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Set VaR against worst-case scenario analysis, describing the latter first.

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

The idea

Worst-case scenario analysis asks a different question from VaR. Rather than the loss that will be exceeded with probability 1 − c on a single day, it looks at the distribution of the worst loss over a longer period such as N days. If daily losses are independent with distribution function F, the probability that the worst loss over N days is below x is as follows. Prob = F(x)ᴺ. A daily 99 percent VaR is expected to be breached about 2.5 times in 250 days. The chance of at least one breach is 1 − 0.99²⁵⁰ = 92 percent, so the worst day in a year usually exceeds the VaR by some margin. VaR gives a threshold but not the size of the tail loss, while worst-case analysis focuses on severity. The mistake is to read VaR as the maximum loss.

Formulas

Distribution of the worst daily loss over N days

P(worst loss over N days ≤ x) = F(x)^(N)

F(x)
the probability that one day's loss is at most x
N
the number of days

Daily losses must be independent with the same distribution. The worst loss over a year is typically well beyond a one-day 99% VaR.

Probability of at least one VaR breach

P(at least one breach in N days) = 1 − c^(N)

c
the daily VaR confidence level
N
the number of days

Independent days and a correct VaR. At 99% over 250 days it is about 92%; reporting c raised to N gives the chance of no breach instead.

A practice question

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

How does worst-case scenario analysis differ in its question from a VaR calculation?

  1. It averages the losses that exceed the VaR threshold on a single day of trading.Why not: This describes expected shortfall, not worst-case analysis.
  2. It repeats the VaR calculation with a higher confidence level on the same single day.Why not: A higher confidence level is still a single-period percentile.
  3. It measures the loss under one management-specified event with no attached probability.Why not: This describes a stress test; worst-case analysis has a probability distribution for the worst loss.
  4. It examines the distribution of the worst loss over N days rather than the loss exceeded with probability 1 - c on one day. - correct

VaR is a percentile of a single period's distribution. Worst-case analysis asks how bad the single worst outcome over a horizon of N periods is likely to be, which brings in the length of the horizon.

A harder one

A bank reports a one-day 99% VaR. Assuming daily losses are independent and the VaR is correct, the probability that at least one day in the next 250 trading days has a loss exceeding the VaR is closest to:

  1. 8.1%Why not: This is 0.99^250, the probability of no exceedance at all, rather than at least one.
  2. 91.9% - correct
  3. 22.2%Why not: Uses 25 days instead of 250 in the calculation 1 - 0.99^n.
  4. 1.0%Why not: Uses the single-day exceedance probability and ignores that there are 250 independent chances.

The probability of no exceedance in 250 days is 0.99^250 = 0.0811. The probability of at least one is 1 - 0.0811 = 91.9%. The expected number of exceedances is 250 x 1% = 2.5.

The hardest tier

Daily losses on a USD 10 million position are independent and normal with zero mean and a daily standard deviation of 1.5%. The probability that the worst daily loss over 250 days is below x is F(x)^250. The median worst daily loss solves F(x)^250 = 0.5. Standard normal quantiles: N^-1(0.99) = 2.326; N^-1(0.99723) = 2.774; N^-1(0.998) = 2.878; N^-1(0.999) = 3.090. The median worst daily loss is closest to:

  1. USD 349,000Why not: Uses the 99% one-day VaR as if it were the typical worst loss over the year.
  2. USD 432,000Why not: Takes F(x) = 1 - 0.5/250 = 0.998, an approximation that over-corrects the tail probability.
  3. USD 464,000Why not: Uses a daily 99.9% quantile, a conventional level that does not solve F(x)^250 = 0.5.
  4. USD 416,000 - correct

F(x) = 0.5^(1/250) = 0.99723, so x = 2.774 x 1.5% x 10 million = USD 416,000. This is about 19% above the one-day 99% VaR of USD 349,000, which shows how the horizon raises the worst-case figure.

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