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Pinpoint the strengths and weaknesses of non-parametric estimation techniques.

FRM® Part II · Market Risk Measurement and Management · Non-Parametric Approaches · FRM2.MR.2.d

The idea

Non-parametric methods such as historical simulation, with or without bootstrapping and weighting, have clear strengths. They are intuitive and easy to implement. They assume no distribution for returns, so skewness and fat tails in the data are retained. They need no covariance matrix, and they handle non-linear positions such as options by revaluing them in each historical scenario. Their weaknesses all come from relying on the sample. Results depend on the chosen window, and the past is assumed to represent the future. Equal weights respond slowly to regime shifts. Large losses leaving the window cause abrupt falls in VaR, the ghost or drop-off effect. Extreme quantiles are limited by the worst loss observed and by the few tail observations. Longer holding periods leave very few independent observations, and new instruments lack history. The usual exam error is crediting these methods with the ability to extrapolate beyond the data or to react quickly.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following is a recognised strength of non-parametric (historical simulation) estimation of VaR?

  1. It can estimate losses far beyond the worst loss in the sampleWhy not: The estimate is confined to the observed data and cannot exceed the worst loss observed.
  2. It reacts immediately to a sudden rise in market volatilityWhy not: With equal weights, a volatility shock takes time to dominate the window, so the response is slow.
  3. It needs very little historical data, even at high confidence levelsWhy not: High confidence levels leave very few tail observations, so large samples are needed.
  4. It assumes no shape for the return distribution, so skewness and fat tails in the sample carry into the estimate - correct

Because the distribution is taken from the data, non-normal features are retained and no covariance matrix or distribution has to be estimated. The cost is dependence on the sample.

A harder one

A bank uses a 500-day window of equally weighted historical simulation for 99% VaR, which is the fifth largest loss. The seven largest losses in the window (USD million) are 8.0, 6.5, 5.5, 5.0, 4.4, 4.1 and 3.9. The 8.0 loss is the oldest observation in the window and drops out tomorrow, replaced by a small loss. Nothing about current market conditions has changed. The new 99% VaR is:

  1. USD 4.4 millionWhy not: Assumes the VaR is unchanged because the dropped loss lay beyond the quantile, but dropping it moves every rank up by one.
  2. USD 5.0 millionWhy not: Moves the rank the wrong way, taking the fourth largest of the original list as the new fifth largest.
  3. USD 4.1 million - correct
  4. USD 3.9 millionWhy not: Moves two ranks instead of one; only the 8.0 loss leaves the window.

The remaining losses in order are 6.5, 5.5, 5.0, 4.4, 4.1, so the fifth largest is 4.1 and VaR falls from 4.4. This is the ghost or drop-off effect: VaR changes when old observations leave the window, not because risk changed.

The hardest tier

A risk manager has 500 daily returns and wants a 10-day 99% VaR using non-overlapping 10-day returns from this sample. How many observations are expected to lie beyond the 99% quantile?

  1. 5Why not: Applies 1% to the 500 daily observations and ignores the 10-day horizon.
  2. 50Why not: This is the number of non-overlapping 10-day returns in the sample, not the number in the 1% tail.
  3. 0.5 - correct
  4. 0.05Why not: Divides by the horizon a second time after already forming 50 non-overlapping periods.

500 daily returns give 500/10 = 50 non-overlapping 10-day returns, and 1% of 50 is 0.5, less than a single observation. The 99% estimate cannot rest on a real tail observation, a weakness of historical simulation at longer horizons.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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