Continuously Increasing Whole Life Insurance: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3L - Section 31

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3L, authored by Mr. Stolyarov. This is Section 31 of the Study Guide. See an index of all sections by following the link in this paragraph.

As in Section 21, the following is defined to be the present-value function.

z_t = Z = b_tv_t

z_t = Z is the present value, at policy issue, of the benefit payment.

b_tis the benefit function.

v_tis the discount function. v is the one-year discount factor by which a sum of money payable one year from now is multiplied to get its present value today. If the annual effective interest rate is r, then v = 1/(1+r).

A countinuously increasing whole life insurance policy will pay n unit in benefits if death occurs at time n, irrespective of whether n is a whole number or not. The following functions characterize continuously increasing whole life insurance.

b_t = t for t ≥ 0;

v_t = v^t for t ≥ 0;

Z = Tv^Tfor T ≥ 0.

The actuarial present value of a continuously increasing whole life insurance policy is denoted as (ĪĀ)_x and can be found using the following formula:

(ĪĀ)_x = ₀^∞∫t*v^t*_tp_x*μ_x(t)dt = ₀^∞∫ _s│Ā_x ds

Meaning of variables:

_tp_x = probability that life (x) will survive for t more years.

μ_x(t) = force of mortality that life (x) will experience at age (x + t).

_s│Ā_x= the actuarial present value of an s-year deferred life insurance policy that pays one unit in benefits.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. pp. 106-107.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L31-1. An s-year deferred life insurance policy on the life of a 3-year-old rabbitskinned rabbit has the following actuarial present value: e^-0.02s. Find the actuarial present value of a continuously increasing whole life insurance policy on the life of a 3-year-old rabbitskinned rabbit.

Solution S3L31-1. We use the formula (ĪĀ)_x = ₀^∞∫ _s│Ā_x ds. We are given _s│Ā₃ = e^-0.02s.