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# P1.T4.810. Spectral risk measures, especially Expected Shortfall (ES) (Dowd Ch.2)

#### Nicole Seaman

##### Director of FRM Operations
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Learning objectives: Explain and calculate Expected Shortfall (ES), and compare and contrast VaR and ES. Describe spectral risk measures, and explain how VaR and ES are special cases of spectral risk measures. Describe how the results of scenario analysis can be interpreted as coherent risk measures.

Questions:

810.1. In the event of certain operational loss, the severity of the loss ranges from three to 18. As luck would have it, the loss distribution is similar to the roll of three dice.

For example, in the event of a loss occurrence, the probability of a loss of either three (the minimum loss) or 18 (the maximum loss) is given by approximately (1/6)^3 = 0.4630%; the probability of a loss of four or 17 is 3/6^3 = 1.489%. Which are nearest, respectively, to the 95.0% value at risk (VaR) and 95.0% expected shortfall (ES) for the severity of this loss?

a. 14.5 (95.0% VaR) and 15.5 (95.0% ES)
b. 15.0 (95.0% VaR) and 16.4 (95.0% ES)
c. 15.5 (95.0% VaR) and 17.1 (95.0% ES)
d. 16.0 (95.0% VaR) and 18.0 (95.0% ES)

810.2. The following table shows the quantile (aka, inverse cumulative distribution function) and probability density function (pdf) for certain confidence levels among the standard normal distribution. The common 95.0% and 99.0% are highlighted:

Assume an asset's (arithmetic) return is normally distributed with a daily volatility of $20.00. Which is nearest to its one-day 99.0% expected shortfall (ES)? a.$0.53
b. $1.24 c.$46.53
d. \$53.30

810.3. Each of the following is true EXCEPT which is false?

a. VaR is a type (special case) of spectral measure
b. ES is a type (special case) of spectral measure
c. VaR and ES are both special cases of the general risk measure
d. ES meets the "weekly increasing" condition by assigning the same weight to all tail losses, but this is a sub-optimal parametrization of risk-aversion

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