Downside Loss Aversion and Portfolio Management Robert Jarrow a and Feng Zhao b September 2005 Abstract: Downside lossaverse 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. a Jarrow is from Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853 ([email protected]). b Zhao is from Rutgers Business School, Rutgers University, Newark, NJ 07102 ([email protected]). We thank David Hsieh (the editor), the associate editor and an anonymous referee for the helpful suggestions. We are responsible for any remaining errors.
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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 ([email protected]). bZhao is from Rutgers Business School, Rutgers University,
Newark, NJ 07102 ([email protected]). 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
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
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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,
[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
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.