MATH 403A, fall 98, CHAPTER 4 Section 4.2

Insurances payable at the moment of death

Definitions: Z = the present value, at policy issue, of the benefit payment.

Z = b_{T} v_{T}

where b_{t} is the benefit function and v_{t} = v^{t} is the discount factor.

These are the basic life insurances types with benefits payable at the time of death:

1. Whole Life insurance: A benefit of $1 is paid at the time of death. In this case, b_{t} = 1 if t ≥ 0 and v_{t} = v^{t}. The actuarial present value is:

\overline{A}_{x}^{1} = E[Z] = ∫_{0}^{∞} v^{t} f_{T} (t)dt = ∫_{0}^{∞} _{t}p_{x} μ_{x}(t)dt

2. n-year term life insurance: provides a payment only if the insured dies within the n–year term of an insurance commencing at issue. If a unit is paid at the moment of death of (x), then b_{t} = 1 if t ≤ n, 0 otherwise, and v_{t} = v^{t}. The actuarial present value is:

\overline{A}_{x:\overline{n}|}^{1} = E[Z] = ∫_{0}^{n} v^{t} f_{T} (t)dt = ∫_{0}^{n} _{t}p_{x}μ_{x}(t)dt

3. n–year pure endowment: provides for a payment at the end of n years if and only if the insured survives at least n years from the time of policy issue. Then b_{t} = 0 if t ≤ n and 1 if t > n and v_{t} = v^{n}. The actuarial present value is:

A_{x:\overline{n}^{1}|} = E[Z] = ∫_{n}^{∞} v^{t}f_{T} (t)dt = v^{n} _{n}p_{x}

4. n–year endowment insurance: provides for a payment either at the time of death, if death occurs within the n-year term, or at the end of n years if the insured survives the n years term from the time of policy issue. Then b_{t} = 1 if t ≥ 0 and v_{t} = v^{t} if t ≤ n and v_{t} = v^{n} if t > n. The actuarial present value is:

\overline{A}_{x:\overline{n}|} = \overline{A}_{x:\overline{n}|}^{1} = A_{x:\overline{n}^{1}|} = E[Z] = ∫_{0}^{n} v^{t}f_{T} (t)dt + v^{n} _{n}p_{x} 5. m–year deferred insurance: provides for a benefit following

the death of the insured only if the insured dies at least m years following policy issue. Then b_{t} = 1 if t > m and v_{t} = v^{t}. The actuarial present value is:

m|\overline{A}_{x} = ∫_{m}^{∞} v^{t} _{t}p_{x} μ_{x}(t)dt.

Rule of moments: helpful to compute variances.

E[Z^{j}]@δ_{t} = E[Z]@jδ_{t}.

A case to remember: constant force of mortality:

\overline{A}_{x} = \frac{μ}{μ + δ} .

PROBLEMS

1. The probability density function of T (x) is assumed to be f_{T} (t) = 1/80 if 0 ≤ t ≤ 80 and 0 otherwise. At a force of interest δ, calculate for Z = the present value random variable for a whole life insurance of unit amount issued to (x): (a) the actuarial present value, (b) the variance, (c) the 90th percentile η_{Z}^{0.9}.

2. For the example above, determine the distribution function and the probability density function of Z.

3. Assume each of 100 independent lives is age x, is subject to a constant force of mortality μ = .04, and is insured for a death benefit amount of 10 units, payable at the moment of death.

The benefit payments are to be withdrawn from an investment fund earning δ = .06. Calculate the minimum amount at t = 0 so that the probability is approximately .95 that sufficient funds will be on hand to withdraw the benefit payment at the death of each individual.

4. Consider a 5 year deferred whole life insurance payable at the moment of death of (x). The individual is subject to a constant force of mortality μ = .04. For the distribution of the present value of the benefit payment δ = .10. Calculate (a) the expectation, (b) the variance, (c) display the distribution function, (d) calculate the median.

5. You are given that μ_{x} = .08 for all x ≥ 0. You are also given that δ_{t} = .06 for all t ≥ 0. Find \overline{A}_{40:\overline{25}|}.

6. Z is the present value for a 10–year term life insurance with a benefit of 1 payable at the moment of death issued to a life aged 35. Mortality follows a deMoivre’s law with ω = 110. The force of interest is .07. Find the variance of Z.

7. A 15–year endowment insurance with a benefit of 1 payable at the moment of death is issued to a life aged 55. Mortality follows a deMoivre’s law with ω = 100. The force of interest is .05. Find the net single premium for the insurance.

8. Z is the present value for a 20–year endowment insurance with a benefit of 1 payable at the moment of death issued to a life aged 47. The force of mortality is constant .06 for all ages, and the force of interest is a constant .07. Find the variance of Z.

9. Textbook problems: 4.2, 4.8.