SOA Exam ALTAM Sample Question 19

Person P and Person R, each age 40, buy a fully discrete, last survivor insurance with a sum insured of 100,000

You are given:

(i) Premiums are payable while at least one life is alive, for a maximum of 20 years.

(ii) Mortality of each follows the Standard Ultimate Life Table (SULT).

(iii) $i = 0.05$

(iv) With independent future lifetimes, $\ddot{a}_{40:40:\overline{20}|} = 12.9028$ .

Part a (2 points)

Show that the annual net premium assuming that the future lifetimes are independent is 620 to the nearest 10. You should calculate the value to the nearest 1.

Part b (1 point)

State two reasons why couples may have dependent future lifetimes.

Part c (3 points)

The insurer decides that premiums and policy values for this policy will be determined using a mortality model incorporating dependency.

You are given the following information about this model:

(i) The future lifetimes for the first 20 years are not independent.

(ii) If both lives survive 20 years, it is assumed that the future lifetimes from that time will be independent, and will follow the Standard Ultimate Life Table.

(iii) The mortality of each of P and R, individually, follows the Standard Ultimate Life Table, whether the other is alive or dead.

(iv) $\ddot{a}_{40:40:\overline{10}|} = 8.0703$ ; $\ddot{a}_{40:40:\overline{20}|} = 12.9254$ ; ${}_{20}E_{40:40} = 0.35912$ ; ${}_{10}E_{50:50} = 0.59290$ .

(v) $A_{\overline{50:50}} = 0.13441$

(vi) ${}_{10}p_{40:40} = 0.9866$ , ${}_{10}p_{\overline{40:40}} = 0.9980$

Use the dependent mortality model for the rest of this question.

(i) Show that $A_{40:40} = 0.158$ to the nearest 0.001. You should calculate the value to the nearest 0.0001.

(ii) Show that ${}_{10}E_{40:40} = 0.606$ to the nearest 0.01. You should calculate the value to the nearest 0.0001.

(iii) Show that $\ddot{a}_{50:50:\overline{10}|} = 8.02$ to the nearest 0.01. You should calculate the value to the nearest 0.0001.

Part d (1 point)

Show that the annual net premium is 645 to the nearest 5. You should calculate the value to the nearest 0.1.

Part e (3 points)

Let ${}_kL$ denote the net future loss random variable at time $k$ for the insurance.

(i) Calculate $E\left[{}_{10}L\right]$ given that only P is alive at time 10.

(ii) Calculate $E\left[{}_{10}L\right]$ given that both P and R are alive at time 10.

(iii) Calculate $E\left[{}_{10}L\right]$ given that at least one of P and R is alive at time 10.

Part f (1 point)

Because the insurer is not informed of the first death for these policies, the actuary decides that they should hold the same reserve for all policies in force. Explain which, if any, of the policy values in part (e) would be a suitable value for the time 10 reserve for each in-force policy.

(LTAM, Spring 2019, Question 4) (This is Question 4 from the Spring 2019 Written Answer exam.)

Answer this question and see the worked solution