The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 391 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 391

ON THE APPLICABILITY OF EXTREME VALUE

THEORY IN FINANCE

In financial applications, EVT for fat-tailed processes has been applied to

questions of risk management and portfolio optimization, especially port-

folios with exposure to credit risk.

We can illustrate the importance of fat-tailed processes in credit risk

management using an example prepared by Srichander Ramaswamy^35

Exhibit 13.4 shows the credit risk of a 23-corporate bond portfolio

under different modeling assumptions. Risk values in the first column

are computed considering default losses under the assumption that joint

asset return distribution is normal. Values in the second column are

computed under the same distributional assumptions but consider not

only default losses but also the losses incurred due to rating migration.

The values in the third column are computed under the assumption that

the joint distribution of asset returns is a multivariate t with 8 degrees

of freedom.

The risk measures considered are Unexpected Loss (UL) measured

by the standard deviation in the second row, credit risk Value-at-Risk

(CrVaR) in the third row, and Expected Shortfall Risk (ESR) in the

fourth row. (We will discuss these measures in Chapter 22, where we

cover risk management.) The Expected Loss tabulated in the first row is

a measure of credit cost and not of risk.

As explained in Chapter 22, under the assumption of multivariate

normality, the three risk measures UL, VaR, and ES are equivalent; how-

ever, if we drop this assumption, the three risk measures are no longer

equivalent. Observe, in particular, that moving from a multivariate nor-

EXHIBIT 13.4 Portfolio Credit Risk Measures Under Different Modeling

Assumptions

Default Mode Migration Mode Migration Mode

and Multivariate and Multivariate and Multivariate

Description Normal Normal t-Distributed

Expected loss 13.9 bp 34.1 bp 34.0 bp

Unexpected loss 65.9 bp 88.9 bp 105.1 bp

CrVaR at 90% confidence 0.0 bp 102.9 bp 96.6 bp

ESR at 90% confidence 139.0 bp 240.3 bp 256.2 bp

(^35) This illustration is adapted from his book, Managing Credit Risk in Corporate
Bond Portfolios: A Practitioner’s Guide (Hoboken, NJ: John Wiley & Sons, 2004).