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FRM Part II · FRM Exam Part II · Empirical Properties of Correlation: How Do Correlations Behave in the Real World?

A discrete mean-reverting correlation model gives ρ(t+1) − ρ(t) = a(μ − ρ(t)) with a = 0.25 per month, μ = 0.40 and ρ(0) = 0.80, ignoring shocks. What is the expected correlation after 3 months?

The expected correlation is about 0.5688. Each month the gap to the 0.40 mean shrinks by 25 percent, so the initial 0.40 gap becomes 0.40 times 0.75 cubed, which is 0.16875, giving 0.40 plus 0.16875.

  1. A0.4000
  2. B0.5688Correct
  3. C0.6000
  4. D0.4750

Explanation

The gap shrinks by factor (1 − a) = 0.75 each month. Gap0 = 0.40; after 3 months = 0.40 × 0.75³ = 0.40 × 0.421875 = 0.16875. So ρ = 0.40 + 0.16875 = 0.56875 ≈ 0.5688. Linear reduction of 0.10 per month would give 0.50, and 0.60 comes from only two months of... a single step (0.80 − 0.10 − ... ) error.

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