# P2.T5.22.4. Semi-parametric historical simulation value at risk (HS VaR)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Compare and contrast the age-weighted, the volatility-weighted, the correlation-weighted, and the filtered historical simulation approaches. Identify advantages and disadvantages of non-parametric estimation methods.

Questions:

22.4.1. Sally sorted her portfolio's daily loss/profit (L/P) over last year's 250 trading days and the worst five days experienced the following losses: 9.60, 7.40, 5.50, 4.00, and 3.00. These are sorted in L/P format and denominated in millions of USD dollars; e.g., the single worst daily loss over the last year was a loss of 9.60 million, the fourth-worst daily was a loss of 4.00 million. Given the sample of 250 days, she observes that the basic (aka, simple) historical simulation one-day value at risk (basic VaR) with 99.0% confidence is $5.50 million. Her colleague observes that, although there are 250 days in the sample, the three worst losses occurred with the most recent two weeks. Specifically, the worst loss of$9.60 million occurred seven (7) days ago; the second-worst loss of $7.40 million occurred five (5) days ago, and the third-worst loss of$5.50 million occurred only three days ago. Sally wants to retrieve the age-weighted HS VaR but she does not want to do any interpolation; she wants to keep it relatively simple. If Sally assumes a rate of decay of 0.850 as given by lambda, λ = 0.850, what is the age-weighted 99.0% HS VaR?

a. 4.00 million
b. 5.50 million
c. 7.40 million
d. 9.60 million

22.4.2. Peter manages a large portfolio and wants to generate a one-day value at risk (VaR) but he has the following preferences with respect to the approach:
• He would like the approach to incorporate the cross-sectional correlation structure on a given day
• He would like to be able to incorporate autocorrelations in a long-term forecast that will extend over at least 20 trading days
• If the current regime suggests, he would like to be able to produce VaR estimates that exceed the greatest loss in the historical sample
Given these preferences, which of the following approaches is BEST?

a. Basic HS
b. Bootstrap HS
c. Filtered HS
d. Volatility-weighted HS

22.4.3. Barbara is preparing a presentation to her board's risk committee. The presentation will display VaRs at multiple horizons and confidence levels for several portfolios. This question concerns one of the firm's new portfolios. The new portfolio only has 63 (daily L/P) observations, and these occurred during a relatively stable (aka, low volatility) regime. Of course, Barbara realizes that many experts prefer to have a sample of at least two years or 500 observations. The board wants to see value at risk (VaR) at very high confidence levels including 99.50%, 99.90%, and 99.99%. Finally, VaRs for other portfolios are presented using non-parametric methods because they are "intuitive and conceptually simple," as Dowd(†) explains.

In this situation, among the following approaches, what is probably her BEST approach?

a. Order statistics historical simulation (OHS) refined with a Richardson extrapolation
b. If semi-parametric refinements are insufficient, she should recommend a parametric approach
c. Bootstrap historical simulation with a pseudo-random (as opposed to naively random) number generator
d. Age-weighted historical simulation with an Epanechinikov or triangular (rather than Gaussian) kernel estimator kernel