FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models
In a Vasicek model with k = 0.20, σ = 1.00% per year, the current short rate r0 = 3.00% and long-run mean θ = 5.00% (risk-neutral). Ignoring the risk premium, what is the expected short rate in 5 years?
The expected rate is about 4.26%. Mean reversion closes the gap between 3% and 5% by a factor of e^(−0.2×5) = e^(−1) ≈ 0.368, leaving a 0.736% gap below the 5% long-run mean, so the rate is 5% − 0.736% ≈ 4.26%.
- A4.26%Correct
- B3.63%
- C5.00%
- D4.00%
Explanation
E[r(t)] = θ + (r0 − θ)e^(−kt) = 5% − 2%·e^(−1) = 5% − 2%(0.3679) = 5% − 0.736% = 4.264%. Using 3.63% would result from e^(−1) applied incorrectly to the gap (3% + 0.63%). 4.00% ignores the decay, taking half the gap.
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