GARP.FRM.PQ.P1 Insurance Premium Payment

[chiayu]

I need help with the example related to insurance premium payment in FRM Part I Book Financial Markets and Products Chapter 2. Any help would be greatly appreciated!

Assume that interest rates for all maturities are 4% per year (with semiannual compounding) and premiums are paid once a year at the beginning of the year. What is an insurance company's break-even premium for $100,000 of term life insurance for a man of average health aged 90?

If the term insurance lasts one year, the expected payout is 0.168352 * $100,000 = $16,835 ( 0.168352 is the probability of death within 1 year for a man aged 90) .
Assume that the payout occurs halfway through the year, the premium is $16,835 discounted for six month, which is $16,835/1.02=$16,505.
Suppose next that the term insurance lasts two year. In this case, the present value of expected payout in the first year is $16,505 as before. My question is why the present value of expected payout in the first year is $16,505 as before?

I think the present value of expected payout in the first year is $16,181, which is $16,505/1.02, because the present value of expected payout in the first year is calculated by discounting $16,505 at time 1,which is the 6th month of the 1st year, to time 0.

 

[David Harper CFA FRM]

Hi @chiayu Hmmm. The second scenario concerns a man who has just turned 90 years old and buys term life insurance over the next two years, until he reaches his 92nd birthday. (I guess you are in Hull's 4th edition rather than 5th edition for some reason). On the contingent payout side of the equality:

• Per the (4th edition) table, there is a 16.835% (conditional death) probability that he dies during his 90th year; per assumption, in six months from today, T(0) + 0.5 years.
• The unconditional probability that he dies during his 91st year = (1 - 16.835%)*18.5486% = 15.4259061%. This is a probability from the perspective of T(0); i.e., at the beginning of his 90th birthday

The present value is to time T(0), on the man's 90th birthday. From this perspective, there are two future contingent payouts: one in six months, halfway thru his 90th year, and another in 1.5 years, halfway thru his 91st year. So the PV to T(0) = (16.835%*$100,000)/(1+4%/2)^(0.5 years*2) +(15.4259061%*$100,000)/(1+4%/2)^(1.5 years*2). Let me know if that isn't clarifying, thanks!