Actuarial Notes

SOA Exam ALTAM Spring 2025, Question 4

Study guide: Exam ALTAM

You are pricing a special fully discrete whole life insurance on a life age 60. The death benefit is 100,000 during the first 20 years and 200,000 thereafter.

You are given:

  • Mortality follows the Standard Ultimate Mortality Model (SUMM).
  • The only expenses are commissions, which are 50% of premium in year 1, and 5% of premiums thereafter.
  • i = 0.05
  • Level gross premiums are determined using the equivalence principle.

Part a (2 points)

Calculate the annual gross premium.

Part b (2 points)

The insurer decides to include lapses in the pricing model, as illustrated in the following multiple state model.

You are given:

  • There are no benefits payable upon lapse.
  • The force of mortality is unchanged, that is, \mu_x^{01} = A + Bc^x where A = 0.00022, B = 2.7 \times 10^{-6}, and c = 1.124.

(i) State the meaning of "lapse supported business".

(ii) State one disadvantage to the insurer of issuing lapse supported policies.

Part c (3 points)

You have calculated these actuarial values incorporating a lapse assumption:

x \ddot{a}_x^{00} A_x^{01} {}_{10}p_x^{00}
60 8.8023 0.10330 0.49694
70 8.7967 0.24061 0.54578
80 6.9365 0.41889 0.37068

(i) Show that the gross premium is 1,850 to the nearest 10. You should calculate the value to the nearest 0.1.

(ii) Show that the gross premium policy value at the end of year 10 is 22,650 to the nearest 50. You should calculate the value to the nearest 1.

Part d (3 points)

You are given that \mu_{70}^{02} = 0.04 and \mu_{70.5}^{02} = 0.045.

(i) Use Euler's forward method with step size h = 0.5 to show that p_{70}^{00} = 0.95 to the nearest 0.01. You should calculate the value to the nearest 0.001.

(ii) Use Euler's forward method with step size h = 0.5 to show that p_{70}^{01} = 0.01 to the nearest 0.01. You should calculate the value to the nearest 0.001.

(iii) Using recursion, calculate the gross premium policy value at time 11.

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