For a 30 -year term life insurance of 100,000 on (45), you are given:

(i) The death benefit is payable at the moment of death

(ii) Mortality follows the Standard Ultimate Life Table

(iii) [math]\delta=0.05[/math]

(iv) Deaths are uniformly distributed over each year of age

Calculate the [math]95^{\text {th }}[/math] percentile of the present value of benefits random variable for this insurance.

• 30,200
• 31,200
• 35,200
• 36,200
• 37,200

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: C

The earlier the death (before year 30), the larger the loss. Since we are looking for the [math]95^{\text {th }}[/math] percentile of the present value of benefits random variable, we must find the time at which [math]5 \%[/math] of the insureds have died. The present value of the death benefit for that insured is what is being asked for.

[[math]] \begin{aligned} & l_{45}=99,033.9 \Rightarrow 0.95 l_{45}=94,082.2 \\ & l_{65}=94,579.7 \\ & l_{66}=94,020.3 \end{aligned} [[/math]]

So, the time is between ages 65 and 66, i.e., time 20 and time 21.

[[math]] \begin{aligned} & l_{65}-l_{66}=94,579.7-94,020.3=559.4 \\ & l_{65+t}-l_{66}=94,579.7-94,082.2=497.5 \end{aligned} [[/math]]

[math]497.5 / 559.4=0.8893[/math]

The time just before the last [math]5 \%[/math] of deaths is expected to occur is: [math]20+0.8893=20.8893[/math]

The present value of death benefits at this time is:

[math]100,000 e^{-20.8893(0.05)}=35,188[/math]

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.