ASM PE 1 #7

[V1per41]

A continuous deferred life annuity pays 1 per year starting at time 10. If death occurs before time 10, the single benefit premium is refunded without interest at the moment of death. You are given:

(i)
(ii)

Calculate the single benefit premium for this annuity.
_____________________________

How I set it up:

Since the premium is refunded without interest I figured that if you set the premium equal to the premium times the probability that the person dies within 10 years + the deferred annuity, then easily solve for P:



However, the book sets up the answer as follows:



It seems that this second method would be if the premium was refunded with interest. If I'm wrong (probably), and this is how it's solved, then how would you solve it if the premium is refunded + interest?

[Gandalf]

Originally Posted by V1per41

How I set it up:

Since the premium is refunded without interest I figured that if you set the premium equal to the premium times the probability that the person dies within 10 years + the deferred annuity

If you are trying the basic approach APV of premiums = APV of benefits (as you should are, and should be) then is the APV of the premium refund equal to the premium times the probability of a refund?

"Without interest" means that the benefit, if death occurs in the 10 years, is P.

[LifeIsAPoissonProcess]

The interest in the first half of your equation is referring to the insurer's valuation of their payment in the first 10 years, not the amount of the premium.

The fact that P is constant, like normal term insurance, means that the value of the benefit is continually decreasing over time. When you pay your single benefit premium, the insurer is getting a 5% return on your money continually until they pay the premium for the death. Just because you aren't earning any interest on your premium payment doesn't mean they aren't!

[V1per41]

Originally Posted by LifeIsAPoissonProcess

The interest in the first half of your equation is referring to the insurer's valuation of their payment in the first 10 years, not the amount of the premium.

The fact that P is constant, like normal term insurance, means that the value of the benefit is continually decreasing over time. When you pay your single benefit premium, the insurer is getting a 5% return on your money continually until they pay the premium for the death. Just because you aren't earning any interest on your premium payment doesn't mean they aren't!

So how is the problem solved if the insured gets the premium refunded + interest?

[Gandalf]

Like this

Originally Posted by V1per41

Because now the benefit, P with interest, discounted to issue, is P.

[LifeIsAPoissonProcess]

Originally Posted by V1per41

So how is the problem solved if the insured gets the premium refunded + interest?

Originally Posted by Gandalf

Like this

Because now the benefit, P with interest, discounted to issue, is P.

Yep, looks like that was actually what you already solved for, although your equation also represents the solution to the original problem if i=0.

[LifeIsAPoissonProcess]

Also you can think about it like this:

You have term insurance with n = infinity (whole life insurance) with a death benefit of e^.05t. We know the SBP for a whole life insurance is u/u+delta. But if the benefit also grows exponentially at the rate of delta, then the growth rate and delta cancel out to make the SBP u/u or 1.

We can check this through some sample lifetimes. If I die at time 1, my benefit is e^.05, while the accumulated value of my SBP is also e^.05. If I die at time 5, my benefit is e^.25, while the accumulated value of my SBP is also e^.25. No matter when you die, your SBP exactly covers your death benefit. Therefore, the benefit, for a term insurance where the premium is refunded with interest equal to the force of interest, doesn't depend on when you die, but if you die, which is perfectly modeled by your wrong answer.