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PSTAT 172B - ACTUARIAL STATISTICS II - QUIZ 2 PART 1

MAY 1, 2025

Show all work, do not just write an answer. For questions that use a calculator, make sure to write down the formula you are using that is entered into the calculator. Please make your final results easy to read. Good luck!

Problem 1. Consider the state of a person aged (40) follows the sickness-death model with the following transition forces:

Consider a disability income policy issued to 50 healthy individuals of age (40) whose lives are independent:

• The product pays a disability benefit of 5,000/yr paid continuously while the insured is in state 1

• The product pays a lump sum of 50,000 immediately after dead.

a) [3 points] Compute the probability that at least 3 policies pay any benefit within the first 5 years.

b) [2 points] Using the CLT, estimate the probability that more than 27 policies pay a benefit within the first 5 years.

c) [3 points] Write out all of the following kolmogorov equations, with the corresponding boundary conditions:

d) [3 points] Use newton’s method with a h = 0.5 step size to approximate

Problem 2. Consider the state of a person follows the following permanent disability model with the following constant forces of transition:

Consider a disability insurance product issued to a healthy life aged (30) that:

• Pays 10,000/year while the policyholder is disabled, payable at a a continuous rate.

• Pays 100,000 immediately after the insured dies.

• The force of interest is ω = 0.04

a) [3 points] Find the net single premium (NSP) under the equivalence principle.

b) [4 points] Assume instead that the policy has continuous net premiums payable while the insured is healthy. Calculate the annual premium P under the equiv. principle.

c) [2 points] Find where: