Chapter5: TIME VALUE OF MONEY 2: ANALYZING ANNUITY CASH FLOWS
5.2 Present Value of Multiple Cash Flows
The future value concept is very useful to understand how to build wealth for the future. The present value concept will help you most particularly for personal applications such as evaluating loans (like car and mortgage loans) and business applications (like determining the value of business opportunities).
Finding the Present Value of Several Cash Flows
CALCULATOR
The first cash flow is already in year zero, so its value will not change. We will discount the second cash flow one year and the third cash flow two years. Using the present value equation from the previous chapter, the present value of today’s payment is simply $100 ÷ (1 + 0.07)^{0} = $100. Similarly, the present value of the next two cash flows are $125 ÷ (1 + 0.07)^{1} = $116.82 and $150 ÷ (1 + 0.07)^{2} = $131.02, respectively. Therefore, the present value of these cash flows is $347.84 (= $100 + $116.82 + $131.02).
Putting these three individual present value equations together would yield:
The general equation for discounting multiple and varying cash flows is:
In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can differ from the others in terms of size and timing.
Present Value of Level Cash Flows
We denote the present value of an annuity as PVA. If we group the common level cash flow as PMT, the equation becomes
And finally, reduced to a simpler equation:

Suppose that someone makes $100 payments at the end of each year for five years. If interest rates are 8 percent per year, the present value of this annuity stream is computed using equation 54 as:
CALCULATOR
The time line for these payments and present value appears as:

The present value of a cash flow made far into the future is not very valuable today. That’s why doubling the number of years in the table from 20 to 40 only increased the present value by approximately 30 percent. Figure 5.2 shows how the present value of $100 annuity payments declines for the cash flows made later in time, especially at higher discount rates. The $100 cash flow in year 20 is worth less than $15 today if we use a 10 percent discount rate; they’re worth more than double, at nearly $38 today, if we use a discount rate of 5 percent. The figure also shows how quickly present value declines with a higher discount rate relative to a lower rate. As we showed above, the present values of the annuities in the figure are the sums of the present values shown. Since the present values for the 10 percent discount rate are smaller, the present value of an annuity is smaller as interest rates rise.

Present Value of Multiple Annuities
p. 151We begin by showing the salary cash flows with the time line:
First create a $27 million, 10year annuity. Here are the associated cash flows:
Now create a $–2 million, sixyear annuity:
Notice that creating the $− 2 million annuity also resulted in the third annuity of $− 4 million for four years. This time line shows three annuities. If you add the cash flows in any year, the sum is ARod’s salary for that year. Now we can find the present value of each annuity using equation 54 three times.
CALCULATOR
p. 152Adding the value of the three annuities reveals that the present value of ARod’s salary was $158.67 million (= $181.17 − $9.25 − $13.25). Adding in the $10 million signing bonus produces a contract value of $168.67 million. So, the present value of ARod’s contract turns out to be quite considerable, but you might not call him the $252 million man!^{2}
Perpetuity—A Special Annuity
The value of an investment like this is the present value of all future annuity payments. As the cash flow continues indefinitely, we can’t use equation 54. Luckily, mathematicians have figured out that when the number of periods, N, in equation 54 goes to infinity, the equation reduces to a very simple one:
For example, the present value of an annual $100 perpetuity discounted at 10 percent is $1,000 (= $100 ÷ 0.10). Compare this to the present value of a $100 annuity of 40 years as shown in Table 5.2. The 40year annuity’s value is $977.91. You’ll see that extending the payments from 40 years to an infinite number of years adds only $22.09 (= $1,000 − $977.91) of value. This demonstrates once again how little value today is placed on cash flows paid many years into the future.

^{1}The contract actually contains some complications like incentives to play well and salary deferral. We ignore those complicating factors here.
^{2}Rodriguez opted out of the contract after the 2007 season and then resigned with the New York Yankees with a new contract.
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