Nicole Seaman

Director of CFA & FRM Operations
Staff member
Subscriber
Learning objectives: Describe coherent risk measures. Estimate risk measures by estimating quantiles. Evaluate estimators of risk measures by estimating their standard errors. Interpret quantile-quantile (QQ) plots to Identify the characteristics of a distribution.

Questions:

22.1. Peter fit his portfolio's historical return dataset with a parametric distribution. He is able to retrieve the inverse cumulative distribution function (inverse CDF), analogous to Excel's NORM.S.INV(.) function that returns an inverse CDF; e.g., NORM.S.INV(0.990) = 2.326. However, he lacks a corresponding, analytical function for the 9X% expected shortfall (ES).

Let's assume the following variables: C is the confidence level; S is the significance level given by S = (1 - C); N is the number of slices, and L = S/N is the size of each slice. If Peter wants to approximate the expected shortfall (ES), what can he do?

a. Compute the summation of N tail VaRs at confidence levels given by the sequence {C+L, C+2*L, ..., C+N*L}
b. Compute the average (N-1) tail VaRs at confidence levels given by the sequence {C+L, C+2*L, ..., C+(N-1)*L}
c. Compute the average (N+1) tail VaRs at confidence levels given by the sequence {C-L/N, C-2*L/N, ..., C-(N-1)*L/N}
d. It is inappropriate to attempt to estimate the expected shortfall (ES) as a function of a vector of tail VaRs


22.2. The general risk measure, M(ϕ) as given by a weighted average of the quantiles:

P2.T5.22.2.png


In this general risk measure, ϕ(p) is the weighting (aka, risk aversion) function. Which of the following is true?

a. All cases of this general risk measure are coherent
b. If the weights applied to the tail-loss quantile are unequal, this measure is incoherent
c. If the weights applied to the tail-loss quantiles are weakly increasing, this measure is coherent
d. The weakness of this measure is that we cannot represent different degrees of risk aversion


22.3. Although it is common in risk measurement to estimate value at risk (VaR) and expected shortfall (ES) metrics, it is less common to estimate their standard errors. However, arguably, the standard errors (SE) of risk measures are almost as important as the measures themselves. Sometimes an analytical solution is available. When an analytical solution to SE estimation is not readily available, a practical and useful technique is bootstrapping. As Dowd explains, "we bootstrap a large number of estimators from the given distribution function ... and we estimate the standard error of the sample of bootstrapped estimators. Even better, we can also use a bootstrapped sample of estimators to estimate a confidence." -- Source: Kevin Dowd, Measuring Market Risk, 2nd Edition (John Wiley & Sons, 2005), Chapter 3.

In regard to the standard error of risk measures, each of the following is true EXCEPT which is false?

a. The standard error (SE) is a measure of the precision of an estimate
b. The standard error (SE) of a coherent estimator is asymptotically normal
c. The standard error (SE) decreases (aka, narrows) as the sample size increases
d. The standard error (SE) of VaR can be bootstrapped but a drawback is the requirement of a parametric distribution and the generally implied mathematical complexity

Answers here:
 
Last edited by a moderator:
Top