FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models
In a Vasicek model with k = 0.40, a risk manager holds a 5-year zero-coupon bond. Using the first-order sensitivity of price to the short rate, P = A(T)e^(−B(T)r) with B(T) = (1 − e^(−kT))/k, and ignoring convexity, what is the approximate percentage price change if the short rate rises by 50 basis points? (e^(−2) = 0.1353)
The bond falls by about 1.08%. The Vasicek rate sensitivity of the zero price is B(T) = (1 − e^(−kT))/k, which is 2.1617 for T = 5 and k = 0.4. Multiplying by the 0.50% rate rise gives −1.08%, well below the 2.50% implied by full-maturity duration.
- A−1.08%Correct
- B−2.50%
- C−0.43%
- D−0.17%
Explanation
B(5) = (1 − 0.1353)/0.4 = 2.1617. The price change is about −B × Δr = −2.1617 × 0.5% = −1.08%. Using B = T = 5 gives −2.50%, which ignores mean reversion. Omitting the division by k gives −0.43%. Using only e^(−kT)/k gives −0.17%.
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