MATH 403A, fall 98, CHAPTER 4 Section 4.2, 4.3

Varing Benefits Insurance

Annually increasing whole life insurance: provides 1 at the moment of death if death occurs in the first year, 2 at the moment of death if death occurs in the second year, 3 at the moment of death if death occurs in the third year, etc.

Actuarial present value (expected value of the present value):

(I\overline{A})_{x} = E(Z) = ∫_{0}^{∞}⌊t + 1⌋ v^{t} f_{T (x)}(t)dt

If increases occur more or less frequent than once a year, we have

(I^{(m)}\overline{A})_{x} = E(Z) = ∫_{0}^{∞} \frac{⌊tm + 1⌋}{m} v^{t} f_{T (x)}(t)dt

If increases occur continuously,

(\overline{I}\overline{A})_{x} = E(Z) = ∫_{0}^{∞} t v^{t} f_{T (x)}(t)dt

For decreasing benefits we have an n–tern insurance, where the payment is n if death occurs in the first year, n − 1 in the second year and so on.

(D\overline{A})_{x} = E(Z) = ∫_{0}^{n}(n − ⌊t⌋) v^{t} f_{T (x)}(t)dt

Note that

(I\overline{A})_{{1 \atop x}:\overline{n}|} + (D\overline{A})_{x} = n + 1

Insurances payable at the end of the year of death

Whole Life insurance: A benefit of $1 is paid at the end of the year of death. The actuarial present pavue is

A_{x} = E[Z] = ∑_{0}^{∞} v^{k+1} _{k}p_{x} q_{x+k}

In term of the table function

l_{x} A_{x} = ∑_{0}^{∞} v^{k+1} d_{x+k}

PROBLEMS 1. A group of 100 lives aged 30 set up a fund to pay 1,000 at the

end of the year of death of each member to a designated survivor. The mutual agreement is to pay into the fund an amount equal to the whole life insurance actuarial present value calcualted on the basis of the illustrative life table at 6% interest. The members, not selected by an insurance company, decide to use this population table as the basis of their plan. The actual experience of the fund is one death in each of the second and fifth years, interest income is at 6% in the first year, 6 1/2% in the second and third years,7% in the fourth and fifth years. What is the difference, at the end of the first 5 years, between the expected size of the fund as determined at the inception of the plan and the actual fund?

2. Z is the present value random variable for a discrete n–year term insurance of amount 1 issued to a life aged x. Y is the present value random variable for a discrete n–year pure endowment of amount 1 issued to a life aged x. You are given that

2A_{x^{1}:\overline{n}|} = .40 Var[Z] = .15
2A_{x:\overline{n}|^{1}} = .16 Var[Y] = .12

Find the variavce of the present value random variable for a discrete n–year endowment insurance of amount 1 issued to a life aged x.

3. Under the assumptions of (a) UDD (b) ordinary compound interest (c) benefit timing must be at the moment of death for the continuous insurance and at the end of the year od death for the discrete. (that is pure endowment doe not work) (d) the benefit under the insurance whose net single premium is on the left side must be level within each year (but not necessarily between years), discuss

\overline{A} = \frac{i}{δ} A

4. You are given that μ_{x} = .08 for all x ≥ 0. You are also given that δ_{t} = .06 for all t ≥ 0. Find the ratio of Ā_{40} to A_{40}.

5. You are give UDD and that

Ā_{x:\overline{n}|} = .67 A_{x^{1}:\overline{n}|} = .40 \frac{i}{δ} = 1.05 _{n}p_{x} = v^{n}.

Find v^{n}.

6. True or false: Under UDD (ĪĀ)_{40} = (i/δ)(IA)_{40}.

7. Textbook problems: 4.10, 4.14, 4.16, 4.17 4.18.