Mapping Zero-coupon Bonds to Risk Factors
We earlier learned that VaR calculation of a complex portfolio can be simplified by decomposing the instruments into building blocks, or primitive instruments, which are further mapped to a small set of risk factors. Zero-coupon bonds are one such primitive instrument apart from spot FX positions, equity positions, and futures/forwards. In this article we will learn about mapping cash flows for a zero-coupon bond.
Consider a 5-year Treasury bond with semi-annual payments. This bond has 10 cash flows that are sensitive to different parts of the term structure. The first coupon is sensitive to the 6-month interest rate, the next coupon is sensitive to the one-year interest rate, and the last (10th) payment will be sensitive to the 5-year zero-coupon interest rate.
For the purpose of mapping each cash flow, the risk manager will need to identify a set of zero-coupon bonds at different maturities. J.P. Morgan suggests 14 different maturities, namely, 1 month, 3 month, 6 month, and 1, 2, 3, 4, 5, 7, 9, 10, 15, 20, and 30 years.
For our 5-year Treasury bond, 6 out of 10 cash flows will directly map to the benchmark maturities, while the others will be converted into an equivalent cash flows suiting one of the benchmark maturities. For example, the cash flow at 2.5 years will be mapped into two equivalent cash flows, one with a maturity of one year and another with two years. The risk sensitivity of cash flow at 2.5 years will be equivalent to the two cash flows, one at 2 years, and the other at 3 years. In other words, the original cash flow at 2.5 years is mapped to two cash flows (at 2 and 3 years).
A simple way to do this is to perform a linear interpolation. However, in reality the method for such transformation to equivalent cash flow can be complex. Even interest rate futures are used to calibrate the curve to the market.
One way to do is to transform the original cash flow into two cash flows with the same value and modified duration. Another way is to use two weights α and (1-α) and divide the cash flow such that the return volatility of the portfolio of two equivalent cash flows is the same as the return volatility of the original cash flow.