wk2dis-talk

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time value of money 1:

analyzing single cash flows

( I )n business and personal life, cash flows of different types are paid and received in the future. Your company can contract to build and ship its product to a foreign buyer for a $10 million single payment in two years. You may have a car loan and a $300 per month level payment over the next four years. It may be that you will pay a series of uneven tuition payments over the next couple of years as tuition changes. Whether the future entails single, level, or uneven cash flows, we need a method for comparing them when paid at different points in

time.

Both this chapter and the next illustrate time value of money (TVM) calculations, which we will use throughout the rest of this book. We hope you will see what powerful tools they are for making financial decisions. Whether you're managing the financial or other functional area of a business or making decisions in your personal life, being able to make TVM calculations will help you make financially sound decisions.

This background will also allow you to understand why CEOs, CFOs, and other professionals make the decisions that they do. Together, this chapter and the next will present all aspects of TVM. Since some students find this topic intimidating, we split the topic into two chapters as a way of providing more examples and practice opportunities. As you see the examples and work the practice problems, we believe that you will find that TVM is not difficult.

Factors to consider when making time value of money decisions include:

· Size of the cash flows.

· Time between the cash flows.

· Rate of return we can earn.

The title of this chapter refers to the time value of money. But why might money change values, and why does it depend on time? Consider that $100 can buy you an assortment of food and drinks today. Will you be able to buy those same items in five years with the same $100? Probably not. Inflation might cause these items to cost $120. If so, in terms of buying "stuff," the dollar would have lost value over the five years. If you don't need to spend your money today, putting it in your mattress will only cause it to lose value over time. Instead, there are banks that would like to use your money and pay you back later, with interest. This interest is your compensation to offset the money's decline in value. Each dollar will be worth less in the future, but you'll get more dollars. So you'll be able to buy the same items as before.

The basic idea behind the time value of money is that $1 today is worth more than $1 promised next year.

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But how much more? Is $1 today worth $1.05 next year? $1.08? $1.12? The answer varies depending on current interest rates. This chapter describes the time value of money concept and provides the tools needed to analyze single cash flows at different points in time.

LEARNING GOALS

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1 Create a cash flow time line.

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2 Compute the future value of money.

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3 Show how the power of compound interest increases wealth.

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4 Calculate the present value of a payment made in the future.

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5 Move cash flows from one year to another.

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6 Apply the Rule of 72.

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7 Compute the rate of return realized on selling an investment.

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8 Calculate the number of years needed to grow an investment.

viewpoints

business APPLICATION

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As the production manager of Head Phone Gear, Inc., you have received an offer from the supplier who provides the wires used in headsets. Due to poor planning, the supplier has an excess amount of wire and is willing to sell $500,000 worth for only $450,000. You already have one year's supply on hand. It would cost you $2,000 to store the wire until Head Phone Gear needs it next year. What implied interest rate would you be earning if you purchased and stored the wire? Should you make the purchase?

ORGANIZING CASH FLOWS LG4-1

Managing cash flow timing is one of the most important tasks in successfully operating a business. A helpful tool for organizing our analysis is the time line, which shows the magnitude of cash flows at different points in time, such as monthly, quarterly, semiannually, or yearly. Cash we receive is called an inflow, and we denote it with a positive number. Cash that leaves us, such as a payment or contribution to a deposit, is an outflow designated with a negative number.

The following time line illustrates a $100 deposit you made at a bank that pays 5 percent interest. In one year, the $100 has become $105. Given that interest rate, having $100 now (in year 0) has the same value as having

$105 in one year.

Here's a simple example: Suppose you allowed the bank to rent your $100 for a year at a cost of 5 percent, or $5. This cost is known as the interest rate.

Interest rates will affect you throughout your life, both in business and in your personal life. Companies borrow money to build factories and expand into new locations and markets. They expect the future revenues generated by these activities to more than cover the interest payments and repay the loan. People borrow money on credit cards and obtain loans for cars and home mortgages. They expect their purchases to give them the satisfaction in the future that compensates them for the interest payments charged on the loan. Understanding the dynamics between interest rates and cash inflows and outflows over time is key to financial success. The best place to start learning these concepts lies in understanding how money grows over time.

time out!

4-1 Why is a dollar worth more today than a dollar received one year from now?

4-2 Drawing on your past classes in accounting, explain why time lines must show one negative cash flow and one positive cash flow.

4-3 Set up a time line, given a 6 percent interest rate, with a cash inflow of $200 today and a cash outflow of

$212 in one year.

FUTURE VALUE

The $105 one-time cash flow that your bank credits to your account in one year is known as a future value (FV) of $100 in one year at a 5 percent annual interest rate. If interest rates were higher than 5 percent, then the future value of your $100 would also be higher. If you left your money in the bank for more than one year, then its future value would continue to grow over time. Let's see why.

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personal APPLICATION

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Payday lending has become a multibillion-dollar industry across the United States in just a few years. It provides people with short-term loans and gets its name from the fact that the loan is to be paid back at the borrower's next payday. Anthony is short a few hundred dollars and his next paycheck is two weeks away. For a $300 loan, Anthony must pay a $50 "fee" in advance and repay the $300 loan in two weeks. What implied interest rate would Anthony pay for this two-week period? Is this a good deal?

But what if Anthony can’t pay the loan back on

time? Scan the QR code for an extended look. Turn to the back of the book for solutions to these

applications.

Single-Period Future Value LG4-2

Computing the future value of a sum of money one year from today is straightforward: Add the interest earned to today's cash flow. In this case:

We computed the $5 interest figure by multiplying the interest rate by today's cash flow ($100 x 5%). Note that

in equations, interest rates appear in decimal format. So we use 0.05 for 5 percent:

$100 + ($100 x 0.05) = $105

Note that this is the same as:

$100 x (1 + 0.05) = $105

We need the 1 in the parentheses to recapture the original deposit and the 0.05 is for the interest earned. We can generalize this computation to any amount of today's cash flow. In the general form of the future value equation, we call cash today present value, or PV. We compute the future value one year from now, called FV1, using the interest rate, i:

Notice that this is the same equation we used to figure the future value of your $100. We've simply made it generic so we can use it over and over again. The 1 subscript means that we are calculating for only one period- in this case, one year. If interest rates were 6 percent instead of 5 percent per year, for instance, we could use equation 4-1 to find that the future value of $100 in one year is $106 [= $100 x (1 + 0.06)].

Of course, the higher the interest rate, the larger the future value will be. Table 4.1 shows the interest cost and future value for a sample of different cash flows and interest rates. Notice from the first two lines of the table that, while the difference in interest earned between 5 percent and 6 percent ($1) doesn't seem like much on a

$100 deposit, the difference on a $15,000 deposit (the following two lines) is substantial ($150).

Compounding and Future Value LG4-1

After depositing $100 for one year, you must decide whether to take the $105 or leave the money at the bank for another year to earn another 5 percent (or whatever interest rate the bank currently pays). In the second year at the bank, the deposit earns 5 percent on the $105 value, which is $5.25 (= $105 x 0.05). Importantly, you get more than the $5 earned the first year, which would be a simple total of $110. The extra 25 cents earned in the second year is interest on interest that was earned in the first year. We call this process of earning interest both on the original deposit and on the earlier interest payments compounding.

time line A graphical representation showing the size and timing of cash flows through time.

inflow Cash received, often from income or sale of an investment.

outflow Cash payment, often a cost or the price of an investment or deposit.

interest rate The cost of borrowing money, denoted as a percent.

future value (FV) The value of an investment after one or more periods.

present value (PV) The amount a future cash flow is worth today.

compounding The process of adding interest earned every period on both the original investment and the reinvested earnings.

TABLE 4.1 Higher Interest Rates and Cash Flows Lead to Higher Future Values

So, let's illustrate a $100 deposit made for two years at 5 percent in the following time line:

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The question mark denotes the amount we want to solve for. As with all TVM problems, we simply have to identify what element we're solving for; in this case, we're looking for the FV. To compute the two-year compounded future value, simply use the 1-year equation (4-1) twice.

$100 x (1 + 0.05) x (1 + 0.05) = $110.25

So the future value of $100 deposited today at 5 percent interest is $110.25 in period 2. You can see that this represents $10 of interest payments generated from the original $100 ($5 each year) and $0.25 of interest earned in the second year on previously earned interest payments. The $5 of interest earned every year on the original deposit is called simple interest. Any amount of interest earned above the $5 in any given year comes from compounding. Over time, the new interest payments earned from compounding can become substantial. The multiyear form of equation 4-1 is the future value in year N, shown as:

We can solve the 2-year deposit problem more directly using equation 4-2 as $110.25 = $100 x (1.05)2. Here, solving for FV in the equation requires solving for only one unknown. In fact, all TVM equations that you will encounter only require figuring out what is unknown in the situation and solving for that one unknown factor.

( FIGURE 4.1 The Future Value of $100 )

Small differences in interest rates can really add up over time!

We can easily adapt equation 4-2 to many different future value problems. What is the future value in 30 years of that $100 earning 5 percent per year? Using equation 4-2, we see that the future value is $100 x (1.05)30 =

$432.19. The money has increased substantially! You have made a profit of $332.19 over and above your original $100. Of this profit, only $150 (= $5 x 30 years) came from simple interest earned on the original deposit. The rest, $182.19 (= $332.19 - $150), is from the compounding effect of earning interest on previously earned interest.

Remember that the difference between earning 5 percent and 6 percent in interest on the $100 was only $1 the first year. So what is the future value difference after 15 years? Is it $15? No, as Figure 4.1 shows, the difference in future value substantially increases over time. The difference is $31.76 in year 15 and $142.15 in year 30.

The Power of Compounding Compound interest is indeed a powerful tool for building wealth. Albert Einstein, the German-born American physicist who developed the special and general theories of relativity and won the Nobel Prize for Physics in 1921, is supposed to have said, "The most powerful force in the universe is compound interest."1 Figure 4.2 illustrates this point. It shows the original $100 deposited, the cumulative interest earned on that deposit, and the cumulative interest-on-interest earned. By the 27th year, the money from the interest-on-interest exceeds the interest earned on the original deposit. By the 40th year, interest-on-interest contributes more than double the interest on the deposit. The longer money can earn interest, the greater the compounding effect.

Earning higher interest rates on the investment for additional time periods magnifies compounding power. Consider the future value of $100 deposited at different interest rates and over different time periods as shown in Table 4.2. The future value of $100 earning 5 percent per year for five years is $127.63, for a gain of $27.63. Would you double your gain by simply investing that same $100 at double the interest rate, 10 percent? No, because compounding changes the nature of the investment so that your money grows exponentially, not in a simple linear relationship. The future value of $100 in five years at 10 percent is $161.05. The $61.05 gain is more than double the gain of $27.63 earned at 5 percent. Tripling the interest rate to 15 percent shows a gain of

$101.14 that is nearly quadruple the gain earned at 5 percent.

simple interest Interest earned only on the original deposit.

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( FIGURE 4.2 Interest Earned on Prior Interest at a 5 Percent Rate )

The money from interest-on-interest will eventually exceed the interest from the original deposit.

The same effect occurs when we increase the time. When the deposit earns 10 percent per year for five years, the gain is $61.05. When we double the amount of time to 10 years, the gain more than doubles to $159.37. If we double the time again to 20 years, the gain increases not to just $318.74 (= $159.37 x 2) but to $572.75. At 10 percent for 30 years, the gain on $100 is a whopping $1,644.94. Interest rates and time are both important factors in compounding! These relationships are illustrated in Figure 4.3.

TABLE 4.2 Compounding Builds Wealth Over Time

time out!

4-4 How does compounding help build wealth (or increase debt) over time?

4-5 Why does doubling the interest rate or time quickly cause more than a doubling of the future value?

Compounding at Different Interest Rates Over Time Interest rates have varied over time. In the past half-century, banks have offered depositors rates lower than 1 percent and as high as double digits. They've also charged interest from about 5.5 percent to 21.6 percent to consumers for various kinds of loans. Let's look at how to compute future value when rates change, so that money earns interest at multiple interest rates over time. In our first example in this chapter, your deposit of $100 earned 5 percent interest. Now consider the future value when the bank announces it will pay 6 percent interest in the second year. How much will you earn now? We can illustrate the question with this time line:

We already know that the $100 deposit will grow to $105 at the end of the first year. This $105 will then earn 6 percent in the second year and have a value of $111.30 (= $105 x 1.06).

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( FIGURE 4.3 The Impact of Time and the Magnitude of the Interest Rate )

The future value differences between compounding interest rates expand exponentially over time. Future value of $100 deposited at 5%, 10%, and 15% interest rates

EXAMPLE

4-1 Graduation Celebration Loan LG4-3

Scan the code or log in to Connect for access to the interactive guided examples

Dominic is a fourth-year business student who wants to go on a graduation celebration/vacation in Mexico but he has no money to pay for the trip. After the vacation, Dominic will start his career. His job will require moving to a new town and buying professional clothes. He asked his parents to lend him $1,500, which he figures he will be able to pay back in three years. His parents agree to lend him the money, but they will charge 7 percent interest per year. What amount will Dominic need to pay back? How much interest will he pay? How much of what he pays is interest-on-interest?

SOLUTION:

Dominic will have to pay:

Of the $1,837.56 he owes his parents, $337.56 (= $1,837.56 - $1,500) is interest. We can illustrate this time-value problem in the following time line.

Compare this compound interest with simple interest. Simple interest would be 7 percent of $1,500 (which is $105) per year. The 3-year cost would then be $315 (= 3 x $105). The difference between the compound interest of $337.56 and the total simple interest of $315 is the interest-on-interest of $22.56.

Similar to Problems 4­3, 4­4, 4­5, 4­6, 4­21, 4­22, 4­33, 4­34, self­test problem 1

discounting The process of finding present value by reducing future values using the discount, or interest, rate.

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EXAMPLE 4-2

Celebration Loan with Payback Incentive LG4-3

Scan the code or log in to Connect for access to the interactive guided examples

Reexamine the loan Dominic was seeking from his parents in the previous example. His parents want to give him an

incentive to pay off the loan as quickly as possible. They structure the loan so they charge 7 percent interest the first year and increase the rate 1 percent each year until the loan is paid. How much will Dominic owe if he waits three years to pay off the loan? Say that in the third year he considers whether to pay off the loan or wait one more year. How much more will he pay if he waits one more year?

SOLUTION:

For a payment in the third year, Dominic will pay interest of 7 percent the first year, 8 percent the second year, and 9 percent the third year. He will have to pay:

The cash flow time line is:

Even worse, if he waits until the fourth year, he will pay one year of interest at 10 percent. The total payment will be

FV4 = $1,889.41 x 1.10 = $2,078.35

Because of both the escalating interest rate and the compounding effect, Dominic must make timely and increasing payments the longer he delays in paying off the loan. Deciding in the third year to put off the payment an extra year would cost him an additional $188.94 (= $2,078.35 - $1,889.41).

Similar to Problems 4­7, 4­8

If we put the two steps together into one equation, the solution appears as $111.30 = $100 x 1.05 x 1.06. From this you should not be surprised that a general equation for future value of multiple interest rates is:

Note that the future value equation 4-2 is a special case of the more general equation 4-3. If the interest rate every period is the same, we can write equation 4-3 as equation 4-2.

PRESENT VALUE LG4-4

We asked earlier what happens when you deposit $100 cash in the bank to earn 5 percent interest for one year- the bank pays you a $105 future value. However, we could have asked the question in reverse. That is, if the bank will pay $105 in one year and interest rates are 5 percent, how much would you be willing to deposit now, to receive that payment in a year? Here, we start with a future value and must find the present value-a different kind of calculation called discounting.

Discounting

While the process of a present value growing over time into the future is called compounding, the process of figuring out how much an amount that you expect to receive in the future is worth today is discounting. Just as compounding significantly increases the present value into the future, discounting significantly decreases the value of a future amount to the present. Since discounting is the reverse of compounding, we can rearrange equation 4-1 to solve for the present value of a cash flow received one year in the future.

the

Math Coach on...

Using a Financial Calculator

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Financial, or business, calculators are programmed to perform the time value of money equations we develop in this chapter and the next. The two most common types of inexpensive financial calculators that can perform such functions are the Hewlett-Packard 10B II Business Calculator and the Texas Instruments BA II (Plus or Professional). Among many useful financial shortcuts these calculators have five specific financial buttons. The relevant financial buttons for time value of money (TVM) calculations are listed below. The HPIOBII calculator buttons look like this:

1. N (for the number of periods),

2. I/YR (for the interest rate),

3. PV (for present value),

4. PMT (for a constant payment every period), and

5. FV (for future value).

Notice that the TI BA II Plus financial calculator buttons appear to be very similar:

A common, more sophisticated and expensive calculator is the TI-83. This calculator has a menu system that includes the financial functions as shown:

To get to the TVM menu, select APPLICATIONS and then choose FINANCE, and finally 1 TVM SOLVER on the previous screens.

Setting Up Your Calculator These calculators come from the factory with specific settings. You will find it useful to change two of them. The first is to set the number of digits shown after the decimal point on the calculator display. The factory setting is for two digits. However, consider a problem in which we use a 5.6 percent interest rate. The decimal version of this percentage is 0.056, which a two-digit display would round to 0.06. It's less worrisome to set the calculator to display the number of digits necessary to show the right number; this is called a floating point display. To set a floating point display for the HP calculator, press the

color button, then the DISP button, and finally the decimal (.) button. To set the display for a floating point decimal on the TI calculator, push the 2ND button, followed by the FORMAT button, followed by the 9 button, and finally the ENTER button.

The second change you'll want to make is to set the number of times the calculator compounds each period. The settings may be preset to 12 times per period. Reset this to one time per period. To change the HP calculator to compound once per period, push the 1 button, then the color button, and finally the P/YR button. On the TI calculator, simply push the 2ND button, the P/Y button, the number one, and the ENTER button. These new settings will remain in the calculator until you either change them or remove the calculator's batteries.

Using Your Calculator

The calculators compute time-value problems in similar ways. Enter the cash flows into the time-value buttons (PV, PMT, and FV) consistent with the way they are shown in a time line. In other words, cash inflows should be positive and cash outflows negative. Thus, PV and FV cash flows are nearly always opposite in sign. Enter interest rates (I) in the percentage form, not the decimal form. Also enter the number of periods in the problem (N).

Consider our earlier example of the $100 deposit for two years earning a 5 percent interest rate.

1. To set the number of years, press 2 and then the N button.

2. To set the interest rate, press 5 and then the I button. (Note that interest rates are in percentage format for using a financial calculator and in decimal format for using the equations.)

3. To enter the current cash flow: press 100, then make it negative by pressing the +/- button, then press the PV button.

4. We won't use the PMT button, so enter 0 and then the PMT button.

5. To solve for future value, press the compute button (CPT) [for the TI] and then the FV button [press the FV button only for the HP].

6. Solution: the display should show FV = 110.25

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Note that the answer is positive, consistent with an inflow and the time line diagram. These values remain in the TVM registers even after the calculator is turned off. So when you start a new problem, you should clear out old values first. For the HP calculator, clear the registers by pressing the shift/orange key before pressing

C. You can clear the BAII Plus calculator using 2ND and CLR TVM.

You'll notice that throughout this book we use the equations in the main text to solve time value of money problems. We provide the calculator solutions in the margins.

How much would you be willing to deposit now to receive a certain payment in a year?

Suppose the bank is going to pay $105 in one year and interest rates are 5 percent. Then the present value of the payment is $105/1.05 = $100. Present values are always smaller than future values (as long as interest rates are greater than zero!), and the difference between what an investment is worth today and what it's worth when you're supposed to redeem it gets larger as the interest rate increases. Likewise, if the amount of time until the expected payment date increases, the difference will also increase in value.

Discounting Over Multiple Periods

Discounting over multiple periods is the reverse process of compounding over multiple periods. Knowing this, we can find the general equation for present value by rearranging the terms in equation 4-2 to form:

The interest rate, i, which we use to calculate present value, is often referred to as the discount rate. How much is a $100 payment to be made in the future worth today? Of course, it depends on how far into the future you expect to receive the payment and the discount rate used. If you receive a $100 cash flow in five years, then its present value is $78.35, discounted at 5 percent:

The time line looks like this:

If the discount rate rises to 10 percent, the present value of our $100 to be paid to us in five years is only $62.09 today. At a 15 percent interest rate, the present value declines to less than half the future cash flow: $49.72. Higher interest rates discount future cash flows more quickly and dramatically. You can see this principle illustrated in Figure 4.4.

discount rate The interest rate used to discount future cash flow(s) to the present.

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( FIGURE 4.4 Present Value of a $100 Cash Flow Made in the Future )

The higher the discount rate, the more quickly the cash flow value falls.

EXAMPLE 4-3

Buy Now and Don't Pay for Two Years LG4-4

Scan the code or log in to Connect for access to the interactive guided examples

Suppose that a marketing manager for a retail furniture company proposes a sale. Customers can buy now but don't have to pay for their furniture purchases for two years. From a time value of money perspective, selling furniture at full price with payment in two years is equivalent to selling furniture at a sale, or discounted, price with immediate payment. If interest rates are 7.5 percent per year, what is the equivalent sale price of a $1,000 sleeper-sofa when the customer takes the full two years to pay for it?

SOLUTION:

The time line for this problem is:

Using equation 4-5, the present value computation is

In this case, the marketing proposal for delaying payment for two years is equivalent to selling the $1,000 sleeper-sofa for a sale price of $865.33, or a 13.5 percent discount. When stores promote such sales, they often believe that customers will not be able to pay the full amount at the end of the two years and then must pay high interest rate charges and late fees. Customers who do pay on time are getting a good deal.

Similar to Problems 4­9, 4­10, 4­11, 4­12, self­test problem 2

Moving right from point A in the figure, notice that if interest rates are 0 percent, the present value will equal the future value. Also note from the curved lines that when the discount rate is greater than zero, the discounting to present value is not linear through time. The higher the discount rate,

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the more quickly the cash flow value falls. If the discount rate is 10 percent, a $100 cash flow that you would receive in 25 years is worth less than $10 today, as shown at point B in the figure. With a 15 percent discount rate, the $100 payment to be received in 33 years, at point C, is worth less than $1 today.

Discounting with Multiple Rates LG4-4, LG4-1

We can also discount a future cash flow at different interest rates per period. We find the general form of the equation for present value with multiple discount rates by rearranging equation 4-3:

Suppose that we expect interest rates to increase over the next few years, from 7 percent this year, to 8 percent next year, to 8.5 percent in the third year. In this environment, how would we work out the present value of a future $2,500 cash flow in year 3? The time line for this problem is

Using equation 4-6 shows that the present value is $1,993.90:

time out!

4-6 How are interest rates in the economy related to the way people value future cash payments? 4-7 Explain how discounting is the reverse of compounding.

USING PRESENT VALUE AND FUTURE VALUE LG4-5

Moving Cash Flows

As managers analyze investment projects, debt management, and cash flow, they frequently find it useful to move cash flows to different points in time. While you may be planning to keep money deposited in the bank for three years when you will buy a car, life often has a way of altering plans. What type of car might you purchase if the money earns interest for only two years, or for four years? How is a corporate financial forecast affected if the firm needs to remodel a factory two years sooner than planned? Moving cash flows around in time is important to businesses and individuals alike for sound financial planning and decision making.

Moving cash flows from one point in time to another requires us to use both present value and future value equations. Specifically, we use the present value equation for moving cash flows earlier in time, and the future value cash flows for moving cash flows later in time. For example, what's the value in year 2 of a $200 cash flow to be received in three years, when interest rates are 6 percent? This problem requires moving the $200 payment in the third year to a value in the second year, as shown in the time line:

Since the cash flow is to be moved one year earlier in time, we use the present value equation:

When interest rates are 6 percent, a $188.68 payment in year 2 equates to a $200 payment in year 3.

What about moving the $200 cash flow to year 5? Since this requires moving the cash flow later in time by two years, we use the future value equation. In this case, the equivalent of $200 in the third year is a fifth-year payment of:

Table 4.3 illustrates how we might move several cash flows. At an 8 percent interest rate, a $1,000 cash flow due in year 5 compounded to year 10 equals $1,469.33. We could also discount that same $1,000 cash flow to a value of $793.83 in year 2. At an 8 percent interest rate, the three cash flows ($793.83 in year 2, $1,000 in year 5, and $1,469.33 in year 10) become equivalent. Table 4.3 illustrates the movement of other cash flows given different interest rates and time periods.

Moving cash flows from one year to another creates an easy way to compare or combine two cash flows. Would you rather receive $150 in year 2 or $160 in year 2? Since both cash flows occur in the same year, the comparison is straightforward. But we can't directly add or compare cash flows in different years until we consider their time value. We can compare cash flows in different years by moving one cash flow to the same time as the other using the present value or future value equations. Once you have the value of each cash flow in the same year, you can directly compare or combine them.

Rule of 72 An approximation for the number of years needed for an investment to double in value.

TABLE 4.3 Equivalent Cash Flows in Time

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Rule of 72 Albert Einstein is also credited with popularizing compound interest by introducing a simple mathematical approximation for the number of years required to double an investment. It's called the Rule of 72.

The Rule of 72 illustrates the power of compound interest. How many years will it take to double money deposited at 6 percent per year? Using the Rule of 72, we find the answer is 12 years (= 72/6). A higher interest rate causes faster increases in future value. A 9 percent interest rate allows money to double in just eight years (= 72/9). Remember that this rule provides only a mathematical approximation. It's more accurate with lower interest rates. After all, with a 72 percent interest rate, the rule predicts that it will take one year to double the money (= 72/72). However, we know that it actually takes a 100 percent rate to double money in one year.

We can also use the Rule of 72 to approximate the interest rate needed to double an investment in a specific amount of time. What rate do we need to double an investment in 5 years? Rearranging equation 4-7 shows that the rate needed is 14.4 percent (= 72/5) per year.

time out!

4-8 In Example 4-4, could Timber, Inc., have performed its analysis by moving the $175,000 to year 2 and comparing? Would the firm then have made the same decision?

4-9 At what interest rate (and number of years) does the Rule of 72 become too inaccurate to use?

COMPUTING INTEREST RATES LG4-7, LG4-1

Time value of money calculations come in handy when we know two cash flows and need to find the interest rate. The investment industry often uses this analysis. Solving for the interest rate, or rate of return,2 can answer questions like, "If you bought a gold coin for $350 three years ago and sell it now for $475, what rate of return have you earned?" The time line for this problem looks like this:

EXAMPLE 4-4

Pay Damages or Appeal? LG4-5

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Timber, Inc., lost a lawsuit in a business dispute. The judge ordered the company to pay the plaintiff $175,000 in one year. Timber's attorney advises Timber to appeal the ruling. If so, Timber will likely lose again and will still have to pay the $175,000. But by appealing, Timber moves the $175,000 payment to year 2, along with the attorney's fee of

$20,000 for the extra work. The interest rate is 7 percent. What decision should Timber make?

SOLUTION:

Timber executives must decide whether to pay $175,000 in one year or $195,000 in two years. To compare the two choices more directly, move the payment in year 2 to year 1 and then compare it to $175,000. Timber should choose to

make the smaller payment. The computation is:

The value in year 1 of a year 2 payment of $195,000 is $182,242.99, which is clearly more than the

$175,000 year 1 payment. So Timber should not appeal and should pay the plaintiff $175,000 in one year (and may want to look for another attorney).

Similar to Problems 4­23, 4­24, 4­25, 4­26, 4­27, 4­28, 4­41

In general, computing interest rates is easiest with a financial calculator. To compute the answer using the time- value equations, consider how the cash flows fit into the future value equation 4-2:

Rearranging gives:

To solve for the interest rate, i, take the third root of both sides of the equation. To do this, take 1.357 to the 1/3 power using the yx button on your calculator.3 Doing this leads to:

If you buy a gold coin for $350 and sell it three years later for $475, you earn a 10.7 percent return per year. Time is an important factor in computing the return that you're earning per year. Turning a $100 investment into

$200 is a 100 percent return. If your investments earn this much in two years, then they earned a 41.42 percent rate of return per year [$100 x (1.4142)2 = $200]. Table 4.4 shows the annual interest rate earned for doubling an investment over various time periods. Notice how compounding complicates the solution: It's not as simple as just dividing by the number of years. Getting a 100 percent return in two years means earning 41.42 percent per year, not 50 percent per year. Table 4.4 also shows the Rule of 72 interest rate estimate.

Return Asymmetries

Suppose you bought a gold coin for $700 last year and now the market will pay you only $350. Clearly, the investment earned a negative rate of return. Use a financial calculator or a time-value equation to verify that this is a return of -50 percent. You lost half your money! So, in order to break even and get back to $700, you need to earn a positive 50 percent, right? Wrong. Note that to get from $350 to $700, your money needs to double! You need a 100 percent return to make up for a 50 percent decline. Similarly, you need a gain of 33.33 percent to make up for a 25 percent decline. If your investment declines by 10 percent, you'll need an 11.11 percent gain to offset the loss. In general, only a higher positive return can offset any given negative return.

TABLE 4.4 Interest Rate per Year to Double an Investment

( 20 /28 )

( 21 /28 )

finance at work II: investments

TVM Caveat Emptor

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Not making your payments on time can get very expensive. Reconsider the furniture selling experience in Example 4-3. The store has given customers the opportunity to buy the sleeper-sofa today and not pay the $1,000 price for two years. But what happens if you forget to pay on time? Indeed, many people do forget. Others simply haven't saved $1,000 and can't make the payment. The fine print in these deals provides the penalties for late payment. For example, the late clause might require a 10 percent annually compounded interest rate to apply to any late payment-retroactive to the sale date. Thus, being one day late with the payment will automatically incur an interest charge for the entire two years of $210 (=

$1,000 x 1.12 - $1,000), which is in addition to the original $1,000 still owed, of course.

The impact of making late payments can also show up later when you apply for credit cards, auto loans, and other credit. Credit rating agencies gather information on us from companies, banks, and landlords to grade our payment history. If you consistently make late payments on your apartment, credit card, or electric bill, these agencies will give you a poor grade. The higher your grade, called a credit score, the more likely you are to be able to get a loan and pay a lower interest rate on that loan. People with lower credit scores may not be able to borrow money and when they can, they will pay higher interest rates on their credit cards and auto loans. Paying a higher interest rate can really cost you a lot of money. Remember the future value differences between interest rates illustrated in Table 4.2 and Figure 4.3. Those people who have really bad credit scores can't get loans from banks and merchants and have to rely on payday lending places that charge enormously high rates, as highlighted in this chapter's Personal Application Viewpoint.

Making a late payment might not seem like a big deal at the time, but it can really cost you!

Want to know more?

Key Words to Search for Updates: credit report, credit ratings, credit score

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Consider the future value problem of Example 4-1. The spreadsheet solution is the same as the TVM calculator solution. Note that since the PV is listed as a positive number, the FV output is a negative number.

The inputs to the function can be directed to other cells, like the rate, nper, and pv in the this illustration. Or the input can be the actual number, like pmt and type.

This spreadsheet solves for the interest rate in the preceding example in the text. Just like using the TVM calculator, the PV and FV must be of opposite signs to avoid getting an error message.

Log in to Connect to watch instructional videos on using spreadsheets. Also note that the solutions to all the examples in the book are illustrated using spreadsheets in videos that can be accessed in Connect or by scanning the accompanying QR codes.

time out!

4-10 Say you double your money in three years. Explain why the rate of return is NOT 33.3 percent per year.

4-11 Show that you must earn a 25 percent return to offset a 20 percent loss.

EXAMPLE 4-5

Growth in Staffing Needs LG4-8

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Say that you are the sales manager of a company that produces software for human resource departments. You are planning your staffing needs, which depend on the volume of sales over time. Your company currently sells $350 million of merchandise per year and has grown 7 percent per year in the past. If this growth rate continues, how long will it be before the firm reaches $500 million in sales? How long before it reaches $600 million?

SOLUTION:

You could set up the following time line to illustrate the problem:

As shown in the margin, $350 million of sales growing at 7 percent per year will reach $500 million in five years and three months. To reach $600 million will take just two weeks short of eight years.

Similar to Problems 4­31, 4­32, self­test problem 4

SOLVING FOR TIME LG4-8, LG4-7

Sometimes you may need to determine the time period needed to accumulate a specific amount of money. If you know the starting cash flow, the interest rate, and the future cash flow (the sum you will need), you can solve the time value equations for the number of years that you will need to accumulate that money. Just as with solving for different interest rates, solving for the number of periods is complicated and requires using natural logarithms.4 Many people prefer to use a financial calculator to solve for the number of periods.

When interest rates are 9 percent, how long will it take for a $5,000 investment to double? Finding the solution with a financial calculator entails entering

• I = 9

• PV = -5,000

· PMT = 0

• FV = 10,000

The answer is 8.04 years, or eight years and two weeks. The Rule of 72 closely approximates the answer, which predicts eight years (= 72/9).

time out!

4-12 In Example 4-5, how long will it take your company to double its sales? 4-13 In what other areas of business can these time-value concepts be used?

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Your Turn...

Questions

1. List and describe the purpose of each part of a time line with an initial cash inflow and a future cash outflow. Which cash flows should be negative and which positive? Why? (LG4­1)

2. How are the present value and future value related? (LG4­2)

3. Would you prefer to have an investment earning 5 percent for 40 years or an investment earning 10 percent for 20 years? Explain. (LG4­3)

4. How are present values affected by changes in interest rates? (LG4­4)

5. What do you think about the following statement? "I am going to receive $100 two years from now and

$200 three years from now, so I am getting a $300 future value." How could the two cash flows be compared or combined? (LG4­5)

6. Show how the Rule of 72 can be used to approximate the number of years to quadruple an investment. (LG4­6)

7. Without making any computations, indicate which of each pair has a higher interest rate: (LG4­7)

a. $100 doubles to $200 in five years or seven years.

b. $500 increases in four years to $750 or to $800.

c. $300 increases to $450 in two years or increases to $500 in three years.

8. A $1,000 investment has doubled to $2,000 in eight years because of a 9 percent rate of return. How much longer will it take for the investment to reach $4,000 if it continues to earn a 9 percent rate? (LG4­8)

Problems

BASIC PROBLEMS

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4-1 Time Line Show the time line for a $500 cash inflow today, a $605 cash outflow in year 2, and a 10 percent interest rate. (LG4­1)

4-2 Time Line Show the time line for a $400 cash outflow today, a $518 cash inflow in year 3, and a 9 percent interest rate. (LG4­1)

4-3 One Year Future Value What is the future value of $500 deposited for one year earning an 8 percent interest rate annually? (LG4­2)

4-4 One Year Future Value What is the future value of $400 deposited for one year earning an interest rate of 9 percent per year? (LG4­2)

4-5 Multiyear Future Value How much would be in your savings account in 11 years after depositing $150 today if the bank pays 8 percent per year? (LG4­3)

4-6 Multiyear Future Value Compute the value in 25 years of a $1,000 deposit earning 10 percent per year. (LG4­3) 4-7 Compounding with Different Interest Rates A deposit of $350 earns the following interest rates:

a. 8 percent in the first year.

b. 6 percent in the second year.

c. 5.5 percent in the third year.

What would be the third year future value? (LG4­3)

4-8 Compounding with Different Interest Rates A deposit of $750 earns interest rates of 9 percent in the first year and 12 percent in the second year. What would be the second year future value? (LG4­3)

4-9 Discounting One Year What is the present value of a $350 payment in one year when the discount rate is 10 percent? (LG4­4)

4-10 Discounting One Year What is the present value of a $200 payment in one year when the discount rate is 7 percent? (LG4­4)

4-11 Present Value What is the present value of a $1,500 payment made in nine years when the discount rate is 8 percent? (LG4­4)

4-12 Present Value Compute the present value of an $850 payment made in ten years when the discount rate is 12 percent. (LG4­4)

4-13 Present Value with Different Discount Rates Compute the present value of $1,000 paid in three years using the following discount rates: 6 percent in the first year, 7 percent in the second year, and 8 percent in the third year. (LG4­ 4)

4-14 Present Value with Different Discount Rates Compute the present value of $5,000 paid in two years using the following discount rates: 8 percent in the first year and 7 percent in the second year. (LG4­4)

4-15 Rule of 72 Approximately how many years are needed to double a $100 investment when interest rates are 7 percent per year? (LG4­6)

4-16 Rule of 72 Approximately how many years are needed to double a $500 investment when interest rates are 10 percent per year? (LG4­6)

4-17 Rule of 72 Approximately what interest rate is needed to double an investment over five years? (LG4­6)

4-18 Rule of 72 Approximately what interest rate is earned when an investment doubles over 12 years? (LG4­6) 4-19 Rates over One Year Determine the interest rate earned on a $1,400 deposit when $1,800 is paid back in

one year. (LG4­7)

4-20 Rates over One Year Determine the interest rate earned on a $2,300 deposit when $2,900 is paid back in one year. (LG4­7)

INTERMEDIATE PROBLEMS

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4-21 Interest-on-Interest Consider a $2,000 deposit earning 8 percent interest per year for five years. What is the future value, and how much total interest is earned on the original deposit versus how much is interest earned on interest? (LG4­3)

4-22 Interest-on-Interest Consider a $5,000 deposit earning 10 percent interest per year for ten years. What is the future value, how much total interest is earned on the original deposit, and how much is interest earned on interest? (LG4­3)

4-23 Comparing Cash Flows What would be more valuable, receiving $500 today or receiving $625 in three years if interest rates are 7 percent? Why? (LG4­5)

4-24 Comparing Cash Flows Which cash flow would you rather pay, $425 today or $500 in two years if interest rates are 10 percent? Why? (LG4­5)

4-25 Moving Cash Flows What is the value in year 3 of a $700 cash flow made in year 6 if interest rates are 10 percent? (LG4­5)

4-26 Moving Cash Flows What is the value in year 4 of a $1,000 cash flow made in year 6 if interest rates are 8 percent? (LG4­5)

4-27 Moving Cash Flows What is the value in year 10 of a $1,000 cash flow made in year 3 if interest rates are 9 percent? (LG4­5)

4-28 Moving Cash Flows What is the value in year 15 of a $250 cash flow made in year 3 if interest rates are 11 percent? (LG4­5)

4-29 Solving for Rates What annual rate of return is earned on a $1,000 investment when it grows to $1,800 in six years? (LG4­7)

4-30 Solving for Rates What annual rate of return is earned on a $5,000 investment when it grows to $9,500 in five years? (LG4­7)

4-31 Solving for Time How many years (and months) will it take $2 million to grow to $5 million with an annual interest rate of 7 percent? (LG4­8)

4-32 Solving for Time How long will it take $2,000 to reach $5,000 when it grows at 10 percent per year? (LG4­8)

ADVANCED PROBLEMS

4-33 Future Value At age 30 you invest $1,000 that earns 8 percent each year. At age 40 you invest $1,000 that earns 12 percent per year. In which case would you have more money at age 60? (LG4­2)

4-34 Future Value At age 25 you invest $1,500 that earns 8 percent each year. At age 40 you invest $1,500 that earns 11 percent per year. In which case would you have more money at age 65? (LG4­2)

4-35 Solving for Rates You invested $2,000 in the stock market one year ago. Today, the investment is valued at $1,500.

What return did you earn? What return would you need to get next year to break even overall? (LG4­7)

4-36 Solving for Rates You invested $3,000 in the stock market one year ago. Today, the investment is valued at $3,750. What return did you earn? What return would you suffer next year for your investment to be valued at the original

$3,000? (LG4­7)

4-37 Solving for Rates What annual rate of return is earned on a $4,000 investment made in year 2 when it grows to $6,500 by the end of year 7? (LG4­7)

4-38 Solving for Rates What annual rate of return is implied on a $2,500 loan taken next year when $3,500 must be repaid in year 4? (LG4­7)

4-39 General TVM Ten years ago, Hailey invested $2,000 and locked in a 9 percent annual interest rate for 30

years (ending 20 years from now). Aidan can make a 20-year investment today and lock in a 10 percent interest rate. How much money should he invest now in order to have the same amount of money in 20 years as Hailey? (LG4­2, LG4­4)

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4-40 General TVM Ten years ago, Hailey invested $3,000 and locked in an 8 percent annual interest rate for 30 years (ending 20 years from now). Aidan can make a 20-year investment today and lock in a 10 percent interest rate. How much money should he invest now in order to have the same amount of money in 20 years as Hailey? (LG4­2, LG4­4)

4-41 Moving Cash Flows You are scheduled to receive a $500 cash flow in one year, a $1,000 cash flow in two years, and pay an $800 payment in three years. If interest rates are 10 percent per year, what is the combined present value of these cash flows? (LG4­5)

4-42 Spreadsheet Problem Oil prices have increased a great deal in the last decade. The following table shows the average oil price for each year since 1949. Many companies use oil products as a resource in their own business operations (like airline firms and manufacturers of plastic products). Managers of these firms will keep a close

watch on how rising oil prices will impact their costs. The interest rate in the PV/FV equations can also be interpreted as a growth rate in sales, costs, profits, and so on (see Example 4-5).

a. Using the 1949 oil price and the 1969 oil price, compute the annual growth rate in oil prices during those 20 years.

b. Compute the annual growth rate between 1969 and 1989 and between 1989 and 2012.

c. Given the price of oil in 2012 and your computed growth rate between 1989 and 2012, compute the future price of oil

in 2015 and 2020.

1 No one seems to know exactly what he said, when he said it, or to whom. Similar statements commonly attributed to Einstein are:

(1) compound interest is the greatest wonder of the universe, (2) compound interest is the ninth wonder of the world, and (3) it is the greatest mathematical discovery of all time. If he did not say any of these things, he (or someone else) should have!

2 The terms interest rate and rate of return are referring to the same thing. However, it is a common convention to refer to interest rate when you are the one paying the cash flows and refer to rate of return when you are the one receiving the cash flows.

3 The general equation for computing the interest rate is

4 The equation for solving for the number of periods is

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