Present Value for Annuity Cash Flows

Suppose we were examining an asset that promised to pay $500 at the end of each of the next three years. The cash flows from this asset are in the form of a three-year, $500 annuity. If we wanted to earn 10 percent on our money, how much would we offer for this annuity?

From the previous section, we know that we can discount each of these $500 payments back to the present at 10 percent to determine the total present value:

Present value = ($500/1.10 + (500/1.12) + (500/1.13) = ($500/1.1) + (500/1.21) + (500/1.331) = $454.55 + 413.22 + 375.66 = $1,243.43

This approach works just fine. However, we will often encounter situations in which the number of cash flows is quite large. For example, a typical home mortgage calls for monthly payments over 30 years, for a total of 360 payments. If we were trying to determine the present value of those payments, it would be useful to have a shortcut.

Because the cash flows of an annuity are all the same, we can come up with a very useful variation on the basic present value equation. It turns out that the present value of an annuity of C dollars per period for t periods when the rate of return or interest rate is r is given by:

Annuity present value = C X

1 - Present value factor

The term in parentheses on the first line is sometimes called the present value interest factor for annuities and abbreviated PVIFAfr t).

The expression for the annuity present value may look a little complicated, but it isn't difficult to use. Notice that the term in square brackets on the second line, 1/(1 + r)t, is the same present value factor we've been calculating. In our example from the beginning of this section, the interest rate is 10 percent and there are three years involved. The usual present value factor is thus:

Ross et al.: Fundamentals I III. Valuation of Future I 6. Discounted Cash Flow I I © The McGraw-Hill of Corporate Finance, Sixth Cash Flows Valuation Companies, 2002

Edition, Alternate Edition

CHAPTER 6 Discounted Cash Flow Valuation 167

To calculate the annuity present value factor, we just plug this in:

Annuity present value factor = (1 - Present value factor)/r

Just as we calculated before, the present value of our $500 annuity is then:

Annuity present value = $500 X 2.48685 = $1,243.43

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