MATH 403A, fall 98, CHAPTER 6 Benefit Premiums

The benefit premium is the actuarial present value for the insurance divided by the actuarial present value representing the pattern of the premium payments. Note: Benefit Premium means that the premiums are net.

Let L be the insurer loss function, the random variable of the present value of the benefits to be paid by the insurer less the annuity of premiums to be paid by the insured.

The equivalence principle has the requirement that

E(L) = 0

or

E(present value of benefits) = E(present value of premiums)

Benefit Premiums are those satisfying this equation.

In general

P = \frac{A}{a}

For fully continuous Benefit Premiums:

Plan  Premium
Whole life insurance  P̄(Ā_{x}) = Ā_{x}/ā_{x}
n–year term insurance  P̄(Ā_{x^{1}:\overline{n}|}) = Ā_{x^{1}:\overline{n}|}/ā_{x:\overline{n}|}
n –year endowment insurance  P̄(Ā_{x:\overline{n}|}) = Ā_{x:\overline{n}|}/ā_{x:\overline{n}|}
h–payment whole life insurance  _{h}P̄(Ā_{x}) = Ā_{x}/ā_{x:\overline{h}|}
h–payment n–year endowment insurance _{h}P̄(Ā_{x:\overline{n}|}) = Ā_{x:\overline{n}|}/ā_{x:\overline{h}|}
n–year pure endowment  P̄(Ā_{x:\overline{n}|^{1}}) = Ā_{x:\overline{n}|1}/ā_{x:\overline{n}|}

For fully Discrete Annual Benefit Premiums:

Plan  Premium
Whole life insurance  P (A_{x}) = A_{x}/ä_{x}
n–year term insurance  P (A_{x^{1}:\overline{n}|}) = A_{x^{1}:\overline{n}|}/ä_{x:\overline{n}|}
n –year endowment insurance  P (A_{x:\overline{n}|}) = A_{x:\overline{n}|}/ä_{x:\overline{n}|}
h–payment whole life insurance  _{h}P (A_{x}) = A_{x}/ä_{x:\overline{h}|}
h–payment n–year endowment insurance _{h}P (A_{x:\overline{n}|}) = A_{x:\overline{n}|}/ä_{x:\overline{h}|}
n–year pure endowment  P (A_{x:\overline{n}|^{1}}) = A_{x:\overline{n}|1}/ä_{x:\overline{n}|}

Variance:

The Loss function is L = Z − πY, where π is the premium and Z and Y are as in the prevoius chapters.

So, in the continuous case

L = Z − π \frac{1 − Z}{δ} = \frac{π}{δ} + (1 + \frac{π}{δ})Z

Hence Var(L) = (1 + π/δ)^{2}Var(Z). In the dicrete case

L = Z − π \frac{1 − Z}{d} = \frac{π}{d} + (1 + \frac{π}{d})Z

Hence Var(L) = (1 + π/d)^{2}Var(Z).

Problems

1. You are given that A_{x^{1}:\overline{n}|} = .155, _{n}E_{x} = .305, and d = .06. Find the difference between the largest and the smallest of the following premiums

P (Ā_{x^{1}:\overline{n}|}) P_{x^{1}:\overline{n}|} and P̄(Ā_{x^{1}:\overline{n}|})

2. Mortality follows DeMoivre’s law with w = 105. The effective annual rate of interest is 10%. Find P (Ā_{45:\overline{20}|}.

3. L is the loss random variable 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(L).

4. L is the present value random variable for a fully discrete 5–year endowment insurance of 1 per year issued to (x). You are given

A_{x^{1}:\overline{5}|} = .21 ^{2}A_{x^{1}:\overline{5}|} = .18 i = .055 _{5}q_{x} = .25

Level benefit premiums are determined by the equivalence principle. Find Var(L).

5. A 20–year continuously increasing term insurance issued to (35) has a benefit at time t of b_{t} = 1000t. It is paid for by a continuous level premiums at a rate of P per year determined by the equivalence principle. You are given that the force of mortality is constant .075 and that the force of interest is constant .05. Find P.

6. A special 10–year term insurance policy issued to (35) provides for a refund of the first 5 years’s benefit premiums at the end of the 10–year period if the policyholder lives through the 10–year period. The death benefit is a level amount of 100,000 payable at the end of the year of death. level benefit premiums are payable at the begining of each year for 10 years. You are given

A_{35^{1}:\overline{10}|} = .04 ä_{35:\overline{10}|} = 5.51 i = .16

Find the level annual benefit premium.