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 Consider the cash flows that we showed in the very beginning of the chapter: you deposit $100 today, followed by a $125 deposit next year, and a $150 deposit at the end of the second year. In the previous situation, we sought the future value when interest rates are 7 percent. Instead of future value, we compute the present value of these three cash flows. The time line for this problem appears as: (K) 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 values 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: (K) The general equation for discounting multiple and varying cash flows is: (K)   (K)

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.
p. 102
 time out! 51   Describe how compounding affects the future value computation of an annuity.  52   Reconsider your original retirement plan example to invest $4,500 per year for 40 years. Now consider the result if you don't contribute anything for four years (years 19 to 22) while your child goes to college. How many annuity equations will you need to find the future value of your 401(k) in this situation? 
  (K) 

Present Value of Level Cash Flows You will find that this present value of an annuity concept will have many business and personal applications throughout your life. Most loans are set up so that the amount borrowed (the present value) is repaid through level payments made every period (the annuity). Lenders will examine borrowers' budgets and determine how much each borrower can afford as a payment. The maximum loan offered will be the present value of that annuity payment. We derived the equation for the present value of an annuity from the general equation for the present value of multiple cash flows, equation 53. Since each cash flow is the same, and the borrower pays the cash flows every period, we can write the equation as: (K) We denote the present value of an annuity as PVA. If we group the common level cash flow as PMT, the equation becomes (K) And finally, reduced to a simpler equation: (K) 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: (K) The time line for these payments and present value appears as: (K) Notice that although five payments of $100 each were made, $500 total, the present value is only $399.27. As we've noted previously, the span of time over which the borrower pays the annuity and the interest rate for discounting strongly affect present value computations. When you borrow money from the bank, the bank views the amount they lend as the present value of the annuity they receive over time from the borrower. Consider the examples in Table 5.2. A $50 deposit made every year for 20 years is discounted to $573.50 with a 6 percent discount rate. Doubling the annual cash flow to $100 also doubles the present value to $1,146.99. But extending the time period does not impact the present value as much as you might expect. Making $100 payments for twice the amount of time—40 years—does not double the present value. As you can see in Table 5.2, the present value increases less than 50 percent to only $1,504.63! If the discount rate increases from 6 percent to 10 percent on the 40year annuity, the present value will shrink to $977.91. 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; it's 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.   (K)

p. 103

Value of Payments Your firm needs to buy additional physical therapy equipment that costs $20,000. The equipment manufacturer will give you the equipment now if you will pay $6,000 per year for the next four years. If your firm can borrow money at a 9 percent interest rate, should you pay the manufacturer the $20,000 now or accept the fouryear annuity offer of $6,000? SOLUTION: We can find the cost of the fouryear, $6,000 annuity in present value terms using equation 54: (K) The cost of paying for the equipment over time is $19,438.32. This is less, in present value terms, than paying $20,000 cash. The firm should take the annuity payment plan.  (K) 

(K) TABLE 5.2 Impact of Magnitude of the Annuity, Number of Years Invested, and Interest Rate on PV 
(K) Present Value of Multiple Annuities Just as we can combine annuities to solve various future value problems, we can also combine annuities to solve some present value problems with changing level cash flows. Consider Alex Rodriguez's (ARod's) baseball contract in 2000 with the Texas Rangers. This contract made ARod into the “$252 million man.” The contract was structured so that the Rangers paid ARod a $10 million signing bonus, $21 million per year in 2001 through 2004, $25 million per year in 2005 and 2006, and $27 million per year in 2007 through 2010.^{1} Note that adding the signing bonus to the annual salary equals the $252 million figure. However, Rodriguez will receive the salary in the future. Using an 8 percent discount rate, what is the present value of ARod's contract? We begin by showing the salary cash flows with the time line: (K) First create a $27 million, 10year annuity. Here are the associated cash flows: (K) Now create a $−2 million, sixyear annuity: (K) 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. (K) FIGURE 5.2 Present Value of Each Annuity Cash Flow 

(K) The reported values for many sports contracts may be misleading in present value terms. 
(K) (K) (K) Adding 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}
p. 105 Perpetuity—A Special Annuity A perpetuity is a special type of annuity with a stream of level cash flows that are paid forever. These arrangements are called perpetuitiesAn annuity with cash flows that continue forever. because payments are perpetual. Assets that offer investors perpetual payments are preferred stocks and British 2½% Consolidated Stock, a debt referred to as consolsInvestment assets structured as perpetuities.. 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.   (K) 
 time out! 53   How important is the magnitude of the discount rate in present value computations? Do significantly higher interest rates lead to significantly higher present values?  54   Reconsider the physical therapy equipment example. If interest rates are only 7 percent, should you pay the upfront fee or the annuity? 
  (K) 

