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FRM Part I · FRM Exam Part I · Calculating and Applying VaR

A risk manager simulates a two-asset portfolio using Monte Carlo with correlated normal returns. Asset volatilities are 20% and 30% with correlation 0.5. To generate correlated shocks from independent standard normals z1 and z2 using Cholesky decomposition, which pair of shocks should be used for the two assets?

Use ε1 = z1 and ε2 = 0.5·z1 + 0.866·z2. Cholesky sets the second shock as ρ times the first plus the square root of one minus ρ squared times the independent draw, preserving unit variance and correlation of 0.5.

  1. Aε1 = z1 and ε2 = 0.5·z1 + 0.866·z2Correct
  2. Bε1 = z1 and ε2 = 0.5·z1 + 0.5·z2
  3. Cε1 = 0.5·z1 and ε2 = z2
  4. Dε1 = z1 and ε2 = 0.25·z1 + 0.75·z2

Explanation

Cholesky gives ε1 = z1 and ε2 = ρ·z1 + √(1-ρ²)·z2. With ρ = 0.5, √(1-0.25) = 0.866. Variance of ε2 = 0.25 + 0.75 = 1 and covariance with ε1 = 0.5, as required. Using 0.5 on z2 gives variance 0.5, which is wrong.

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