A man purchases a life insurance policy on his 40th birthday. The policy will pay 5000 if he dies before his 50th birthday and will pay 0 otherwise. The length of lifetime, in years from birth, of a male born the same year as the insured has the cumulative distribution function

[[math]] F(t) = \begin{cases} 1-\exp(\frac{1-1.1^t}{1000}), \, t \gt 0 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

Calculate the expected payment under this policy.

• 333
• 348
• 421
• 549
• 574

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

Solution: B

[[math]] \begin{align*} \operatorname{P}(\textrm{40 year old man dies before age 50}) &= \operatorname{P}(T \lt 50 | T \gt 40) \\ &= \frac{\operatorname{P}(40 \lt T \lt 50)}{\operatorname{P}(T \gt 40)} \\ &= \frac{F (50) − F (40)}{1 − F (40)} \\ &= \frac{ \exp(\frac{1-1.1^{40}}{1000}) -\exp(\frac{1-1.1^{50}}{1000}) }{1-1 + \exp(\frac{1-1.1^{40}}{1000})} \\ &= \frac{0.9567 − 0.8901}{0.9567} \\ &= 0.0696. \end{align*} [[/math]]

And the expected benefit equals 5000(0.0696) = 348.

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