For a special 2-year term insurance policy on [math](x)[/math], you are given:

(i) Death benefits are payable at the end of the half-year of death

(ii) The amount of the death benefit is 300,000 for the first half-year and increases by 30,000 per half-year thereafter

(iii) [math]\quad q_{x}=0.16[/math] and [math]q_{x+1}=0.23[/math]

(iv) [math]\quad i^{(2)}=0.18[/math]

(v) Deaths are assumed to follow a constant force of mortality between integral ages

(vi) [math]Z[/math] is the present value random variable for this insurance

Calculate [math]\operatorname{Pr}(Z\gt277,000)[/math].

• 0.08
• 0.11
• 0.14
• 0.18
• 0.21

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

Answer: D

Half-year	PV of Benefit	PV > 277, 000 if and only if (x) dies in the 2nd or 3rd half years.
2	[math]300,000 v^{0.5}=(300,000)(1.09)^{-1}=275,229[/math]
3	[math]330,000 v^{1}=(330,000)(1.09)^{-2}=277,754[/math]
4	[math]390,000 v^{2}=(390,000)(1.09)^{-4}=276,286[/math]

Under CF assumption, [math]{ }_{0.5} p_{x}={ }_{0.5} p_{x+0.5}=(0.84)^{0.5}=0.9165[/math] and [math]{ }_{0.5} p_{x+1}={ }_{0.5} p_{x+1.5}=(0.77)^{0.5}=0.8775[/math] Then the probability of dying in the [math]2^{\text {nd }}[/math] or [math]3^{\text {rd }}[/math] half-years is [math]\left({ }_{0.5} p_{x}\right)\left(1-{ }_{0.5} p_{x+0.5}\right)+\left(p_{x}\right)\left(1-{ }_{0.5} p_{x+1}\right)=(0.9165)(0.0835)+(0.84)(0.1225)=0.1794[/math]

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