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Risk Management with Stress Testing: Implications for

Portfolio Selection and Asset Pricing

Gordon J. Alexander

University of Minnesota

Alexandre M. Baptista

The George Washington University

June 22, 2006


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Risk Management with Stress Testing: Implications for

Portfolio Selection and Asset Pricing

Abstract

Stress Testing (ST) is often used by banks and securities firms to set risk exposure

limits. Accordingly, we examine a model with an agent who faces K binding ST constraints

and another who does not. We obtain four results. First, the constrained agents optimal

portfolio exhibits (K+2)-fund separation. Second, the effect of the constraints on the optimal

portfolio is identical to that of an adjustment in the expected returns of the risky securities

that tends to lower them, thereby increasing the optimal portfolios weight in the riskfree

security (or the minimum variance portfolio when this security is not available). Third, the

market portfolio is inefficient. Fourth, a securitys expected return is affected by both its

systematic risk and its idiosyncratic returns in the states used in the constraints. Thus, we

provide further motivation to the literature in which security prices are not solely driven bysystematic risk.

JEL classification: G11; G12; D81

Keywords: Stress Testing; Portfolio Selection; Asset Pricing; Idiosyncratic Returns


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1. INTRODUCTION

Over the past ten years, some investors have suffered huge losses due to extreme events.

For example, Barings Bank failed in 1995, Long Term Capital Management (LTCM) collapsed

in 1998, and Enron went bankrupt in 2001. Furthermore, the terrorist attacks in the U.S.

(2001), Spain (2004), and the U.K. (2005) have tremendously affected financial markets.

Since the occurrence of these events, the importance of risk management has been ex-

tensively recognized by banks and securities firms when deciding the amount of risk they

are willing to take. Moreover, bank regulators now put an emphasis on risk management

practices in attempting to reduce the fragility of banking systems.

Of the risk management tools currently available, Value-at-Risk (VaR) and Stress Testing

(ST) have emerged as two of the most popular. For example, under the Basle Capital Accord,

VaR is used in setting the minimum capital requirement associated with a banks exposure

to market risk. Furthermore, the Committee on the Global Financial System (2005, pp. 1,

15) of the Bank for International Settlements and Scholes (2000) note that ST is often used

by banks and securities firms to set risk exposure limits.

While the previous literature examines the impact of using VaR as a risk management

tool on portfolio selection and asset pricing (see, e.g., Alexander and Baptista (2002)), it

has yet to similarly explore the impact of using ST constraints. Our paper fills this gap

in the literature by providing parsimonious characterizations of (1) optimal portfolios in

the presence of ST constraints and (2) equilibrium security expected returns in a two-agent

economy where one agent faces these constraints and the other does not.

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An examination of the impact of ST constraints is of particular interest for several reasons.

First, as noted earlier, financial institutions often use them to set risk exposure limits. Second,

even when a financial institution holds capital in excess of the minimum capital requirements

as determined by the Basle Capital Accord, the ST constraints may still be binding. This

can happen since ST captures extreme events where losses can be very large. Third, since

there is empirical evidence that equity returns have skewness and kurtosis (see, e.g., Harvey

and Siddique (2000) and Dittmar (2002)), an ST constraint is a device that can be utilized to

control portfolio skewness and kurtosis. Fourth, the use of ST constraints can be motivated

by the goal of limiting losses arising from the need to unwind positions in markets that

become illiquid as a result of extreme events such as the LTCM collapse. Finally, since

the return distributions of portfolios can resemble those of certain options strategies (see,

e.g., Merton (1981) and Jagannathan and Korajczyk (1986)), the use of ST constraints can

mitigate the option-like features in these distributions.

In investigating the impact of ST constraints, we use Markowitzs (1952, 1959) mean-

variance model. There are important reasons for doing so. First, this model is the corner-

stone of portfolio theory. Second, it is widely used in practice to (1) determine optimal asset

allocations, (2) measure gains from international diversification, and (3) evaluate portfolio

performance. Third, since a set of ST constraints captures some measure of risk beyond vari-

ance (i.e., the returns in extreme events), a mean-variance objective function is of interest.

Finally, the model has been extensively used in the banking literature.

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Regardless on whether a riskfree security is available, we obtain four main results. First,

the constrained agents optimal portfolio exhibits (K + 2)-fund separation, where K is the

number of binding ST constraints. While this result might not seem to be particularly

surprising, Alexander and Baptista (2006a) have shown that under certain conditions a

constrained agents optimal portfolio still exhibits two-fund separation in the presence of

a binding risk management constraint.

Second, the effect of the constraints on the optimal portfolio is identical to that of an

adjustment in the expected returns of all risky securities that tends to lower them. Hence,

the constraints tend to increase the optimal portfolios weight in the riskfree security (or the

minimum variance portfolio when this security is not available).

Third, the market portfolio is inefficient. The reason why this result holds is that only

two of the K+ 2 funds required for the constrained agents optimal portfolio are efficient.

Fourth, a securitys expected return is affected by both its systematic risk (i.e., beta)

and its idiosyncratic returns in each one of those states that are used in the constraints.

Specifically, securities with negative (positive) idiosyncratic returns in these states have rel-

atively high (low) expected returns in equilibrium. The intuition for this result is straight-

forward. Due to the constraints, it is costly (beneficial) for the constrained agent to hold

securities with negative (positive) idiosyncratic returns in these states. Accordingly, the

agent requires them to have relatively high (low) expected returns.

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Next, we illustrate our theoretical results with a simple example. In this example, there

are five securities (one of them is riskfree), two agents (each one has half of the wealth in

the economy), and two binding ST constraints. Our main findings are as follows. First,

since there are two constraints, the constrained agents optimal portfolio exhibits four-fund

separation. Second, the portfolios weights in the two inefficient funds are notable, which

cause it to have an efficiency loss of 3.79%. Third, since the market portfolios weights in

these funds are smaller than those of the constrained agents optimal portfolio, the market

portfolio has an efficiency loss of only 0.63%. Fourth, the effect of the constraints on the

optimal portfolio is identical to that of a downward adjustment in the expected returns

of the risky securities ranging from 2.81% to 9.23%. Thus, the weight of the constrained

agents optimal portfolio in the riskfree security is substantially larger than that of the

unconstrained agents optimal portfolio. Fifth, the cost of ST as measured by the reduction

in the certainty equivalent return incurred by selecting the constrained portfolio is 3.16%.

Finally, the component of a securitys expected return that arises from its idiosyncratic

returns can be notable. This component is 2.42% for one of the securities that has negative

idiosyncratic returns in both of the states used in the constraints. However, the component

is 1.25% for one of the other securities since it has positive idiosyncratic returns in these

states. In sum, by exploring the implications of ST constraints, we contribute to the literature

in which security prices are not solely driven by systematic risk.

Previous theoretical papers in this literature have recognized the importance of idiosyn-

cratic risk. For example, Levy (1978), Merton (1987), and Malkiel and Xu (2002) develop

models in which security expected returns depend on both systematic and idiosyncratic risk.

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The empirical literature has also examined the importance of idiosyncratic risk. Campbell,

Lettau, Malkiel, and Xu (2001) and Malkiel and Xu (2003) show that the risk of individual

stocks has noticeably increased over time. Goyal and Santa-Clara (2003) provide evidence of

a significant positive relation between (1) a measure of idiosyncratic risk given by the equal-

weighted average stock risk and (2) the market return. However, Bali, Cakici, Yan, and

Zhang (2005) find no evidence of a significant relation between the value-weighted average

stock risk and the market return. Moreover, Guo and Savickas (2006) show that when the

value-weighted average stock risk and aggregate stock market risk are jointly used to forecast

the market return, there is a significant negative relation between the value-weighted average

stock risk and the market return. Ang, Hodrick, Xing, and Zhang (2006a,b) examine the

cross-sectional relation between the idiosyncratic risk of individual stocks and their returns,

and find that U.S. and other developed-market stocks with higher idiosyncratic risk earn

lower average returns.

Two features of our work differ from the idiosyncratic risk literature. First, the reason

why idiosyncratic returns affect expected returns in our model (i.e., the ST constraints) is,

to the best of our knowledge, novel. Second, our model captures the effects of idiosyncratic

returns in some states (i.e., those used in the ST constraints) on asset pricing while the

models in the literature capture the effects of idiosyncratic risk on asset pricing. Hence, the

model is able to generate equilibrium expected returns consistent with the evidence provided

by Ang, Hodrick, Xing, and Zhang (2006a,b) that stocks with high idiosyncratic risk earn

low average returns. For example, consider a security with (1) high idiosyncratic risk and

(2) positive idiosyncratic returns in the states used in the ST constraints. Consistent with

empirical evidence, our model predicts that such a security has a relatively low expected

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return.

Since our work derives the effects of a certain measure of risk beyond variance on asset

pricing, it is important to emphasize that there is an extensive literature that examines the

effects of skewness and kurtosis. In a seminal paper, Kraus and Litzenberger (1976) develop

a model that captures the effect of unconditional skewness. Lim (1989) tests their model

using stock returns and provides evidence that skewness is priced. More recently, Harvey and

Siddique (2000) explore a model where conditional skewness is priced and present empirical

evidence that it is helpful in explaining the cross-sectional variation in stock returns. Finally,

Dittmar (2002) develops a framework in which agents are averse to kurtosis.

Our work differs from this literature in two respects. First, the importance of a measure

of risk beyond variance arises in our model due to the existence of ST constraints while it

arises in other models from agents having either a preference for right-skewed portfolios over

left-skewed portfolios or an aversion to kurtosis. Second, our model captures the effects of

the returns in just those states used in the ST constraints on asset pricing while other models

use the returns in all states to capture the effects of skewness and kurtosis.

Also related to our work are papers that investigate portfolio selection when an agent

keeps his or her wealth above a floor. For example, Black and Perold (1992) and Grossman

and Vila (1992) examine the case when the floor is non-stochastic, while Grossman and Zhou

(1993) and Cvitanic and Karatzas (1995) examine the case when the floor is stochastic. Our

work differs from these papers in several important ways. First, we examine the impact of a

set of ST constraints rather than that of a floor. Second, we derive equilibrium results in

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the presence of ST constraints, while the aforementioned papers focus on portfolio selection

when an agent keeps his or her wealth above a floor. Finally, we use the mean-variance model

rather than the expected utility maximization continuous-time model.

The paper proceeds as follows. Section 2 characterizes optimal portfolios and equilib-

rium security expected returns when a riskfree security is present. Section 3 examines the

case when a riskfree security is absent. Section 4 provides an example that illustrates our

theoretical results and Section 5 concludes. The Appendix contains the proofs.

2. THE MODEL

2.1. Securities

Consider an economy where uncertainty is represented by S states. There are J risky

securities and a riskfree security with return R . The returns of the risky securities are given

by a J S matrix R, with R denoting the return of security j in state s. Let R be the

J 1 expected return vector and V be the J J variance-covariance matrix associated with

R. Let R [R R ] . Suppose that (1) there is no arbitrage; (2) rank(V) = J

so that there are no redundant securities; and (3) rank([1 R R R ]) = J for

any set ofJ 2 distinct states, s ,...,s , where 1 denotes the J 1 vector [1 1] .

A portfolio is a (J+ 1) 1 vector w with w = 1 w 1, where w [w w ] .Let R denote the return of portfolio w in state s. Let R and denote, respectively, the

expected return and variance of w.

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2.2. Agents

There are two agents with a mean-variance objective function U : R R R defined

by

U(R, ) = R

2 , (1)

where > 0. The unconstrained agent has = . The constrained agent has = , and

is restricted to select a portfolio w that satisfies K binding Stress Testing (ST) constraints:

R T , s = s ,...,s ,

where (i) 1 K J 2, (ii) s ,...,s are distinct states, and (iii) T ,...,T are possibly

distinct bounds in states s ,...,s , respectively.

2.3. Portfolio Selection Implications

Next, we characterize the optimal portfolios of unconstrained and constrained agents.

2.3.1. Unconstrained Agent

A portfolio is on the mean-variance boundary if there is no other portfolio with the same

expected return and a smaller variance. A portfolio is efficient if it lies on this boundary and

has an expected return equal to or greater than R . Otherwise, the portfolio is inefficient.

Let A 1 V R, B R V R, and C 1 V 1. Let w [0 0 1] and

w 0 denote the riskfree and tangency portfolios. Portfolios w and ware useful to characterize the unconstrained agents optimal portfolio.

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Theorem 1. The unconstrained agents optimal portfolio is

w = (1 )w + w , (2)

where .

Theorem 1 says that the unconstrained agents optimal portfolio w exhibits two-fund

separation. Since these funds are w and w , w is efficient. Note that if R is smaller

(larger) than A/C, then w is efficient (inefficient) and is positive (negative). Figure 1

illustrates Theorem 1 when > 0. Note that w , w , and w , represented by, respectively,

points f, t, and u, lie on the line representing the efficient frontier.

2.3.2. Constrained Agent

Let F 1 V R and w

0

where s = s ,...,s . Portfolios

w , ...,w are useful to characterize the constrained agents optimal portfolio.

Theorem 2. The constrained agents optimal portfolio is

w =

1

w + w +

w , (3)

where , for s = s ,...,s , and ,..., are Lagrange multipliers

associated with the ST constraints.

Theorem 2 says that the constrained agents optimal portfolio w exhibits (K+ 2)-fund

separation. Since these funds are w , w , and w ,...,w where the last K funds are

inefficient, w is also inefficient.

Figure 1 illustrates Theorem 2 when K = 1 and < 0. Note that w and w ,

represented by, respectively, points s and c, lie below the efficient frontier. Using Equation

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(3), we have

w = (1 )w + w , (4)

where + and w . Note that w , represented by point x, lies on the

dashed hyperbola representing combinations of w and w . As Equation (4) implies, w

lies on the dotted line representing combinations of w and w .

To isolate the impact of the ST constraints on the optimal portfolio from that of, assume

that = . Using Theorems 1 and 2, we have = . Thus, the constraints do not affect

the optimal portfolios weight in fund w . Furthermore, since the fund weights of each of the

optimal portfolios sum to one, we have = (1 ) 1 . That is,the sum of w s weights in funds w ,...,w is equal to the difference between the weights

of w and w in fund w .

The following result further explores the effect of the constraints on the optimal portfolio.

Theorem 3. Suppose that = . The constrained agents optimal portfolio when the

expected return vector is R coincides with the unconstrained agents optimal portfolio when

the expected return is R R +

(R 1R ).

Using Theorem 3, the effect of the ST constraints on the optimal portfolio is identical

to that of an adjustment in the expected return vector R. Since > 0 for s = s ,...,s ,

security js adjusted expected return R is smaller (larger) than its expected return R if

R is smaller (larger) than R for s = s ,...,s . In practice, a state is chosen to be in an

ST constraint only if there are some securities with negative returns in the state. Hence, the

case when R R for j = 1,...,J and s = s ,...,s is not plausible.

In the case when R < R for j = 1,...,J and s = s ,...,s , the expected returns

of all risky securities are adjusted downward. Thus, fund w is relatively more attractive.

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Consequently, the optimal portfolios weight in w tends to be larger in the presence of the

constraints.

Consider now the case when R < R < R for some distinct securities j , j

{1,...,J} and states s, s {s ,...,s }. In this case, (1) the expected return of at least

one risky security can be adjusted downward and (2) the expected return of at least one

risky security can be adjusted upward. Thus, fund w can be either relatively more or less

attractive. Consequently, there is no clear tendency for the optimal portfolios weight in w

in the presence of the constraints.

2.4. Asset Pricing Implications

Letting m R denote the market portfolio and denote the covariance between

security j and m, security js beta is / . Furthermore, let be the constrained

agents fraction of the wealth in the economy, where 0 < < 1.

Theorem 4. In equilibrium, the market portfolio is inefficient. Furthermore, security js

expected return is

R = R + (R R )

[(R R ) (R R )] , (5)

where

/

(1 )/ + /, k = 1,...,K, (6)

is a positive constant.

Theorem 4 says that in equilibrium the market portfolio is inefficient. The intuition for

this result is straightforward. Since (1) the unconstrained agents optimal portfolio is efficient,

(2) the constrained agents optimal portfolio is inefficient, and (3) the market portfolio is a

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combination of these optimal portfolios, the market portfolio is inefficient. Figure 1 illustrates

that when K = 1, the market portfolio, represented by point m, lies below the efficient frontier

on the dashed hyperbola representing combinations of w and w .

Equation (5) indicates that security js expected return depends on K + 2 terms. The

first two terms, R and (R R ), are identical to those contained in the CAPM.

However, the last K terms, which are subtracted from the sum of the first two terms, are not

present in the CAPM. Each one of these K terms is given by the product of: (1) security js

idiosyncratic return in state s , (R R ) (R R ), and (2) the risk premium on

the idiosyncratic return in state s , , where k {1,...,K}. We refer to the sum of these

K terms as the idiosyncratic return adjustment.

Since > 0 for k = 1,...,K, whether security js expected return is larger than, equal

to, or smaller than that in the CAPM depends on its idiosyncratic returns in states s ,...,s .

First, if security js idiosyncratic return is zero in states s ,...,s , then its expected return

is equal to that in the CAPM. Second, if security js idiosyncratic return is (1) either zero or

positive in states s ,...,s , and (2) positive for some state s {s ,...,s }, then its expected

return is smaller than that in the CAPM. Third, if security js idiosyncratic return is (1)

either zero or negative in states s ,...,s , and (2) negative for some state s {s ,...,s },

then its expected return is larger than that in the CAPM. Finally, if security js idiosyncratic

return is negative in some state s {s ,...,s } and positive in some state s {s ,...,s },

then its expected return can be smaller than, equal to, or larger than that in the CAPM.

The intuition for why securities with negative (positive) idiosyncratic returns in states

s ,...,s have relatively high (low) expected returns in equilibrium is straightforward. Due

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to the constraints, it is costly (beneficial) for the constrained agent to hold securities with

negative (positive) idiosyncratic returns in these states. Accordingly, the agent requires them

to have relatively high (low) expected returns.

Using Equation (6), the risk premium on the idiosyncratic return in state s depends on

two constants: (1) and (2) . The first constant measures the constrained

agents marginal cost (in terms of utility) arising from marginally decreasing the bound T

(i.e., tightening the ST constraint in state s ). The second constant measures the constrained

agents fraction of the risk-aversion-adjusted wealth in the economy. Note that if = ,

then this constant is equal to .

3. ABSENCE OF A RISKFREE SECURITY

Suppose now that there is no riskfree security. Let w and w denote, re-

spectively, the minimum variance portfolio and the portfolio on the mean-variance boundary

with expected return B/A.


Next, we characterize the optimal portfolios of unconstrained and constrained agents.


In the absence of a riskfree security, a portfolio is efficient if it lies on the mean-variance

boundary and has an expected return equal to or greater than A/C (i.e., that of w ).

Theorem 5. The unconstrained agents optimal portfolio is

w = (1 )w + w , (7)

where .

Theorem 5 says that the unconstrained agents optimal portfolio w exhibits two-fund

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separation. Since these funds are w and w , w is efficient. Figure 2 illustrates Theorem 5.

Note that w , w , and w , represented by, respectively, points a, b, and u, lie on the upper

part of the thick hyperbola representing the mean-variance boundary.


Let w for s = s ,...,s . Portfolios w ,...,w are useful to characterize the

constrained agents optimal portfolio.

Theorem 6. The constrained agents optimal portfolio is

w = 1 w + w + w , (8)where , for s = s ,...,s , and ,..., are Lagrange multipliers associ-

ated with the ST constraints.

Theorem 6 says that the constrained agents optimal portfolio w exhibits (K+ 2)-fund

separation. Since these funds are w , w , and w ,...,w where the last K funds are

inefficient, w is also inefficient.

Figure 2 illustrates Theorem 6 when K = 1 and < 0. Note that w and w ,

represented by, respectively, points s and c, lie below the efficient frontier. Using Equation

(8), we have

w = (1 )w + w , (9)

where + and w . Note that w , represented by point x, lies on the

dashed hyperbola representing combinations of w and w . As Equation (9) implies, w

lies on the dotted hyperbola representing combinations of w and w .

To isolate the impact of the ST constraints on the optimal portfolio from that of, assume

that = . Using Theorems 5 and 6, we have = . Thus, the constraints do not affect

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the optimal portfolios weight in fund w . Furthermore, since the fund weights of each of the

optimal portfolios sum to one, we have

= (1 )

1

. That is,

the sum of w s weights in funds w ,...,w is equal to the difference between the weights

of w and w in fund w .

The following result further explores the effect of the constraints on the optimal portfolio.

Theorem 7. Suppose that = . The constrained agents optimal portfolio when the

expected return vector is R coincides with the unconstrained agents optimal portfolio when

the expected return is R R + R .Using Theorem 7, the effect of the ST constraints on the optimal portfolio is identical

to that of an adjustment in the expected return vector R. Since > 0 for s = s ,...,s ,

security js adjusted expected return R is smaller (larger) than its expected return R if

R is smaller (larger) than zero for s = s ,...,s . Note that the case when R 0 for

j = 1,...,J and s = s ,...,s is not plausible since in practice a state is chosen to be in an

ST constraint only if there are some securities with negative returns in the state.

In the case when R < 0 for j = 1,...,J and s = s ,...,s , the expected returns of

all risky securities are adjusted downward. Thus, fund w is relatively more attractive.

Consequently, the optimal portfolios weight in w tends to be larger in the presence of the

constraints.

Consider now the case when R < 0 < R for some distinct securities j , j {1,...,J}

and states s, s {s ,...,s }. In this case, (1) the expected return of at least one security can

be adjusted downward and (2) the expected return of at least one security can be adjusted

upward. Thus, fund w can be either relatively more or less attractive. Consequently, there

is no clear tendency for the optimal portfolios weight in w in the presence of the constraints.

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We now characterize the equilibrium.

Theorem 8. In equilibrium, the market portfolio is inefficient. Furthermore, security js

expected return is

R = R + (R R) R R , (10)where

R 1 (1 )A/ A/

F /

(1 )/ + / 1

C(11)

and is a positive constant as defined in Equation (6).

Theorem 8 says that the market portfolio is inefficient. The intuition for this result is

similar to that of Theorem 4. Figure 2 illustrates that the market portfolio, represented by

point m, lies below the efficient frontier on the thin hyperbola representing combinations of

wand

w.

Equation (10) indicates that security js expected return depends on K+ 2 terms. The

first two terms are related to those in the Black model. However, the last K terms, which

are subtracted from the sum of the first two terms, are not present in the Black model. Each

one of these K terms is given by the product of: (1) the idiosyncratic return of security j

in state s , R R , and (2) the risk premium on the idiosyncratic return in state s ,

, where k {1,...,K}.

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Since > 0 for k = 1,...,K, securities with negative (positive) idiosyncratic returns in

states s ,...,s have relatively high (low) expected returns in equilibrium. The intuition for

this result is similar to that presented for Theorem 4.

4. EXAMPLE

In this section, we illustrate our theoretical results using a simple example.

4.1. Securities

There are four risky securities (j = 1, 2, 3, 4) and a riskfree security (j = 5). The risky

securities are assumed to be in positive net supply, while the riskfree security is assumed

to be in zero net supply. Panel (a) of Table 1 presents their expected returns and standard

deviations. For simplicity, assume that the correlation coefficient between the returns of each

pair of distinct risky securities is equal to 0.4. As explained shortly, the constrained agent

faces two ST constraints which use states s and s . Panel (b) of Table 1 provides the security

returns in these states. These returns are notably negative, but nevertheless plausible. For

example, the Nasdaq index declined by 27.1% and 22.9% in, respectively, October 1987 and

November 2000. It is important to emphasize that the qualitative results in our example do

not depend on the assumptions that are imposed on (1) the existence of a riskfree security,

(2) the net supply of the riskfree security, (3) the distribution of security returns, and (4)

the number of ST constraints.

4.2. Agents

The unconstrained and constrained agents have the objective function defined by Equa-

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tion (1) with = = = 2. The constrained agent faces two ST constraints:

R T , s = s , s ,

where T = 3% and T = 4%. The reason why we have T > T is that the returns of

three out of the four risky securities in state s are larger in absolute terms than those in

state s . Nevertheless, examples with similar results can be constructed when T T .


Table 2 presents the optimal portfolios of unconstrained and constrained agents.


The first row of panel (a) indicates that the unconstrained agents optimal portfolio w

is characterized by weights = 36.93% and = 136.93% in, respectively, funds w and

w . The first row of panel (b) shows that w s weights in the risky securities range from

28.50% (security 4) to 41.42% (security 1). The first row of panel (d) says that w s expected

return and standard deviation are, respectively, 16.32% and 25.07%. Moreover, w s returns

in states s and s , at 21.02% and 25.74%, are notably negative.


The second row of panel (a) indicates that the constrained agents optimal portfolio w

is characterized by weights = 36.93%, = 136.93%, = 35.13%, and = 38.74%

in, respectively, funds w , w , w , and w . Note that the weights of w and w in fund w

are identical. Furthermore, w s weight in fund w is not only notably larger than w s, it

is also positive. The intuition for this result is simple. Since w s returns in states s and

s are positive, a relatively large weight in w is useful to meet the ST constraints. Further

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intuition can be seen in panel (c), which provides the adjustment in expected returns with the

same effect on the optimal portfolio as the constraints. Observe in line 2 that the expected

return of all risky securities are notably adjusted downward. Since w is now relatively more

attractive, the constraints cause the optimal portfolios weight in w to increase substantially.

The second row of panel (b) shows that w s weights in the risky securities range from

13.13% (security 1) to 44.52% (security 2). Note that w s weights in securities 1, 3, and 4

are smaller than those of w , while that in security 2 is larger. The intuition for these results

is simple. The returns of securities 1, 3, and 4 in states s and s are larger in absolute terms

than those of security 2. Thus, the ST constraints force a reduction of the weights in securities

1, 3, and 4. Panel (c) provides further intuition. The largest adjustment in the expected

return occurs for securities 1 (9.23%), 3 (5.24%), and 4 (3.31%). Consequently, the

ST constraints lead to a reduction of the weights in these securities. Since security 1 has a

relatively small adjusted expected return (5.77%), w s weight in this security is negative.

In contrast, since security 2 has a relatively large adjusted expected return (10.39%), w s

weight in this security is positive.

The second row of panel (d) says that w s expected return and standard deviation are,

respectively, 8.79% and 13.84%. Thus, they are notably smaller than those of w . It can be

seen that w s returns in states s and s meet the ST constraints. Since w is inefficient,

of particular interest is its efficiency loss, denoted by , which we measure in terms of

standard deviation. That is, w s efficiency loss is the increase in standard deviation arising

from selecting w instead of the efficient portfolio with the same expected return. Note that

w s efficiency loss is = 3.79%.

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4.3.3. Cost of Stress Testing

Next, we assess the cost of ST. In doing so, we measure the cost of ST by the reduction

in the certainty equivalent return incurred by the agent who selects the constrained opti-

mal portfolio instead of the unconstrained optimal portfolio. Since the certainty equivalent

returns of the unconstrained and constrained optimal portfolios are, respectively, 10.03%

and 6.87%, the cost of ST is 3.16%. That is, the protection provided by imposing the ST

constraints is procured at the cost of a reduction in the certainty equivalent return of 3.16%.


Suppose that the constrained agents fraction of the wealth in the economy is = 50%.

4.4.1. Market Portfolio

The third row of panel (a) indicates that the market portfolio m is characterized by

weights = 136.93%, = 17.56%, and = 19.37% in, respectively, funds w , w ,

and w . The third row of panel (b) shows that ms weights in the risky securities range

from 14.15% (security 1) to 37.58% (security 2). The third row of panel (d) says that ms

expected return and standard deviation are, respectively, 12.55% and 18.20%. Thus, they

are smaller (larger) than those of w (w ). Moreover, ms returns in states s and s are

smaller (larger) in absolute terms than those of w (w ). Finally, ms efficiency loss is only

= 0.63%. The reason why we obtain this small efficiency loss is that ms weights in funds

w and w are substantially smaller than that in fund w .

4.4.2. Security Expected Returns

Panel (a) of Table 3 shows the risk premia. First, the CAPMs risk premium is R R =

8.81%. Second, the risk premium of the idiosyncratic return in state s is = 8.69% so

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that a securitys expected return increases by 8.69 basis points per percentage point of its

idiosyncratic return in state s . Third, the risk premium of the idiosyncratic return in state

s is = 7.35% so that a securitys expected return increases by 7.35 basis points per

percentage point of its idiosyncratic return in state s .

Panel (b) presents characteristics of the risky securities. The first row shows that their

betas range from 0.82 (security 4) to 1.22 (security 2). The second and third rows pro-

vide their idiosyncratic returns in states s and s . While securities 1 and 3 have negative

idiosyncratic returns in both states, security 2 has positive ones. Security 4 has a negative

idiosyncratic return in state s , but a positive one in state s .

Panel (c) decomposes security expected returns in four terms. The first three rows provide

the two terms contained in the CAPM and their sum, i.e., the CAPM expected return. As

expected, this sum is higher when the security beta is higher. The next three rows provide

the two terms that are not present in the CAPM and their sum, i.e., the idiosyncratic return

adjustment. Since securities 1 and 3 have negative idiosyncratic returns in states s and

s , the idiosyncratic return adjustment is negative for these securities. The idiosyncratic

return adjustment for security 1 (2.42%) is larger in absolute terms than that for security

3 (0.76%) because the idiosyncratic returns of the former are also larger in absolute terms

than those of the latter as shown in panel (b). Since security 2 has positive idiosyncratic

returns in states s and s , the idiosyncratic return adjustment for this security is positive

(1.25%). Finally, since security 4 has a negative idiosyncratic return in state s , but a positive

one in state s , the idiosyncratic return adjustment for this security is close to zero (0.13%).

In sum, the idiosyncratic return adjustments can be notably positive or negative.

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5. CONCLUSION

Over the past ten years, some investors have suffered huge losses due to extreme events

(e.g., the Barings Bank, LTCM, and Enron failures). In order to prevent such losses, ST

is now commonly utilized as a risk management tool. For example, the Committee on the

Global Financial System (2005, pp. 1, 15) of the Bank for International Settlements and

Scholes (2000) note that ST is often used by banks and securities firms to set risk exposure

limits. Accordingly, our paper examines a simple model to explore the portfolio selection

and asset pricing implications of ST constraints.

Regardless on whether a riskfree security is available, we obtain four main results. First,

the constrained agents optimal portfolio exhibits (K + 2)-fund separation, where K is the

number of binding ST constraints. Second, the effect of the constraints on the optimal

portfolio is identical to that of an adjustment in the expected returns of all risky securities

that tends to lower them. Hence, the constraints tend to increase the optimal portfolios

weight in the riskfree security (or the minimum variance portfolio when this security is not

available). Third, the market portfolio is inefficient. Fourth, a securitys expected return is

affected by both its systematic risk (i.e., beta) and its idiosyncratic returns in each one of

those states that are used in the constraints. Specifically, securities with negative (positive)

idiosyncratic returns in these states have relatively high (low) expected returns in equilibrium.

Next, we illustrate our theoretical results with a simple example. In this example, there

are five securities (one of them is riskfree), two agents (each of them has half of the wealth

in the economy), and two binding ST constraints. Our main findings are as follows. First,

since there are two constraints, the constrained agents optimal portfolio exhibits four-fund

separation. Second, the portfolios weights in the two inefficient funds are notable, which

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cause it to have an efficiency loss of 3.79%. Third, since the market portfolios weights in

these funds are smaller than those of the constrained agents optimal portfolio, the market

portfolio has an efficiency loss of only 0.63%. Fourth, the effect of the constraints on the

optimal portfolio is identical to that of a downward adjustment in the expected returns

of the risky securities ranging from 2.81% to 9.23%. Thus, the weight of the constrained

agents optimal portfolio in the riskfree security is substantially larger than that of the

unconstrained agents optimal portfolio. Fifth, the cost of ST as measured by the reduction

in the certainty equivalent return incurred by selecting the constrained portfolio is 3.16%.

Finally, the component of a securitys expected return that arises from its idiosyncratic

returns can be notable. This component is 2.42% for one of the securities that has negative

idiosyncratic returns in both of the states used in the constraints. However, the component

is 1.25% for one of the other securities since it has positive idiosyncratic returns in these

states. In sum, by exploring the implications of ST constraints, we provide further motivation

to the literature in which security prices are not solely driven by systematic risk.

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APPENDIX

Proof of Theorem 1. Observe that w solvesmax R +

w

(R

1

R )

2w Vw

. (12)

A first-order condition for w to solve problem (12) isR 1R Vw = 0. (13)

Since rank(V) = J, Equation (13) implies that

w =V (R 1R )

. (14)

Using the definition of w and Equation (14), we have Equation (2).

Proof of Theorem 2. Note that the ST constraints are assumed to bind. Hence, w solvesmax R + w (R 1R )

2w Vw (15)

s.t. w (R 1R ) = T R , s = s ,...,s . (16)

A first-order condition for w to solve problem (15) subject to constraints (16) isR 1R Vw + (R 1R ) = 0, (17)

where ,..., are Lagrange multipliers associated with these constraints. Since rank(V) =

J, Equation (17) implies that

w = V (R 1R ) + V (R 1R ) . (18)Using the definition of w , w ,...,w , and Equation (18), we have Equation (3).

We now find ,..., . Premultiplying Equation (18) by (R 1R ) and using Equa-

tion (16), we have

T R =I +

I

, s = s ,...,s , (19)

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where I (R 1R ) V (R 1R ) and I (R 1R ) V (R 1R ). It follows

from Equation (19) that

I = [I + (T + R ) ] , s = s ,...,s . (20)Let X [ ] , U [R 1R R 1R ], Y U V U , and

Z [[I + (T + R ) ] [I + (T + R ) ]] . Using Equation (20), we have

Y X =Z . Hence, X =Y Z .

Proof of Theorem 3. The desired result follows from Equations (14) and (18).

Proof of Theorem 4. We begin by showing that the market portfolio is inefficient. Using

Equations (2) and (3), we have

m =

1

w + w , (21)

where

=

(1 ) +

+

, s = s ,...,s . (22)

Since (1) = 0 for s = s ,...,s , (2) portfolio w is inefficient for s = s ,...,s , and (3)

rank([1 R R R ]) = K+ 2, Equation (21) implies that m is also inefficient.

Next, we show that Equation (5) holds. Using Equation (21) and the definitions of wand

w , we have

[ ] = Vm =

1

R 1R

A CR

+

R 1R

F CR

.

(23)

It follows from Equation (23) that

= m Vm =

1

R R

A CR

+

R R

F CR

. (24)

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The fact that = and Equations (23) and (24) imply that

R = R + (R R )

[(R R ) (R R )]

1

A CR

F CR

.

(25)

Using Equations (22) and (25), we have Equation (5). Note that > 0 since > 0.

Proof of Theorem 5. Observe that w solves

max w R

2w Vw (26)

s.t. w 1 = 1. (27)

A first-order condition for w to solve to problem (26) subject to constraint (27) is

R Vw + 1 = 0, (28)

where is the Lagrange multiplier associated with this constraint. Since rank(V) = J,

Equation (28) implies that

w =V R + V 1

. (29)

Using the definition of w and w , and Equation (29), we have Equation (7).

We now find . Premultiplying Equation (29) by 1 and using constraint (27), we have

1 = . Hence, = .

Proof of Theorem 6. Note that the ST constraints are assumed to bind. Hence, w solves

max w R

2w Vw (30)

s.t. w 1 = 1 (31)

w R = T , s = s ,...,s . (32)

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A first-order condition for w to solve problem (30) subject to constraints (31) and (32) is

R Vw + 1 +

R = 0, (33)

where , ,..., are Lagrange multipliers associated with these constraints. Since rank(V) =

J, Equation (33) implies that

w =V R + V 1 +

V R

. (34)

Using the definition of w , w , w ,...,w and Equation (34), we have Equation (8).

We now find , ,..., . Premultiplying Equation (34) by 1 and using constraint

(31), we have

1 =A + C+

F

. (35)


C+

F = A. (36)

Premultiplying Equation (34) by R and using constraint (32), we have

T =L + F +

L

, s = s ,...,s , (37)

where L R V R and L R V R . It follows from Equation (37) that

F + L = T L , s = s ,...,s . (38)Let X [ ] , U [1 R R ], Y U V U , and

Z [ A (T + L ) (T + L )] . Using Equations (36) and (38), we

have Y X =Z . Hence, X =Y Z .

Proof of Theorem 7. The desired result follows from Equations (29) and (34).

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Proof of Theorem 8. We begin by showing that the market portfolio m is inefficient.

Using Equations (7) and (8), we have

m = 1 w + w + w , (39)where

= (1 ) + , (40)

= , s = s ,...,s . (41)

Since (1) = 0 for s = s ,...,s , (2) portfolio w is inefficient for s = s ,...,s , and (3)

rank([1 R R R ]) = K+ 2, Equation (39) implies that m is also inefficient.

Next, we show that Equation (10) holds. Using Equation (39) and the definitions of w ,

w , and w , we have

[ ] = Vm =

1

1

C+

R

A+

R

F. (42)


= m Vm =

1

1

C+

R

A+

R

F. (43)

The fact that = and Equations (42) and (43) imply that

R = 1

A

C+ R

1

A

C

R R

A

F. (44)

Using Equations (40), (41), and (44), we have Equation (10). Note that > 0 since

> 0.

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Figure 1. Optimal portfolios and the market portfolio in the presence of a

riskfree security

This figure illustrates the unconstrained and constrained agents optimal portfolios (w and

w

) and the market portfolio (m

), represented by, respectively, points u, c, and m, inthe presence of a riskfree security. Also shown are w , w , w , and w , represented by,

respectively, points f, t, s , and x. The line represents combinations of w and w (i.e.,

the efficient frontier). The dashed hyperbola represents combinations ofw and w . The

dotted line represents combinations ofw and w .

f

t

s

u

c

mx

R

1

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Figure 2. Optimal portfolios and the market portfolio in the absence of a

riskfree security

This figure illustrates the unconstrained and constrained agents optimal portfolios (w and

w

) and the market portfolio (m

), represented by, respectively, points u, c, and m, inthe absence of a riskfree security. Also shown are w , w , w , and w , represented by,

respectively, points a, b, s , and x. The thick hyperbola represents combinations of w and

w (i.e., the mean-variance boundary). The dashed hyperbola represents combinations ofw

and w . The dotted hyperbola represents combinations ofw and w . The thin hyperbola

represents combinations ofw and w .

a

bs

u

c

m

x

R

1

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Table 1. Parameters used in the example

This table presents the parameters used in the example of Section 6. There are four risky

securities (j = 1, 2, 3, 4) and a riskfree security (j = 5). Panel (a) shows their annualized

expected returns (R , j = 1, ..., 5) and standard deviations ( , j = 1, ..., 5). Panel (b) showsthe monthly returns in the states used in stress testing (R and R , j = 1, ..., 5). All

numbers are reported in percentage terms.

(a) Annualized expected returns and standard deviations

j 1 2 3 4 5

R 15.00 13.20 12.00 10.80 3.74

27.71 25.98 22.52 20.78 0.00

(b) Monthly returns in the states used in stress testing

R 28.00 8.00 11.00 10.00 0.31

R 29.00 9.00 22.00 10.00 0.31

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Table 2. Optimal portfolios and the market portfolio

This table presents the unconstrained and constrained agents optimal portfolios (w andw )

and the market portfolio (m). Panel (a) provides the portfolios fund weights. The weights

in funds w and w are denoted by, respectively, and . The weights in funds w and

w are denoted by, respectively, and . Panel (b) provides the portfolios security

weights. Panel (c) provides the adjustment in expected returns with the same effect on the

optimal portfolio as the ST constraints and the adjusted expected returns (R , j = 1,..., 4).

This adjustment depends on the Lagrange multipliers associated with the constraints (

and ). Panel (d) provides summary statistics on the optimal portfolios and the market

portfolio. Portfolio ws efficiency loss is denoted by . The rest of the notation is defined

in Table 1. All numbers are reported in percentage terms.

(a) Fund weights

w 36.93 136.93

w 36.93 136.93 35.13 38.74

m 136.93 17.56 19.37

(b) Security weights

j 1 2 3 4 5

w

41.42 30.65 36.36 28.50

36.93w 13.13 44.52 6.24 25.44 36.93

m 14.15 37.58 21.30 26.97

(c) Adjustment in expected returns

j 1 2 3 4

R 15.00 13.20 12.00 10.80

(R R ) + (R R ) 9.23 2.81 5.24 3.31

R 5.77 10.39 6.76 7.49

(d) Summary statistics

j R R R

w 16.32 25.07 21.02 25.74

w 8.79 13.84 3.00 4.00 3.79

m 12.55 18.20 12.01 14.87 0.63

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Table 3. Risk premia, security characteristics, and decomposition of

security expected returns

Panel (a) provides the risk premia. The risk premia on idiosyncratic returns in states s

and s are denoted by, respectively, and . Panel (b) presents security characteristics.

The security betas are denoted by , j = 1, ..., 4. The rest of the notation is defined in

Tables 1 and 2. Panel (c) decomposes security expected returns. With the exception of the

idiosyncratic returns in states s and s , all numbers are annualized. With the exception of

the betas, all numbers are reported in percentage terms.

(a) Risk premia

R R 8.81

8.69

7.35

(b) Security characteristics

j 1 2 3 4

1.00 1.22 0.85 0.82

(R R ) (R R ) 15.95 6.66 0.83 0.26

(R R ) (R R ) 14.08 9.14 9.40 2.07

(c) Decomposition of security expected returns

j 1 2 3 4

R 3.74 3.74 3.74 3.74

(R R ) 8.84 10.71 7.50 7.19

CAPM expected return 12.58 14.45 11.24 10.93

[(R R ) (R R )] 1.39 0.58 0.07 0.02

[(R R ) (R R )] 1.03 0.67 0.69 0.15

Idiosyncratic return adjustment

2.42 1.25

0.76 0.13

R 15.00 13.20 12.00 10.80

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