Value at Risk
Mark Casella, a partner at Coopers and Lybrand, and Gifford Fong, president of Gifford Fong Associates, explain how to combine valueatrisk measures.
Analyzing VAR
Mark Casella and Gifford Fong
Valueatrisk 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.
