S, now age 65 , purchased a 20 -year deferred whole life annuity-due of 1 per year at age 45. You are given:

(i) Equal annual premiums, determined using the equivalence principle, were paid at the beginning of each year during the deferral period

(ii) Mortality at ages 65 and older follows the Standard Ultimate Life Table

(iii) [math]\quad i=0.05[/math]

(iv) [math]\quad Y[/math] is the present value random variable at age 65 for S's annuity benefits

Calculate the probability that [math]Y[/math] is less than the actuarial accumulated value of S's premiums.

• 0.35
• 0.37
• 0.39
• 0.41
• 0.43

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

Answer: C

Let [math]C[/math] be the annual contribution, then [math]C=\frac{{ }_{20} E_{45} \ddot{a}_{65}}{\ddot{a}_{45: 20}}[/math]

Let [math]K_{65}[/math] be the curtate future lifetime of (65). The required probability is

[[math]] \operatorname{Pr}\left(\frac{C \ddot{a}_{45: 20}}{{ }_{20} E_{45}}\gt\ddot{a}_{\overline{K_{65}+1}}\right)=\operatorname{Pr}\left(\frac{{ }_{20} E_{45} \ddot{a}_{65}}{\ddot{a}_{45: 20}} \frac{\ddot{a}_{45: 20}}{{ }_{20} E_{45}}\gt\ddot{a}_{\overline{K_{65}+1}}\right)=\operatorname{Pr}\left(\ddot{a}_{65}\gt\ddot{a}_{\overline{K_{65}+1}}\right)=\operatorname{Pr}\left(13.5498\gt\ddot{a}_{\overline{K_{65}+1}}\right) [[/math]]

Thus, since [math]\ddot{a}_{\overline{21}}=13.4622[/math] and [math]\ddot{a}_{\overline{22}}=13.8212[/math] we have

[[math]] \operatorname{Pr}\left(\ddot{a}_{K_{65}+1}\lt13.5498\right)=\operatorname{Pr}\left(K_{65}+1 \leq 21\right)=1-_{21} p_{65}=1-\frac{l_{86}}{l_{65}}=1-\frac{57,656.7}{94,579.7}=0.390 [[/math]]

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