You are given:

(i) [math]\delta_{t}=0.06, \quad t \geq 0[/math]

(ii) [math]\quad \mu_{x}(t)=0.01, \quad t \geq 0[/math]

(iii) [math]\quad Y[/math] is the present value random variable for a continuous annuity of 1 per year, payable for the lifetime of [math](x)[/math] with 10 years certain

Calculate [math]\operatorname{Pr}(Y\gt\mathrm{E}[Y])[/math].

• 0.705
• 0.710
• 0.715
• 0.720
• 0.725

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

Answer: A

[math]E(Y)=\bar{a}_{10}+e^{-\delta(10)} e^{-\mu(10)} \bar{a}_{x+10}[/math]

[math]=\frac{\left(1-e^{-0.6}\right)}{0.06}+e^{-0.7} \frac{1}{0.07}[/math]

[math]=14.6139[/math]

[math]Y \gt E(Y) \Rightarrow\left(\frac{1-e^{-0.06 T}}{0.06}\right)\gt14.6139[/math]

[math]\Rightarrow T \gt34.90[/math]

[math]\operatorname{Pr}[Y \gt E(Y)]=\operatorname{Pr}(T\gt34.90)=e^{-34.90(0.01)}=0.705[/math]

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