an upper bound s 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 program-

ming 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

minX∈Z

LPMn(a;FX)

s.t. E(X) ≥ µ.

(12)

Parallel to M-V analysis, M-LPM analysis can be studied by considering a

11

two-parameter efficient frontier (see Bawa (1975), Bawa and Lindenberg (1977),

and Harlow and Rao (1989)). The M-LPMn efficient frontier is the solution to

expression (12) for different (a, µ). If in (12), instead of X ∈ Z we have X ∈ Z0,

the efficient frontier is generated by only the risky assets. Different from the

M-V frontier, the M-LPMn 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 expres-

sion (3).

Corollary 3 Under expression (3), for a fixed a, the solution to (10) is on

the M-LPM1 efficient frontier for some (µ,LPM1(a;FX)). Conversely, for any

(µ,LPM1(a;FX)), 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,

DRMX = b · LPM1(a;FX).

From Proposition 1, any solution to (10) with the utility function (3) solves

(11), which degenerates to the M-LPM1 optimization problem.

Conversely, we show that the tangent point between the M-LPM1 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

the utility function (3) has the following form

E[X]− b · LPM1(a;FX) = U0 for some constant U0.

This is a straight line with intercept U0 and slope b. From Proposition 10 in the

appendix, the M-LPM1 frontier is convex in the mean. Therefore, for any point

(E[X∗],LPM1(a;FX∗)) on the M-LPM1 frontier there is a tangent indifference

curve for some U0 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 compu-

tation procedure and then present the results.

3.1 The Computational Procedure

To compute the solution to expression (12) in the general case, analytic expres-

sions for LPMn 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 func-

tion estimated from Monte-Carlo simulations. Alternatively, optimization of

larger asset portfolios could use the newly-developed “simulated optimization”

13

techniques (see Fu (2002)).

Denoting R = (R1, . . . , RN )0 as the vector of asset returns, the computation

of DRM involves the simulation of l copies ofnR(j)

olj=1. Given the portfolio

weights {wi}Ni=1 , we can compute l copies of the portfolio returns©X(j)

ªlj=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

1l

lXj=1

y(X(j))−Ey(X)

=⇒d Normal¡0, σ2 (y(X))

¢.

The accuracy of the estimate can be controlled by choosing the appropriate

simulation length l. In our case, we let l = 4× 106. The resulting accuracy for

our simulated objective function with a unit variance is 5×10−4. 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 10−5.

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

of the portfolio weights:

4w0 = |wMV − wMLPM |/w0

where the normalization represents an equally weighted portfolio, w0 = 1N . We

construct three measures to quantify the differences in the optimal portfolio

weights: sup¡4w0

¢, mean

¡4w0¢and median

¡4w0¢.

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

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 sub-

ject 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 LPM1 as the

downside risk measure. Although not reported here, we repeated our computa-

tions using LPM2, with qualitatively similar results. In general, the difference

is that higher order LPMs impose higher penalties on portfolios with larger

losses in the tail of the distribution.

3.2.1 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 = (1N −Θ) ·A+Θ · 0N

where the non-default return distribution, A are log-Normal with mean µA and

variance ΣA, 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

lios, we assume that the asset returns are independent. Thus, the correlation

among assets can be represented by ΣA, a diagonal matrix, and Θ, a vector of

independent Bernoulli random variables8.

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 ratings9(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 1.11. 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

8This can be generalized to the Bernoulli mixture. Let Ψ be a M × 1 vector with M ≤ Nand functions Ti : RM → [0, 1] for i = 1, ...,N. The Bernoulli mixture Θ is such that

P (Θi = 1|Ψ) = Ti(Ψ)

for i = 1, ..., N.9To 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

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, the M-V and M-

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 10−5. 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

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.

3.2.2 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.

lnSt+4t − lnSt = µ4 t+ σεt+4t

p4t+

M(λ4t)Xm=1

Jm

where µ and σ are constants, and εt+4t is an Normal(0, 1) random variable.

The number of jumps M that occur during the interval 4t is a Poisson random

10Note that the M-V frontier is concave in the mean return. The locus of the M-LPMportfolios is not necessarily concave in M-V space, although it is in M-LPM space. Whenthe LPM constraint is constant, the constrained M-V frontier is concave, but not when theconstraint varies across the mean return levels.

19

variable with parameter λ4 t. The jump size is given by Jm. We assume that

Jm 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 year11 .

We let the five stocks have Gaussian jump sizes with mean -0.9, -0.5, 0, 0.5,

0.7, respectively12 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

11We also set the average jump frequency to be once every two years and once every tenyears, with similar results. When the jumps are less frequent, i.e. less skewed return distrib-utions, 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 zeroif the M-V portfolios have zero LPM.12Because 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, Longstaffand Pan (2003) used symmetric jump sizes.

20

four representative (one plus) mean return levels for the optimal portfolios are

1.07, 1.17, 1.27, and 1.37. 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 compo-

nent 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.

Figures 2.B shows the difference between the M-V portfolios with and with-

out the LPM constraints in M-V space. For the reasons discussed earlier, we

set the constraint equal to 10−5. Similar to the high-yield bonds scenario, the

unconstrained M-V and constrained M-V portfolios coincide when the LPM con-

straint 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

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 well-

studied 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 Mea-

sures of Risk,” Mathematical Finance, v9, n3 (July), 203-28.

[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, 371-405.

22

[3] Bawa, V.S. (1975), ”Optimal rules for ordering uncertain prospects,” Jour-

nal of Financial Economics, 2, 95-121.

[4] Bawa, V.S. (1978), “Safety first, Stochastic dominance and optimal portfo-

lio selection,” Journal of Financial and Quantitative Analysis, 13, 255-271.

[5] Bawa, V.S. and Lindenberg, E.B. (1977), Capital market equilibrium in a

mean-lower partial moment framework,” Journal of Financial Economics,

5, 189-200.

[6] Bielecki, T. and M. Rutkowski (2001), Credit Risk: Modeling, Valuation

and Hedging, Springer Verlag, New York.

[7] Carty, L. V. and D. Lieberman(1997), “Historical Default Rates of Corpo-

rate Bond Issuers, 1920 — 1996", Moody’s Investors Service, Global Credit

Research, Jan., 1997

[8] Christoffersen, P.F. and Diebold, F.X. (1997), “Optimal Prediction Under

Asymmetric Loss,” Econometric Theory, Volume 13, 808-817.

[9] Cuoco, D., He, H. and Issaenko, S. (2002), “Optimal Dynamic Trading

Strategies with Risk Limits,” SSRN working paper.

[10] Fishburn, P.C. (1977), “Mean-risk analysis with risk associated with below-

target returns,” The American Economic Review, 116-126.

[11] Frey, R. and A. J. McNeil (2002), “VaR and expected shortfall in portfolios

of dependent credit risks: Conceptual and practical insights,” Journal of

Banking & Finance, 26, 1317-1334.

23

[12] Fu, M. (2002), “Optimization for Simulation: Theory vs. Practice" (Feature

Article), INFORMS Journal on Computing, Vol.14, No.3, 192-215.

[13] Granger, C. W. J. (1969), “Prediction with a Generalized Cost of Error

Function,” Operational Research Quarterly, 20, 199-207.

[14] Grootveld, H. and Hallerbach W. (1999), “Variance vs downside risk: Is

there really that much difference?” European Journal of Operational Re-

search, 114, 304-319.

[15] Harlow, W. V. and Rao, R.K.S. (1989), “Asset pricing in a generalized

mean-lower partial moment framework,” Journal of Financial and Quan-

titative Analysis, 3, 285-311.

[16] Jarrow, R. (2002). “Put Option Premiums and Coherent Risk Measures,”

Mathematical Finance, v12, n2 (April), 135-42.

[17] Kahneman, D. and Tversky, A.(1979), “Prospect Theory: An analysis of

decision under risk,” Econometrica, XLVII, 263-291.

[18] Krokhmal, P., Palmquist, J. and Uryasev, S. (2002), “Portfolio Optimiza-

tion with Conditional Value-at-Risk Objective and Constraints”, The Jour-

nal of Risk, V. 4, # 2.

[19] Leland, H. (1999), “Beyond Mean-Variance: Performance Measurement in

a Nonsymmetrical World,” Financial Analysts Journal, January/February,

27-36.

24

[20] Liu, J., Longstaff, F. and Pan, J. (2003), “Dynamic Asset Allocation with

Event Risk?”, Journal of Finance, February, v58, n1, 231-259.

[21] Meyer J. (1987), “Two-Moment Decision Models and Expected Utility

Maximization,” The American Economic Review, June, 421-430.

[22] Pedersen, C. S.(2002), “Derivatives and Downside Risk,” Derivatives Use,

Trading and Regulation (forthcoming).

[23] Rockafellar, R.T. and Uryasev S. (2002), “Conditional Value-at-Risk for

General Loss Distributions”, Journal of Banking and Finance, 26/7, 1443-

1471

[24] Steinbach, M. C. (2001), “Markowitz Revisited: Mean-Variance Models in

Financial Portfolio Analysis,” SIAM ,vol. 43, 31-85.

[25] Tasche, D. (2002), “Expected shortfall and beyond,” Journal of Banking &

Finance, 26, 1519-1533.

[26] Tversky, A. and Kahneman, D. (1992), “Advances in Prospect Theory: Cu-

mulative Representation of Uncertainty”, Journal of Risk and Uncertainty,

5, 297-323.

[27] Varian, H. (1974). “Equity, Envy and Efficiency,” Journal of Economic

Theory, 9, 63-91.

[28] Zellner, A. (1986), “Bayesian Estimation and Prediction Using Asymmetric

Loss Functions,” Journal of the American Statistical Association, 81, 446-

451.

25

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 -0.5. Panel B reports the three distance measures defined over the portfolio weights. We compare the normalized differences of the vector of portfolio weights, ∆w0 = | wM-V – wM-LPM|/w0, where the normalization corresponds to an equally weighted portfolio with w0 = 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 1.11. Panel A: Simulation Parameters and Bonds Statistics

Bond 1 2 3 4 5

Default Prob. 0% 0.3% 1% 3% 7%

Mean 1.0300 1.0355 1.0479 1.0823 1.1439

Std. Dev. 0.1007 0.1155 0.1456 0.2145 0.3285

5% Quantile 0.8733 0.8796 0.8933 0.9263 0

LPM1(0.5) 0 0.0015 0.005 0.015 0.035

Panel B: Distance Measures between the M-V and M-LPM Optimal Portfolios

Mean Return Level 1.05 1.07 1.09 1.11

Sup(∆w0) 2.36 1.88 1.31 0.78

Mean(∆w0) 1.16 0.99 0.90 0.35

Median(∆w0) 0.96 0.88 0.88 0.36

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 -0.5. 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, ∆w0 = | wM-V – wM-LPM|/w0, where the normalization corresponds to an equally weighted portfolio with w0 = 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 1.37. Panel A: Simulation Parameters and Stocks Statistics

Stock 1 2 3 4 5

Mean 1.472 1.193 1.063 1.263 1.538

Std. Dev. 0.985 0.532 0.160 1.090 2.671

5% Quantile 0.167 0.372 0.821 0.540 0.443 LPM1(0.5) 0.0454 0.0146 0.0000 0.0006 0.0069

Panel B: Distance Measures Between the M-V and M-LPM Optimal Portfolios

Mean Return Level 1.07 1.17 1.27 1.37

Sup(∆w0) 0.101 1.019 1.480 2.483

Mean(∆w0) 0.043 0.507 0.932 1.831

Median(∆w0) 0.026 0.427 0.851 2.093

Panel A: High-Yield Bonds

1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

(1) Mean Return 1.05

1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

(2) Mean Return 1.07

1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

(3) Mean Return 1.09

1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

(4) Mean Return 1.11

Panel B: Stocks with Gaussian Jumps

1 2 3 4 5-0.2

00.20.40.60.8

(1) Mean Return 1.07

1 2 3 4 5-0.2

00.20.40.60.8

(2) Mean Return 1.17

1 2 3 4 5-0.2

00.20.40.60.8

(3) Mean Return 1.27

1 2 3 4 5-0.2

00.20.40.60.8

(4) Mean Return 1.37

Figure 1: Comparison of Optimal Portfolio Weights (Black: M-V; White: M-LPM1) 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.

0.05 0.1 0.15 0.2 0.25 0.30.9

0.95

1

1.05

1.1

1.15

Standard Deviation

Mea

n

Panel A: High-Yield Bonds

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.7

0.8

0.9

1

1.1

1.2

1.3

1.4Panel B: Stocks with Gaussian Jumps

Standard Deviation

Mea

n

Figure2: M-V Frontiers from the Simulation Studies with LPM1 Constraints

(Solid: M-V portfolios; Dash: M-V portfolios with LPM1 constraints) This figure plots the M-V frontier and the LPM1-constrained M-V frontiers. The thick dashed line represents a constant LPM1 constraint set at 10-5. The thin dashed line represents a variable LPM1 constraint set at the minimum LPM1 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.