SOA Exam ALTAM Sample Question 16

Study guide: Exam ALTAM

An insurer issues a 20-year term insurance to (50) which pays a death benefit of 500,000, and an additional 100,000 on diagnosis of a critical illness (CI).

The insurer uses the Markov model illustrated below to value the CI benefit.

You are given the following information:

(i) Premiums are payable continuously while the policyholder is healthy.

(ii) The annual net premium rate is 4,850.

(iii) Death and CI benefits are payable immediately on transition.

(iv) $i = 0.05$ .

(v) ${}_tV^{(j)}$ denotes the net premium policy value at time $t$ , conditional on being in State $j$ at that time.

(vi) The following values from the model illustrated above:

$x$	$\bar{a}_x^{00}$	$\bar{A}_x^{01}$	$\bar{A}_x^{02}$	$\bar{A}_x^{12}$	${}_{10}p_x^{00}$	${}_{10}p_x^{01}$	${}_{10}p_x^{11}$
60	10.1729	0.34249	0.39077	0.47904	0.75055	0.13135	0.75283
70	6.5690	0.49594	0.54335	0.62237

Part a (3 points)

(i) Show that $\bar{A}_{60:\overline{10}|}^{02} = 0.0902$ to the nearest 0.0001. You should calculate the value to the nearest 0.00001.

(ii) Show that ${}_{10}V^{(0)} = 21{,}850$ to the nearest 50. You should calculate the value to the nearest 1.

(iii) Show that ${}_{10}V^{(1)} = 95{,}700$ to the nearest 10. You should calculate the value to the nearest 1.

Part b (3 points)

You are given that $\mu_{60}^{01} = 0.00818$ ; $\mu_{60}^{02} = 0.00724$ ; $\mu_{60}^{12} = 0.01811$ .

(i) Calculate $\dfrac{d}{dt}\,{}_tV^{(0)}$ at $t = 10$ .

(ii) Calculate $\dfrac{d}{dt}\,{}_tV^{(1)}$ at $t = 10$ .

(iii) Explain why ${}_tV^{(1)}$ is decreasing while ${}_tV^{(0)}$ is increasing at $t = 10$ .

Part c (2 points)

The policy terms are revised for new policies such that the CI benefit is paid six months after diagnosis, conditional on the policyholder surviving to that time, and conditional on the diagnosis occurring within the 20-year term.

Write down an integral expression for the expected present value at issue of the revised CI benefit.

Part d (2 points)

One reason for the change is that mortality is very high in the first year after a CI diagnosis.

(i) Define the Markov property.

(ii) Explain whether high mortality in the period immediately after a CI diagnosis is consistent with the Markov property.

(LTAM, Spring 2022, Question 2)

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SOA Exam ALTAM Sample Question 16

Part a (3 points)

Part b (3 points)

Part c (2 points)

Part d (2 points)

Concepts tested

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