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P2.T5.701. Value at risk (VaR) and expected shortfall (ES)

Nicole Seaman

Director of FRM Operations
Staff member
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701.1. A highly risky asset with an very high expected return exhibits the following characteristics:
  • Incredibly high expected return of 26.0% per annum
  • But this high expected return comes at the cost of high DAILY volatility of 1.0% per day
  • Current portfolio value: USD 10,000,000
  • Trading days: 20 per month and 250 per year
We are interested in the absolute value at risk (aVaR) where the worst expected loss is expressed as a positive number. In the case of this asset, for example, the 95.0% confident one-day normal aVaR is given by -26.0%/250 + 1.0%*1.645 = +1.541%. In regard to a 95.0% confident VaR, each of the following statements about this asset is true EXCEPT which is false?

a. The one-year (absolute) 95.0% confident normal VaR is nearly zero
b. The one-year (absolute) 95.0% confident lognormal VaR is nearly zero
c. As we extend the time horizon of the 95.0% confident VaR, the normal VaR never peaks and the lognormal VaR (as usual) cannot become negative
d. Over any short-term horizon, specifically less than ten days, the difference between the normal and lognormal 95.0% confident VaR is less than ten basis points (0.10%)

701.2. Donald the risk manager collects 300 trading days of profit and loss (P/L) data and plots the daily outcomes in a histogram. The loss tail of the histogram is plotted below, but this histogram only displays the worst 21 losses out of 300 days in the full window. Specifically, in this case, the worst loss among the 300 days was a loss of -14.5%; the second worst loss was -12.8%; and the third worst loss was -12.1%. Consequently, there is no bar plotted at the interval (-13%, 14%) which is between -13% and -14%.

Which of the following is MOST LIKELY the 99.0% confident expected shortfall (ES)?

a. 14.0%
b. 13.1%
c. 12.5%
d. 10.7%

701.3. Betty the analyst has collected a dataset and seeks to fit a relatively simple univariate distribution to the data. She hopes the data is approximately normal. To test this hypothesis, she generates a quantile-quantile plot (QQ plot) using a standard normal distribution as the reference. This QQ plot is displayed below.

Among the following choices, which distribution is probably the best fit for this data?

a. Binomial
b. Poisson
c. Lognormal
d. Normal but with a large (non standard) variance

Answers here: