Actuarial Notes

SOA Exam ALTAM Sample Question 7

Study guide: Exam ALTAM

An actuarial student in your department is using the following 3-state Markov model to price a one-year consumer warranty for a television.

The product has the following features:

(i) The warranty is sold at the time that a consumer purchases the television and pays a replacement cost of 1000 at the end of the half-year during which the television becomes Broken Beyond Repair.

(ii) Premiums are payable at the beginning of each half-year, but are waived if the television is not Working at the time a premium is due.

(iii) All televisions are in the Working state at time of purchase.

Part a (2 points)

Write down the Kolmogorov forward differential equations with associated boundary conditions (initial conditions) for {}_tp_0^{00}, {}_tp_0^{01}, and {}_tp_0^{02} under this model.

Part b (2 points)

The forces of transition in this model for a television purchased t years ago are:

\mu_t^{01} = 0.5 + 0.6t

\mu_t^{10} = 0.2

\mu_t^{12} = 2^t

The student has applied the Forward Euler approximation to the Kolmogorov forward differential equations with a step size of h = 0.5 to calculate probabilities for this model. You have verified that, using this approach, {}_{0.5}p_0^{00} = 0.75.

Using the student's approach:

(i) Calculate {}_{0.5}p_0^{01}.

(ii) Show that the probability that a television will become Broken Beyond Repair within a year of purchase is 0.18 to the nearest 0.01. You should calculate the probability to the nearest 0.001.

Part c (2 points)

Using the probabilities from the student's approach and i^{(2)} = 8\%:

(i) Calculate the actuarial present value at issue of the replacement cost payments for this policy.

(ii) Calculate the semi-annual net premium for this policy.

Part d (1 point)

Suggest a change to the student's approach that would improve the accuracy of the probability calculations.

(MLC, Spring 2014, Question 2)

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Concepts tested

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