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Work out VaR from historical simulation of past data.

FRM® Part II · Market Risk Measurement and Management · Estimating Market Risk Measures: An Introduction and Overview · FRM2.MR.1.a

The idea

Historical simulation (HS) estimates VaR without assuming any distribution. It treats past profit and loss (P&L) as a sample from the distribution of future P&L. Apply each past day's asset returns to today's holdings, sort the results from worst to best, and read off the loss that only a fraction 1 − confidence of the observations exceed. Let n be the number of observations and c the confidence level. The tail size then follows from the sample size and the confidence level, as in Tail = n × (1 − c). With 500 observations and 95 percent confidence, the tail holds 25 observations. VaR is a single order statistic, not the whole tail. Under a common convention it is the 25th-worst loss, while other conventions use the 26th-worst or interpolate. State the quantile convention, and keep loss and P&L signs consistent when you sort. Common mistakes are using the 95th-worst observation, forgetting that past P&L reflects old positions, and expecting an estimate beyond the worst loss in the window. Exam point: HS is nonparametric and simple, but it depends entirely on how many tail observations the window contains.

Formulas

Number of observations in the historical-simulation tail

tail count = n × (1 − c)

n
the number of observations in the window
c
the confidence level

Equal weights, one observation per day. VaR is a single order statistic at the edge of this tail, not its average, and the quantile convention (25th-worst or 26th-worst of 500 at 95%) must be stated.

A practice question

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

A risk manager estimates VaR by historical simulation. Which statement best describes the method?

  1. It uses the empirical distribution of past P&L as the distribution of future P&L and reads VaR from a quantile of it. - correct
  2. It fits a normal distribution to past P&L and derives VaR from the sample mean and standard deviation.Why not: Describes the parametric (variance-covariance) method, not historical simulation, which assumes no distribution.
  3. It simulates many random price paths from a fitted stochastic process and takes a quantile of the results.Why not: Describes Monte Carlo simulation, which generates scenarios from a model rather than reusing actual past outcomes.
  4. It weights recent observations more heavily and so needs an estimated decay factor to be applied.Why not: Describes an age-weighted variant; basic historical simulation gives every observation in the window equal weight.

Basic historical simulation is nonparametric: the sorted past P&L observations stand in for the future P&L distribution and VaR is read from the appropriate quantile, with equal weights and no distributional assumption.

A harder one

A position is worth USD 40 million. Over the last 100 trading days its six worst daily losses, as percentages of position value, were 3.8%, 3.1%, 2.9%, 2.6%, 2.4%, 2.2% (largest first). Using historical simulation at 95 percent confidence, with VaR set midway between the 5th-worst and 6th-worst losses, the one-day VaR is closest to:

  1. USD 0.96 millionWhy not: Uses the 5th-worst loss alone and ignores the stated midpoint convention between the 5th and 6th observations.
  2. USD 1.18 millionWhy not: Averages the five worst losses, which is an expected shortfall type calculation, not a quantile.
  3. USD 0.92 million - correct
  4. USD 1.52 millionWhy not: Uses the single worst loss, which would correspond to a much higher confidence level than 95 percent.

With 100 observations and a 5 percent tail, VaR lies between the 5th and 6th worst losses: (2.4 + 2.2)/2 = 2.30 percent. Applied to USD 40 million this gives 2.30% x 40 = USD 0.92 million.

The hardest tier

A risk manager holds a USD 25 million position and has 500 daily returns. The 5th-worst return is -2.90 percent and the 25th-worst return is -1.84 percent. Taking the 25th-worst return as the 95 percent one-day quantile, and assuming returns are independent and identically distributed, the 10-day 95 percent VaR using the square-root-of-time rule is closest to:

  1. USD 1.45 million - correct
  2. USD 0.46 millionWhy not: Reports the one-day VaR and omits the square-root-of-time scaling to ten days.
  3. USD 2.29 millionWhy not: Uses the 5th-worst return, which is the 99 percent quantile, not the 95 percent quantile.
  4. USD 4.60 millionWhy not: Scales by ten instead of the square root of ten, which would require perfectly correlated daily losses.

A 5 percent tail of 500 observations is 25 observations, so the one-day 95 percent VaR is 1.84% x 25 = USD 0.460 million. Multiplying by the square root of 10 (3.162) gives USD 1.45 million.

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