SOA Exam ALTAM Spring 2023, Question 1

You are given the following information about a portfolio of in-force policies.

(i) Policyholders are either Healthy (State 0), Temporarily Disabled (State 1), Permanently Disabled (State 2), or Dead (State 3).

(ii) At time 0, there are 14 Healthy lives, 0 Temporarily Disabled lives, and 1 Permanently Disabled life.

(iii) All 15 lives are independent.

(iv) Transitions are modeled as a homogeneous Markov model.

(v) The annual transition matrix is

 $P = \begin{pmatrix} 0.80 & 0.10 & 0.05 & 0.05 \\ 0.20 & 0.50 & 0.10 & 0.20 \\ 0 & 0 & 0.70 & 0.30 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

(vi) The two-year transition matrix, but with some missing values denoted by $-\,-$ , is

 ${}_2P = \begin{pmatrix} 0.660 & -\,- & 0.085 & -\,- \\ 0.260 & 0.270 & 0.130 & -\,- \\ 0 & 0 & -\,- & -\,- \\ 0 & 0 & 0 & 1 \end{pmatrix}$

Part a (2 points)

Calculate the missing values in the two-year transition matrix.

Part b (2 points)

Calculate the probability that exactly one policyholder will die within 2 years.

Part c (1 point)

Calculate the expected number of policyholders who will die within 2 years.

Part d (4 points)

The policies provide the following benefits:

A benefit of 500,000 is paid at the end of the year that a policyholder becomes permanently disabled, if they are still alive at that time.
A benefit of 1,000,000 is paid at the end of the year of death (whether they were disabled or not).

The insurer values the benefits at $i = 0.06$ .

(i) (1.5 points) Show that the expected present value of the benefits payable in the next 2 years to the one Permanently Disabled life is 469,920 to the nearest 10. You should calculate the value to the nearest 1.

(ii) (2.5 points) Calculate the expected present value of all the benefits payable in the next 2 years for the 14 Healthy lives.

Part e (1 point)

Describe one disadvantage of the assumption of homogeneity of the Markov model for valuing this portfolio.

Answer this question and see the worked solution