Volatility as a Measure of Risk Exposure

Corporate managers often like to summarize risk exposure with a single number. For

this reason, the standard deviation is often used to summarize the risk impact of a col-

lection of factor betas. Extending the analysis of Chapter 6 to correlated factors, the

formula for the variance of the factor risk of an investment is

KK

² cov(F˜, F˜)

^mnmn

m1n1

where

factor beta on factor m

m

factor beta on factor n

n

The volatility (that is, standard deviation) of the cash flow or value owing to factor risk

is the square root of this.

Example 22.1 illustrates how to implement this formula.

Example 22.1:Computing Factor-Based Volatility fora Cash Flow

Assume that a firm has a cash flow one year from now (in US$ millions) that follows the

factor model

˜302F˜ 4F˜ ˜

C

currint

where the currency factor, ˜, is the percentage change in the ¥/US$ exchange rate over

F

curr

the next year and the interest rate factor, ˜, is the percentage change in three-month

F

int

LIBOR from now until one year from now.Assume that the variance of the currency factor

is estimated to be .011, the variance of the interest rate factor is approximately .022, and

the covariance between the two is .004.What is the factor-based volatility of the cash flow?

Answer:Using the variance formula above, the factor based variance is

²

4(.011) 8(.004) 8(.004)16(.022).332

The square root of this number, the volatility, is .576 (expressed in US$ millions).

Grinblatt1565Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1565Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

777

Clearly, the estimates of variances and covariances of the risk factors are critical

for obtaining a good estimate of the volatility. One procedure for estimating the covari-

ances and variances of risk factors is to compute historical variances and covariances.

However, financial institutions that make use of volatility recognize that variances

andcovariances tend to change over time, so they have developed more sophisticated

estimation procedures.

J. P. Morgan, for example, in its RiskMetrics™covariance matrix (website:

http://www.jpmorgan.com), forecasts variances as weighted averages of the previous

variance forecast and of the latest deviation from the forecast. This model is a special

case of a procedure, Generalized Autoregressive Conditional Heteroskedastic (GARCH)

estimation, developed in the economics statistics literature.³