# P2.T5.22.3 Historical simulation approaches to value at risk (VaR) and expected shortfall (ES)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Apply the bootstrap historical simulation approach to estimate coherent risk measures. Describe historical simulation using non-parametric density estimation.

Questions:

22.3.1. Which of the following is an essential difference between BASIC historical simulation and BOOTSTRAP historical simulation; i.e., one of them involves this feature or element but the other does not?

a. Random number generator
b. Non-parametric methodology
c. Parametric distributional assumption
d. Actual observed (i.e., historical) loss data

22.3.2. When we estimate value or risk (VaR) or expected shortfall (ES), we often stop at the point estimate. Due to the inconvenience, we rarely (most of us!) include a confidence interval around the point estimate of the risk measure. Kevin Dowd gives us two approaches to the estimation of confidence intervals (CIs) when we have generated VaR or ES with historical simulation. One of those methods is an order-statistics approach that applies the theory of order statistics. Dowd says that "one of the most promising methods is to apply the theory of order statistics."(†)

Which of the following statements is TRUE about order statistics when applied to the estimation of confidence intervals around VaR and/or ES?

a. Requires a large sample
b. Difficult to implement in practice due to advanced mathematics
c. Premised on the Poisson distribution where lambda choice is arbitrary
d. Can be applied to any parametric or non-parametric value at risk (VaR) or expected shortfall (ES)

22.3.3. Peter has collected the historical sample of daily profit and loss (P/L) observations for his firm's new equity portfolio but his dataset only contains 250 days. Ideally, he would prefer 1,000 observations because the board asked for a presentation that displays value at risk (VaR) at confidence levels of (99.0%, 99.1%, ..., 99.8%, and 99.9%). The board also expressed a preference for a non-parametric approach; apparently, some directors are concerned that parametric approaches might be too difficult to communicate and understand. In addition to granular VaR confidence levels (e.g., 99.1%), if possible, the board would like to see confidence INTERVALS around the VaR estimates in order to gauge their precision; e.g., a 9X% confidence interval around the 99.1% VaR.

Therefore, Peter considers the following four approaches: basic historical simulation (BHS), bootstrapped historical simulation (BOOT), historical simulation using non-parametric density estimation (NPDE), and age-weighted historical simulation (AWHS). Given the realities of his relatively small, empirical dataset, which of the following statements is TRUE?

a. Peter can easily eliminate the BHS approach because basic historical simulation cannot even retrieve a 99.10% (or 99.20% etc.) VaR quantile due to the fact that 1/250 is greater than 0.10%
b. Peter has a dilemma (i.e., none of the four approaches is feasible) as the construction of confidence levels around VaR estimates will require him to employ a parametric (or semi-parametric at least) approach
c. Peter has a dilemma (i.e., no approach is good here) simply because his sample size is too small to meet the board's preferences under any of the four approaches
d. Non-parametric density estimation (NPDE) can retrieve VaRs at granular confidence levels (e.g., 99.10%) as if the distribution is a continuous probability density function (pdf)