(40) wins the SOA lottery and will receive both:

• A deferred life annuity of [math]K[/math] per year, payable continuously, starting at age [math]40+\stackrel{\circ}{e}_{40}[/math] and
• An annuity certain of [math]K[/math] per year, payable continuously, for [math]\stackrel{\circ}{e}_{40}[/math] years

You are given:

(i) [math]\mu=0.02[/math]

(ii) [math]\delta=0.01[/math]

(iii) The actuarial present value of the payments is 10,000

Calculate [math]K[/math].

• 214
• 216
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Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: A

[math]\stackrel{\circ}{e}_{40}=\frac{1}{\mu}=50[/math] So, receive [math]K[/math] for 50 years guaranteed and for life thereafter.

[math]10,000=K\left[\bar{a}_{\overline{50}}+{ }_{50 \mid} \bar{a}_{40}\right][/math]

[math]\bar{a}_{\overline{50}}=\int_{0}^{50} e^{-\delta t}=\frac{1-e^{-50 \delta}}{\delta}=\frac{1-e^{-50(0.01)}}{0.01}=39.35[/math]

[math]{ }_{50} \bar{a}_{40}={ }_{50} E_{40} \bar{a}_{40+50}=e^{-(\delta+\mu) 50} \frac{1}{\mu+\delta}=e^{-1.5} \frac{1}{0.03}=7.44[/math]

[math]K=\frac{10,000}{39.35+7.44}=213.7[/math]

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