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Level Cash Flows: Perpetuities and Annuities

Frequently, you may need to value a stream of equal cash flows. For example, a home mortgage might require the homeowner to make equal monthly payments for the life of the loan. For a 30-year loan, this would result in 360 equal payments. A 4-year car loan might require 48 equal monthly payments. Any such sequence of equally spaced, level cash flows is called an annuity. If the payment stream lasts forever, it is called a perpetuity.

HOW TO VALUE PERPETUITIES

Some time ago the British government borrowed by issuing perpetuities. Instead of repaying these loans, the British government pays the investors holding these securities a fixed annual payment in perpetuity (forever).

The rate of interest on a perpetuity is equal to the promised annual payment C divided by the present value. For example, if a perpetuity pays $10 per year and you can buy it for $100, you will earn 10 percent interest each year on your investment. In general,

Interest rate on a perpetuity = cash payment present value r = C PV

We can rearrange this relationship to derive the present value of a perpetuity, given the interest rate r and the cash payment C:

PV of perpetuity = C = cash payment r interest rate

Suppose some worthy person wishes to endow a chair in finance at your university. If the rate of interest is 10 percent and the aim is to provide $100,000 a year forever, the amount that must be set aside today is

Present value of perpetuity = C = $100,000 = $1,000,000 r .10

Two warnings about the perpetuity formula. First, at a quick glance you can easily confuse the formula with the present value of a single cash payment. A payment of $1 at the end of 1 year has a present value 1/(1 + r). The perpetuity has a value of 1/r. These are quite different.

Second, the perpetuity formula tells us the value of a regular stream of payments starting one period from now. Thus our endowment of $1 million would provide the university with its first payment of $100,000 one year hence. If the worthy donor wants to provide the university with an additional payment of $100,000 up front, he or she would need to put aside $1,100,000.

Sometimes you may need to calculate the value of a perpetuity that does not start to make payments for several years. For example, suppose that our philanthropist decides to provide $100,000 a year with the first payment 4 years from now. We know that in Year 3, this endowment will be an ordinary perpetuity with payments starting at the end of 1 year. So our perpetuity formula tells us that in Year 3 the endowment will be worth $100,000/r. But it is not worth that much now. To find today`s value we need to multiply by the 3-year discount factor. Thus, the ¬delayed ­ perpetuity is worth

$100,000 1 1 = $1,000,000 1 = $751,315 r (1 + r)3 (1.10)3

ANNUITY Equally spaced level stream of cash flows.

PERPETUITY Stream of level cash payments that never ends