Question

When calculating premiums on life insurance products insurance companies often use life tables which enable the probability of a person dying in any age interval to be calculated.

The following table gives the number out of 100,000 females who are still alive during each five-year period of life between the age of 20 to 60 (inclusive):

Out of 100,000 females born

                       

Exact age (years)                                          Number alive at exact age

20                                                                                                99,150

25                                                                                                 98,983

30                                                                                                 98,781

35                                                                                                 98,545

40                                                                                                 98,182

45                                                                                                 97,628

50                                                                                                96,831

55                                                                                               95,585

60                                                                                               93,718

Suppose a 30 year old female on her 30th birthday purchases a one million dollar, five-year term life policy from an insurance company. That is, the insurance company must pay her estate $1 million if she dies within the next five years.

(a) Determine the insurance company’s expected payout on this policy.

(b) What would be the minimum you would expect the insurance company to charge her for this policy? Give a brief explanation of your answer.

(c) What would the expected payout be if the same policy were taken out by a female on her 40th birthday?

Answer 1

(a) Insurance company's expected payout is equal the "amount paid to the woman if the woman dies" times the "probability that the woman dies within the next 5 years".

Thus, Amount = $1,000,000

Probability = (No. of woman who died between ages 30 and 35)/(No. of woman alive at age 30)

Now, No. of woman who died between 30 and 35 = No. of woman alive at age 30 - No. of woman alive at age 35

= 98781 - 98545 = 236

Thus, Probability = 236/98781 = 0.00238912341

Thus, Expected payout = $1,000,000 * 0.00238912341 = $2389.12

(b) The minimum we expect the insurance company to charge is the expected payout calculated in part (a) ,i.e., $2389.12. This is the minimum value the insurance company needs to charge to avoid almost certain loss. This is the expected payout to the policyholder but the insurance company needs to add some contingency margin so that it doesn't suffer any losses because of the difference between observed losses and expected losses. In addition, the company needs to add the expenses incurred by them in respect of this policy because of which the actual amount charged will be significantly higher than the expected payout.

(c) The calculations are the same as in part (a) with only changes in figures during the probability calculation.

Insurance company's expected payout is equal the "amount paid to the woman if the woman dies" times the "probability that the woman dies within the next 5 years".

Thus, Amount = $1,000,000

Probability = (No. of woman who died between ages 40 and 45)/(No. of woman alive at age 40)

Now, No. of woman who died between 40 and 45 = No. of woman alive at age 40 - No. of woman alive at age 45

= 98,182 - 97,628 = 554

Thus, Probability = 554/98,182 = 0.00564258214

Thus, Expected payout = $1,000,000 * 0.00564258214 = $5642.58