Annually Increasing Whole Life Insurance: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3L - Section 30

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 30 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).

An annually increasing whole life insurance policy will pay 1 unit in benefits if death occurs during the first year, 2 units in benefits if death occurs during the second year, and n units in benefits if death occurs during the nth year. The following functions characterize annually increasing whole life insurance.

b_t = └t + 1┘ for t ≥ 0;

v_t = v^t for t ≥ 0;

Z = └T+ 1┘v^Tfor T ≥ 0.

The function└x┘is the greatest integer function, where└x┘is the greatest integer that is less than or equal to x.

The actuarial present value of annually increasing whole life insurance is denoted as (IĀ)_x and can be found via the following formula:
E[Z] = (IĀ)_x =₀^∞∫└t + 1┘v^t*_tp_x*μ_x(t)dt

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. p. 105.

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

Problem S3L30-1. The lives of magenta hippopotami follow the survival function s(x) = e^-0.09x. The present-value discount factor among magenta hippopotami is computed in an unusual manner such that v^t = 1/└t + 1┘.Sumatopoppih the Magenta Hippopotamus is 3 years old and has an annually increasing whole life insurance policy that pays a benefit of n Golden Hexagons (GH) if death occurs in the nth year of the policy. Find the actuarial present value of such a policy.