SOA Exam ALTAM Spring 2025, Question 2

Study guide: Exam ALTAM

You are analyzing an insurance policy with disability income benefits and critical illness benefits. You use the following multiple state model:

You are given the following information:

The policy is issued to Healthy individuals only.
The level premium is paid continuously while the policyholder is Healthy.
A disability income annuity of 10,000 per year is paid continuously while the policyholder is Disabled.
A critical illness benefit of 120,000 is paid immediately on becoming Critically Ill.
A death benefit of 100,000 is paid immediately on Death, provided that the policyholder has not already received a critical illness benefit.
The force of interest is $\delta = 0.05$ .
Transition intensities for all ages $x$ are:

$\mu_x^{01} = 0.02, \qquad \mu_x^{12} = 0.04, \qquad \mu_x^{03} = 0.05, \qquad \mu_x^{13} = 0.10.$

Part a (1 point)

Explain whether this model satisfies the Markov property.

(i) Write down the Kolmogorov forward differential equations for ${}_tp_x^{00}$ and ${}_tp_x^{01}$ with the associated boundary conditions.

(ii) Show that ${}_tp_x^{01} = \dfrac{2}{7}\left(e^{-0.07t} - e^{-0.14t}\right)$ .

(i) Show that $\bar{a}_x^{01} = 0.88$ to the nearest 0.01. You should calculate the value to the nearest 0.0001.

(ii) Show that $\bar{A}_x^{02} = 0.04\,\bar{a}_x^{01}$ .

You are given that the expected present value of the death benefit for this policy is 50,438.60.

Calculate the annual level premium of this policy using the equivalence principle.

State the Thiele differential equation for ${}_tV^{(1)}$ , the State 1 policy value at time $t$ .

The insurer decides to include an additional death benefit of 50,000 in this policy, payable when the policyholder dies while Critically Ill.

Sketch a diagram of a revised model incorporating this change.