SOA Exam ALTAM Spring 2023, Question 2

Study guide: Exam ALTAM

Consider the following multi-state model for modeling joint life insurance policies issued to a couple $(x)$ and $(y)$ :

Part a (1 point)

State two reasons why a couple may have dependent future lifetimes.

Part b (1 point)

State the conditions for $(x)$ and $(y)$ to have independent future lifetimes, in terms of $\mu_{x+t:y+t}^{01}$ , $\mu_{x+t:y+t}^{02}$ , $\mu_{y+t}^{13}$ , $\mu_{x+t}^{23}$ , and $\mu_{x+t:y+t}^{03}$ .

Part c (3 points)

(i) Write down the Kolmogorov forward differential equation, with the associated boundary condition, for ${}_tp_{x:y}^{00}$ .

(ii) Use the Kolmogorov forward differential equation in part (i) to show that

 ${}_tp_{x:y}^{00} = \exp\left(-\int_0^t \left(\mu_{x+u:y+u}^{01} + \mu_{x+u:y+u}^{02} + \mu_{x+u:y+u}^{03}\right)du\right)$

Part d (2 points)

A couple, $(x)$ who is age 40 and $(y)$ who is age 50, buys a fully discrete last survivor whole life insurance with a sum insured of 100,000.

You are given:

(i) The couple's future lifetimes are independently distributed.

(ii) Premiums are payable while at least one life is alive for a maximum of 10 years.

(iii) Mortality follows the Standard Ultimate Life Table (SULT).

(iv) $i = 0.05$

Show that the annual net premium is 1,220 to the nearest 10. You should calculate the value to the nearest 1.

Part e (2 points)

(i) Show that the policy value at time 10, if $(y)$ is alive but $(x)$ is dead, is 29,000 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Show that the policy value at time 10, if both $(x)$ and $(y)$ are alive, is 15,900 to the nearest 100. You should calculate the value to the nearest 1.

Part f (2 points)

The insurer decides to incorporate a common shock risk in the joint life model, although each life's marginal individual mortality still follows the SULT.

(i) State with reasons whether the policy value in (e)(i) would increase, decrease, or stay the same as a result of the common shock risk.

(ii) State with reasons whether the policy value in (e)(ii) would increase, decrease, or stay the same as a result of the common shock risk.

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