MATH 403A, fall 98, CHAPTERS 7 & 8 Benefit Reserves

The benefit reserve at any time is the actuarial present value of the benefits at that time less the actuarial present value of the remaining benefit premiums at that time.

In this chapter, the prospective loss function is measured at time t. Since you only calculate reserves for policyholders that haven’t died yet, the random variable must be based on mortality going forward from time t. It will be denoted by _{t}L.

As before, _{t}L = Z_{t} − πY_{t}

The distribution and density functions of this random variable are:

F_{_{t}L}(y) = 1 − F_{T (x+t)} (−\frac{1}{δ} log \frac{δy + P̄(Ā_{x})}{δ + P̄(Ā_{x})})

f_{_{t}L}(y) = \frac{1}{δy + P̄(Ā_{x})}f_{T (x+t)} (−\frac{1}{δ} log \frac{δy + P̄(Ā_{x})}{δ + P̄(Ā_{x})})

Problems

1. _{5}L is the loss random variable 5–years after issue for a fully continuous 20–year endowment insurance with benefite of 4000 issued to a life aged 45. Premiums are determined by the equivalence principle. Mortality follows deMoivre’s law with w = 95. You are given that δ = .075. Find Var(_{5}L|T > 5).

2. _{1}L is the loss random variable one year after issue for a fully discrete n–year endowment insurance of 1. You are given the following

k = 0 k = 1
A_{x+k:\overline{n−k}|} .7439 .7703
2A_{x+k:\overline{n−k}|} .5622 .6007

The effective annual interest rate is 5.4%. Find the variance of 1L given survival to time 1.

3. A 4–year fully discrete increasing term insurance issued to (x) has death benefit of 1 in the first year, 2 in the second year, 3 in the third year and 4 in the forth year. Benefit premiums are payable at the beginning of each year in a decreasing pattern: 4π in the first year, 3π in the second, 2π in the third and π in the forth.

You are given that v = .9 and the following:

\begin{matrix}k & _{k|}q_{x} & _{k}p_{x} \\ 0 & .10 & 1.00 \\ 1 & .15 & .90 \\ 2 & .20 & .75 \\ 3 & .25 & .55 \end{matrix}

Find the difference between:

2L given that death occurs at time 2.3 (measured from issue), and

1L given that death occurs at time 3.3 (measured from issue).

4. Find the variace of _{1}L given that life survives at least to time 1, in the previous example.

5. A fully continuous whole life insurance of 1 is issued to a life aged 20 who is subject to deMoivre’s law of mortality with w = 100. The force of interest is .06. At the time that the policy is issued, the benefit reserve is calculated for each future integral age. Given that the policy holder survives her expected age at death, find the probability that the prospective loss random variable at that time takes on a value greater than the reserve at that time.

6. Find the p.d.f. for the loss random variable in the above problem.