SOA Exam ALTAM Fall 2023, Question 2

Study guide: Exam ALTAM

An insurance company offers the following range of options for critical illness (CI) insurance riders attached to a life insurance policy. In each case the critical illness benefit would be paid immediately upon diagnosis of a covered critical illness.

Option A: Partially accelerated death benefit

Option B: Fully accelerated death benefit

Option C: Additional lump sum benefit on CI diagnosis

Part a (2 points)

For each of the options above, state with reasons which of the following models could be used to calculate premiums and reserves.

Model 1:

Model 2:

Model 3:

Part b (1 point)

A special 10-year term insurance policy issued to a Healthy life age 50 pays 100,000 immediately upon death of the policyholder, and an additional lump sum benefit of 50,000 immediately upon diagnosis of critical illness. Premiums are payable annually while the policyholder is Healthy. You are given that the insurer uses Model 3 for analyzing this policy.

(i) Write down an integral expression for the probability that a new policyholder dies before the policy expires.

(ii) Write down an integral expression for the expected present value of the death benefit.

Part c (2 points)

The insurer values the contract payments using force of interest of $\delta = 0.05$ . You are given the following table of actuarial functions under Model 3, at $\delta = 0.05$ .

$x$	${}_{10}p_x^{00}$	${}_{10}p_x^{01}$	$\mu_x^{0\bullet}$	$\bar{a}_x^{00}$	$\bar{A}_x^{01}$	$\bar{A}_x^{02}$	$\bar{A}_x^{12}$
50	0.90880	0.04152	0.00562	13.1491	0.21846	0.21878	0.30227
60	0.75019	0.13565	0.01542	10.0781	0.33692	0.33714	0.41861

Show that $\ddot{a}_{50:\overline{10}|}^{00} = 7.8$ to the nearest 0.1, using Woolhouse's 3-term formula. You should calculate the value to the nearest 0.001.

Part d (3 points)

Show that the annual net premium is 495 to the nearest 5. You should calculate the value to the nearest 0.1.

Part e (2 points)

You are given the following additional information for a policy in force at time 5:

$t$	${}_tV^{(0)}$	${}_tV^{(1)}$	$\mu_{50+t}^{01}$	$\mu_{50+t}^{02}$
5	702.6	5322.0	0.004265	0.004835
6	714.3	4562.4	0.004845	0.005234

Calculate $\dfrac{d}{dt}\,{}_tV^{(0)}$ at time 5, immediately after the premium payment.

Part f (2 points)

Sketch the graph of ${}_tV^{(0)}$ as a function of $t$ for $5 \le t \le 6$ . You should mark key numerical values on each axis.

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