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# P1.T2.714. Lognormal distribution (Miller Chapter 4)

#### Nicole Seaman

##### Director of FRM Operations
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Learning objective: Distinguish the key properties among the following distributions: ... lognormal distribution

Questions:

714.1. Consider a stock with an initial price of $60.00, an expected return of 9.0% per annum, and a volatility of 10.0% per annum. The continuously compounded return is assumed to be normally distributed; i.e., the log return LN[S(t)/S(0)] is approximately normal. Therefore, the future stock price has a lognormal distribution. Which is nearest to an estimate of the worst expected stock price in four (4) years with 95.0% confidence; put another way, what is the price level, in four years, that we expect the stock to at least exceed with 95.0% confidence? a.$49.83
b. $52.25 c.$60.67
d. \$70.12

714.2. Peter the Risk Analyst wants to characterize an operational loss severity process with a relatively simple and recognizable probability distribution. His first choice is the lognormal distribution. Peter has several criteria for his distribution. The lognormal distribution does support, or at least does not violate, each of the following criteria EXCEPT which of the following criteria is not met by the lognormal distribution?

a. He requires a genuine probability distribution
b. His expected loss (aka, mean) is non-negative and the standard deviation is significantly greater than one
c. He wants summation stability: his simulation will add several independent randoms variables and he wants their sum to maintain the same distribution
d. He wants positive skew (losses are positive values; aka, L/P format) where the mean is greater than the median, and he does not want a theoretical limit on the maximum loss

714.3. Consider a stock with an an expected return of 12.0% per annum and a volatility of 10.0% per annum. The distributional assumption is the same geometric Brownian motion assumed by the Black-Scholes-Merton: log returns are normal such that prices are lognormal. Which is nearest to the 95.0% two-sided confidence interval for the AVERAGE return realized over nine (9) years?

a. Between 3.0% and 20.0%
b. Between 5.0% and 18.0%
c. Between 7.0% and 17.0%
d. Between 9.0% and 15.0%