P2.T5.22.10. Value at risk (VaR) mapping

Nicole Seaman

Director of FRM Operations
Staff member
Subscriber
Learning objectives: Summarize how to map a fixed-income portfolio into positions of standard instruments. Describe how mapping of risk factors can support stress testing. Explain how VaR can be computed and used relative to a performance benchmark. Describe the method of mapping forwards, forward rate agreements, interest rate swaps, and options.

Questions:

22.10.1. Ralph manages a fixed-income portfolio and he has asked his colleague Mary (who recently earned the Financial Risk Manager designation) to design a value at risk (VaR) mapping process so that they can stress test the portfolio. The portfolio is focused on high-quality corporate bonds; it is a large portfolio with many positions, including mostly coupon-bearing bonds but some zero-coupon bonds.

To map this fixed-income portfolio, Ralph and Mary are considering three approaches: principal mapping, duration mapping, and cash-flow mapping. In regard to these three VaR mapping approaches, each of the following statements is true EXCEPT which is false?

a. In all three cases, the primitive instrument(s) is a zero-coupon bond
b. Principal mapping is simple but overstates the true risk because it ignores coupons
c. The VaR returned by duration mapping must be less than the VaR returned by principal mapping
d. The diversified VaR returned by cash flow mapping is an increasing function of the pairwise correlations between the risk factors

22.10.2. An advantage of value at risk (VaR) mapping is that its design lends itself to stress testing and performance benchmarking against an index. Consider both extensions of VaR mapping:

I. Stress testing a fixed-income portfolio by shocking the risk factors, which are vertices on the spot rate term structure
II. Performance benchmarking a fixed-income portfolio by tracking it against a major, well-known U.S. bond index

In regard to these VaR extensions, which of the following statements is TRUE?

a. The stress test is equivalent to retrieving an undiversified value at risk (VaR) at a sufficiently high confidence level
b. A key disadvantage of stress testing is that it cannot capture imperfect correlations across different sectors of the yield curve
c. In performance benchmarking, minimizing the absolute market risk by definition will also minimize the relative market risk
d. The variance improvement in performance benchmarking can be measured by 1 - (TE-VaR/absolute VAR), which is analogous to R^2 in a linear regression

22.10.3. A firm can engage in value at risk (VaR) mapping for a variety of positions and purposes. The more common reference instruments in a VaR mapping approach include, but are not limited to the following:
• Forward foreign exchange (FX) contract
• Consumption commodity forward contract
• Vanilla interest rate swap
• Stock option portfolio
In regard to mapping these instruments, which of the following statements is TRUE?

a. The forward FX contract is one of the most difficult instruments to map
b. The consumption commodity forward contract is one of the easiest instruments to map
c. The vanilla interest rate swap should be mapped to the dot product of its diffusion vector and its amortization vector
d. The stock option portfolio can be mapped in many ways, from simple (delta-normal) to complex (multiple Greek risk factors)