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Value at Risk

Mark Casella, a partner at Coopers and Lybrand, and Gifford Fong, president of Gifford Fong Associates, explain how to combine value-at-risk measures.

Analyzing VAR

Mark Casella and Gifford Fong

Value-at-risk has been widely embraced as a summary risk measure by regulators and practitioners alike. While serving a useful role, VAR is based on specific assumptions concerning the probability and time horizon selected. Depending on the makeup of the portfolio, modest changes in these assumptions can produce materially different values for VAR. It is sensible, therefore, to augment any VAR analysis with specific scenarios of risk factor change to provide added perspective on the dynamic character of the overall risk.

While many of the characteristics of transactions and their portfolios combine proportionately, a number of important risk measures cannot be as easily combined. For example, Table 1 gives a decomposition of volatility. Each component displayed contributes to total volatility but the sequence of component ordering will affect their respective relative contribution.

Table1

Source Interest Rate Risk Other Market Risks Derivatives Risks Specific Risks Foreign Exchange Risks Total Risks
Volatility 10% 2% 3% 1% 6% 22%

Although the risks are measured by the standard deviation, and standard deviations do not add, the component risks in Table 1 do add up to the total risk. This is accomplished by calculating the components resulting from the increment, which are derived from that source of risk on top of the previous subtotal. This makes the decomposition dependent on the order in which the components are listed, but makes the components meaningful: 3 percent derivative risk component in Table 1, for instance, means that the risk as a result of the fund's derivative holdings add 3 percent to the 12 percent price variability resulting from market factors.

The VARs can be calculated for individual securities, portfolio sectors and the total portfolio, as well as by the sources of risk. This can lead to a useful breakdown such as in Table 2.

Table2

Security Interest Rate Risk $ Other Market Risks $ Derivatives Risks $ Specific Risks $ Foreign Exchange Risks $ Total Risk $
Sector A
Security 1 103,400 19,500 52,100 5,700   133,100
Security 2 85,600     2,300   86,700
Total 189,000 19,500 52,100 6,100   217,500
Sector B
Security 3
Security 4
Total
Portfolio Total 2,358,100 311,700 827,700 63,300 556,900 3,581,900

The numbers in the table do not necessarily add up, either down or across. The reason that they do not add up for the sectors and the total portfolio is that the VAR as a result of, say, interest rate risk may come from rising interest rates for one security (as for most bonds), while it comes from declining interest rates for another security, such as an interest only or a short position in futures. The reason the numbers do not add up across the sources of risk is that events of a given probability, say 1 percent, do not add up: An interest rate change that can happen with 1 percent likelihood when considered alone is not the same as the change that would happen together with, say, an exchange rate movement for a joint 1 percent probability.

A number of scenarios of risk factor change can be analyzed and values produced for each scenario similar to Table 2. This range of outcomes provides an enhanced perspective on the underlying risks.

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