Perpetuity Rework Problem 1 under the assumption that Brad’s estate can make an investment...

Perpetuity Rework Problem 1 under the assumption that Brad’s estate can make an investment at 10% compounded annually.

Problem 1

Perpetuity Brad’s will includes an endowment to Dalhousie University that is to provide each year after his death, forever, a $500 prize for the top student in the business mathematics class, MATH 1115. Brad’s estate can make an investment at 5% compounded annually to pay for this endowment. Adapt the solution of Example 1 to determine how much this endowment will cost Brad’s estate.

EXAMPLE 1

Finding the Sum of an Infinite Geometric Sequence

A rich woman would like to leave $100,000 a year, starting now, to be divided equally among all her direct descendants. She puts no time limit on this bequeathment and is able to invest for this long-term outlay of funds at 2% compounded annually. How much must she invest now to meet such a long-term commitment?

Solution:

Let us write R = 100,000, set the clock to 0 now, and measure time in years from now. With these conventions we are to account for payments of R at times 0, 1, 2, 3,…, k,… by making a single investment now. (Such a sequence of payments is called a perpetuity.) The payment now simply costs her R. The payment at time 1 has a present value of R(1.02)−1. (See Chapter 5.) The payment at time 2 has a present value of R(1.02)−2. The payment at time 3 has a present value of R(1.02)−3, and, quite generally, the payment at time k has a present value of R(1.02)−k. Her investment now must exactly cover the present value of all these future payments. In other words, the investment must equal the sum

We recognize the infinite sum as that of a geometric series, with first term a = R = 100,000 and common ratio r = (1.02)−1. Since |r| = (1.02)−1<1, we can evaluate the required investment as

In other words, an investment of a mere $5,100,000 now will allow her to leave $100,000 per year to her descendants forever!