# P2.T5.22.1 Basic historical simulation value at risk (HS VaR), lognormal VaR, and expected shortfall (ES)

#### David Harper CFA FRM

##### David Harper CFA FRM
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Learning objectives: Estimate VaR using a historical simulation approach. Estimate VaR using a parametric approach for both normal and lognormal return distributions. Estimate the expected shortfall given profit and loss (P/L) or return data.

Questions:

22.1.1. Peter has collected the daily loss/profit data for his firm's largest fund over the last year. The full year included 250 trading days. Where the units are millions of dollars, the worst ten days were the following: {39.0, 38.0, 33.0, 31.0, 26.0, 23.0, 19.0, 15.0, 12.0, and 9.0}. Therefore, the single worst loss was -$39.0 million. Because his data format is L(+)/P(-) rather than the mathematically natural P(+)/L(-), this worst loss is expressed as +39.0. The gains, which are not here provided, are expressed as negatives; for example, in L(+)/P(-) format, a gain of +8.0 million is expressed as -8.0. For this 250-day sample, Peter's calculates the one-day historical simulation value at risk (HS VaR) at both the 97.5% and 99.0% confidence levels. His HS VaR is the basic HS (aka, unweighted or equally-weighted HS) where each day in the historical window is assigned the same weight. If X equals the one-day 97.5% VaR and Y equals the 95.0% one-day VaR, what is (X-Y)? Put another way but equivalently, what is the difference between these one-day 97.5% and 99.0% HS VaRs? a. -3.0 b. 9.0 c. 14.0 d. 21.0 22.1.2. Bertha is a risk manager who estimates the market risk of one of her firm's portfolios. Because geometric returns are time-additive and arithmetic returns are cross-additive, she measures performance with both arithmetic and geometric returns. Her assumptions are gathered below: • The portfolio value is$100.0 million and she assumes there are 250 trading days per year.
• Using geometric returns, the average return +12.0% per annum with a standard deviation of 28.0%
• Using arithmetic returns, the average return +14.0% per annum with a standard deviation of 32.0%
She wants to measure the one-day 95.0% value at risk (VaR) under both approaches: assuming arithmetic returns are normal (aka, normal VaR) and assuming geometric returns are normal; aka, lognormal VaR. She assumes the returns are serially independent. Which of the following is nearest to the difference between the one-day 95.0% normal VaR and the one-day 95.0% lognormal VaR?

a. The one-day 95% normal VaR is approximately $37,700 less than the one-day 95% lognormal VaR b. The one-day 95% normal VaR is approximately$380,100 less than the one-day 95% lognormal VaR
c. The one-day 95% normal VaR is approximately $29,400 greater than the one-day 95% lognormal VaR d. The one-day 95% normal VaR is approximately$448,800 greater than the one-day 95% lognormal VaR

22.1.3. Consider a portfolio that holds three junk bonds. The bonds are independent and identical (i.i.d.). Each has a default probability (PD) of 10.0%. The face value of each bond is $100.00. Therefore, the portfolio's value is$300.00. For convenience, we will assume their coupons equal their yields such that they are always priced at par. Further, we will assume a loss given default (LGD) of 100% (i.e., there will be no recovery) and a single period (such that horizon is not a factor). What is the portfolio's 95.0% expected shortfall (ES)?

a. $100.00 b.$158.00
c. \$300.00
d. Ambiguous