actuary2099
05-06-2006, 07:22 AM
Hey guys i feel so bad asking these questions but i am getting no help from my lecturer and i am understanding some stuff..can u please help me with the following questions..keep in mind i am a hard worker and not a slacker plz..thanks so much guys...
Q(1) Suppose that the force of interest delta=0.08 and that tPx = 1-(t^3/125) , 0<=t<=5
(a) A whole-life insurance policy with unit benefit paid at the moment of death is paid for by a cont. annuity of benefit premiums being paid at the rate of P per period.Calculate the value of P.
(b) Suppose that a policyholder in part(a) wanted to surrender his policy at time t=2.What would be the fair surrender value for his policy?
Q(2) Suppose that the density function f of T(x) is f(t)=4(t^2)exp(-2t), t>0
(a)Let Z be the present value of a whole-life insurance policy paying a benefit of exp(rt) at the moment of death, the force of interest being (delta).Derive an expression for E(Z) and show that, for integer k such that 2-ktimes(delta)-kr>0, E(Z^k)=8/(2+ktimes(delta)-kr)^3. The density function of a Gamma(alpha,beta) random variable is given (the hint is that the integral of the gamma between 0 and 1 is equal to 1).
(B) If 100 such policies are sold and benefits are paid out of a fund established at t=0, find the size of the fund such that the probability is 0.95 that sufficient funds will be on hand to pay out benefits.Use r=0.05, and delta=0.08
Q(1) Suppose that the force of interest delta=0.08 and that tPx = 1-(t^3/125) , 0<=t<=5
(a) A whole-life insurance policy with unit benefit paid at the moment of death is paid for by a cont. annuity of benefit premiums being paid at the rate of P per period.Calculate the value of P.
(b) Suppose that a policyholder in part(a) wanted to surrender his policy at time t=2.What would be the fair surrender value for his policy?
Q(2) Suppose that the density function f of T(x) is f(t)=4(t^2)exp(-2t), t>0
(a)Let Z be the present value of a whole-life insurance policy paying a benefit of exp(rt) at the moment of death, the force of interest being (delta).Derive an expression for E(Z) and show that, for integer k such that 2-ktimes(delta)-kr>0, E(Z^k)=8/(2+ktimes(delta)-kr)^3. The density function of a Gamma(alpha,beta) random variable is given (the hint is that the integral of the gamma between 0 and 1 is equal to 1).
(B) If 100 such policies are sold and benefits are paid out of a fund established at t=0, find the size of the fund such that the probability is 0.95 that sufficient funds will be on hand to pay out benefits.Use r=0.05, and delta=0.08