Value at Risk as a Measure of Risk Exposure

Perhaps the most popular way to measure risk exposure today is value at risk (VAR),

defined as the worst loss possible under normal market conditionsfor a given time

horizon. For example, an investment position that loses a maximum of $100 million

over the next year, no more than 1 percent of the time, will be viewed by some man-

agers as having a value at risk of $100 million for the next year.

Value at risk is determined by the time interval under consideration as well as by

what the manager regards as normal market conditions. Aposition with a value at risk

of $100 million over the next year will have considerably less value at risk over a

shorter horizon, say over the next month. Similarly, a manager who considers abnor-

mally bad market conditions to be those that occur less than 5 percent of the time will

have less value at risk than a manager who is willing to ignore only those losses that,

because of their astonishing magnitude, occur less than 1 percent of the time.

The importance of both the significance level (5 percent or 1 percent as the typ-

ical thresholds for determining abnormal market conditions) and the time horizon are

illustrated when representing value at risk in a diagram using the distribution of prof-

its and losses. Exhibit 22.1 illustrates the value at risk at the 5 percent significance

level for a transaction with zero expected profit. The time horizon affects the shape

of the distribution curve. The longer the time horizon, the more uncertain the prof-

its, and the more spread out is the normal distribution curve. This should shift point

A—the boundary of the 5 percent area under the curve’s left tail—to the left, increas-

ing VAR.The threshold for the area in the tail (5 percent versus 1 percent) matters,

too, as a shift to a 1 percent tail as the threshold moves point Ato the left, thereby

increasing VAR.

Value at risk is the standard methodology used for measuring the risk to the value

of a portfolio of derivatives or other securities. There is a regulatory impetus for this.

In late 1996, the Bank for International Settlements, the U.S. Securities and Exchange

Commission, and the Federal Reserve Board proposed that the institutions they

supervise use this risk measure as a standard for certain activities. An analogous

methodology applied to cash flows, known as cash flow at risk (CAR), is becoming

an increasingly important concept for corporations.

Estimating VARand CARfrom Standard Deviations.VARand CARare simple

translations of the standard deviation if the value or cash flow is normally distributed.

For example,

³However, until recently GARCH estimation was virtually impossible to implement when there are

five or more risk factors.

Grinblatt1567Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1567Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

778Part VIRisk Management

EXHIBIT22.1Probability Distribution and Value at Risk

Probability

Critical value

5%	A0
		Profit
	Value at
	risk

VAR(5% significance level) 1.65

where is the standard deviation of the value. The same formula applies to CAR,except

that is the standard deviation of the cash flow.

The 1.65 in the preceding equation is obtained from a normal distribution table,

such as that found in Table A.5 in Appendix Aat the end of this text. In particular, note

that N( 1.65) is approximately .05 in such a table. More generally, let xbe the value

or cash flow at which the probability that a normally distributed value or cash flow

with a mean of zero and a standard deviation of is less than ppercent; that is, N(x)

p%. Then, VARor CARis x.

Example 22.2 illustrates how to transform a to a CAR.

Example 22.2:ComputingCARfrom Standard Deviations Assuming a

NormalDistribution

In Example 22.1, the standard deviation of the cash flow was approximately $576,200.What

is the cash flow at risk at the 5 percent significance level assuming that the cash flow is nor-

mally distributed?

Answer:1.65($576,200)$950,720.

Estimating VARorCARUsing Simulation.When CARis estimated, simulation usu-

ally is preferred to the standard deviation formula as an estimation procedure. Given

prespecified factor betas, the cash flow is then simulated from the factor equation for

the factor values observed over a given historical period. The CARis then the differ-

ence between the average cash flow and the fifth percentile outcome over the historical

period.

Grinblatt1569Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1569Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

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