An investor invests 100 dollars in a stock. Each month, the investment has probability 0.5 of increasing by 1.10 dollars and probability 0.5 of decreasing by 0.90 dollars. The changes in price in different months are mutually independent.

Calculate the probability that the investment has a value greater than 91 dollars at the end of month 100.

• 0.63
• 0.75
• 0.82
• 0.94
• 0.97

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

Solution: E

Let [math]X_k[/math] be the random change in month [math]k[/math]. Then [math]\operatorname{E}(X_k) = (0.5)(1.1) + 0.5(−0.9) = 0.1[/math] and [math]\operatorname{Var}(X_k) = 0.5(1.1)^2 + 0.5(−0.9)^2 − (0.1)^2 = 1.[/math] Let [math]S = \sum_{k=1}^{100}X_k [/math]. Then, [math]\operatorname{E}(S) = 1000(0.1) = 10[/math] and [math]\operatorname{Var}(S) = 100(1) = 100 [/math]. Finally,

[[math]] \operatorname{P}(100 + S \gt 91) = \operatorname{P}(S \gt -9) = \operatorname{P}( Z \gt \frac{-9-10}{\sqrt{100}} = -1.9 ) = 0.9713. [[/math]]

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