SOA Exam ALTAM Spring 2024, Question 3

NED Life sells special whole life insurances of 100,000 that include a sickness benefit. You are given:

Pricing and valuation is based on the Standard Sickness-Death Model.
In the Standard Sickness-Death Model:

$\mu_x^{01} = a_1 + b_1 e^{c_1 x}; \quad \mu_x^{02} = a_2 + b_2 e^{c_2 x}; \quad \mu_x^{10} = b_1 e^{c_1(110-x)}; \quad \mu_x^{12} = 1.4\mu_x^{02}$

where $a_1 = 4 \times 10^{-4}$ ; $b_1 = 3.47 \times 10^{-6}$ ; $c_1 = 0.138$ ; $a_2 = 5 \times 10^{-4}$ ; $b_2 = 7.85 \times 10^{-5}$ (as printed); $c_2 = 0.087$

Editor's note The SOA's model solution notes that $b_2$ was printed as $7.85 \times 10^{-5}$ when it should have been $7.58 \times 10^{-5}$ , the Standard Sickness-Death Model's value; the figures in parts (a) and (b)(i) follow from $7.58 \times 10^{-5}$ . With the printed $7.85 \times 10^{-5}$ , ${}_{10}V^{(1)}$ is about 161,460, not 162,300.

Death benefits are payable at the moment of death.
Sickness benefits are payable continuously at a rate of 12,000 per year while Sick.
Level premiums are payable continuously while the insured is Healthy.
All insureds are age 55 and Healthy at purchase.
$i = 0.05$
The net premium rate is 6,622 per year.
${}_tV^{(j)}$ denotes the net premium policy value for a policy in state $j$ at time $t$ .
${}_{10}V^{(0)} = 44{,}250$

Part a (1 point)

Show that ${}_{10}V^{(1)} = 162{,}300$ to the nearest 100. You should calculate the value to the nearest 1.

Part b (3 points)

(i) Show that $\left.\dfrac{d}{dt}\,{}_tV^{(1)}\right|_{t=10} = -1950$ to the nearest 25. You should calculate the value to the nearest 1.

(ii) Using Euler's backward method with a step size of $h = 0.25$ , estimate ${}_{9.75}V^{(1)}$ .

Part c (1 point)

NED introduces a new product, called Product Q, with the same features as the original policy except that:

Death benefits are payable at the end of the quarter of death.
Premiums are payable quarterly if the insured is Healthy at the start of the quarter.
A sickness benefit of 3,000 is payable at the beginning of each quarter if the insured is Sick then.

You are given that:

(i) $A^{(4)02}_{55} = 0.39127$ ; $A^{(4)02}_{65} = 0.53233$ ; $A^{(4)12}_{65} = 0.56465$

(ii) $\ddot{a}^{(4)00}_{55} = 10.2478$ ; $\ddot{a}^{(4)01}_{55} = 2.3057$ ; $\ddot{a}^{(4)10}_{65} = 0.0395$ ; $\ddot{a}^{(4)11}_{65} = 8.9373$

(iii) ${}_{10}V^{(0)} = 43{,}792$

Show that the annualized net premium is 6,520 to the nearest 10. You should calculate the value to the nearest 1.

Part d (1 point)

Show that ${}_{10}V^{(1)}$ for this new policy is 163,450 to the nearest 25. You should calculate the value to the nearest 1.

Part e (2 points)

You are also given:

$x$	${}_{0.25}p_x^{01}$	${}_{0.25}p_x^{02}$	${}_{0.25}p_x^{10}$	${}_{0.25}p_x^{12}$
64.75	0.00674	0.00547	0.00043	0.00764

Using backwards recursion with a step size of $h = 0.25$ , calculate:

(i) ${}_{9.75}V^{(0)}$

(ii) ${}_{9.75}V^{(1)}$

Part f (2 points)

NED Life believes policyholders care more about the sickness benefits in the early years and the death benefits thereafter. They introduce a new variation of Product Q, called Product Q+:

The quarterly net premium for Product Q+ is the same as for Product Q.
Death and sickness benefits in the first 10 years for Product Q+ are the same as for Product Q.
No sickness benefits are paid after 10 years under Product Q+.
The death benefit after 10 years for Product Q+ is increased to $F$ .

Calculate $F$ .

Part g (1 point)

The head of marketing proposes that since the premiums and benefits of Product Q and Product Q+ are the same for the first 10 years, any policyholder request to switch between the policies within the 10 years should be granted. State with reasons whether you agree or disagree with the proposal.

Answer this question and see the worked solution