For an annuity-due that pays 100 at the beginning of each year that (45) is alive, you are given:

(i) Mortality for standard lives follows the Standard Ultimate Life Table

(ii) The force of mortality for standard lives age [math]45+t[/math] is represented as [math]\mu_{45+t}^{\text {SULT }}[/math]

(iii) The force of mortality for substandard lives age [math]45+t, \mu_{45+t}^{S}[/math], is defined as:

[[math]] \mu_{45+t}^{S}= \begin{cases}\mu_{45+t}^{S U L T}+0.05, & \text { for } 0 \leq t\lt1 \\ \mu_{45+t}^{S U L T}, & \text { for } t \geq 1\end{cases} [[/math]]

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

Calculate the actuarial present value of this annuity for a substandard life age 45.

• 1700
• 1710
• 1720
• 1730
• 1740

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

Answer: A

[math]\ddot{a}_{45}^{S}=1+v p_{45}^{S} \ddot{a}_{46}^{S U L T}[/math]

[math]p_{45}^{S}=e^{-\int_{0}^{1} \mu_{45+t}^{S} d t}=e^{-\int_{0}^{1}\left(\mu_{45+t}^{S U L T}+0.05\right) d t}=e^{-\int_{0}^{1}\left(\mu_{45+t}^{S U L T}\right) d t} e^{-\int_{0}^{1}(0.05) d t}=p_{45}^{S U L T} \cdot e^{-0.05}=\left(\frac{98,957.6}{99,033.9}\right) e^{-0.05}=0.9504966[/math]

[math]\ddot{a}_{45}^{S}=1+v p_{45}^{S} \ddot{a}_{46}^{S U L T}=1+(1.05)^{-1}(0.9504966)(17.6706)=17.00[/math]

[math]100 \ddot{a}_{45}^{S}=1700[/math]

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