SOA Exam ALTAM Sample Question 25

ABC Insurance uses the following model for calculating premiums and policy values for joint life insurance policies.

The forces of transition are as follows, where $\mu_z^s$ is the force of mortality at age $z$ under the Standard Ultimate Life Table (SULT).

$\mu_{x:y}^{01} = \mu_y^s - 0.005; \qquad \mu_{x:y}^{02} = \mu_x^s; \qquad \mu_x^{13} = \mu_x^s + 0.0025; \qquad \mu_y^{23} = \mu_y^s + 0.005$

Part a (1 point)

State with reasons whether the future lifetimes of $(x)$ and $(y)$ are independent under this model.

Part b (1 point)

Show that ${}_tp_{x:y}^{00} = e^{\lambda t}\,{}_tp_x^s\,{}_tp_y^s$ , where ${}_tp_x^s$ and ${}_tp_y^s$ are from the SULT, and where $\lambda > 0$ is a constant which you should specify.

Part c (1 point)

In the following table, you are given annuity values calculated using SULT mortality, at different forces of interest, and assuming independent future lifetimes for joint life functions.

$\delta$	3.50%	3.75%	4.00%	4.25%	4.50%
$\bar{a}_{40:50}^s$	19.3199	18.6530	18.0216	17.4235	16.8564
$\bar{a}_{40}^s$	22.4243	21.4880	20.6129	19.7939	19.0268
$\bar{a}_{50}^s$	20.0391	19.3174	18.6358	17.9917	17.3825

Determine $\bar{a}_{x:y}^{00}$ at $\delta = 4\%$ , for $x = 40$ and $y = 50$ , based on the ABC Insurance model.

Part d (3 points)

(i) Write down an integral expression for the expected present value of a payment of 1 immediately on the first death of $(x)$ and $(y)$ , using the multiple state model notation.

(ii) Show that, for $x = 40$ and $y = 50$ , at $\delta = 4\%$ , the expected present value of a payment of 1 immediately on the first death, under the ABC Insurance joint life mortality model, is 0.227 to the nearest 0.001. You should calculate the value to the nearest 0.00001.

Part e (1 point)

ABC Insurance issues a fully continuous joint life policy of 500,000 on $(x)$ , who is 40, and $(y)$ , who is 50.

Calculate the annual rate of net premium for this policy using the ABC Insurance joint life mortality model, with $\delta = 4\%$ .

Part f (1 point)

Exactly 10 years after issue $(y)$ dies while $(x)$ survives. At that time, $(x)$ uses the sum insured to provide a continuous annuity paying $X$ per year while she survives. The annuity amount is determined using the equivalence principle with $\delta = 4\%$ .

Calculate $X$ .

(This is Question 4 from the Spring 2022 Written Answer exam.)

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