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FRM Part II · FRM Exam Part II · Non-parametric Approaches

A firm applies age-weighted historical simulation with decay factor λ = 0.98 to 100 daily observations, where observation i = 1 is the most recent. The weight on observation i is λ^(i-1)(1-λ)/(1-λ^100). The most recent loss is the single worst loss in the sample. Approximately what weight does this observation receive, compared with the equal weight of 1%? (Use 0.98^100 ≈ 0.133.)

The weight is about 2.3%. For the most recent observation, λ^0 is 1, so the weight is 0.02 divided by (1 minus 0.133), roughly 0.0231. This exceeds the 1% equal weight, so recent losses matter more to the VaR quantile.

  1. AAbout 1.0%, same as equal weighting
  2. BAbout 2.3%, because (1-0.98)/(1-0.133) ≈ 0.023Correct
  3. CAbout 2.0%, because the (1-λ^100) term is ignored
  4. DAbout 13.3%, because it is the worst loss

Explanation

For i = 1, weight = (1-0.98)/(1-0.133) = 0.02/0.867 = 0.0231. Option C ignores the normalisation term that makes weights sum to one. The result is more than twice the equal weight of 1%, so recent large losses push VaR up faster.

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