Insurance companies A and B each earn an annual profit that is normally distributed with the same positive mean. The standard deviation of company A’s annual profit is one half of its mean. In a given year, the probability that company B has a loss (negative profit) is 0.9 times the probability that company A has a loss.

Calculate the ratio of the standard deviation of company B’s annual profit to the standard deviation of company A’s annual profit.

• 0.49
• 0.90
• 0.98
• 1.11
• 1.71

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let [math]X[/math] and [math]Y[/math] represent the annual profits for companies A and B, respectively and m represent the common mean and s the standard deviation of [math]Y[/math]. Let [math]Z[/math] represent the standard normal random variable.

Then because X’s standard deviation is one-half its mean,

[[math]] \operatorname{P}(X \lt 0) = \operatorname{P} \left( \frac{X-m}{0.5m} \lt \frac{0-m}{0.5m}\right) = \operatorname{P}(Z \lt 2) = 0.0228. [[/math]]

Therefore company B’s probability of a loss is 0.9(0.0228) = 0.02052. Then,

[[math]] 0.02052 = \operatorname{P}(Y \lt 0) = \operatorname{P} \left( \frac{Y-m}{s} \lt \frac{0-m}{s} \right) = \operatorname{P}(Z \lt -m/s). [[/math]]

From the tables, –2.04 = –m/s and therefore s = m/2.04. The ratio of the standard deviations is (m/2.04)/(0.5m) = 0.98.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.