SOA Exam ALTAM Spring 2024, Question 2

Study guide: Exam ALTAM

An insurer models the mortality of couples, denoted $(x)$ and $(y)$ , using the Markov multiple-state model described in the following diagram:

Part a (1 point)

Your colleague states that this model assumes that the future lifetimes of $(x)$ and $(y)$ are independent. State with reasons whether or not your colleague is correct.

Part b (3 points)

(i) Write down an integral expression for ${}_tp^{01}_{x:y}$ using multiple state model notation.

(ii) Write down an integral expression for ${}_tq_{\overset{1}{x}:y}$ using multiple state model notation.

(iii) Describe in words the event which has probability ${}_tq_{\overset{1}{x}:y} - {}_tp^{01}_{x:y}$ .

Part c (2 points)

A couple, (50) and (60), purchase an annuity that pays 10,000 per year continuously while (60) is alive and (50) is dead. The insurer will refund 50% of the net single premium without interest at the moment of death of (60) if (60) dies before (50).

The insurer uses the following assumptions for premiums and policy values:

The future lifetimes are independent.
Mortality follows the Standard Ultimate Mortality Model.
Deaths of (50) and (60) are uniformly distributed over each year of age.
$i = 0.05$
$\bar{A}_{x:y} \approx \dfrac{i}{\delta}A_{x:y}$ ; $\quad \bar{A}_{50:\overset{1}{60}} = 0.24898$ ; $\quad \bar{A}_{55:\overset{2}{65}} = 0.06091$

Show that the net single premium is 7250 to the nearest 25. You should calculate the value to the nearest 1.

Part d (2 points)

Calculate the net premium policy value at the end of 5 years if both lives are still alive.

Part e (1 point)

At the end of 10 years only the older life, now age 70, is alive and decides to remarry. The survivor requests to apply the policy value as the net single premium for a continuous last survivor life annuity with the new spouse, commencing immediately. State with reasons whether the company would be likely to approve the request.

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