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FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Drift

In a Ho-Lee model with constant volatility σ = 1.00% per year and no mean reversion, the drift is λ(t) = 0.20% per year for all t. The current short rate is 3.00%. Using the model's expected short rate, what is the expected short rate in 5 years, and what is the standard deviation of the short rate at that horizon?

The expected short rate is 4.00%, from 3.00% plus 0.20% a year over five years. The standard deviation is σ times the square root of time, 1.00% × √5, about 2.24%. Scaling volatility linearly by 5 would be incorrect.

  1. A4.00% expected; standard deviation 2.24%Correct
  2. B3.00% expected; standard deviation 2.24%
  3. C4.00% expected; standard deviation 5.00%
  4. D3.20% expected; standard deviation 1.00%

Explanation

E[r(5)] = 3.00% + 0.20%×5 = 4.00%. Standard deviation = σ√T = 1.00%×√5 = 2.236% ≈ 2.24%. Using σT = 5.00% is wrong because variance grows linearly with time, not the standard deviation.

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