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FRM Part I · FRM Exam Part I · Measuring and Monitoring Volatility

A risk manager builds an EWMA covariance matrix for many risk factors. Which design choice ensures the resulting matrix is positive semi-definite, so that portfolio variances can never be negative?

Use the same decay factor λ for all variances and covariances. A common λ makes the EWMA matrix a weighted sum of return outer products, which is positive semi-definite. Pair-specific decay factors or mismatched windows can break this and produce negative portfolio variances.

  1. AUse the same decay factor λ for every variance and covariance elementCorrect
  2. BChoose a separate λ for each pair to best fit that pair's history
  3. CUse a longer window for covariances than for variances
  4. DEstimate each correlation separately and then combine them with the volatilities

Explanation

With a single common λ, the matrix is a weighted sum of outer products of return vectors, which guarantees positive semi-definiteness. Pair-specific decay factors or mismatched windows can produce matrices with negative eigenvalues and hence possibly negative portfolio variances.

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