Re: a Chapter 6 Life Contingencies question

james.smieszkal@zurichus.com
Wed, 6 Oct 1999 09:47:30 -0500

A 25-year mortgage of 100,000 issued to (40) is to be repaid with equal
annual payments at the end of each year. A 25-year term insurance has a
death benefit which will pay off the mortgage at the end of the year of
death, including the payment then due.

You are given:

i) i = .05

ii) the 25-year term life annuity due for (40) is 14 (notation: a "double
dot" sub 40:"angle"25 = 14)

iii) 25_q_40 =.2

Calculate the net annual premium for this insurance.
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The APV of the insurance is the sum of the PV of a mortgage payment (at time t)
times the probability that the insured WILL NOT be able to make the payment.

APV = X * SUM(from t=1 to t=25) v^t * tq40 where X is the annual payment of
the mortgage.

The 25 year annuity immediate (certain) has the following PV:

a25| = SUM(from t=1 to t=25) v^t OR SUM(from t=1 to t=25) v^t * (tq40 +
tp40)

Now, we need to change the life annuity due to a life annuity immediate:

a(due)40:25| = SUM(from t=0 to t=24) v^(t+1) * tp40

a(imm)40:25| = a(due)40:25| -- 1 + (25p40 * v^25) (IOW, stripping
the t=0 payment and adding a payment at t=25)

a(imm)40:25| = SUM(from t=1 to t=25) v^t * tp40

a25| - a(imm)40:25| = [SUM(from t=1 to t=25) v^t * (tq40 + tp40)] - [SUM(from
t=1 to t=25) v^t * tp40]
= SUM(from t=1 to t=25) v^t * tq40
= APV/X

P=APV / a(due)40:25| Premiums are still collected at the beginning of the
year.