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# P2.T5.707. Historical simulation and lognormal value at risk (VaR) (Dowd)

#### Nicole Seaman

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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.

Questions:

707.1. A mutual fund's daily returns for the last 300 trading days is plotted on this histogram. Additionally, the worst 20 daily returns are sorted explicitly below the histogram:

Under the basic historical simulation approach, which of the following is the BEST one-day 99.0% value at risk (VaR)?

a. 0.710
b. 0.840
c. 0.800
d. 0.900

707.2. Consider the following asset over three periods that starts with an initial price of $20.00 and pays a$2.00 dividend each period. In addition to the asset's price and dividends over three periods, we also show its per-period arithmetic (aka, simple) return:

a. Geometric returns cannot be greater than arithmetic returns
b. The geometric returns in each period are approximately +22.31%, -9.10% and 35.14%
c. If the asset instead paid zero dividends, then the geometric return would equal the arithmetic return
d. The three-period geometric return is conveniently the sum of per-period geometric returns which, in this case, is about +22.31% -9.10% +35.14% = 48.36%

707.3. A risk manager is estimating the market risk of a portfolio using both the normal distribution and the lognormal distribution assumptions. The manager gathers the following data on the portfolio:
• Annual mean: 12.0%
• Annual volatility: 37.0%
• Current portfolio value: USD $1.1 million due to 12,500 shares at a current price of$80.00 per share
• Trading days in a year: 250
Which of the following statements is correct? (This question inspired by GARP's 2017 Practice Exam Part 2, Question #2 despite its tedious nature)

a. Lognormal 95% VaR is greater than than normal 95% VaR at the one-day holding period by about 2.43%
b. Lognormal 95% VaR is less than than normal 95% VaR at the one-year (250 days) holding period by about 3.90%
c. Lognormal 99% VaR is greater than normal 99% VaR at the one-day holding period by about 4.04%
d. Lognormal 99% VaR is less than normal 99% VaR at the one-year (252 days) holding period by about 21.75%