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Walk through how the historical simulation approach is used to compute both VaR and ES.

FRM® Part I · Valuation and Risk Models · Calculating and Applying VaR · FRM1.VR.2.b

The idea

Historical simulation applies the risk-factor changes observed on each of the last N days, typically 500 to 1,000, to today's portfolio and revalues it to give N hypothetical profit and loss outcomes. These are ranked from worst to best. VaR at confidence c is the loss at the (1 − c) percentile, so with 500 scenarios the 99 percent VaR is around the fifth worst loss. Expected shortfall is the average of the losses in the tail beyond that point, for example the mean of the five worst, and is never smaller than VaR. The method needs no distributional assumption and handles non-linear positions through full revaluation. However, it weights all days equally, reacts slowly to changing volatility and is limited by the sample. The common error is to apply past P&L rather than past risk-factor changes.

A practice question

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

Which description best fits the way historical simulation computes a one-day VaR?

  1. Fit a normal distribution to the portfolio's historical returns and multiply the standard deviation by the z-value for the confidence level.Why not: This is the parametric (variance-covariance) approach; historical simulation uses the empirical distribution.
  2. Apply the risk-factor changes seen on each of the past N days to today's portfolio, revalue, and read the loss percentile. - correct
  3. Draw random risk-factor values from a specified stochastic process and revalue the whole portfolio in each trial.Why not: This describes Monte Carlo simulation, which relies on a model rather than on actual past changes.
  4. Rank the actual daily profits and losses the firm booked over the past N days and take the percentile.Why not: Uses past P&L from past positions instead of applying past risk-factor changes to today's holdings.

Historical simulation keeps today's positions fixed and replays N days of actual market moves on them. The N hypothetical P&L values are ranked and the percentile read off.

A harder one

A bank uses 500 historical scenarios. It takes the 99% VaR as the 5th worst loss and the ES as the average of the 5 worst losses. The six worst losses in USD million are 5.8, 5.1, 4.6, 4.2, 3.9 and 3.7. Which pair of figures is correct?

  1. VaR is 3.7 and ES is 4.55.Why not: Off by one: uses the 6th worst loss as VaR and averages six losses.
  2. VaR is 4.72 and ES is 3.9.Why not: Swaps the two measures; ES can never be smaller than VaR at the same confidence level.
  3. VaR is 3.9 and ES is 5.80.Why not: Reports the single worst loss as ES instead of averaging the tail.
  4. VaR is 3.9 and ES is 4.72. - correct

The 5th worst loss is 3.9. ES = (5.8 + 5.1 + 4.6 + 4.2 + 3.9) / 5 = 23.6 / 5 = 4.72. The ES averages the whole tail, so it sits above VaR.

The hardest tier

A bank applies volatility-scaled historical simulation: each historical return is multiplied by (current daily volatility / volatility on that day). Current daily volatility is 2.0%. Four historical days gave these (return, volatility then): Day 1 (-3.0%, 1.0%); Day 2 (-2.4%, 1.6%); Day 3 (-1.8%, 0.5%); Day 4 (-4.0%, 2.5%). For a position of USD 20 million, which day gives the worst scenario and what is the loss?

  1. Day 4, a loss of about USD 0.80 million.Why not: Ranks on the raw historical returns without volatility scaling.
  2. Day 3, a loss of about USD 1.44 million. - correct
  3. Day 4, a loss of about USD 1.00 million.Why not: Inverts the scaling ratio (historical volatility / current volatility), which scales the wrong way.
  4. Day 3, a loss of about USD 0.36 million.Why not: Identifies the right day but applies the raw return instead of the volatility-scaled return.

Scaled returns: Day 1 -6.0%, Day 2 -3.0%, Day 3 -1.8 x 2.0/0.5 = -7.2%, Day 4 -4.0 x 2.0/2.5 = -3.2%. The worst is Day 3 and the loss is 20 million x 7.2% = USD 1.44 million.

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