credit risk optimization with conditional VaR criterion.pdf

CVaR

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Digital Object Identiﬁer (DOI) 10.1007/s101070000201

Math. Program., Ser. B 89: 273–291 (2001)

Fredrik Andersson · Helmut Mausser · Dan Rosen · Stanislav Uryasev

Credit risk optimization with Conditional Value-at-Risk

criterion

Received: November 1, 1999 / Accepted: October 1, 2000

Published online December 15, 2000 –  Springer-Verlag 2000

Abstract. This paper examines a new approach for credit risk optimization. The model is based on the

Conditional Value-at-Risk (CVaR) risk measure, the expected loss exceeding Value-at-Risk. CVaR is also

known as Mean Excess, Mean Shortfall, or Tail VaR. This model can simultaneously adjust all positions

in a portfolio of ﬁnancial instruments in order to minimize CVaR subject to trading and return constraints.

The credit risk distribution is generated by Monte Carlo simulations and the optimization problem is solved

effectively by linear programming. The algorithm is very efﬁcient; it can handle hundreds of instruments

and thousands of scenarios in reasonable computer time. The approach is demonstrated with a portfolio of

emerging market bonds.

1. Introduction

Risk management is a core activity in asset allocation conducted by banks, insurance

and investment companies, or any ﬁnancial institution that evaluates risks. This paper

examines a new approach for minimizing portfolio credit risk. Credit risk is the risk

of a trading partner not fulﬁlling their obligations in full on the due date or at any

time thereafter. Losses can result both from counterparty default, and from a decline

in market value stemming from the credit quality migration of an issuer or counter-

party. Traditionally used tools for assessing and optimizing market risk assume that the

portfolio return-loss is normally distributed. With this assumption, the two statistical

measures, mean and standard deviation, can be used to balance return and risk. The

optimal portfolio is selected on the “efﬁcient frontier”, the set of portfolios that have the

best mean-variance proﬁle [10]. In other words, this is the set of Pareto optimal points

with two conﬂicting criteria: mean and variance.

Although this traditional approach has proven to be quite useful in various applica-

tions, it is inadequate for credit risk evaluations because credit losses are characterized

by a large likelihood of small earnings, coupled with a small chance of losing a large

amount of the investment. Thus the loss distributions are, in general, heavily skewed

and standard optimization tools developedfor market risk are inadequate. This, together

F. Andersson: Ementor, Stortorget 1, 111 29 Stockholm, Sweden,

e-mail: fredrik.andersson@ementor.se, web: http://www.ementor.se

H. Mausser, D. Rosen: Algorithmics, Inc., 185 Spadina Avenue, Toronto, Ontario M5T 2C6, Canada,

web: http://www.algorithmics.com

S. Uryasev: University of Florida, Dept. of Industrial and Systems Engineering, PO Box 116595, 303 Weil

Hall, Gainesville, FL 32611-6595, e-mail: uryasev@ise.ufl.edu,

web: http://www.ise.ufl.edu/uryasev

Mathematics Subject Classiﬁcation (1991): 20E28, 20G40, 20C20

274 Fredrik Andersson et al.

with the lack of historical data to estimate credit correlations, poses signiﬁcant modeling

challenges, compared to market risk modeling and optimization.

To cope with skewed return-loss distributions, we consider Conditional Value-at-

Risk (CVaR) as the risk measure. This measure is also called Mean Excess Loss, Mean

Shortfall, or Tail VaR. By deﬁnition, β-CVaR is the expected loss exceeding β-Value-at-

Risk (VaR), i.e., it is the mean value of the worst (1 − β) ∗ 100% losses. For instance, at

β = 0.95, CVaR is the average of the 5% worst losses. CVaR is a currency-denominated

measure of signiﬁcant undesirable changes in the value of the portfolio.

CVaR maybe compared with thewidely accepted VaR riskperformancemeasure for

which various estimation techniques have been proposed, see, e.g., [7,15]. VaR answers

the question: what is the maximum loss with the conﬁdence level β ∗ 100% over a given

time horizon? Thus, its calculation reveals that the loss will exceed VaR with likelihood

(1 − β) ∗ 100%, but no information is provided on the amount of the excess loss, which

may be signiﬁcantly greater. Mathematically, VaR has serious limitations. In the case of

a ﬁnite numberof scenarios, it is a nonsmooth, nonconvex,and multiextremum function

[11] (with respect to positions), making it difﬁcult to control and optimize. Also, VaR

has some other undesirable properties, such as the lack of sub-additivity [1,2].

By contrast, CVaR is considered a more consistent measure of risk than VaR. CVaR

supplements the information provided by VaR and calculates the quantity of the excess

loss. Since CVaR is greater than or equal to VaR, portfolios with a low CVaR also

have a low VaR. Under quite general conditions, CVaR is a convex function with

respect to positions [17], allowing the construction of efﬁcient optimization algorithms.

In particular, it has been shown in [17], that CVaR can be minimized using linear

programming (LP) techniques. The minimum CVaR approach [17] is based on a new

representation of the performance function that allows the simultaneous calculation of

VaR and minimization of CVaR. A simple description of the approach for minimization

of CVaR and optimization problems with CVaR constraints can be found in [21]. Since

CVaR can be minimized by LP algorithms, a large number of instruments and scenarios

can be handled. LP techniques are routinely used in ﬁnancial planning applications, see,

for instance, paper [4] which applies a penalty approach to controlling risks. However,

comparing to the paper [4], we directly handle the quantile-based constraints with

a speciﬁed conﬁdence level.

Similar measures as CVaR have been earlier introduced in the stochastic program-

ming literature, although not in ﬁnancial mathematics context. The conditional expecta-

tion constraints and integrated chance constraints described in [14] may serve the same

purpose as CVaR. The reader interested in other applications of optimization techniques

in ﬁnance area can ﬁnd relevant papers in [22].

In credit risk evaluations, we are interested in the losses experienced in the event of

counterparty default or credit quality migration in the course of a day, a year or other

standardized period. Several approaches are available for estimating credit risk [5,6,13,

18,19]. Probably, the most inﬂuential contribution in this ﬁeld has been J.P. Morgan’s

CreditMetrics methodology [5].

Bucay and Rosen [3] conducted a case study and applied the CreditMetrics method-

ology to a portfolio of bonds issued in emerging markets. The portfolio consists of 197

bonds, issued by 86 obligors in 29 countries. Bond maturities range from a few months

to 98 years and the portfolio duration is approximately ﬁve years. The mark-to-market

Credit risk optimization with Conditional Value-at-Risk criterion 275

value of the portfolio is $8.8 billion. Mausser and Rosen [12] applied the regret op-

timization framework to minimize the credit risk of this portfolio. In this paper, we

analyzed the same portfolio and minimized the credit risk using the Minimum CVaR

approach. We have used the dataset of Monte Carlo scenarios generated at Algorithmics

Inc. [3,12]. Using the CreditMetrics methodology, a large number of scenarios is cal-

culated based on credit events such as defaults and credit migrations. By evaluating the

portfolio for each scenario, the loss distribution is generated. Monte Carlo simulation

tools are widely used for evaluating credit risk and other risks of portfolios containing

non-linear instruments, such as options (see, for instance, [11,17,15]).

The optimization analysis conducted for the portfolio of bonds may be brieﬂy sum-

marized as follows. First, each single position (a position corresponds to an obligor)

is optimized. This provides the best hedge, i.e. the position that gives the minimum

CVaR when holding the other positions ﬁxed. Then, all portfolio positions are simultan-

eously adjusted to minimize the portfolio CVaR under trading and budget constraints.

In this framework, the CVaR-return efﬁcient frontier of the portfolios is also calculated.

As mentioned above, the minimization of CVaR automatically leads to a signiﬁcant

improvement of VaR.

The remainder of this paper is organized as follows. First, we present the minimum

CVaR approach [17]. Then, following [3,12], we describe the bond portfolio. In this

section, we give a brief description of the CreditMetrics methodology used to calcu-

late the portfolio loss distribution. Then, we describe the optimization model and its

parameters. Finally, we present the analysis and concluding remarks.

2. Minimum CVaR approach

This section describes the approach to minimization of CVaR. Let f : IR

× IR

→ R

be the loss function which depends upon the control vector x ∈ IR

and the random

vector y ∈ IR

. We use bold face for the vectors to distinguish them from scalars. We

considerthat therandomvectory hasthe probabilitydistributionfunction p : IR

→ IR .

However, the existence of the density is not critical for the considered approach, this

assumption can be relaxed. Denote by (x, a) the probability function

(x,α) =



f(x,y)≤ α

p(y) dy , (1)

which, by deﬁnition, is the probability that the loss function f(x, y) does not exceed

some threshold value α. The VaR (or percentile) function α(x,β) is deﬁned as follows

α(x,β) = min{α ∈ IR : (x,α) ≥ β} . (2)

Let us consider the following CVaR performancefunction (x) which is the conditional

expectedvalueof the loss f(x, y) underthe conditionthat it exceeds the quantile α(x,β),

i.e.,

(x) = (1 − β)

−1



f(x,y)≥ α(x,β)

f(x, y) p(y) dy . (3)

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