SOA Exam ALTAM Fall 2023, Question 1

Study guide: Exam ALTAM

XYZ Insurance issues fully discrete joint life 20-year endowment insurance policies to couples, each age 65, with independent future lifetimes. The sum insured is 100,000, payable at the end of the year of the first death, or at time 20 if both individuals survive to that time.

Premiums are payable annually while both lives survive, for a maximum of 20 years.

You are given the following assumptions for premiums and reserves:

(i) Acquisition expenses are 500 plus 20% of the first year's premium.

(ii) Renewal expenses are 5% of each subsequent premium.

(iii) Mortality follows the Standard Ultimate Mortality Model.

(iv) Gross premiums are calculated using the equivalence principle.

(v) $i = 0.05$

(vi) Reserves are gross premium policy values.

Part a (1 point)

Calculate ${}_{20}E_{65:65}$ .

Part b (2 points)

Show that the gross premium for the policy is 4,750 to the nearest 10. You should calculate the value to the nearest 1.

Part c (1 point)

Show that the gross premium policy value at time 10 is 38,250 to the nearest 10. You should calculate the value to nearest 1.

Part d (2 points)

At time 10, XYZ Insurance had 100 identical and independent policies in force. During the 11th policy year, the actual experience was as follows:

(i) The interest rate earned was 5.5%.

(ii) Renewal expenses were 5% of the premium.

(iii) There were 4 claims.

(iv) Each claim had a termination expense of 100.

You are given that the gross premium policy value at time 11 is 42,811.92.

Calculate the total gain in the 11th policy year for the portfolio.

Part e (4 points)

Calculate the gain by source in the 11th policy year, in the order of (1) Interest, (2) Mortality, and (3) Expenses.

Part f (2 points)

At time 19, there were 40 policies remaining in force. During the 20th policy year, a pandemic occurred, such that the actual experience was:

The interest rate earned was 5.0%.
Renewal expenses were 5% of the premium.
There were 35 claims.
There were no termination expenses.

(i) Show that the total gain in the 20th policy year for the portfolio is zero.

(ii) Explain why there is no mortality loss despite the pandemic.

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