An insurance company sells special fully discrete two-year endowment insurance policies to smokers (S) and non-smokers (NS) age [math]x[/math]. You are given:

(i) The death benefit is 100,000 ; the maturity benefit is 30,000

(ii) The level annual premium for non-smoker policies is determined by the equivalence principle

(iii) The annual premium for smoker policies is twice the non-smoker annual premium

(iv) [math]\quad \mu_{x+t}^{\mathrm{NS}}=0.1, t\gt0[/math]

(v) [math]\quad q_{x+k}^{\mathrm{s}}=1.5 q_{x+k}^{\mathrm{Ns}}[/math] for [math]k=0,1[/math]

(vi) [math]\quad i=0.08[/math]

Calculate the expected present value of the loss at issue random variable on a smoker policy.

• -30,000
• -29,000
• -28.000
• -27.000
• -26.000

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

Answer: A

[math]q_{x}^{\mathrm{NS}}=q_{x+1}^{\mathrm{NS}}=1-e^{-0.1}=0.095[/math]

Then the annual premium for the non-smoker policies is [math]P^{\mathrm{NS}}[/math], where

[[math]] \begin{aligned} P^{\mathrm{NS}}\left(1+v p_{x}^{\mathrm{NS}}\right) & =100,000 v q_{x}^{\mathrm{NS}}+100,000 v^{2} p_{x}^{\mathrm{NS}} q_{x+1}^{\mathrm{NS}}+30,000 v^{2} p_{x}^{\mathrm{NS}} p_{x+1}^{\mathrm{NS}} \\ P^{\mathrm{NS}} & =\frac{100,000(0.926)(0.095)+100,000(0.857)(0.905)(0.095)+30,000(0.857)(0.905)^{2}}{1+(0.926)(0.905)} \\ P^{\mathrm{NS}} & =20,251 \end{aligned} [[/math]]

And then [math]P^{\mathrm{S}}=40,502[/math].

[[math]] \begin{aligned} q_{x}^{S}=q_{x+1}^{S}= & 1.5\left(1-e^{-0.1}\right)=0.143 \\ E P V\left(L^{\mathrm{S}}\right)= & 100,000 v q_{x}^{\mathrm{S}}+100,000 v^{2} p_{x}^{\mathrm{S}} q_{x+1}^{\mathrm{S}}+30,000 v^{2} p_{x}^{\mathrm{S}} p_{x+1}^{\mathrm{S}}-P^{\mathrm{S}}-P^{\mathrm{S}} v p_{x}^{\mathrm{S}} \\ = & 100,000(0.926)(0.143)+100,000(0.857)(0.857)(0.143) \\ & \quad+30,000(0.857)(0.857)^{2}-40,502-40,502(0.926)(0.857) \\ = & -30,017 \end{aligned} [[/math]]

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