Skip to content

FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Volatility and Distribution

A normal model has short rate 3%, σ = 1.2% per year, and no drift. Using the normal distribution, what is the approximate probability that the rate is negative after 4 years? Use N(−1.25) ≈ 0.106 and N(−2.5) ≈ 0.006, N(−0.625) ≈ 0.266.

After four years the standard deviation is 1.2% times the square root of 4, or 2.4%. The 3% rate is 1.25 standard deviations above zero, so the probability of a negative rate is N(−1.25), about 10.6%.

  1. A10.6%Correct
  2. B0.6%
  3. C26.6%
  4. DZero, because rates cannot be negative in a normal model

Explanation

Standard deviation after 4 years = 1.2% × √4 = 2.4%. Rate 3% is 3/2.4 = 1.25 standard deviations above zero, so P(r<0) = N(−1.25) ≈ 10.6%. Using 4 × 1.2% = 4.8% as the horizon deviation gives 0.625 and 26.6%; ignoring the square-root scaling is the error.

Did you get it right without looking?

One question tells you little. A timed set on The Art of Term Structure Models: Volatility and Distribution shows your real accuracy, how long you take and where you lose marks.

Practise The Art of Term Structure Models: Volatility and DistributionFree diagnostic mock

More The Art of Term Structure Models: Volatility and Distribution questions