SOA Exam ALTAM Fall 2025, Question 5

Study guide: Exam ALTAM

NED Insurance Company uses the following multi-state model to price and reserve term life insurance policies offering an additional benefit for lives in Nursing Home care.

Let $P_x$ denote the one-year probability transition matrix for a person age $x$ . You are given:

$P_{80} = \begin{bmatrix} 0.80 & 0.15 & 0.05 \\ 0.40 & 0.40 & 0.20 \\ 0 & 0 & 1 \end{bmatrix} \quad\text{and}\quad P_{81} = \begin{bmatrix} 0.70 & 0.20 & 0.10 \\ 0.30 & 0.40 & 0.30 \\ 0 & 0 & 1 \end{bmatrix}$

You are also given that all transitions occur before the year end.

A Healthy life age 80 purchases a 2-year policy with the following benefits:

A death benefit of 100,000 payable at the end of the year of death.
A benefit of 50,000 payable at the end of each year if the policyholder is in a Nursing Home at that time.
Level gross annual premiums are payable conditional on the policyholder being in State 0 at the payment date.

You are also given:

$v = 0.95$
The expense to issue the policy is 500. This would be considered a pre-contract cost in a profit testing exercise.
The maintenance expense for the policy is 300 at the beginning of each year including the first.

Part a (3 points)

(i) Calculate the 2-year transition matrix for a life age 80.

(ii) Show that the expected present value of benefits for this policy is 33,000 to the nearest 1000. You should calculate the value to the nearest 1.

Part b (1 point)

The gross annual premium for this policy is calculated using the equivalence principle. Show that the gross annual premium is 19,400 to the nearest 100. You should calculate the value to the nearest 1.

Part c (2 points)

The gross premium policy value at time $t$ for a policy in state $j$ is denoted ${}_tV^{(j)}$ .

(i) Show that ${}_1V^{(0)}$ is $-100$ to the nearest 100. You should calculate the value to the nearest 1.

(ii) Show that ${}_1V^{(1)}$ is 48,000 to the nearest 1000. You should calculate the value to the nearest 1.

Part d (5 points)

NED conducts a profit test of the contract using the following assumptions:

The earned interest is 12%.
The hurdle rate is 10%.
Reserves are the gross premium policy values.

All other assumptions are the same as the premium basis above.

The expected profit at time $t$ per policy in state $j$ at time $t-1$ is denoted by $Pr_t^{(j)}$ .

(i) Show that $Pr_1^{(0)}$ is 1800 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Calculate $Pr_2^{(0)}$ and $Pr_2^{(1)}$ .

(iii) Calculate the profit signature for this policy.

(iv) Calculate the profit margin for this policy.

Part e (1 point)

Explain why the profit margin for this contract will increase if NED increases its reserves at time 1.

Answer this question and see the worked solution