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

A risk manager uses kernel density estimation on a sample of daily returns to estimate the left-tail density. She is deciding on the bandwidth (smoothing parameter). Which outcome is most likely if she chooses a bandwidth that is far too large?

An excessively large bandwidth oversmooths the density, creating bias and washing out real features like fat tails and skewness. A very small bandwidth would instead produce a spiky, high-variance estimate. The bandwidth therefore trades off bias against variance.

  1. AThe density estimate becomes oversmoothed, with high bias that can understate the true tail features such as fat tails and skewCorrect
  2. BThe density estimate becomes spiky and reproduces each individual observation, giving high variance
  3. CThe estimate converges exactly to a normal distribution with the sample mean and variance
  4. DThe estimate becomes unbiased but cannot be integrated to one

Explanation

A large bandwidth averages over many observations, flattening features and adding bias, which can hide tail behaviour. A very small bandwidth gives the spiky, high-variance estimate described in the second option. Neither choice forces normality, and a proper kernel density still integrates to one.

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