Regarding problem 17 on the Fall 98 exam. The key here is to assume (it is not
specifically stated in the problem) that mortality over time (T) and the
constant interest rate (I) are independent random variables.
m=mu, d=delta
f(t)=m*e^-mt 0<t<inf is the pdf for the future lifetime.
f(i)= 10 0<i<0.10 is the pdf for the constant interest rate.
Since T and I are independent, f(t,i)=f(t)*f(i). f(t,i)=10*m*e^-mt
The APV for a life insurance is the expected value of v^t.
E[u(t,i)]= integral u(t,i)*f(t,i)dtdi (over the whole support)
E[v^t]=Dbl Int. (1+i)^-t * 10 * m * e^-mt didt (limits of int are 0 to inf
and 0 to 0.1
Now I'll go through the given answers.
First, answer B is just answer A evaluated. These answers are wrong because they
do not evaluate E[v^t]; they just use the expected value of the interest rate in
the evaluation.
C is wrong because 0.10 is out front when 10 should be.
D is wrong because nothing is front when 10 should be.
E is correct. The integration is done over T (keeping I constant), and this is
just like evaluating a regular APV for whole life insurance. With constant force
of mortality, APV=m/(m+d). Replacing ln(1+i) for d and replacing the multiple of
10 originally in the pdf, the expression in E equals E[v^t] after integrating
over T.
Hope this helps.