MY NUMBER 4

5

The Time Value of Money

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Learning Objectives

 Explain what the time value of money is and why it is so important in the field of finance.

 Explain the concept of future value, including the meaning of the terms principal, simple interest, and compound interest, and use the future value formula to make business decisions.

 Explain the concept of present value, how it relates to future value, and use the present value formula to make business decisions.

 Discuss why the concept of compounding is not restricted to money, and use the future value formula to calculate growth rates.

When you purchase an automobile from a dealer, the decision of whether to pay cash or finance your purchase can affect the price you pay. For example, automobile manufacturers often offer customers a choice between a cash rebate and low-cost financing. Both alternatives affect the cost of purchasing an automobile; but one alternative can be worth more than the other.

To see why, consider the following. In June 2010, as the end of the model year approached, the automobile manufacturer General Motors wanted to reduce its inventory of 2010 Yukon sport utility vehicles (SUVs) before it introduced the 2011 models. In an effort to increase sales of the 2010 Yukon SUVs, the company offered consumers a choice between (1) receiving $3,000 off the base price of $38,020 if they paid cash and (2) receiving 0 percent financing on a five-year loan if they paid the base price. For someone who had enough cash to buy the car outright and did not need the cash for some other use, the decision of whether to pay cash or finance the purchase of a Yukon depended on the rate of return they could earn by investing the cash. On the one hand, if it was possible to earn only a 1 percent interest rate by investing in a certificate of deposit at a bank, the buyer was better off paying cash for the Yukon. On the other hand, if it was possible to earn 5 percent, the buyer was better off taking the financing. With a 3.36 percent rate of return, the buyer would have been largely indifferent between the two alternatives. In  Chapters 5  and  6  you will learn how to calculate the rate of return at which the buyer would be indifferent in a situation like this.

As with most business transactions, a crucial element in the analysis of the alternatives offered by General Motors is the value of the expected cash flows. Because the cash flows for the two alternatives take place in different time periods, they must be adjusted to account for the time value of money before they can be compared. A car buyer wants to select the alternative with the cash flows that have the lowest value (price). This chapter and the next provide the knowledge and tools you need to make the correct decision. You will learn that at the bank, in the boardroom, or in the showroom, money has a time value—dollars today are worth more than dollars in the future—and you must account for this when making financial decisions.

CHAPTER PREVIEW

Business firms routinely make decisions to invest in productive assets to earn income. Some assets, such as plant and equipment, are tangible, and other assets, such as patents and trademarks, are intangible. Regardless of the type of investment, a firm pays out money now in the hope that the value of the future benefits (cash inflows) will exceed the cost of the asset. This process is what value creation is all about—buying productive assets that are worth more than they cost.

The valuation models presented in this book will require you to compute the present and future values of cash flows. This chapter and the next one provide the fundamental tools for making these calculations.  Chapter 5  explains how to value a single cash flow in different time periods, and  Chapter 6  covers valuation of multiple cash flows. These two chapters are critical for your understanding of corporate finance.

We begin this chapter with a discussion of the time value of money. We then look at future value, which tells us how funds will grow if they are invested at a particular interest rate. Next, we discuss present value, which answers the question “What is the value today of cash payments received in the future?” We conclude the chapter with a discussion of several additional topics related to time value calculations.

5.1 THE TIME VALUE OF MONEY

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In financial decision making, one basic problem managers face is determining the value of (or price to pay for) cash flows expected in the future. Why is this a problem? Consider as an example the popular Mega MillionsTM lottery game. 1  In Mega Millions, the jackpot continues to build up until some lucky person buys a winning ticket—the payouts for a number of jackpot winning tickets have exceeded $100 million.

If you won $100 million, headlines would read “Lucky Student Wins $100 Million Jackpot!” Does this mean that your ticket is worth $100 million on the day you win? The answer is no. A Mega Millions jackpot is paid either as a series of 26 payments over 25 years or as a cash lump sum. If you win “$100 million” and choose to receive the series of payments, the 26 payments will total $100 million. If you choose the lump sum option, Mega Millions will pay you less than the stated value of $100 million. This amount was about $50 million in June 2010. Thus, the value, or market price, of a “$100 million” winning Mega Millions ticket is really about $50 million because of the time value of money and the timing of the 26 cash payments. An appropriate question to ask now is, “What is the time value of money?”

 Take an online lesson on the time value of money from  TeachMeFinance.com  at  http://teachmefinance.com/timevalueofmoney.html .

Consuming Today or Tomorrow

The time value of money is based on the idea that people prefer to consume goods today rather than wait to consume similar goods in the future. Most people would prefer to have a large-screen TV today than to have one a year from now, for example. Money has a time value because a dollar in hand today is worth more than a dollar to be received in the future. This makes sense because if you had the dollar today, you could buy something with it—or, instead, you could invest it and earn interest. For example, if you had $100,000, you could buy a one-year bank certificate of deposit paying 5 percent interest and earn $5,000 interest for the year. At the end of the year, you would have $105,000 ($100,000 + $5,000 = $105,000). The $100,000 today is worth $105,000 a year from today. If the interest rate was higher, you would have even more money at the end of the year.

time value of money

the difference in value between a dollar in hand today and a dollar promised in the future; a dollar today is worth more than a dollar in the future

BUILDING INTUITION THE VALUE OF MONEY CHANGES WITH TIME

The term time value of money reflects the notion that people prefer to consume things today rather than at some time in the future. For this reason, people require compensation for deferring consumption. The effect is to make a dollar in the future worth less than a dollar today.

Based on this example, we can make several generalizations. First, the value of a dollar invested at a positive interest rate grows over time. Thus, the further in the future you receive a dollar, the less it is worth today. Second, the trade-off between money today and money at some future date depends in part on the rate of interest you can earn by investing. The higher the rate of interest, the more likely you will elect to invest your funds and forgo current consumption. Why? At the higher interest rate, your investment will earn more money.

In the next two sections, we look at two views of time value—future value and present value. First, however, we describe time lines, which are pictorial aids to help solve future and present value problems.

Time Lines as Aids to Problem Solving

Time lines are an important tool for analyzing problems that involve cash flows over time. They provide an easy way to visualize the cash flows associated with investment decisions. A time line is a horizontal line that starts at time zero and shows cash flows as they occur over time. The term time zero is used to refer to the beginning of a transaction in time value of money problems. Time zero is often the current point in time (today).

time zero

the beginning of a transaction; often the current point in time

Exhibit 5.1  shows the time line for a five-year investment opportunity and its cash flows. Here, as in most finance problems, cash flows are assumed to occur at the end of the period. The project involves a $10,000 initial investment (cash outflow), such as the purchase of a new machine, that is expected to generate cash inflows over a five-year period: $5,000 at the end of year 1, $4,000 at the end of year 2, $3,000 at the end of year 3, $2,000 at the end of year 4, and $1,000 at the end of year 5. Because of the time value of money, it is critical that you identify not only the size of the cash flows, but also the timing.

If it is appropriate, the time line will also show the relevant interest rate for the problem. In  Exhibit 5.1  this is shown as 5 percent. Also, note in  Exhibit 5.1  that the initial cash flow of $10,000 is represented by a negative number. It is conventional that cash outflows from the firm, such as for the purchase of a new machine, are treated as negative values on a time line and that cash inflows to the firm, such as revenues earned, are treated as positive values. The 2$10,000 therefore means that there is a cash outflow of $10,000 at time zero. As you will see, it makes no difference how you label cash inflows and outflows as long as you are consistent. That is, if all cash outflows are given a negative value, then all cash inflows must have a positive value. If the signs get “mixed up”—if some cash inflows are negative and some positive—you will get the wrong answer to any problem you are trying to solve.

EXHIBIT 5.1 Five-year Time Line for a $10,000 Investment

Time lines help us to correctly identify the size and timing of cash flows—critical tasks in solving time value problems. This time line shows the cash flows generated over five years by a $10,000 investment in a situation where the relevant interest rate is 5 percent.

Financial Calculator

We recommend that students purchase a financial calculator for this course. A financial calculator will provide the computational tools to solve most problems in the book. A financial calculator is just an ordinary calculator that has preprogrammed future value and present value algorithms. Thus, all the variables you need to make financial calculations exist on the calculator keys. To solve problems, all you have to do is press the proper keys. The instructions in this book are generally meant for Texas Instruments calculators, such as the TI BAII Plus. If you are using an HP or Sharp calculator, consult the user's manual for instructions.

It may sound as if the financial calculator will solve problems for you. It won't. To get the correct answer to textbook or real-world problems, you must first analyze the problem correctly and then identify the cash flows (size and timing), placing them correctly on a time line. Only then will you enter the correct inputs into the financial calculator.

A calculator can help you eliminate computation errors and save you a great deal of time. However, it is important that you understand the calculations that the calculator is performing. For this reason we recommend that when you first start using a financial calculator that you solve problems by hand and then use the calculator's financial functions to check your answers.

To help you master your financial calculator, throughout this chapter, we provide helpful hints on how to best use the calculator. We also recognize that some professors or students may want to solve problems using one of the popular spreadsheet programs. In this chapter and a number of other chapters, we provide solutions to several problems that lend themselves to spreadsheet analysis. In solving these problems, we used Microsoft ExcelTM. The analysis and basic commands are similar for other spreadsheet programs. We also provide spreadsheet solutions for additional problems on the book's Web site. Since spreadsheet programs are very commonly used in industry, you should make sure to learn how to use one of these programs early in your studies and become proficient with it before you graduate.

> BEFORE YOU GO ON

1. Why is a dollar today worth more than a dollar one year from now?

2. What is a time line, and why is it important in financial analysis?

5.2 FUTURE VALUE AND COMPOUNDING

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The future value (FV) of an investment is what the investment will be worth after earning interest for one or more time periods. The process of converting the initial amount into future value is called compounding. We will define this term more precisely later. First, though, we illustrate the concepts of future value and compounding with a simple example.

future value (FV)

the value of an investment after it earns interest for one or more periods

Single-Period Investment

Suppose you place $100 in a bank savings account that pays interest at 10 percent a year. How much money will you have in one year? Go ahead and make the calculation. Most people can intuitively arrive at the correct answer, $110, without the aid of a formula. Your calculation could have looked something like this:

This approach computes the amount of interest earned ($100 × 0.10) and then adds it to the initial, or principal, amount ($100). Notice that when we solve the equation, we factor out the $100. Recall from algebra that if you have the equation y = c + (c × x), you can factor out the common term c and get y = c × (1 + x). By doing this in our future value calculation, we arrived at the term (1 + 0.10). This term can be stated more generally as (1 + i), where i is the interest rate. As you will see, this is a pivotal term in all time value of money calculations.

Let's use our intuitive calculation to generate a more general formula. First, we need to define the variables used to calculate the answer. In our example $100 is the principal amount (P0), which is the amount of money deposited (invested) at the beginning of the transaction (time zero); the 10 percent is the simple interest rate (i); and the $110 is the future value (FV1) of the investment after one year. We can write the formula for a single-period investment as follows:

Looking at the formula, we more easily see mathematically what is happening in our intuitive calculation. P0 is the principal amount invested at time zero. If you invest for one period at an interest rate of i, your investment, or principal, will grow by (1 + i) per dollar invested. The term (1 + i) is the future value interest factor—often called simply the future value factor—for a single period, such as one year. To test the equation, we plug in our values:

Good, it works!

Two-Period Investment

We have determined that at the end of one year (one period), your $100 investment has grown to $110. Now let's say you decide to leave this new principal amount (FV1) of $110 in the bank for another year earning 10 percent interest. How much money would you have at the end of the second year (FV2)? To arrive at the value for FV2, we multiply the new principal amount by the future value factor (1 + i). That is, FV2 = FV1 × (1 + i). We then substitute the value of FV1 (the single-period investment value) into the equation and algebraically rearrange terms, which yields FV2 = P0 × (1 + i)2. The mathematical steps to arrive at the equation for FV2 are shown in the following; recall that FV1 = P0 × (1 + i):

The future value of your $110 at the end of the second year (FV2) is as follows:

Another way of thinking of a two-period investment is that it is two single-period investments back-to-back. From that perspective, based on the preceding equations, we can represent the future value of the deposit held in the bank for two years as follows:

Turning to  Exhibit 5.2 , we can see what is happening to your $100 investment over the two years we have already discussed and beyond. The future value of $121 at year 2 consists of three parts. First is the initial principal of $100 (first row of column 2). Second is the $20 ($10 + $10 = $20) of simple interest earned at 10 percent for the first and second years (first and second rows of column 3). Third is the $1 interest earned during the second year (second row of column 4) on the $10 of interest from the first year ($10 × 0.10 = $1.00). This is called interest on interest. The total amount of interest earned is $21 ($10 + $11 = $21), which is shown in column 5 and is called compound interest.

EXHIBIT 5.2 Future Value of $100 at 10 Percent

With compounding, interest earned on an investment is reinvested so that in future periods, interest is earned on interest as well as on the principal amount. Here, interest on interest begins accruing in year 2.

We are now in a position to formally define some important terms already mentioned in our discussion. The principal is the amount of money on which interest is paid. In our example, the principal amount is $100. Simple interest is the amount of interest paid on the original principal amount. With simple interest, the interest earned each period is paid only on the original principal. In our example, the simple interest is $10 per year or $20 for the two years. Interest on interest is the interest earned on the reinvestment of previous interest payments. In our example, the interest on interest is $1. Compounding is the process by which interest earned on an investment is reinvested so that in future periods, interest is earned on the interest previously earned as well as the principal. In other words, with compounding, you are able to earn compound interest, which consists of both simple interest and interest on interest. In our example, the compound interest is $21.

principal

the amount of money on which interest is paid

simple interest

interest earned on the original principal amount only

interest on interest

interest earned on interest that was earned in previous periods

compounding

the process by which interest earned on an investment is reinvested, so in future periods interest is earned on the interest as well as the principal

compound interest

interest earned both on the original principal amount and on interest previously earned

The Future Value Equation

Let's continue our bank example. Suppose you decide to leave your money in the bank for three years. Looking back at equations for a single-period and two-period investment, you can probably guess that the equation for the future value of money invested for three years would be:

With this pattern clearly established, we can see that the general equation to find the future value after any number of periods is as follows:

which is often written as:

where:

Let's test our general equation. Say you leave your $100 invested in the bank savings account at 10 percent interest for five years. How much would you have in the bank at the end of five years? Applying Equation 5.1 yields the following:

Exhibit 5.2  shows how the interest is earned on a year-by-year basis. Notice that the total compound interest earned over the five-year period is $61.05 (column 5) and that it is made up of two parts: (1) $50.00 of simple interest (column 3) and (2) $11.05 of interest on interest (column 4). Thus, the total compound interest can be expressed as follows:

The simple interest earned is $100 × 0.10 = $10.00 per year, and thus, the total simple interest for the five-year period is $50.00 (5 years × $10.00 = $50.00). The remaining balance of $11.05 ($61.05 − $50.00 = $11.05) comes from earning interest on interest.

 CNNMoney's Web site has a savings calculator at  http://cgi.money.cnn.com/tools/savingscalc/savingscalc.html .

A helpful equation for calculating the simple interest can be derived by using the equation for a single-period investment and solving for the term FV1 − P0, which is equal to the simple interest. 2  The equation for the simple interest earned (SI) is:

where:

Thus, the calculation for simple interest is: 3

Exhibit 5.3  shows graphically how the compound interest in  Exhibit 5.2  grows. Notice that the simple interest earned each year remains constant at $10 per year but that the amount of interest on interest increases every year. The reason, of course, is that interest on interest increases with the cumulative interest that has been earned. As more and more interest is earned, the compounding of interest accelerates the growth of the interest on interest and therefore the total interest earned.

EXHIBIT 5.3 How Compound Interest Grows on $100 at 10 Percent

The amount of simple interest earned on $100 invested at 10 percent remains constant at $10 per year, but the amount of interest earned on interest increases each year. As more and more interest builds, the effect of compounding accelerates the growth of the total interest earned.

EXHIBIT 5.4 Future Value of $1 for Different Periods and Interest Rates

The higher the interest rate, the faster the value of an investment will grow, and the larger the amount of money that will accumulate over time. Because of compounding, the growth over time is not linear but exponential—the dollar increase in the future value is greater in each subsequent period.

An interesting observation about Equation 5.1 is that the higher the interest rate, the faster the investment will grow. This fact can be seen in  Exhibit 5.4 , which shows the growth in the future value of $1.00 at different interest rates and for different time periods into the future. First, notice that the growth in the future value over time is not linear, but exponential. The dollar value of the invested funds does not increase by the same dollar amount from year to year. It increases by a greater amount each year. In other words, the growth of the invested funds is accelerated by the compounding of interest. Second, the higher the interest rate, the more money accumulated for any time period. Looking at the right-hand side of the exhibit, you can see the difference in total dollars accumulated if you invest a dollar for 10 years: At 5 percent, you will have $1.63; at 10 percent, you will have $2.59; at 15 percent, you will have $4.05; and at 20 percent, you will have $6.19. Finally, as you should expect, if you invest a dollar at 0 percent for 10 years, you will only have a dollar at the end of the period.

The Future Value Factor

To solve a future value problem, we need to know the future value factor, (1 + i) n . Fortunately, almost any calculator suitable for college-level work has a power key (the yx  key) that we can use to make this computation. For example, to compute (1.08)10, we enter 1.08, press the yx  key and enter 10, and press the = key. The number 2.159 should emerge. Give it a try with your calculator.

Alternatively, we can use future value tables to find the future value factor at different interest rates and maturity periods.  Exhibit 5.5  is an example of a future value table. For example, to find the future value factor (1.08)10, we first go to the row corresponding to 10 years and then move along the row until we reach the 8 percent interest column. The entry is 2.159, which is identical to what we found when we used a calculator. This comes as no surprise, but we sometimes find small differences between calculator solutions and future value tables due to rounding differences. Exhibit A.1 at the end of the book provides a more comprehensive version of  Exhibit 5.5 .

Future value tables (and the corresponding present value tables) are rarely used today, partly because they are tedious to work with. In addition, the tables show values for only a limited number of interest rates and time periods. For example, what if the interest rate on your $100 investment was not a nice round number such as 10 percent but was 10.236 percent? You would not find that number in the future value table. In spite of their shortcomings, these tables were very commonly used in the days before financial calculators and spreadsheet programs were readily available. You can still use them—for example, to check the answers from your computations of future value factors.

EXHIBIT 5.5 Future Value Factors

To find a future value factor, simply locate the row with the appropriate number of periods and the column with the desired interest rate. The future value factor for 10 years at 8 percent is 2.159.

Applying the Future Value Formula

Next, we will review a number of examples of future value problems to illustrate the typical types of problems you will encounter in business and in your personal life.

The Power of Compounding

Our first example illustrates the effects of compounding. Suppose you have an opportunity to make a $5,000 investment that pays 15 percent per year. How much money will you have at the end of 10 years? The time line for the investment opportunity is:

where the $5,000 investment is a cash outflow and the future value you will receive in 10 years is a cash inflow.

 You can find a compound interest calculator at  SmartMoney.com :  http://www.smartmoney.com/compoundcalc .

We can apply Equation 5.1 to find the future value of $5,000 invested for 10 years at 15 percent interest. We want to multiply the original principal amount (PV) times the appropriate future value factor for 10 years at 15 percent, which is (1 + 0.15)10; thus:

Now let's determine how much of the interest is from simple interest and how much is from interest on interest. The total compound interest earned is $15,227.79 ($20,227.79 − $5,000.00 = $15,227.79). The simple interest is the amount of interest paid on the original principal amount: SI = P0 × i = $5,000 × 0.15 = $750 per year, which over 10 years is $750 × 10 = $7,500. The interest on interest must be the difference between the total compound interest earned and the simple interest: $15,227.79 − $7,500 = $7,727.79. Notice how quickly the value of an investment increases and how the reinvestment of interest earned—interest on interest—impacts that total compound interest when the interest rates are high.

APPLICATION 5.1 LEARNING BY DOING

The Power of Compounding

PROBLEM: Your wealthy uncle passed away, and one of the assets he left to you was a savings account that your great-grandfather had set up 100 years ago. The account had a single deposit of $1,000 and paid 10 percent interest per year. How much money have you inherited, what is the total compound interest, and how much of the interest earned came from interest on interest?

APPROACH: We first determine the value of the inheritance, which is the future value of $1,000 retained in a savings account for 100 years at a 10 percent interest rate. Our time line for the problem is:

To calculate FV100, we begin by computing the future value factor. We then plug this number into the future value formula (Equation 5.1) and solve for the total inheritance. Once we have computed FV100, we calculate the total compound interest and the total simple interest and find the difference between these two numbers, which will give us the interest earned on interest.

SOLUTION:

First, we find the future value factor:

Then we find the future value:

Your total inheritance is $13,780,612. The total compound interest earned is this amount less the original $1,000 investment, or $13,779,612:

The total simple interest earned is calculated as follows:

The interest earned on interest is the difference between the total compound interest earned and the simple interest:

That's quite a difference!

As Learning by Doing Application 5.1 indicates, the relative importance of interest earned on interest is especially great for long-term investments. For many people, retirement savings include the longest investments they will make. As you might expect, interest earned on interest has a great impact on how much money people ultimately have for their retirement. For example, consider someone who inherits and invests $10,000 on her 25th birthday and earns 8 percent per year for the next 40 years. This investment will grow to:

BUILDING INTUITION COMPOUNDING DRIVES MUCH OF THE EARNINGS ON LONG-TERM INVESTMENTS

The earnings from compounding drive much of the return earned on a long-term investment. The reason is that the longer the investment period, the greater the proportion of total earnings from interest earned on interest. Interest earned on interest grows exponentially as the investment period increases.

by the investor's 65th birthday. In contrast, if the same individual waited until her 35th birthday to invest the $10,000, she would have only:

when she turned 65.

Of the $116,618.65 difference in these amounts, the difference in simple interest accounts for only $8,000 (10 years × $10,000 × 0.08 = $8,000). The remaining $108,618.65 is attributable to the difference in interest earned on interest. This example illustrates both the importance of compounding for investment returns and the importance on getting started early when saving for retirement. The sooner you start saving, the better off you will be when you retire.

Compounding More Frequently Than Once a Year

Interest can, of course, be compounded more frequently than once a year. In Equation 5.1, the term n represents the number of periods and can describe annual, semiannual, quarterly, monthly, or daily payments. The more frequently interest payments are compounded, the larger the future value of $1 for a given time period. Equation 5.1 can be rewritten to explicitly recognize different compounding periods:

  Moneychimp.com  provides a compound interest calculator at  http://www.moneychimp.com/calculator/compound_interest_calculator.htm .

where m is the number of times per year that interest is compounded and n is the number of periods specified in years.

Let's say you invest $100 in a bank account that pays a 5 percent interest rate semiannually (2.5 percent twice a year) for two years. In other words, the annual rate quoted by the bank is 5 percent, but the bank calculates the interest it pays you based on a six-month rate of 2.5 percent. In this example there are four six-month periods, and the amount of principal and interest you would have at the end of the four periods would be:

It is not necessary to memorize Equation 5.2; using Equation 5.1 will do fine. All you have to do is determine the interest paid per compounding period (i/m) and calculate the total number of compounding periods (m × n) as the exponent for the future value factor. For example, if the bank compounds interest quarterly, then both the interest rate and compounding periods must be expressed in quarterly terms: (i/4) and (4 × n).

If the bank in the above example paid interest annually instead of semiannually, you would have:

at the end of the two-year period. The difference between this amount and the $110.38 above is due to the additional interest earned on interest when the compounding period is shorter and the interest payments are compounded more frequently.

During the late 1960s, the effects of compounding periods became an issue in banking. At that time, the interest rates that banks and thrift institutions could pay on consumer savings accounts were limited by regulation. However, financial institutions discovered they could keep their rates within the legal limit and pay their customers additional interest by increasing the compounding frequency. Prior to this, banks and thrifts had paid interest on savings accounts quarterly. You can see the difference between quarterly and daily compounding in Learning by Doing Application 5.2.

APPLICATION 5.2 LEARNING BY DOING

Changing the Compounding Period

PROBLEM: Your grandmother has $10,000 she wants to put into a bank savings account for five years. The bank she is considering is within walking distance, pays 5 percent annual interest compounded quarterly (5 percent per year/4 quarters per year = 1.25 percent per quarter), and provides free coffee and doughnuts in the morning. Another bank in town pays 5 percent interest compounded daily. Getting to this bank requires a bus trip, but your grandmother can ride free as a senior citizen. More important, though, this bank does not serve coffee and doughnuts. Which bank should your grandmother select?

APPROACH: We need to calculate the difference between the two banks' interest payments. Bank A, which compounds quarterly, will pay one-fourth of the annual interest per quarter, 0.05/4 = 0.0125, and there will be 20 compounding periods over the five-year investment horizon (5 years × 4 quarters per year = 20 quarters). The time line for quarterly compounding is as follows:

Bank B, which compounds daily, has 365 compounding periods per year. Thus, the daily interest rate is 0.000137 (0.05/365 = 0.000137), and there are 1,825 (5 years × 365 days per year = 1,825 days) compounding periods. The time line for daily compounding is:

We use Equation 5.2 to solve for the future values the investment would generate at each bank. We then compare the two.

SOLUTION:

Bank A:

Bank B:

With daily compounding, the additional interest earned by your grandmother is $19.66:

Given that the interest gained by daily compounding is less than $20, your grandmother should probably select her local bank and enjoy the daily coffee and doughnuts. (If she is on a diet, of course, she should take the higher interest payment and walk to the other bank).

It is worth noting that the longer the investment period, the greater the additional interest earned from daily compounding vs. quarterly compounding. For example, if $10,000 was invested for 40 years instead of five years, the additional interest would increase to $900.23. (You should confirm this by doing the calculation.)

Continuous Compounding

We can continue to divide the compounding interval into smaller and smaller time periods, such as minutes and seconds, until, at the extreme, we would compound continuously. In this case, m in Equation 5.2 would approach infinity (∞). The formula to compute the future value for continuous compounding (FV∞) is stated as follows:

where e is the exponential function, which has a known mathematical value of about 2.71828, n is the number of periods specified in years, and i is the annual interest rate. Although the formula may look a little intimidating, it is really quite easy to apply. Look for a key on your calculator labeled ex . If you don't have the key, you still can work the problem.

Let's go back to the example in Learning by Doing Application 5.2, in which your grandmother wants to put $10,000 in a savings account at a bank. How much money would she have at the end of five years if the bank paid 5 percent annual interest compounded continuously? To find out, we enter these values into Equation 5.3:

If your calculator has an exponent key, all you have to do to calculate e 0.25 is enter the number 0.25, then hit the ex  key, and the number 1.284025 should appear (depending on your calculator, you may have to press the equal [=] key for the answer to appear). Then multiply 1.284025 by $10,000, and you're done! If your calculator does not have an exponent key, then you can calculate e 0.25 by inputting the value of e (2.71828) and raising it to the 0.25 power using the yx  key, as described earlier in the chapter.

Let's look at your grandmother's $10,000 bank balance at the end of five years with several different compounding periods: yearly, quarterly, daily, and continuous: 4

Notice that your grandmother's total earnings get larger as the frequency of compounding increases, as shown in column 2, but the earnings increase at a decreasing rate, as shown in column 4. The biggest gain comes when the compounding period goes from an annual interest payment to quarterly interest payments. The gain from daily compounding to continuous compounding is small on a modest savings balance such as your grandmother's. Twenty-two cents over five years will not buy grandmother a cup of coffee, let alone a doughnut. However, for businesses and governments with mega-dollar balances at financial institutions, the difference in compounding periods can be substantial.

EXAMPLE 5.1 DECISION MAKING

Which Bank Offers Depositors the Best Deal?

SITUATION: You have just received a bonus of $10,000 and are looking to deposit the money in a bank account for five years. You investigate the annual deposit rates of several banks and collect the following information:

You understand that the more frequently interest is earned in each year, the more you will have at the end of your five-year investment horizon. To determine which bank you should deposit your money in, you calculate how much money you will have at the end of five years at each bank. You apply Equation 5.2 and come up with the following results. Which bank should you choose?

DECISION: Even though you might expect Bank D's daily compounding to result in the highest value, the calculations reveal that Bank B provides the highest value at the end of five years. Thus, you should deposit your money in Bank B because its higher rate offsets the more frequent compounding at Banks C and D.

Calculator Tips for Future Value Problems

As we have mentioned, all types of future value calculations can be done easily on a financial calculator. Here we discuss how to solve these problems, and we identify some potential problem areas to avoid.

A financial calculator includes the following five basic keys for solving future value and present value problems:

The keys represent the following inputs:

· N is the number of periods. The periods can be years, quarters, months, days, or some other unit of time.

· i is the interest rate per period, expressed as a percentage.

· PV is the present value or the original principal (P0).

· PMT is the amount of any recurring payment.

· FV is the future value.

Given any four of these inputs, the financial calculator will solve for the fifth. Note that the interest rate key i differs with different calculator brands: Texas Instruments uses the I/Y key, Hewlett-Packard an i, %i, or I/Y key, and Sharp the i key.

For future value problems, we need to use only four of the five keys: N for the number of periods, i for the interest rate (or growth rate), PV for the present value (at time zero), and FV for the future value in n periods. The PMT key is not used at this time, but, when doing a problem, always enter a zero for PMT to clear the register. 5

USING EXCEL TIME VALUE OF MONEY

Spreadsheet computer programs are a popular method for setting up and solving finance and accounting problems. Throughout this book, we will show you how to structure and calculate some problems using the Microsoft Excel spreadsheet program. Spreadsheet programs are like your financial calculator but are especially efficient at doing repetitive calculations. For example, once the spreadsheet program is set up, it will allow you to make computations using preprogrammed formulas. Thus, you can simply change any of the input cells, and the preset formula will automatically recalculate the answer based on the new input values. For this reason, we recommend that you use formulas whenever possible.

We begin our spreadsheet applications with time value of money calculations. As with the financial calculator approach, there are five variables used in these calculations, and knowing any four of them will let you calculate the fifth one. Excel has already preset formulas for you to use. These are as follows:

Solving for	Formula
Present Value	= PV (RATE, NPER, PMT, FV)
Future Value	= FV (RATE, NPER, PMT, PV)
Discount Rate	= RATE (NPER, PMT, PV, FV)
Payment	= PMT (RATE, NPER, PV, FV)
Number of Periods	= NPER (RATE, PMT, PV, FV)

To enter a formula, all you have to do is type in the equal sign, the abbreviated name of the variable you want to compute, and an open parenthesis, and Excel will automatically prompt you to enter the rest of the variables. Here is an example of what you would type to compute the future value:

1. =

2. FV

3. (

Here are a few important things to note when entering the formulas: (1) be consistent with signs for cash inflows and outflows; (2) enter the rate of return as a decimal number, not a percentage; and (3) enter the amount of an unknown payment as zero.

To see how a problem is set up and how the calculations are made using a spreadsheet, let's return to Learning by Doing Application 5.2. The spreadsheet for that application is on the left.

To solve a future value problem, enter the known data into your calculator. For example, if you know that the number of periods is five, key in 5 and press the N key. Repeat the process for the remaining known values. Once you have entered all of the values you know, then press the key for the unknown quantity, and you have your answer. Note that with some calculators, including the TI BAII Plus, you get the answer by first pressing the key labeled CPT (compute).

Let's try a problem to see how this works. Suppose we invest $5,000 at 15 percent for 10 years. How much money will we have in 10 years? To solve the problem, we enter data on the keys as displayed in the following calculation and solve for FV. Note that the initial investment of $5,000 is a negative number because it represents a cash outflow. Use the +/− key to make a number negative.

EXHIBIT 5.6 Tips for Using Financial Calculators

Following these tips will help you avoid problems that sometimes arise in solving time value of money problems with a financial calculator.

Use the Correct Compounding Period. Make sure that your calculator is set to compound one payment per period or per year. Because financial calculators are often used to compute monthly payments, some will default to monthly payments unless you indicate otherwise. You will need to consult your calculator's instruction manual because procedures for changing settings vary by manufacturer. Most of the problems you will work in other chapters of the book will compound annually.

Clear the Calculator Before Starting. Be sure you clear the data from the financial register before starting to work a problem because most calculators retain information between calculations. Since the information may be retained even when the calculator is turned off, turning the calculator off and on will not solve this problem. Check your instruction manual for the correct procedure for clearing the financial register of your calculator.

Negative Signs on Cash Outflows. For certain types of calculations, it is critical that you input a negative sign for all cash outflows and a positive sign for all cash inflows. Otherwise, the calculator cannot make the computation, and the answer screen will display some type of error message.

Putting a Negative Sign on a Number. To create a number with a negative sign, enter the number first and then press the “change of sign key.” These keys are typically labeled “CHS” or “+/−”.

Interest Rate as a Percentage. Most financial calculators require that interest rate data be entered in percentage form, not in decimal form. For example, enter 7.125 percent as 7.125 and not 0.07125. Unlike nonfinancial calculators, financial calculators assume that rates are stated as percentages.

Rounding off Numbers. Never round off any numbers until all your calculations are complete. If you round off numbers along the way, you can generate significant rounding errors.

Adjust Decimal Setting. Most calculators are set to display two decimal places. You will find it convenient at times to display four or more decimal places when making financial calculations, especially when working with interest rates or present value factors. Again, consult your instruction manual.

Have Correct BEG or END mode. In finance, most problems that you solve will involve cash payments that occur at the end of each time period, such as with the ordinary annuities discussed in  Chapter 6 . Most calculators normally operate in this mode, which is usually designated as “END” mode. However, for annuities due, which are also discussed in  Chapter 6 , the cash payments occur at the beginning of each period. This setting is designated as the “BEG” mode. Most leases and rent payments fall into this category. When you bought your financial calculator, it was set in the END mode. Financial calculators allow you to switch between the END and BEG modes.

If you did not get the correct answer of $20,227.79, you may need to consult the instruction manual that came with your financial calculator. However, before you do that, you may want to look through  Exhibit 5.6 , which lists the most common problems with using financial calculators. Also, note again that PMT is entered as zero to clear the register.

One advantage of using a financial calculator is that if you have values for any three of the four variables in Equation 5.1, you can solve for the remaining variable at the press of a button. Suppose that you have an opportunity to invest $5,000 in a bank and that the bank will pay you $20,227.79 at the end of 10 years. What interest rate does the bank pay? The time line for our situation is as follows:

We know the values for N (10 years), PV ($5,000), and FV ($20,227.79), so we can enter these values into our financial calculator:

Press the interest rate (i) key, and 15.00 percent appears as the answer. Notice that the cash outflow ($5,000) was entered as a negative value and the cash inflow ($20,227.79) as a positive value. If both values were entered with the same sign, your financial calculator algorithm could not compute the equation, yielding an error message. Go ahead and try it.

> BEFORE YOU GO ON

1. What is compounding, and how does it affect the future value of an investment?

2. What is the difference between simple interest and compound interest?

3. How does changing the compounding period affect the amount of interest earned on an investment?

5.3 PRESENT VALUE AND DISCOUNTING

·

In our discussion of future value, we asked the question “If you put $100 in a bank savings account that pays 10 percent annual interest, how much money would accumulate in one year?” Another type of question that arises frequently in finance concerns present value. This question asks, “What is the value today of a cash flow promised in the future?” We'll illustrate the present value concept with a simple example.

Single-Period Investment

Suppose that a rich uncle gives you a bank certificate of deposit (CD) that matures in one year and pays $110. The CD pays 10 percent interest annually and cannot be redeemed until maturity. Being a student, you need the money and would like to sell the CD. What would be a fair price if you sold the CD today?

From our earlier discussion, we know that if we invest $100 in a bank at 10 percent for one year, it will grow to a future value of $110 = $100 × (1 + 0.10). It seems reasonable to conclude that if a CD has an interest rate of 10 percent and will have a value of $110 a year from now, it is worth $100 today.

More formally, to find the present value of a future cash flow, or its value today, we “reverse” the compounding process and divide the future value ($110) by the future value factor (1 + 0.10). The result is $100 = $110/(1 + 0.10), which is the same answer we derived from our intuitive calculation. If we write the calculations above as a formula, we have a one-period model for calculating the present value of a future cash flow:

The numerical calculation for the present value (PV) from our one-period model follows:

discounting

the process by which the present value of future cash flows is obtained

discount rate

the interest rate used in the discounting process to find the present value of future cash flows

present value (PV)

the current value of future cash flows discounted at the appropriate discount rate

We have noted that while future value calculations involve compounding an amount forward into the future, present value calculations involve the reverse. That is, present value calculations involve determining the current value (or present value) of a future cash flow. The process of calculating the present value is called discounting, and the interest rate i is known as the discount rate. Accordingly, the present value (PV) can be thought of as the discounted value of a future amount. The present value is simply the current value of a future cash flow that has been discounted at the appropriate discount rate.

Just as we have a future value factor, (1 + i), we also have a present value factor, which is more commonly called the discount factor. The discount factor, which is 1/(1 + i), is the reciprocal of the future value factor. This expression may not be obvious in the equation above, but note that we can write that equation in two ways:

These equations amount to the same thing; the discount factor is explicit in the second one.

Multiple-Period Investment

Now suppose your uncle gives you another 10 percent CD, but this CD matures in two years and pays $121 at maturity. Like the other CD, it cannot be redeemed until maturity. From the previous section, we know that if we invest $100 in a bank at 10 percent for two years, it will grow to a future value of $121 = $100 × (1 + 0.10)2. To calculate the present value, or today's price, we divide the future value ($121) by the future value factor (1 + 0.10)2. The result is $100 = $121/(1 + 0.10)2.

If we write the calculations we made as an equation, the result is a two-period model for computing the present value of a future cash flow:

Plugging the data from our example into the equation yields no surprises:

By now, you know the drill. We can extend the equation to a third year, a fourth year, and so on:

The Present Value Equation

Given the pattern shown above, we can see that the general formula for the present value is: 6

where:

Note that Equation 5.4 can be written in slightly different ways, which we will sometimes do in the book. The first form, introduced earlier, separates out the discount factor, 1/(1 + i):

In the second form, DF n  is the discount factor for the nth period: DF n  = 1/(1 + i) n :

Future and Present Value Equations Are the Same

By now, you may have recognized that the present value equation, Equation 5.4, is just a restatement of the future value equation, Equation 5.1. That is, to get the future value (FV n ) of funds invested for n years, we multiply the original investment by (1 + i) n . To find the present value of a future payment (PV), we divide FV n  by (1 + i) n . Stated another way, we can start with the future value equation (Equation 5.1), FV n  = PV × (1 + i) n  and then solve it for PV; the resulting equation is the present value equation (Equation 5.4), PV = FV n /(1 + i) n .

Exhibit 5.7  illustrates the relation between the future value and present value calculations for $100 invested at 10 percent interest. You can see from the exhibit that present value and future value are just two sides of the same coin. The formula used to calculate the present value is really the same as the formula for future value, just rearranged.

Applying the Present Value Formula

Let's work through some examples to see how the present value equation is used. Suppose you are interested in buying a new BMW Sports Coupe a year from now. You estimate that the car will cost $40,000. If your local bank pays 5 percent interest on savings deposits, how much money will you need to save in order to buy the car as planned? The time line for the car purchase problem is as follows:

EXHIBIT 5.7 Comparing Future Value and Present Value Calculations

The future value and present value formulas are one and the same; the present value factor, 1/(1 + i) n , is just the reciprocal of the future value factor, (1 + i) n .

The problem is a direct application of Equation 5.4. What we want to know is how much money you have to put in the bank today to have $40,000 a year from now to buy your BMW. To find out, we compute the present value of $40,000 using a 5 percent discount rate:

If you put $38,095.24 in a bank savings account at 5 percent today, you will have the $40,000 to buy the car in one year.

Since that's a lot of money to come up with, your mother suggests that you leave the money in the bank for two years instead of one year. If you follow her advice, how much money do you need to invest? The time line is as follows:

 SmartMoney's personal finance Web site provides a lot of useful information for day-today finance dealings at  http://www.smartmoney.com/pf/?nav=dropTab .

For a two-year waiting period, assuming the car price will stay the same, the calculation is:

Given the time value of money, the result is exactly what we would expect. The present value of $40,000 two years out is lower than the present value of $40,000 one year out—$36,281.18 compared with $38,095.24. Thus, if you are willing to leave your money in the bank for two years instead of one, you can make a smaller initial investment to reach your goal.

Now suppose your rich neighbor says that if you invest your money with him for one year, he will pay you 15 percent interest. The time line is:

The calculation for the initial investment at this new rate is as follows:

Thus, when the interest rate, or discount rate, is 15 percent, the present value of $40,000 to be received in a year's time is $34,782.61, compared with $38,095.24 at a rate of 5 percent and a time of one year. Holding maturity constant, an increase in the discount rate decreases the present value of the future cash flow. This makes sense because when interest rates are higher, it is more valuable to have dollars in hand today to invest; thus, dollars in the future are worth less.

APPLICATION 5.3 LEARNING BY DOING

European Graduation Fling

PROBLEM: Suppose you plan to take a “graduation vacation” to Europe when you finish college in two years. If your savings account at the bank pays 6 percent, how much money do you need to set aside today to have $8,000 when you leave for Europe?

APPROACH: The money you need today is the present value of the amount you will need for your trip in two years. Thus, the value of FV2 is $8,000. The interest rate is 6 percent. Using these values and the present value equation, we can calculate how much money you need to put in the bank at 6 percent to generate $8,000. The time line is:

SOLUTION:

Thus, if you invest $7,119.97 in your savings account today, at the end of two years you will have exactly $8,000.

The Relations among Time, the Discount Rate, and Present Value

From our discussion so far, we can see that (1) the farther in the future a dollar will be received, the less it is worth today, and (2) the higher the discount rate, the lower the present value of a dollar to be received in the future. Let's look a bit more closely at these relations.

Recall from  Exhibit 5.4  that the future value of a dollar increases with time because of compounding. In contrast, the present value of a dollar becomes smaller the farther into the future that dollar is to be received. The reason is that the present value factor 1/(1 + i) n  is the reciprocal of the future value factor (1 + i) n . Thus, the present value of $1 must become smaller the farther into the future that dollar is to be received. You can see this relation in  EXHIBIT 5.8 , which shows the present value of $1 for various interest rates and time periods. For example, at a 10 percent interest rate, the present value of $1 one year in the future is 90.9 cents ($1/1.10); at two years in the future, 82.6 cents [$1/(1.10)2]; at five years in the future, 62.1 cents [$1/(1.10)5]; and at 30 years in the future, 5.7 cents [$1/(1.10)30]. The relation is consistent with our view of the time value of money. That is, the longer you have to wait for money, the less it is worth today. Exhibit A.2, at the end of the book, provides present value factors for a wider range of years and interest rates.

EXHIBIT 5.8 Present Value Factors

To locate a present value factor, find the row for the number of periods and the column for the proper discount rate. Notice that whereas future value factors grow larger over time and with increasing interest rates, present value factors become smaller. This pattern reflects the fact that the present value factor is the reciprocal of the future value factor.

Exhibit 5.9  shows the present values of $1 for different time periods and discount rates. For example, the present value of $1 discounted at 5 percent for 10 years is 61 cents, at 10 percent it is 39 cents, and at 20 percent, 16 cents. Thus, the higher the discount rate, the lower the present value of $1 for a given time period.  Exhibit 5.9  also shows that, just as with future value, the relation between the present value of $1 and time is not linear but exponential. Finally, it is interesting to note that if interest rates are zero, the present value of $1 is $1; that is, there is no time value of money. In this situation, $1,000 today has the same value as $1,000 a year from now or, for that matter, 10 years from now.

Calculator Tips for Present Value Problems

Calculating the discount factor (present value factor) on a calculator is similar to calculating the future value factor but requires an additional keystroke on most advanced-level calculators. The discount factor, 1/(1 + i) n , is the reciprocal of the future value factor, (1 + i) n . The additional keystroke involves the use of the reciprocal key (1/x) to find the discount factor. For example, to compute 1/(1.08)10, first enter 1.08, press the yx  key and enter 10, then press the equal (=) key. The number on the screen should be 2.159. This is the future value factor. It is a calculation you have made before. Now press the 1/x key, then the equal key, and you have the present value factor, 0.463!

EXHIBIT 5.9 Present Value of $1 for Different Time Periods and Discount Rates

The higher the discount rate, the lower the present value of $1 for a given time period. Just as with future value, the relation between the present value and time is not linear, but exponential.

Calculating present value (PV) on a financial calculator is the same as calculating the future value (FV n ) except that you solve for PV rather than FV n . For example, what is the present value of $1,000 received 10 years from now at a 9 percent discount rate? To find the answer on your financial calculator, enter the following keystrokes:

then solve for the present value (PV), which is −$422.41. Notice that the answer has a negative sign. As we discussed previously, the $1,000 represents an inflow, and the $442.41 represents an outflow.

EXAMPLE 5.2 DECISION MAKING

Picking the Best Lottery Payoff Option

SITUATION: Congratulations! You have won the $1 million lottery grand prize. You have been presented with several payout alternatives, and you have to decide which one to accept. The alternatives are as follows:

· $1 million today

· $1.2 million lump sum in two years

· $1.5 million lump sum in five years

· $2 million lump sum in eight years

You are intrigued by the choice of collecting the prize money today or receiving double the amount of money in the future. Which payout option should you choose?

Your cousin, a stockbroker, advises you that over the long term you should be able to earn 10 percent on an investment portfolio. Based on that rate of return, you make the following calculations:

DECISION: As appealing as the higher amounts may sound, waiting for the big payout is not worthwhile in this case. Applying the present value formula has enabled you to convert future dollars into present, or current, dollars. Now the decision is simple—you can directly compare the present values. Given the above choices, you should take the $1 million today.

Future Value versus Present Value

We can analyze financial decisions using either future value or present value techniques. Although the two techniques approach the decision differently, both yield the same result. Both techniques focus on the valuation of cash flows received over time. In corporate finance, future value problems typically measure the value of cash flows at the end of a project, whereas present value problems measure the value of cash flows at the start of a project (time zero).

Exhibit 5.10  compares the $10,000 investment decision shown in  Exhibit 5.1  in terms of future value and present value. When managers are making a decision about whether to accept a project, they must look at all of the cash flows associated with that project with reference to the same point in time. As  Exhibit 5.10  shows, for most business decisions, that point is either the start (time zero) or the end of the project (in this example, year 5). In  Chapter 6  we will discuss calculation of the future value or the present value of a series of cash flows like that illustrated in  Exhibit 5.10 .

EXHIBIT 5.10 Future Value and Present Value Compared

Compounding converts a present value into its future value, taking into account the time value of money. Discounting is just the reverse—it converts future cash flows into their present value.

> BEFORE YOU GO ON

1. What is the present value and when is it used?

2. What is the discount rate? How does the discount rate differ from the interest rate in the future value equation?

3. What is the relation between the present value factor and the future value factor?

4. Explain why you would expect the discount factor to become smaller the longer the time to payment.

5.4 ADDITIONAL CONCEPTS AND APPLICATIONS

·

In this final section, we discuss several additional issues concerning present and future value, including how to find an unknown discount rate, the time required for an investment to grow by a certain amount, a rule of thumb for estimating the length of time it will take to “double your money,” and how to find the growth rates of various kinds of investments.

Finding the Interest Rate

In finance, some situations require you to determine the interest rate (or discount rate) for a given future cash flow. These situations typically arise when you want to determine the return on an investment. For example, an interesting Wall Street innovation is the zero coupon bond. These bonds are essentially loans that pay no periodic interest. The issuer (the firm that borrows the money) makes a single payment when the bond matures (the loan is due) that includes repayment of the amount borrowed plus all of the interest. Needless to say, the issuer must prepare in advance to have the cash to pay off bondholders.

Suppose a firm is planning to issue $10 million worth of zero coupon bonds with 20 years to maturity. The bonds are issued in denominations of $1,000 and are sold for $90 each. In other words, you buy the bond today for $90, and 20 years from now, the firm pays you $1,000. If you bought one of these bonds, what would be your return on investment?

To find the return, we need to solve Equation 5.1, the future value equation, for i, the interest, or discount, rate. The $90 you pay today is the PV (present value), the $1,000 you get in 20 years is the FV (future value), and 20 years is n (the compounding period). The resulting calculation is as follows:

The rate of return on your investment, compounded annually, is 12.79 percent. Using a financial calculator, we arrive at the following solution:

APPLICATION 5.4 LEARNING BY DOING

Interest Rate on a Loan

PROBLEM: Greg and Joan Hubbard are getting ready to buy their first house. To help make the down payment, Greg's aunt offers to loan them $15,000, which can be repaid in 10 years. If Greg and Joan borrow the money, they will have to repay Greg's aunt the amount of $23,750. What rate of interest would Greg and Joan be paying on the 10-year loan?

APPROACH: In this case, the present value is the value of the loan ($15,000), and the future value is the amount due at the end of 10 years ($23,750). To solve for the rate of interest on the loan, we can use the future value equation, Equation 5.1. Alternatively, we can use a financial calculator to compute the interest rate. The time line for the loan, where the $15,000 is a cash inflow to Greg and Joan and the $23,750 is a cash outflow, is as follows:

SOLUTION:

Using Equation 5.1:

Financial calculator steps:

Finding How Many Periods It Takes an Investment to Grow a Certain Amount

Up to this point we have used variations of Equation 5.1:

to calculate the future value of an investment (FV n ), the present value of an investment (PV), and the interest rate necessary for an initial investment (the present value) to grow to a specific value (the future value) over a certain number of periods (i). Note that Equation 5.1 has a total of four variables. You may have noticed that in all of the previous calculations, we took advantage of the mathematical principal that if we know the values of three of these variables we can calculate the value of the fourth.

The same principal allows us to calculate the number of periods (n) that it takes an investment to grow a certain amount. This is a more complicated calculation than the calculations of the values of the other three variables, but it is an important one for you to be familiar with.

Suppose that you would like to purchase a new cross-country motorcycle to ride on dirt trails near campus. The motorcycle dealer will finance the bike that you are interested in if you make a down payment of $1,175. Right now you only have $1,000. If you can earn 5 percent by investing your money, how long will it take for your $1,000 to grow to $1,175?

To find this length of time, we must solve Equation 5.1, the future value equation, for n.

It will take 3.31 years for your investment to grow to $1,175. If you don't want to wait this long to get your motorcycle, you cannot rely on your investment earnings alone. You will have to put aside some additional money.

Note that because n is an exponent in the future value formula, we have to take the natural logarithm, ln (x), of both sides of the equation in the fourth line of the above series of calculations to calculate the value of n directly. Your financial calculator should have a key that allows you to calculate natural logarithms. Just enter the value in the parentheses and then hit the LN key.

Using a financial calculator, we obtain the same solution.

The Rule of 72

People are fascinated by the possibility of doubling their money. Infomercials on television tout speculative land investments, claiming that “some investors have doubled their money in four years.” Before there were financial calculators, people used rules of thumb to approximate difficult present value calculations. One such rule is the Rule of 72, which was used to determine the amount of time it takes to double the value of an investment. The Rule of 72 says that the time to double your money (TDM) approximately equals 72/i, where i is the rate of return expressed as a percentage. Thus,

Rule of 72

a rule proposing that the time required to double money invested (TDM) approximately equals 72/i, where i is the rate of return expressed as a percentage

Applying the Rule of 72 to our land investment example suggests that if you double your money in four years, your annual rate of return will be 18 percent (i = 72/4 = 18).

Let's check the rule's accuracy by applying the future value formula to the land example. We are assuming that you will double our money in four years, so n = 4. We did not specify a present value or future value amount; however, doubling our money means that we will get back $2 (FV) for every $1 invested (PV). Using Equation 5.1 and solving for the interest rate (i), we find that i = 0.1892, or 18.92 percent. 7

That's not bad for a simple rule of thumb: 18.92 percent versus 18 percent. Within limits, the Rule of 72 provides a quick “back of the envelope” method for determining the amount of time it will take to double an investment for a particular rate of return. The Rule of 72 is a linear approximation of a nonlinear function, and as such, the rule is fairly accurate for interest rates between 5 and 20 percent. Outside these limits, the rule is not very accurate.

Compound Growth Rates

The concept of compounding is not restricted to money. Any number that changes over time, such as the population of a city, changes at some compound growth rate. Compound growth occurs when the initial value of a number increases or decreases each period by the factor (1 + growth rate). As we go through the course, we will discuss many different types of interest rates, such as the discount rate on capital budgeting projects, the yield on a bond, and the internal rate of return on an investment. All of these “interest rates” can be thought of as growth rates (g) that relate future values to present values.

When we refer to the compounding effect, we are really talking about what happens when the value of a number increases or decreases by (1 + growth rate) n . That is, the future value of a number after n periods will equal the initial value times (1 + growth rate) n . Does this sound familiar? If we want, we can rewrite Equation 5.1 in a more general form as a compound growth rate formula, substituting g, the growth rate, for i, the interest rate:

where:

Suppose, for example, that because of an advertising campaign, a firm's sales increased from $20 million in 2009 to more than $35 million in 2012. What has been the average annual growth rate in sales? Here, the future value is $35 million, the present value is $20 million, and n is 3 since we are interested in the annual growth rate over three years. The time line is:

Applying Equation 5.6 and solving for the growth factor (g) yields:

Thus, sales grew nearly 21 percent per year. More precisely, we could say that sales grew at a compound annual growth rate (CAGR) of nearly 21 percent. If we use our financial calculator, we find the same answer:

compound annual growth rate (CAGR)

the average annual growth rate over a specified period of time

Note that we enter $20 million as a negative number even though it is not a cash outflow. This is because one value must be negative when using a financial calculator. It makes no difference which number is negative and which is positive.

APPLICATION 5.5 LEARNING BY DOING

The Growth Rate of the World's Population

PROBLEM: Hannah, an industrial relations major, is writing a term paper and needs an estimate of how fast the world population is growing. In her almanac, she finds that the world's population was an estimated 6.9 billion people in 2010. The United Nations estimates that the population will reach 9 billion people in 2054. Calculate the annual population growth rate implied by these numbers. At that growth rate, what will be the world's population in 2015?

APPROACH: We first find the annual rate of growth through 2054 by applying Equation 5.6 for the 44-year period 2054–2010. For the purpose of this calculation, we can use the estimated population of 6.9 billion people in 2010 as the present value, the estimated future population of 9 million people as the future value, and 44 years as the number of compounding periods (n). We want to solve for g, which is the annual compound growth rate over the 44-year period. We can then plug the 44-year population growth rate in Equation 5.6 and solve for the world's population in 2015 (FV5). Alternatively, we can get the answer by using a financial calculator.

SOLUTION:

Using Equation 5.6, we find the growth rate as follows:

The world's population in 2015 is therefore estimated to be:

Using the financial calculator approach:

APPLICATION 5.6 LEARNING BY DOING

Calculating Projected Earnings

PROBLEM: IBM's net income in 2010 was $14.83 billion. Wall Street analysts expect IBM's earnings to increase by 6 percent per year over the next three years. Using your financial calculator, determine what IBM's earnings should be in three years.

APPROACH: This problem involves the growth rate (g) of IBM's earnings. We already know the value of g, which is 6 percent, and we need to find the future value. Since the general compound growth rate formula, Equation 5.6, is the same as Equation 5.1, the future value formula, we can use the same calculator procedure we used earlier to find the future value. We enter the data on the calculator keys as shown below, using the growth rate value for the interest rate. Then we solve for the future value:

SOLUTION:

Concluding Comments

This chapter has introduced the basic principles of present value and future value. The table at the end of the chapter summarizes the key equations developed in the chapter. The basic equations for future value (Equation 5.1) and present value (Equation 5.4) are two of the most fundamental relations in finance and will be applied throughout the rest of the textbook.

> BEFORE YOU GO ON

1. What is the difference between the interest rate (i) and the growth rate (g) in the future value equation?

SUMMARY OF Learning Objectives

 Explain what the time value of money is and why it is so important in the field of finance.

The idea that money has a time value is one of the most fundamental concepts in the field of finance. The concept is based on the idea that most people prefer to consume goods today rather than wait to have similar goods in the future. Since money buys goods, they would rather have money today than in the future. Thus, a dollar today is worth more than a dollar received in the future. Another way of viewing the time value of money is that your money is worth more today than at some point in the future because, if you had the money now, you could invest it and earn interest. Thus, the time value of money is the opportunity cost of forgoing consumption today.

Applications of the time value of money focus on the trade-off between current dollars and dollars received at some future date. This is an important element in financial decisions because most investment decisions require the comparison of cash invested today with the value of expected future cash inflows. Time value of money calculations facilitate such comparisons by accounting for both the magnitude and timing of cash flows. Investment opportunities are undertaken only when the value of future cash inflows exceeds the cost of the investment (the initial cash outflow).

 Explain the concept of future value, including the meaning of the terms principal, simple interest, and compound interest, and use the future value formula to make business decisions.

The future value is the sum to which an investment will grow after earning interest. The principal is the amount of the investment. Simple interest is the interest paid on the original investment; the amount of simple interest remains constant from period to period. Compound interest includes not only simple interest, but also interest earned on the reinvestment of previously earned interest, the so-called interest earned on interest. For future value calculations, the higher the interest rate, the faster the investment will grow. The application of the future value formula in business decisions is presented in Section 5.2.

 Explain the concept of present value, how it relates to future value, and use the present value formula to make business decisions.

The present value is the value today of a future cash flow. Computing the present value involves discounting future cash flows back to the present at an appropriate discount rate. The process of discounting cash flows adjusts the cash flows for the time value of money. Computationally, the present value factor is the reciprocal of the future value factor, or 1/(1 + i). The calculation and application of the present value formula in business decisions is presented in Section 5.3.

 Discuss why the concept of compounding is not restricted to money, and use the future value formula to calculate growth rates.

Any number of changes that are observed over time in the physical and social sciences follow a compound growth rate pattern. The future value formula can be used in calculating these growth rates.

SUMMARY OF Key Equations

Self-Study Problems

· 5.1 Amit Patel is planning to invest $10,000 in a bank certificate of deposit (CD) for five years. The CD will pay interest of 9 percent. What is the future value of Amit's investment?

· 5.2 Megan Gaumer expects to need $50,000 as a down payment on a house in six years. How much does she need to invest today in an account paying 7.25 percent?

· 5.3 Kelly Martin has $10,000 that she can deposit into a savings account for five years. Bank A pays compounds interest annually, Bank B twice a year, and Bank C quarterly. Each bank has a stated interest rate of 4 percent. What amount would Kelly have at the end of the fifth year if she left all the interest paid on the deposit in each bank?

· 5.4 You have an opportunity to invest $2,500 today and receive $3,000 in three years. What will be the return on your investment?

· 5.5 Emily Smith deposits $1,200 in her bank today. If the bank pays 4 percent simple interest, how much money will she have at the end of five years? What if the bank pays compound interest? How much of the earnings will be interest on interest?

Solutions to Self-Study Problems

· 5.1 Present value of Amit's investment = PV = $10,000

Interest rate = i = 9%

Number of years = n = 5

· 5.2 Amount Megan will need in six years = FV6 = $50,000

Number of years = n = 6

Interest rate = i = 7.25%

Amount needed to be invested now = PV =?

· 5.3 Present value of Kelly's deposit = PV = $10,000

Number of years = n = 5

Interest rate = i = 4%

Compound period (m):

Amount at the end of five years = FV5 =?

· 5.4 Your investment today = PV = $2,500

Amount to be received = FV3 = $3,000

Time of investment = n = 3

Return on the investment = i =?

· 5.5 Emily's deposit today = PV = $1,200

Interest rate = i = 4%

Number of years = n = 5

Amount to be received = FV5 =?

5. Future value with simple interest

Simple interest per year = $1,200 × 0.04 = $48

Simple interest for 5 years = $48 × 5 = $240

FV5 = $1,200 + $240 = $1,440

5. Future value with compound interest

Simple interest = ($1,440 − $1,200) = $240

Interest on interest = $1,459.98 − $1,200 − $240 = $19.98

Critical Thinking Questions

· 5.1 Explain the phrase “a dollar today is worth more than a dollar tomorrow.”

· 5.2 Explain the importance of a time line.

· 5.3 What are the two factors to be considered in time value of money?

· 5.4 Differentiate future value from present value.

· 5.5 Differentiate between compounding and discounting.

· 5.6 Explain how compound interest differs from simple interest.

· 5.7 If you were given a choice of investing in an account that paid quarterly interest and one that paid monthly interest, which one should you choose if they both offer the same stated interest rate and why?

· 5.8 Compound growth rates are exponential over time. Explain.

· 5.9 What is the Rule of 72?

· 5.10 You are planning to take a spring break trip to Cancun your senior year. The trip is exactly two years away, but you want to be prepared and have enough money when the time comes. Explain how you would determine the amount of money you will have to save in order to pay for the trip.

Questions and Problems

· 5.1 Future value: Chuck Tomkovick is planning to invest $25,000 today in a mutual fund that will provide a return of 8 percent each year. What will be the value of the investment in 10 years?

· 5.2 Future value: Ted Rogers is investing $7,500 in a bank CD that pays a 6 percent annual interest. How much will the CD be worth at the end of five years?

· 5.3 Future value: Your aunt is planning to invest in a bank CD that will pay 7.5 percent interest semiannually. If she has $5,000 to invest, how much will she have at the end of four years?

· 5.4 Future value: Kate Eden received a graduation present of $2,000 that she is planning on investing in a mutual fund that earns 8.5 percent each year. How much money can she collect in three years?

· 5.5 Future value: Your bank pays 5 percent interest semiannually on your savings account. You don't expect the current balance of $2,700 to change over the next four years. How much money can you expect to have at the end of this period?

· 5.6 Future value: Your birthday is coming up and instead of other presents, your parents promised to give you $1,000 in cash. Since you have a part-time job and, thus, don't need the cash immediately, you decide to invest the money in a bank CD that pays 5.2 percent quarterly for the next two years. How much money can you expect to earn in this period of time?

· 5.7 Multiple compounding periods: Find the future value of an investment of $100,000 made today for five years and paying 8.75 percent for the following compounding periods:

7. Quarterly.

7. Monthly.

7. Daily.

7. Continuous.

1. 5.8 Growth rates: Joe Mauer, a catcher for the Minnesota Twins, is expected to hit 15 home runs in 2012. If his home-run-hitting ability is expected to grow by 12 percent every year for the following five years, how many home runs is he expected to hit in 2017?

1. 5.9 Present value: Roy Gross is considering an investment that pays 7.6 percent. How much will he have to invest today so that the investment will be worth $25,000 in six years?

1. 5.10 Present value: Maria Addai has been offered a future payment of $750 two years from now. If she can earn 6.5 percent compounded annually on her investment, what should she pay for this investment today?

1. 5.11 Present value: Your brother has asked you for a loan and has promised to pay back $7,750 at the end of three years. If you normally invest to earn 6 percent per year, how much will you be willing to lend to your brother?

1. 5.12 Present value: Tracy Chapman is saving to buy a house in five years. She plans to put 20 percent down at that time, and she believes that she will need $35,000 for the down payment. If Tracy can invest in a fund that pays 9.25 percent annually, how much will she need to invest today?

1. 5.13 Present value: You want to buy some bonds that will have a value of $1,000 at the end of seven years. The bonds pay 4.5 percent interest annually. How much should you pay for them today?

1. 5.14 Present value: Elizabeth Sweeney wants to accumulate $12,000 by the end of 12 years. If the annual interest rate is 7 percent, how much will she have to invest today to achieve her goal?

1. 5.15 Interest rate: You are in desperate need of cash and turn to your uncle, who has offered to lend you some money. You decide to borrow $1,300 and agree to pay back $1,500 in two years. Alternatively, you could borrow from your bank that is charging 6.5 percent interest annually. Should you go with your uncle or the bank?

1. 5.16 Number of periods: You invest $150 in a mutual fund today that pays 9 percent interest annually. How long will it take to double your money?

1. 5.17 Growth rate: Your finance textbook sold 53,250 copies in its first year. The publishing company expects the sales to grow at a rate of 20 percent each year for the next three years and by 10 percent in the fourth year. Calculate the total number of copies that the publisher expects to sell in years 3 and 4. Draw a time line to show the sales level for each of the next four years.

1. 5.18 Growth rate: CelebNav, Inc., had sales last year of $700,000, and the analysts are predicting a good year for the start-up, with sales growing 20 percent a year for the next three years. After that, the sales should grow 11 percent per year for two years, at which time the owners are planning to sell the company. What are the projected sales for the last year before the sale?

1. 5.19 Growth rate: You decide to take advantage of the current online dating craze and start your own Web site. You know that you have 450 people who will sign up immediately and, through a careful marketing research and analysis, determine that membership can grow by 27 percent in the first two years, 22 percent in year 3, and 18 percent in year 4. How many members do you expect to have at the end of four years?

1. 5.20 Multiple compounding periods: Find the future value of an investment of $2,500 made today for the following rates and periods:

20. 6.25 percent compounded semiannually for 12 years.

20. 7.63 percent compounded quarterly for 6 years.

20. 8.9 percent compounded monthly for 10 years.

20. 10 percent compounded daily for 3 years.

20. 8 percent compounded continuously for 2 years.

1. 5.21 Multiple compounding periods: Find the present value of $3,500 under each of the following rates and periods:

21. 8.9 percent compounded monthly for five years.

21. 6.6 percent compounded quarterly for eight years.

21. 4.3 percent compounded daily for four years.

21. 5.7 percent compounded continuously for three years.

1. 5.22 Multiple compounding periods: Samantha plans to invest some money so that she has $5,500 at the end of three years. Which investment should she make given the following choices:

22. 4.2 percent compounded daily.

22. 4.9 percent compounded monthly.

22. 5.2 percent compounded quarterly.

22. 5.4 percent compounded annually.

1. 5.23 Time to grow: Zephyr Sales Company has sales of $1.125 million. If the company expects its sales to grow at 6.5 percent annually, how long will it be before the company can double its sales? Use a financial calculator to solve this problem.

1. 5.24 Time to grow: You are able to deposit $850 in a bank CD today, and you will withdraw the money only once the balance is $1,000. If the bank pays 5 percent interest, how long will it take for the balance to reach $1,000?

1. 5.25 Time to grow: Neon Lights Company is a private company with sales of $1.3 million a year. Management wants to go public but has to wait until the sales reach $2 million. If sales are expected to grow 12 percent annually, when is the earliest that Neon Lights can go public?

1. 5.26 Time to grow: You have just inherited $550,000. You plan to save this money and continue to live off the money that you are earning in your current job. If the $550,000 is everything that you have other than an old car and some beat-up furniture, and you can invest the money in a bond that pays 4.6 percent interest annually, how long will it be before you are a millionaire?

1. 5.27 Growth rates: Xenix Corp had sales of $353,866 in 2011. If management expects its sales to be $476,450 in three years, what is the rate at which the company's sales are expected to grow?

1. 5.28 Growth rate: Infosys Technologies, Inc., an Indian technology company, reported net income of $419 million this year. Analysts expect the company's earnings to be $1.468 billion in five years. What is the expected growth rate in the company's earnings?

1. 5.29 Present value: Caroline Weslin needs to decide whether to accept a bonus of $1,820 today or wait two years and receive $2,100 then. She can invest at 6 percent. What should she do?

1. 5.30 Present value: Congress and the president have decided to increase the Federal tax rate in an effort to reduce the budget deficit. Suppose that Caroline Weslin will pay 35 percent of her bonus to the Federal government for taxes if she accepts the bonus today and 40 percent if she receives her bonus in two years. Will the increase in tax rates affect her decision?

1. 5.31 You have $2,500 you want to invest in your classmate's start-up business. You believe the business idea to be great and hope to get $3,700 back at the end of three years. If all goes according to plan, what will be the return on your investment?

1. 5.32 Patrick Seeley has $2,400 that he is looking to invest. His brother approached him with an investment opportunity that could double his money in four years. What interest rate would the investment have to yield in order for Patrick's brother to deliver on his promise?

1. 5.33 You have $12,000 in cash. You can deposit it today in a mutual fund earning 8.2 percent semiannually, or you can wait, enjoy some of it, and invest $11,000 in your brother's business in two years. Your brother is promising you a return of at least 10 percent on your investment. Whichever alternative you choose, you will need to cash in at the end of 10 years. Assume your brother is trustworthy and both investments carry the same risk. Which one will you choose?

1. 5.34 When you were born your parents set up a bank account in your name with an initial investment of $5,000. You are turning 21 in a few days and will have access to all your funds. The account was earning 7.3 percent for the first seven years, but then the rates went down to 5.5 percent for six years. The economy was doing well in the early 2000s, and your account earned 8.2 percent three years in a row. Unfortunately, the next two years you earned only 4.6 percent. Finally, as the economy recovered, your return jumped to 7.6 percent for the last three years.

34. How much money was in your account before the rates went down drastically (end of year 16)?

34. How much money is in your account now (end of year 21)?

34. What would be the balance now if your parents made another deposit of $1,200 at the end of year 7?

1. 5.35 Sam Bradford, a number 1 draft pick of the St. Louis Rams, and his agent are evaluating three contract options. Each option offers a signing bonus and a series of payments over the life of the contract. Bradford uses a 10.25 percent rate of return to evaluate the contracts. Given the cash flows for each option, which one should he choose?

1. 5.36 Surmec, Inc., reported earnings of $2.1 million last year. The company's primary business is the manufacture of nuts and bolts. Since this is a mature industry, analysts are confident that sales will grow at a steady rate of 7 percent per year. The company's net income equals 23 percent of sales. Management would like to buy a new fleet of trucks but can only do so once the profit reaches $620,000 a year. At the end of what year will they be able to buy the trucks? What will sales and net income be in that year?

1. 5.37 You will be graduating in two years and are thinking about your future. You know that you will want to buy a house five years after you graduate and that you will want to put down $60,000. As of right now, you have $8,000 in your savings account. You are also fairly certain that once you graduate, you can work in the family business and earn $32,000 a year, with a 5 percent raise every year. You plan to live with your parents for the first two years after graduation, which will enable you to minimize your expenses and put away $10,000 each year. The next three years, you will have to live on your own as your younger sister will be graduating from college and has already announced her plan to move back into the family house. Thus, you will be able to save only 13 percent of your annual salary. Assume that you will be able to invest savings from your salary at 7.2 percent. At what interest rate will you need to invest the current savings account balance in order to achieve your goal? Hint: Draw a time line that shows all the cash flows for years 0 through 7. Remember, you want to buy a house seven years from now and your first salary will be in year 3.

Sample Test Problems

· 5.1 Santiago Hernandez is planning to invest $25,000 in a money market account for two years. The account pays interest of 5.75 percent compounded on a monthly basis. How much money will Santiago Hernandez have at the end of two years?

· 5.2 Michael Carter is expecting an inheritance of $1.25 million in four years. If he had the money today, he could earn interest at an annual rate of 7.35 percent. What is the present value of this inheritance?

· 5.3 What is the future value of an investment of $3,000 after three years with compounding at the following rates and frequencies:

3. 8.75 percent compounded monthly.

3. 8.625 percent compounded daily.

3. 8.5 percent compounded continuously.

1. 5.4 Twenty-five years ago, Amanda Cortez invested $10,000 in an account paying an annual interest rate of 5.75 percent. What is the value of the investment today? What is the interest on interest earned on this investment?

1. 5.5 You bought a corporate bond for $863.75 today. In five years the bond will mature and you will receive $1,000. What is the rate of return on this bond?

1  Mega Millions is operated by a consortium of the state lottery commissions in 41 states plus the District of Columbia. To play the game, a player pays one dollar and picks five numbers from 1 to 56 and one additional number from 1 to 46 (the Mega Ball number). Twice a week a machine mixes numbered balls and randomly selects six balls (five white balls and one Mega Ball), which determines the winning combination for that drawing. There are various winning combinations, but a ticket that matches all six numbers, including the Mega Ball number, is the jackpot winner.

2  The formula for a single-period investment is FV1 = P0 + (P0 × i). Solving the equation for FV1 − P0 yields the simple interest, SI.

3  Another helpful equation is the one which computes the total simple interest over several periods (TSI): TSI = Number of periods × SI = Number of periods × (P0 × i).

4  The future value calculation for annual compounding is: FVyearly = $10,000 × (1.05)5 = $12,762.82.

5  The PMT key is used for annuity calculations, which we will discuss in  Chapter 6 .

6  Equation 5.4 can also be written as PV = FV n  × (1+i)−n.

7  Solve Equation 5.1 for i: FV n  = PV × (1 + i) n , where FV n  = $2, PV = $1, and n = 4.

6

Discounted Cash Flows and Valuation

Leon Neal/AFP/Getty Images/NewsCom

Learning Objectives

 Explain why cash flows occurring at different times must be adjusted to reflect their value as of a common date before they can be compared, and compute the present value and future value for multiple cash flows.

 Describe how to calculate the present value and the future value of an ordinary annuity and how an ordinary annuity differs from an annuity due.

 Explain what a perpetuity is and where we see them in business, and calculate the value of a perpetuity.

 Discuss growing annuities and perpetuities, as well as their application in business, and calculate their values.

 Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest rates, and calculate the EAR.

On January 18, 2010, the Board of Directors at Cadbury PLC, the second-largest confectionary company in the world, recommended to its stockholders that they accept a takeover offer from Kraft Foods. The announcement ended a takeover contest that had begun four months earlier and that had taken on many of the characteristics of the hostile takeover contests from the 1980s. By April 2010, Cadbury PLC was no longer an independent company.

Cadbury, founded in 1824 in Birmingham, England, was widely viewed by the British public as a national treasure. The offer from Kraft, an American company, met with widespread opposition from the British public, labor unions, and politicians, as well as the Cad-bury board. It also fueled speculation that Hershey Foods, Nestlé, or both would make a competing friendly offer. In fact, Hershey hired an investment banker and held private talks with Cadbury about a possible deal.

In the end, however, Kraft prevailed by offering a price that neither Hershey nor Nestlé was willing to match. Over the four-month period, Kraft raised its offer from $16.2 billion to $18.9 billion. The final offer represented a 49.6 percent premium over the price at which Cadbury's stock had been trading before the contest began and attracted so much support from key stockholders that the Cadbury board had no choice but to back down from its opposition to the deal. The combination of Kraft and Cadbury brought together well-known Kraft brands such as Oreo cookies, Toblerone chocolates, and Ritz crackers with Cadbury brands such as Trident gum and Dairy Milk chocolates.

In the excitement of such a takeover contest, it is important not to lose sight of the central question: What is the firm really worth? A company invests in an asset—a business or a capital project—because it expects the asset to be worth more than it costs. That's how value is created. The value of a business is the sum of its discounted future cash flows. Thus, the task for Kraft was to estimate the value of the cash flows that Cadbury would generate under its ownership. Whether the $18.9 billion price tag is justified remains to be seen. This chapter, which discusses the discounting of future cash flows, provides tools that help answer the question of what Cadbury is worth to Kraft.

CHAPTER PREVIEW

In  Chapter 5  we introduced the concept of the time value of money: Dollars today are more valuable than dollars to be received in the future. Starting with that concept, we developed the basics of simple interest, compound interest, and future value calculations. We then went on to discuss present value and discounted cash flow analysis. This was all done in the context of a single cash flow.

In this chapter, we consider the value of multiple cash flows. Most business decisions, after all, involve cash flows over time. For example, if Hatteras Hammocks®, a North Carolina-based firm that manufactures hammocks, swings, and rockers, wants to consider building a new factory, the decision will require an analysis of the project's expected cash flows over a number of periods. Initially, there will be large cash outlays to build and get the new factory operational. Thereafter, the project should produce cash inflows for many years. Because the cash flows occur over time, the analysis must consider the time value of money, discounting each of the cash flows by using the present value formula we discussed in  Chapter 5 .

We begin the chapter by describing calculations of future and present values for multiple cash flows. We then examine some situations in which future cash flows are level over time: These involve annuities, in which the cash flow stream goes on for a finite period, and perpetuities, in which the stream goes on forever. Next, we examine annuities and perpetuities in which the cash flows grow at a constant rate over time. These cash flows resemble common cash flow patterns encountered in business. Finally, we describe the effective annual interest rate and compare it with the annual percentage rate (APR), which is a rate that is used to describe the interest rate in consumer loans.

6.1 MULTIPLE CASH FLOWS

·

We begin our discussion of the value of multiple cash flows by calculating the future value and then the present value of multiple cash flows. These calculations, as you will see, are nothing more than applications of the techniques you learned in  Chapter 5 .

Future Value of Multiple Cash Flows

 In  Chapter 5 , we worked through several examples that involved the future value of a lump sum of money invested in a savings account that paid 10 percent interest per year. But suppose you are investing more than one lump sum. Let's say you put $1,000 in your bank savings account today and another $1,000 a year from now. If the bank continues to pay 10 percent interest per year, how much money will you have at the end of two years?

To solve this future value problem, we can use Equation 5.1: FV n  PV (1 i) n . First, however, we construct a time line so that we can see the magnitude and timing of the cash flows. As  Exhibit 6.1  shows, there are two cash flows into the savings plan. The first cash flow is invested for two years and compounds to a value that is computed as follows:

EXHIBIT 6.1 Future Value of Two Cash Flows

This exhibit shows a time line for two cash flows invested in a savings account that pays 10 percent interest annually. The total amount in the savings account after two years is $2,310, which is the sum of the future values of the two cash flows.

The second cash flow earns simple interest for a single period only and grows to:

As  Exhibit 6.1  shows, the total amount of money in the savings account after two years is the sum of these two amounts, which is $2,310 ($1,100 $1,210 $2,310).

Now suppose that you expand your investment horizon to three years and invest $1,000 today, $1,000 a year from now, and $1,000 at the end of two years. How much money will you have at the end of three years? First, we draw a time line to be sure that we have correctly identified the time period for each cash flow. This is shown in  Exhibit 6.2 . Then we compute the future value of each of the individual cash flows using Equation 5.1. Finally, we add up the future values. The total future value is $3,641. The calculations are as follows:

To summarize, solving future value problems with multiple cash flows involves a simple process. First, draw a time line to make sure that each cash flow is placed in the correct time period. Second, calculate the future value of each cash flow for its time period. Third, add up the future values.

Let's use this process to solve a practical problem. Suppose you want to buy a condominium in three years and estimate that you will need $20,000 for a down payment. If the interest rate you can earn at the bank is 8 percent and you can save $3,000 now, $4,000 at the end of the first year, and $5,000 at the end of the second year, how much money will you have to come up with at the end of the third year to have a $20,000 down payment?

EXHIBIT 6.2 Future Value of Three Cash Flows

The exhibit shows a time line for an investment program with a three-year horizon. The value of the investment at the end of three years is $3,641, the sum of the future values of the three separate cash flows.

The time line for the future value calculation in this problem looks like this:

To solve the problem, we need to calculate the future value for each of the cash flows, add up these values, and find the difference between this amount and the $20,000 needed for the down payment. Using Equation 5.1, we find that the future values of the cash flows at the end of the third year are:

At the end of the third year, you will have $13,844.74, so you will need an additional $6,155.26 ($20,000 − $13,844.74 = $6,155.26) at that time to make the down payment.

APPLICATION 6.1 LEARNING BY DOING

Government Contract to Rebuild a Bridge

PROBLEM: The firm you work for is considering bidding on a government contract to rebuild an old bridge that has reached the end of its useful life. The two-year contract will pay the firm $11,000 at the end of the second year. The firm's estimator believes that the project will require an initial expenditure of $7,000 for equipment. The expenses for years 1 and 2 are estimated at $1,500 per year. Because the cash inflow of $11,000 at the end of the contract exceeds the total cash outflows of $10,000 ($7,000 $1,500 $1,500 $10,000), the estimator believes that the firm should accept the job. Drawing on your knowledge of finance from college, you point out that the estimator's decision process ignores the time value of money. Not fully understanding what you mean, the estimator asks you how the time value of money should be incorporated into the decision process. Assume that the appropriate interest rate is 8 percent.

APPROACH: First, construct the time line for the costs in this problem, as shown here:

Second, use Equation 5.1 to convert all of the cash outflows into year-two dollars. This will make all the cash flows comparable. Finally, compare the sum of the cash outflows, stated in year-two dollars, to the $11,000 that you would receive under the contract in year two.

SOLUTION:

Once the future value calculations have been made, the decision is self-evident. With all the dollars stated as year-two dollars, the cash inflow (benefits) is $11,000 and the cash outflow (costs) is $11,285. Thus, the costs exceed the benefits, and the firm's management should reject the contract. If management accepts the contract, the value of the firm will be decreased by $285 ($11,000 − $11,285 = −$285).

Calculator Tip: Calculating the Future Value of Multiple Cash Flows

To calculate the future value of multiple cash flows with a financial calculator, we can use exactly the same process we used in  Chapter 5 . We simply calculate the future value of each of the individual cash flows, write down each computed future value, and add them up.

Alternatively, we can generally use a shortcut. More than likely, your financial calculator has a memory where you can store numbers; refer to your calculator's instruction manual for the keys to use. For the preceding example, you would use your financial calculator's memory (M) as follows: Calculate the future value of the first number, then store the value in the memory (M1); compute the second value, and store it in the memory (M2); compute the third value, and store it in the memory (M3). Finally, retrieve the three numbers from the memory and add them up (M1 + M2 + M3). The advantage of using the calculator's memory is that you eliminate two potential sources of error: (1) writing down a number incorrectly and (2) making a mistake when adding up the numbers.

Present Value of Multiple Cash Flows

In business situations, we often need to compute the present value of a series of future cash flows. We do this, for example, to determine the market price of a bond, to decide whether to purchase a new machine, or to determine the value of a business. Solving present value problems involving multiple cash flows is similar to solving future value problems involving multiple cash flows. First, we prepare a time line so that we can see the magnitude and timing of the cash flows. Second, we calculate the present value of each individual cash flow using Equation 5.4: PV FV n /(1 i) n . Finally, we add up the present values. The sum of the present values of a stream of future cash flows is their current market price, or value. There is nothing new here!

 You can find plenty of future value and present value problems to work out at StudyFinance. com. Go to:  http://www.studyfinance.com/lectures/timevalue/index.mv .

Using the Present Value Equation

Next, we will work through some examples to see how we can use Equation 5.4 to find the present value of multiple cash flows. Suppose that your best friend needs cash and offers to pay you $1,000 at the end of each of the next three years if you will give him $3,000 cash today. You realize, of course, that because of the time value of money, the cash flows he has promised to pay are worth less than $3,000. If the interest rate on similar loans is 7 percent, how much should you pay for the cash flows your friend is offering?

To solve the problem, we first construct a time line, as shown in  Exhibit 6.3 .

EXHIBIT 6.3 Present Value of Three Cash Flows

The exhibit shows the time line for a three-year loan with a payment of $1,000 at the end of each year and an annual interest rate of 7 percent. To calculate the value of the loan today, we compute the present value of each of the three cash flows and then add them up. The present value of the loan is $2,624.32.

Then, using Equation 5.4, we calculate the present value for each of the three cash flows, as follows:

If you view this transaction from a purely business perspective, you should not give your friend more than $2,624.32, which is the sum of the individual discounted cash flows.

Now let's consider another example. Suppose you have the opportunity to buy a small business while you are in school. The business involves selling sandwiches, soft drinks, and snack foods to students from a truck that you drive around campus. The annual cash flows from the business have been predictable. You believe you can expand the business, and you estimate that cash flows will be as follows: $2,000 in the first year, $3,000 in the second and third years, and $4,000 in the fourth year. At the end of the fourth year, the business will be closed down because the truck and other equipment will need to be replaced. The total of the estimated cash flows is $12,000. You did some research and found that a 10 percent discount rate would be appropriate. How much should you pay for the business?

To value the business, we compute the present value of the expected cash flows, discounted at 10 percent. The time line for the investment is:

We compute the present value of each cash flow and then add them up:

This tells us that the present value of the expected cash flows is $9,283.51. If you pay $9,283.51 for the business, you will earn a return of exactly 10 percent. Of course, you should buy the business for the lowest price possible, but you should never pay more than the $9,283.51 value today of the expected cash flows. If you do, you will be paying more than the investment is worth.

Calculator Tip: Calculating the Present Value of Multiple Cash Flows

To calculate the present value of future cash flows with a financial calculator, we use exactly the same process we used in finding the future value, except that we solve for the present value instead of the future value. We can compute the present values of the individual cash flows, save them in the calculator's memory, and then add them up to obtain the total present value.

You should note that from this point forward we will use a different notation. Up to this point, we have used the notation FV n  to represent a cash flow in period n. We have done this to stress that, for n > 0, we were referring to a future value. From this point on, we will use the notation CF n , instead of FV n , because the CF n  notation is more commonly used by financial analysts. When you work through Learning by Doing Application 6.2, you will see the new notation.

APPLICATION 6.2 LEARNING BY DOING

The Value of a Gift to the University

PROBLEM: Suppose that you made a gift to your university, pledging $1,000 per year for four years and $3,000 for the fifth year, for a total of $7,000. After making the first three payments, you decide to pay off the final two payments of your pledge because your financial situation has improved. How much should you pay to the university if the interest rate is 6 percent?

APPROACH: The key to understanding this problem is recognizing the need for a present value calculation. Because your pledge to the university is for future cash payments, the value of the amount you will pay for the remaining two years is worth less than the $4,000 ($1,000 $3,000 $4,000) you promised. If the appropriate discount rate is 6 percent, the time line for the cash payments for the remaining two years of the pledge is as follows:

We now need only calculate the present value of the last two payments.

SOLUTION: The present value calculation for the last two payments is:

A payment of $3,613.39 to the university today (the end of year 3) is a fair payment because at a 6 percent interest rate, it has precisely the same value as paying the university $1,000 at the end of year 4 and $3,000 at the end of year 5. In other words, if you pay the university $3,613.39 and the university invests that amount at 6 percent in a bank, it will be able to withdraw $1,000 in one year and $3,000 in two years.

APPLICATION 6.3 LEARNING BY DOING

Buying a Used Car—Help!

PROBLEM: For a student—or anyone else—buying a used car can be a harrowing experience. Once you find the car you want, the next difficult decision is choosing how to pay for it—cash or a loan. Suppose the cash price you have negotiated for the car is $5,600, but that amount will stretch your budget for the year. The dealer says, “No problem. The car is yours for $4,000 down and payments of $1,000 per year for the next two years. Or you can put $2,000 down and pay $2,000 per year for two years. The choice is yours.” Which offer is the best deal? The interest rate you can earn on your money is 8 percent.

APPROACH: In this problem, there are three alternative streams of cash flows. We need to convert all of the cash flows (CF n ) into today's dollars (present value) and select the alternative with the lowest present value or price. The time line for the three alternatives, along with the cash flows for each, is as follows. (The cash flows at time zero represent the cash price of the car in the case of alternative A and the down payment in the cases of alternatives B and C.)

Now we can use Equation 5.4 to find the present value of each alternative.

SOLUTION:

Alternative A:

Alternative B:

Alternative C:

Once we have converted the three cash flow streams to present values, the answer is clear. Alternative C has the lowest cost, in present value terms, and is the alternative you should choose.

EXAMPLE 6.1 DECISION MAKING

The Investment Decision

SITUATION: You are thinking of buying a business, and your investment adviser presents you with two possibilities. Both businesses are priced at $60,000, and you have only $60,000 to invest. She has provided you with the following annual and total cash flows for each business, along with the present value of the cash flows discounted at 10 percent:

Which business should you acquire?

DECISION: At first glance, business B may look to be the best choice because its un-discounted cash flows for the three years total $110,000, versus $100,000 for A. However, to make the decision on the basis of the undiscounted cash flows ignores the time value of money. By discounting the cash flows, we convert them to current dollars, or their present values. The present value of business A is $85,270 and that of B is $83,810. While both of these investment opportunities are attractive, you should acquire business A if you only have $60,000 to invest. Business A is expected to produce more valuable cash flows for your investment.

> BEFORE YOU GO ON

1. Explain how to calculate the future value of a stream of cash flows.

2. Explain how to calculate the present value of a stream of cash flows.

3. Why is it important to adjust all cash flows to a common date?

6.2 LEVEL CASH FLOWS: ANNUITIES AND PERPETUITIES

·

In finance we commonly encounter contracts that call for the payment of equal amounts of cash over several time periods. For example, most business term loans and insurance policies require the holder to make a series of equal payments, usually monthly. Similarly, nearly all consumer loans, such as auto, personal, and home mortgage loans, call for equal monthly payments. Any financial contract that calls for equally spaced and level cash flows over a finite number of periods is called an annuity. If the cash flow payments continue forever, the contract is called a perpetuity. Most annuities are structured so that cash payments are received at the end of each period. Because this is the most common structure, these annuities are often called ordinary annuities.

Present Value of an Annuity

We frequently need to find the present value of an annuity (PVA). Suppose, for example, that a financial contract pays $2,000 at the end of each year for three years and the appropriate discount rate is 8 percent. The time line for the contract is:

annuity

a series of equally spaced and level cash flows extending over a finite number of periods

perpetuity

a series of level cash flows that continue forever

ordinary annuity

an annuity in which payments are made at the ends of the periods

present value of an annuity (PVA)

the present value of the cash flows from an annuity, discounted at the appropriate discount rate

What is the most we should pay for this annuity? We have worked problems like this one before. All we need to do is calculate the present value of each individual cash flow (CF n ) and add them up. Using Equation 5.4, we find that the present value of the three year annuity (PVA3) at 8 percent interest is:

This approach to computing the present value of an annuity works as long as the number of cash flows is relatively small. In many situations that involve annuities, however, the number of cash flows is large, and doing the calculations by hand would be tedious. For example, a typical 30-year home mortgage has 360 monthly payments (12 months per year × 30 years = 360 months).

Fortunately, our problem can be simplified because the cash flows (CF) for an annuity are all the same (CF1 = CF2... = CF n  = CF). Thus, the present value of an annuity (PVA n ) with n equal cash flows (CF) at interest rate i is the sum of the individual present value calculations:

With some mathematical manipulations that are beyond the scope of this discussion, we can simplify this equation to yield a useful formula for the present value of an annuity:

where:

Notice in Equation 6.1 that 1/(1 i) n  is a term you have already encountered: It is the present value factor. Thus, we can also write Equation 6.1 as follows:

where the term on the right is what we call the PV annuity factor:

It follows that yet another way to state Equation 6.1 is:

 Visit New York Life Insurance Company's Web site to learn more about investment products that pay out annuities:  http://www.newyorklife.com .

Let's apply Equation 6.1 to the example involving a three-year annuity with a $2,000 annual cash flow. To solve for PVA n , we first compute the PV annuity factor for three years at 8 percent. The calculation is made in two steps:

1. Calculate the present value factor for three years at 8 percent:

 Investopedia is a great Web site for a variety of finance topics. For example, you can find a discussion of annuities at  http://www.investopedia.com/articles/03/101503.asp .

2. Use the present value factor to calculate the PV annuity factor:

We now can calculate PVA3 by plugging our values into the equation:

This is almost the same as the $5,154.19 we calculated by hand earlier. The difference is due to rounding.

Annuity Tables: Present Value Factors

Instead of calculating the PV annuity factor by hand, we can use tables that list selected annuity factors.  Exhibit 6.4  contains some entries from such a table, and a more complete set of tables can be found in Appendix A at the end of this book. The annuity table shows the present value of a stream of cash flows that equals $1 a year for n years at different interest rates. Looking at the exhibit, we find that the value for a three-year annuity factor at 8 percent is 2.577, which agrees with our previous calculations.

EXHIBIT 6.4 Present Value Annuity Factors

The table of present value annuity factors shows the present value of $1 to be received each year for different numbers of years and for different interest rates. To locate the desired PV annuity factor, find the row for the appropriate number of years and the column for the appropriate interest rate.

Calculator Tip: Finding the Present Value of an Annuity

There are four variables in a present value of an annuity equation (PVA n , CF, n, and i), and if you know three of them, you can solve for the fourth in a few seconds with a financial calculator. The calculator key that you have not used so far is the PMT (payment) key, which is the key for level cash flows over the life of an annuity.

To illustrate problem solving with a financial calculator, we will revisit the financial contract that paid $2,000 per year for three years, discounted at 8 percent. To find the present value of the contract, we enter 8 percent for the interest rate (i), $2,000 for the payment (PMT), and 3 for the number of periods (N). The key for FV is not relevant for this calculation, so we enter zero into this register to clear it. The key entries and the answer are as follows:

The price of the contract is $5,154.19, which agrees with our other calculations. As discussed in  Chapter 5 , the negative sign on the financial calculator box indicates that $5,154.19 is a cash outflow. 1

APPLICATION 6.4 LEARNING BY DOING

Computing a PV Annuity Factor

PROBLEM: Compute the PV annuity factor for 30 years at a 10 percent interest rate.

APPROACH: First, we calculate the present value factor at 10 percent for 30 years. Then, using this value, we calculate the PV annuity factor.

SOLUTION:

Using this value, we calculate the PV annuity factor to be:

The answer of 9.427 matches the number in  Exhibit 6.4  for a 30 year annuity with a 10 percent interest rate.

We worked through the tedious calculations to show where the numbers come from and how the calculations are made. Financial analysts typically use financial calculators or spreadsheet programs for these calculations. You might check the answer to this problem using your calculator.

Finding Monthly or Yearly Payments

A very common problem in finance is determining the payment schedule for a loan on a consumer asset, such as a car or a home that was purchased on credit. Nearly all consumer loans call for equal monthly payments. Suppose, for example, that you have just purchased a $450,000 condominium in Miami's South Beach district. You were able to put $50,000 down and obtain a 30-year fixed rate mortgage at 6.125 percent for the balance. What are your monthly payments?

In this problem we know the present value of the annuity. It is $400,000, the price of the condominium less the down payment ($450,000 − $50,000 = $400,000). We also know the number of payments; since the payments will be made monthly for 30 years, you will make 360 payments (12 months per year − 30 years = 360 months). Because the payments are monthly, both the interest rate and maturity must be expressed in monthly terms. For consumer loans, to get the monthly interest rate, we divide the annual interest rate by 12. Thus, the monthly interest rate equals 0.51042 percent (6.125 percent per year/12 months per year = 0.51042 percent per month). What we need to calculate is the monthly cash payment (CF) over the loan period. The time line looks like the following:

To find CF (remember that CF1 = CF2 = ... = CF360 = CF), we use Equation 6.1. We need to make two preliminary calculations:

1. First, we calculate the present value factor for 360 months at 0.51042 percent per month

2. Next, we solve for the PV annuity factor:

We can now plug all the data into Equation 6.1 and solve it for CF:

Your mortgage payments will be about $2,430.45 per month.

To solve the problem on a financial calculator takes only a few seconds once the time line is prepared. The most common error students make when using financial calculators is failing to convert all contract variables to be consistent with the compounding period. Thus, if the contract calls for monthly payments, the interest rate and contract duration must be stated in monthly terms.

Having converted our data to monthly terms, we enter into the calculator: N 360 (30 years × 12 months per year = 360 months) months, i = 0.51042 (6.125 percent per year/12 months per year 0.51042 percent per month), PV $400,000, and FV = 0 (to clear the register). Then, pressing the payment button (PMT), we find the answer, which is −$2,430.44. The keystrokes are:

Notice that the hand and financial calculator answers differ by only 1 cent ($2,430.45 − $2,430.44 = $0.01). The answers are so close because when doing the hand calculation, we carried six to eight decimal places through the entire set of calculations. Had we rounded off each number as the calculations were made, the difference between the answers from the two calculation methods would have been about $2.00. The more numbers that are rounded during the calculations, the greater the possible rounding error.

APPLICATION 6.5 LEARNING BY DOING

What Are Your Monthly Car Payments?

PROBLEM: You have decided to buy a new car, and the dealer's best price is $16,000. The dealer agrees to provide financing with a five-year auto loan at 3 percent interest. Using a financial calculator, calculate your monthly payments.

APPROACH: All the problem data must be converted to monthly terms. The number of periods is 60 months (5 years × 12 months per year = 60 months), and the monthly interest charge is 0.25 percent (3 percent per year/12 months per year − 0.25 percent per month). The time line for the car purchase is as follows:

Having converted our data to monthly terms, we enter the following values into the calculator: N 60 months, i = 0.25, PV = $16,000, and FV = 0 (to clear the register). Pressing the payment key (PMT) will give us the answer.

SOLUTION:

Note that since we entered $16,000 as a positive number (because it is a cash inflow to you), the monthly payment of $287.50 is a negative number.

amortizing loan

a loan for which each loan payment contains repayment of some principal and a payment of interest that is based on the remaining principal to be repaid

amortization schedule

a table that shows the loan balance at the beginning and end of each period, the payment made during that period, and how much of that payment represents interest and how much represents repayment of principal

Preparing a Loan Amortization Schedule

Once you understand how to calculate a monthly or yearly loan payment, you have all of the tools that you need to prepare a loan amortization schedule. The term amortization describes the way in which the principal (the amount borrowed) is repaid over the life of a loan. With an amortizing loan, some portion of each month's loan payment goes to paying down the principal. When the final loan payment is made, the unpaid principal is reduced to zero and the loan is paid off. The other portion of each loan payment is interest, which is payment for the use of outstanding principal (the amount of money still owed). Thus, with an amortizing loan, each loan payment contains some repayment of principal and an interest payment. Nearly all loans to consumers are amortizing loans.

A loan amortization schedule is just a table that shows the loan balance at the beginning and end of each period, the payment made during that period, and how much of that payment represents interest and how much represents repayment of principal. To see how an amortization schedule is prepared, consider an example. Suppose that you have just borrowed $10,000 at a 5 percent interest rate from a bank to purchase a car. Typically, you would make monthly payments on such a loan. For simplicity, however, we will assume that the bank allows you to make annual payments and that the loan will be repaid over five years.  Exhibit 6.5  shows the amortization schedule for this loan.

To prepare a loan amortization schedule, we must first compute the loan payment. Since, for consumer loans, the amount of the loan payment is fixed, all the payments are identical in amount. Applying Equation 6.1 and noting from  Exhibit 6.4  that the PV annuity factor for five years at 5 percent is 4.329, we calculate as follows:

EXHIBIT 6.5 Amortization Table for a Five-Year, $10,000 Loan at 5 Percent Interest

A loan amortization table shows how regular payments of principal and interest are applied to repay a loan. The exhibit is an amortization table for a five-year, $10,000 loan with an interest rate of 5 percent and annual payments of $2,309.75. Notice that the interest paid declines with each payment, while the principal paid increases. These relations are illustrated in the pullout graphic in the exhibit.

aThe total annual payment is calculated using the formula for the present value of an annuity, Equation 6.1. The total annual payment is CF in PVA n  = CF × PV annuity factor.

bInterest paid equals the beginning balance times the interest rate.

Alternatively, we enter the values N = 5 years, i = 5 percent, and PV = $10,000 in a financial calculator and then press the PMT key to solve for the loan payment amount. The answer is − $2,309.75 per year. The difference between the two answers results from rounding. For the amortization table calculation, we will use the more precise answer from the financial calculator.

Turning to  Exhibit 6.5 , we can work through the amortization schedule to see how the table is prepared. For the first year, the values are determined as follows:

1. The amount borrowed, or the beginning principal balance (P0), is $10,000.

2. The annual loan payment, as calculated earlier, is $2,309.75.

3. The interest payment for the first year is $500 and is calculated as follows:

4. The principal paid for the year is $1,809.75, calculated as follows:

5. The ending principal balance is $8,190.25, computed as follows:

Note that the ending principal balance for the first year ($8,190.25) becomes the beginning principal balance for the second year ($8,190.25), which in turn is used in calculating the interest payment for the second year:

This calculation makes sense because each loan payment includes some principal repayment. This is why the interest in column 3 declines each year. We repeat the calculations until the loan is fully amortized, at which point the principal balance goes to zero and the loan is paid off.

If we were preparing an amortization table for monthly payments, all of the principal balances, loan payments, and interest rates would have to be adjusted to a monthly basis. For example, to calculate monthly payments for our auto loan, we would make the following adjustments: n 60 = payments (12 months per year × 5 years = 60 months), i = 0.4167 percent (5 percent per year/12 months per year = 0.4167 percent per month), and monthly payment = $188.71.

Note, in  Exhibit 6.5 , the amounts of interest and principal that are paid each year change over time. Interest payments are greatest in the early years of an amortizing loan because much of the principal has not yet been repaid (see columns 1 and 3). However, as the principal balance is reduced over time, the interest payments decline and more of each monthly payment goes toward paying down the principal (see columns 3 and 4).

USING EXCEL LOAN AMORTIZATION TABLE

Loan amortization tables are most easily constructed using a spreadsheet program. Here, we have reconstructed the loan amortization table shown in  Exhibit 6.5  using Excel.

Notice that all the values in the amortization table are obtained by using formulas. Once you have built an amortization table like this one, you can change any of the input variables, such as the loan amount or the interest rate, and all of the other numbers will automatically be updated.

Finding the Interest Rate

Another important calculation in finance is determining the interest, or discount, rate for an annuity. The interest rate tells us the rate of return on an annuity contract. For example, suppose your parents are getting ready to retire and decide to convert some of their retirement portfolio, which is invested in the stock market, into an annuity that guarantees them a fixed annual income. Their insurance agent asks for $350,000 for an annuity that guarantees to pay them $50,000 a year for 10 years. What is the rate of return on the annuity?

As we did when we found the payment amount, we can insert these values into Equation 6.1:

To determine the rate of return for the annuity, we need to solve the equation for the unknown value i. Unfortunately, it is not possible to solve the resulting equation for i algebraically. The only way to solve the problem is by trial and error. We normally solve this kind of problem using a financial calculator or computer spreadsheet program that finds the solution for us. However, it is important to understand how the solution is arrived at by trial and error, so let's work this problem without such aids.

To start the process, we must select an initial value for i, plug it into the right-hand side of the equation, and solve the equation to see if the present value of the annuity stream equals $350,000, which is the left-hand side of the equation. If the present value of the annuity is too large (PVA > $350,000), we need to select a higher value for i. If the present value of the annuity stream is too small (PVA < $350,000), we need to select a smaller value. We continue the trial-and-error process until we find the value of i for which PVA = $350,000.

The key to getting started is to make the best guess we can as to the possible value of the interest rate given the information and data available to us. We will assume that the current bank savings rate is 4 percent. Since the annuity rate of return should exceed the bank rate, we will start our calculations with a 5 percent discount rate. The present value of the annuity is:

That's a pretty good first guess, but our present value is greater than $350,000, so we need to try a higher discount rate. 2  Let's try 7 percent:

The present value of the annuity is still slightly higher than $350,000, so we still need a larger value of i. How about 7.10 percent:

The value is too small, but we now know that i is between 7.00 and 7.10 percent. On the next try, we need to use a slightly smaller value of i—say, 7.07 percent:

Since this value is slightly too high, we should try a number for i that is only slightly greater than 7.07 percent. We'll try 7.073 percent:

The cost of the annuity, $350,000, is now exactly the same as the present value of the annuity stream ($350,000); thus, 7.073 percent is the rate of return earned by the annuity.

It often takes more guesses to solve for the interest rate than it did in this example. Our “guesses” were good because we knew the answer before we started guessing! Clearly, solving for i by trial and error can be a long and tedious process. Fortunately, as mentioned, these types of problems are easily solved with a financial calculator or computer spreadsheet program. Next, we describe how to compute the interest rate or rate of return on an annuity on a financial calculator.

Calculator Tip: Finding the Interest Rate

To illustrate how to find the interest rate for an annuity on a financial calculator, we will enter the information from the previous example. We know the number of periods (N = 10), the payment amount (PMT $50,000), and the present value (PV = −$350,000), and we want to solve for the interest rate (i):

The interest rate is 7.073 percent. Notice that we have used a negative sign for the present value of the annuity contract, representing a cash outflow, and a positive sign for the annuity payments, representing cash inflows. Using the present value formula, you must always have at least one inflow and one outflow. If we had entered both the PV and PMT amounts as positive values (or both as negative values), the calculator would have reported an error since the equation cannot be solved. As we have mentioned before, we could have reversed all of the signs—that is, made cash outflows positive and cash inflows negative—and still gotten the correct answer. Finally, the FV was entered as zero to make sure that the register was cleared.

USING EXCEL CALCULATING THE INTEREST RATE FOR AN ANNUITY

You can also solve for the interest rate using the RATE function in Excel as illustrated below.

APPLICATION 6.6 LEARNING BY DOING

Return on Investments: Good Deal or Bad?

PROBLEM: With some business opportunities you know the price of a financial contract and the promised cash flows, and you want to calculate the interest rate or rate of return on the investment. For example, suppose you have a chance to invest in a small business. The owner wants to borrow $200,000 from you for five years and will make yearly payments of $60,000 at the end of each year. Similar types of investment opportunities will pay 5 percent. Is this a good investment opportunity?

APPROACH: First, we draw a time line for this loan:

To compute the rate of return on the investment, we need to compute the interest rate that equates the initial investment of $200,000 to the present value of the promised cash flows of $60,000 per year. We can use the trial-and-error approach with Equation 6.1, a financial calculator, or a spreadsheet program to solve this problem. Here we will use a financial calculator.

SOLUTION: The financial calculator steps are:

The return on this investment is 15.24 percent, well above the market interest rate of 5 percent. It is a good investment opportunity.

EXAMPLE 6.2 DECISION MAKING

The Pizza Dough Machine

SITUATION: As the owner of a pizza parlor, you are considering whether to buy a fully automated pizza dough preparation machine. Your staff is wildly supportive of the purchase because it would eliminate a tedious part of their work. Your accountant provides you with the following information:

· The cost, including shipping, for the pizza dough machine is $25,000.

· Cash savings, including labor, raw materials, and tax savings due to depreciation, are $3,500 per year for 10 years.

· The present value of the cash savings is $21,506 at a 10 percent discount rate. 3

Given the above data, what should you do?

DECISION: As you arrive at the pizza parlor in the morning, the staff is in a festive mood because word has leaked out that the new machine will save the shop $35,000 and only cost $25,000.

With a heavy heart, you explain that the analysis done at the water cooler by some of the staff is incorrect. To make economic decisions involving cash flows, even for a small business such as your pizza parlor, you cannot compare cash values from different time periods unless they are adjusted for the time value of money. The present value formula takes into account the time value of money and converts the future cash flows into current or present dollars. The cost of the machine is already in current dollars.

The correct analysis is as follows: the machine costs $25,000, and the present value of the cost savings is $21,506. Thus, the cost of the machine exceeds the benefits; the correct decision is not to buy the new dough preparation machine.

Future Value of an Annuity

Generally, when we are working with annuities, we are interested in computing their present value. On occasion, though, we need to compute the future value of an annuity (FVA). Such computations typically involve some type of saving activity, such as a monthly savings plan. Another application is computing terminal values for retirement or pension plans with constant contributions.

future value of an annuity (FVA)

the value of an annuity at some point in the future

We will start with a simple example. Suppose that you plan to save $1,000 at the end of every year for four years with the goal of buying a racing bicycle. The bike you want is a Colnago C50, a top-of-the-line Italian racing bike that costs around $4,500. If your bank pays 8 percent interest a year, will you have enough money to buy the bike at the end of four years?

To solve this problem, we can first lay out the cash flows on a time line, as we discussed earlier in this chapter. We can then calculate the future value for each cash flow using Equation 5.1, which is FVn = PV × (1 + i)n. Finally, we can add up all the cash flows. The time line and calculations are shown in  Exhibit 6.6 . Given that the total future value of the four deposits is $4,506.11, as shown in the exhibit, you should have enough money to buy the bike.

EXHIBIT 6.6 Future Value of a Four-Year Annuity: Colnago C50 Bicycle

The exhibit shows a time line for a savings plan to buy a Colnago C50 bicycle. Under this savings plan, $1,000 is invested at the end of each year for four years at an annual interest rate of 8 percent. We find the value at the end of the four-year period by adding the future values of the separate cash flows, just as in  Exhibits 6.1  and  6.2 .

Future Value of Annuity Equations

Of course, most business applications involve longer periods of time than the Colnago bike example. One way to solve more complex problems involving the future value of an annuity is first to calculate the present value of the annuity, PVA, using Equation 6.1 and then to use Equation 5.1 to calculate the future value of the PVA. In practice, many analyses condense this calculation into a single step by using the future value of annuity (FVA) formula, which we obtain by substituting PVA for PV in Equation 5.1.

where:

We can rearrange Equation 6.2 to write it in terms of the future value factor and the FV annuity factor:

As you would expect, there are tables listing FV annuity factors. Appendix A, at the back of this book, includes a table that shows the future value of a $1 annuity for various interest rates and maturities.

Using Equation 6.2 to compute FVA for the Colnago bike problem is straightforward. The calculation and process are similar to those we developed for PVA problems. That is, we first calculate the FV annuity factor for four years at 8 percent:

We then compute the future value of the annuity by multiplying the constant cash flow (CF) by the FV annuity factor. We plug our computed values into the equation:

This value differs slightly from the one we calculated in  Exhibit 6.6  because of rounding.

Calculator Tip: Finding the Future Value of an Annuity

The procedure for calculating the future value of an annuity on a financial calculator is precisely the same as the procedure for calculating the present value of an annuity discussed earlier. The only difference is that we use the FV (future value) key instead of the PV (present value) key. The PV key is entered as a zero to clear the register.

Let's work the Colnago bicycle problem on a calculator. Recall that we decided to put $1,000 in the bank at the end of each year for four years. The bank pays 8 percent interest. Clear the financial register and make the following entries:

The calculated value of $4,506.11 is the same as in  Exhibit 6.6 .

Perpetuities

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A perpetuity is a constant stream of cash flows that goes on forever. Perpetuities in the form of bonds were used by the British Treasury Department to pay off the debt incurred by the government to finance the Napoleonic wars. These perpetual bonds, called consols, have no maturity date and are still traded in the international bond markets today. They will only be retired when the British Treasury repurchases them all in the open market.

The most important perpetuities in the securities markets today are preferred stock issues. The issuer of preferred stock promises to pay investors a fixed dividend forever unless a retirement date for the perferred stock has been set. If preferred stock dividends are not paid, all previous unpaid dividends must be repaid before any dividends are paid to common stockholders. This preferential treatment is one source of the term preferred stock.

It is worth noting that since, as we discussed in  Chapter 1 , a corporation can have an indefinite life, the expected cash flows from a corporation might also go on forever. When these expected cash flows are constant, they can be viewed as a perpetuity.

From Equation 6.1, we can calculate the present value of a perpetuity by setting n, which is the number of periods, equal to infinity (∞). 4  When that is done, the value of the term 1/(1 + i)∞ approaches 0, and thus the value of a perpetuity that begins next period (PVP) equals:

As you can see, the present value of a perpetuity is the promised constant cash payment (CF) divided by the interest rate (i). A nice feature of the final equation (PVP = CF/i) is that it is algebraically very simple to work with, since it allows us to solve for i directly rather than by trial and error, as is required with Equations 6.1 and 6.2.

For example, suppose you had a great experience during college at the school of business and decided to endow a scholarship fund for finance students. The goal of the fund is to provide the university with $100,000 of financial support each year forever. If the rate of interest is 8 percent, how much money will you have to give the university to provide the desired level of support? Using Equation 6.3, we find that the present value of the perpetuity is:

Thus, a gift of $1.25 million will provide constant annual funding of $100,000 to the university forever.

There is a subtlety here that you should be aware of. In our calculation we made no adjustment for inflation. If the economy is expected to experience inflation, which is generally the case, the real value of the scholarships you are funding will decline each year.

Before we finish our discussion of perpetuities, we should point out that the present value of a perpetuity is typically not very different from the present value of a very long annuity. For example, suppose that instead of funding the scholarship forever, you only plan to fund it for 100 years. If you compute the present value of a 100-year annuity of $100,000 using an interest rate of 8 percent, you will find that it equals $1,249,431.76, which is only slightly less than the $1,250,000 value of the perpetuity. Making your gift a perpetuity would only cost you an additional $568.24. This is because the present value of the cash flows to be received after 100 years is extremely small. The key point here is that cash flows that are to be received far in the future can have very small present values.

APPLICATION 6.7 LEARNING BY DOING

Preferred Stock Dividends

PROBLEM: Suppose that you are the CEO of a public company and your investment banker recommends that you issue some preferred stock at $50 per share. Similar preferred stock issues are yielding 6 percent. What annual cash dividend does the firm need to offer to be competitive in the marketplace? In other words, what cash dividend paid annually forever would be worth $50 with a 6 percent discount rate?

APPROACH: As we have already mentioned, preferred stock is a type of perpetuity; thus, we can solve this problem by applying Equation 6.3. As usual, we begin by laying out the time line for the cash flows:

For preferred stock, PVP is the value of a share of stock, which is $50 per share. The discount rate is 6 percent. CF is the fixed-rate cash dividend, which is the unknown value. Knowing all this information, we can use Equation 6.3 and solve for CF.

SOLUTION:

The annual dividend on the preferred stock would be $3 per share.

Annuities Due

So far we have discussed annuities whose cash flow payments occur at the end of the period, so-called ordinary annuities. Another type of annuity that is fairly common in business is known as an annuity due. Here, cash payments start immediately, at the beginning of the first period. For example, when you rent an apartment, the first rent payment is typically due immediately. The second rent payment is due the first of the second month, and so on. In this kind of payment pattern, you are effectively prepaying for the service.

Exhibit 6.7  compares the cash flows for an ordinary annuity and an annuity due. Note that both annuities are made up of four $1,000 cash flows and carry an 8 percent interest rate. Part A shows an ordinary annuity, in which the cash flows take place at the end of the period, and part B shows an annuity due, in which the cash flows take place at the beginning of the period. There are several ways to calculate the present and future values of an annuity due, and we discuss them next.

annuity due

an annuity in which payments are made at the beginning of each period

EXHIBIT 6.7 Ordinary Annuity versus Annuity Due

The difference between an ordinary annuity (part A) and an annuity due (part B) is that with an ordinary annuity, the cash flows take place at the end of each period, while with an annuity due, the cash flows take place at the beginning of each period. As you can see in this example, the PV of the annuity due is larger than the PV of the ordinary annuity. The reason Is that the cash flows of the annuity due are shifted forward one year and thus are discounted less.

Present Value Method

One way to compute the present value of an annuity due is to discount each individual cash flow to the present, as shown in  Exhibit 6.7 B. Note that since the first $1,000 cash flow takes place now, that cash flow is already in present value terms. The present value of the cash flows for the annuity due is $3,577.

Compare this present value with the present value of the cash flows for the ordinary annuity, $3,312, as calculated in  Exhibit 6.7 A. It should be no surprise that the present value of the annuity due is larger than the present value of the ordinary annuity ($3,577 $3,312), even though both annuities have four $1,000 cash flows. The reason is that the cash flows of the annuity due are shifted forward one year and, thus, are discounted less.

Annuity Transformation Method

An easier way to work annuity due problems is to transform our formula for the present value of an annuity (Equation 6.1) so that it will work for annuity due problems. To do this, we pretend that each cash flow occurs at the end of the period (although it actually occurs at the beginning of the period) and use Equation 6.1. Since Equation 6.1 discounts each cash flow by one period too many, we then correct for the extra discounting by multiplying our answer by (1 i), where i is the discount rate or interest rate.

The relation between an ordinary annuity and an annuity due can be formally expressed as:

This relation is especially helpful because it works for both present value and future value calculations. Calculating the value of an annuity due using Equation 6.4 involves three steps:

1. Adjust the problem time line as if the cash flows were an ordinary annuity.

2. Calculate the present or future value as though the cash flows were an ordinary annuity.

3. Finally, multiply the answer by (1 i).

Let's calculate the value of the annuity due shown in  Exhibit 6.7 B using Equation 6.4, the transformation technique. First, we restate the time line as if the problem were an ordinary annuity; the revised time line looks like the one in  Exhibit 6.7 A. Second, we calculate the present value of the annuity as if the problem involved an ordinary annuity. The value of the ordinary annuity is $3,312, as shown in part A of the exhibit. Finally, we use Equation 6.4 to make the adjustment to an annuity due:

As they should, the answers for the two methods of calculation agree. 5

> BEFORE YOU GO ON

1. How do an ordinary annuity, an annuity due, and a perpetuity differ?

2. Give two examples of perpetuities.

3. What is the annuity transformation method?

6.3 CASH FLOWS THAT GROW AT A CONSTANT RATE

·

So far, we have been examining level cash flow streams. Often, though, management needs to value a cash flow stream that increases at a constant rate over time. These cash flow streams are called growing annuities or growing perpetuities.

Growing Annuity

Financial managers often need to compute the value of multiyear product or service contracts with cash flows that increase each year at a constant rate. These are called growing annuities. For example, you may want to value the cost of a 25-year lease that adjusts annually for the expected rate of inflation over the life of the contract. Equation 6.5 can be used to compute the present value of an annuity growing at a constant rate for a finite time period: 6

growing annuity

an annuity in which the cash flows increase at a constant rate

where:

You should be aware of several important points when applying Equation 6.5. First, the cash flow (CF1) used is not the cash flow for the current period (CF0), but is the cash flow to be received in the next period (t = 1). The relation between these two cash flows is CF1 = CF0 × (1 + g). Second, a necessary condition for using Equation 6.5 is that i > g. If this condition is not met (i ≤ g) the calculations from the equation will be meaningless, as you will get a negative or infinite value for finite positive cash flows. A negative value essentially says that someone would have to pay you money to get you to accept a positive cash flow.

As an example of how Equation 6.5 is applied, suppose you work for a company that owns a number of coffee shops in the New York City area. One coffee shop is located in the Empire State Building, and your boss wants to know how much it is worth. 7  The coffee shop has a 50-year lease, so we will assume that it will be in business for 50 years. It produced cash flows of $300,000 after all expenses this year, and the discount rate used by similar businesses is 15 percent. You estimate that, over the long term, cash flows will grow at 2.5 percent per year because of inflation. Thus, you calculate that the coffee shop's cash flow next year (CF1) will be $307,500, or $300,000 × (1 + 0.025).

Plugging the values from the coffee shop example into Equation 6.5 yields the following result:

The estimated value of the coffee shop is $2,452,128.

Growing Perpetuity

Sometimes cash flows are expected to grow at a constant rate indefinitely. In this case the cash flow stream is called a growing perpetuity. The formula to compute the present value for a growing perpetuity that begins next period (PVP) is as follows:

growing perpetuity

a cash flow stream that grows at a constant rate forever

As before, CF1 is the cash flow occurring at the end of the first period, i is the discount or interest rate, and g is the constant rate of growth of the cash flow (CF). Equation 6.6 is an easy equation to work with, and it is used widely in the valuation of common stock for firms that have a policy and history of paying dividends that grow at a constant rate. It is also widely used in the valuation of entire companies, as we will discuss in  Chapter 18 .

Notice that we can derive Equation 6.6 from Equation 6.5 by setting n equal to ∞. If i is greater than g, as we said it must be, the term [(1 + g)/(1 + i)]∞ is equal to 0, leading to the following result:

This makes sense, of course, since Equation 6.5 describes a growing annuity and Equation 6.6 describes a growing cash flow stream that goes on forever. Notice that both Equations 6.5 and 6.6 are exactly the same as Equations 6.1 and 6.3 when g equals zero.

To illustrate a growing perpetuity, we will consider an example. Suppose that you and a partner, after graduating from college, started a health and athletic club. Your concept included not only providing workout facilities, such as weights, treadmills, and elliptical trainers, but also promoting a healthy lifestyle through a focus on cooking and nutrition. The concept has proved popular, and after only five years, you have seven clubs in operation. Your accountant reports that the firm's cash flow last year was $450,000, and the appropriate discount rate for the club is 18 percent. You expect the firm's cash flows to increase by 5 percent per year, which includes 2 percent for expected inflation. Since the business is a corporation, you can assume it will continue operating indefinitely into the future. What is the value of the firm?

We can use Equation 6.6 to solve this problem. Although the equation is very easy to use, a common mistake is using the current period's cash flow (CF0) and not the next period's cash flow (CF1). Since the cash flow is growing at a constant growth rate, g, we simply multiply CF0 by (1 + g) to get the value of CF1. Thus,

We can then substitute the result into Equation 6.6, which yields a helpful variant of this equation:

Now we can insert the values for the health club into the equation and solve for PVP:

The business is worth $3,634,615.

The growing annuity and perpetuity formulas are useful, and we will be applying them later on in the book. Unfortunately, even though advanced financial calculators have special programs for annuities and perpetuities with constant cash flows, typical financial calculators do not include programs for growing annuities and perpetuities.

> BEFORE YOU GO ON

1. What is the difference between a growing annuity and a growing perpetuity?

6.4 THE EFFECTIVE ANNUAL INTEREST RATE

·

In this chapter and the preceding one, there has been little question about which interest rate to use in a particular computation. In most cases, a single interest rate was supplied. When working with real market data, however, the situation is not so clear-cut. We often encounter interest rates that can be computed in different ways. In this final section, we try to untangle some of the issues that can cause problems.

Why the Confusion?

To better understand why interest rates can be so confusing, consider a familiar situation. Suppose you borrow $100 on your bank credit card and plan to keep the balance outstanding for one year. The credit card's stated interest rate is 1 percent per month. The federal Truth-in-Lending Act requires the bank and other financial institutions to disclose to consumers the annual percentage rate (APR) charged on a loan. The APR is the annualized interest rate using simple interest. It ignores the compound interest associated with compounding periods of less than one year. Thus, the APR is defined as the simple interest charged per period multiplied by the number of periods per year. For the bank credit card loan, the APR is 12 percent (1 percent per month × 12 months = 12 percent).

annual percentage rate (APR)

the simple interest rate charged per period multiplied by the number of periods per year

 Many useful financial calculators, including an APR calculator, can be found at  eFunda.com . Go to  http://www.efunda.com/formulae/finance/apr_calculator.cfm .

At the end of the year, you go to pay off the credit card balance as planned. It seems reasonable to assume that with an APR of 12 percent, your credit card balance at the end of one year would be $112 (1.12 × $100 = $112). Wrong! The bank's actual interest rate is 1 percent per month, meaning that the bank will compound your credit card balance monthly, 12 times over the year. The bank's calculation for the balance due is $112.68 [$100 × (1.01)12 = $112.68]. 8  The bank is actually charging you 12.68 percent per year, and the total interest paid for the one-year loan is $12.68 rather than $12.00. This example raises a question: What is the correct way to annualize an interest rate?

Calculating the Effective Annual Interest Rate

In making financial decisions, the correct way to annualize an interest rate is to compute the effective annual interest rate. The effective annual interest rate (EAR) is defined as the annual interest rate that takes compounding into account. Mathematically, the EAR can be stated as follows:

effective annual interest rate (EAR)

the annual interest rate that reflects compounding within a year

quoted interest rate

a simple annual interest rate, such as the APR

where m is the number of compounding periods during a year. The quoted interest rate is by definition a simple annual interest rate, like the APR. That means that the quoted interest rate has been annualized by multiplying the rate per period by the number of periods per year. The EAR conversion formula accounts for the number of compounding periods and, thus, effectively adjusts the annualized quoted interest rate for the time value of money. Because the EAR is the true cost of borrowing and lending, it is the rate that should be used for making all finance decisions.

We will use our bank credit card example to illustrate the use of Equation 6.7. Recall that the credit card has an APR of 12 percent (1 percent per month). The APR is the quoted interest rate and the number of compounding periods (m) is 12. Applying Equation 6.7, we find that the effective annual interest rate is:

The EAR value of 12.68 percent is the true cost of borrowing the $100 on the bank credit card for one year. The EAR calculation adjusts for the effects of compounding and, hence, the time value of money.

Finally, notice that interest rates are quoted in the marketplace in three ways:

1. The quoted interest rate. This is an interest rate that has been annualized by multiplying the rate per period by the number of compounding periods. The APR is an example. All consumer borrowing and lending rates are annualized in this manner.

2. The interest rate per period. The bank credit card rate of 1 percent per month is an example of this kind of rate. You can find the interest rate per period by dividing the quoted interest rate by the number of compounding periods.

3. The effective annual interest rate (EAR). This is the interest rate actually paid (or earned), which takes compounding into account. Sometimes it is difficult to distinguish a quoted rate from an EAR. Generally, however, an annualized consumer rate is an APR rather than an EAR.

Comparing Interest Rates

When borrowing or lending money, it is sometimes necessary to compare and select among interest rate alternatives. Quoted interest rates are comparable when they cover the same overall time period, such as one year, and have the same number of compounding periods. If quoted interest rates are not comparable, we must adjust them to a common time period. The easiest way, and the correct way, to make interest rates comparable for making finance decisions is to convert them to effective annual interest rates. Consider an example.

Suppose you are the chief financial officer of a manufacturing company. The company is planning a $1 billion plant expansion and will finance it by borrowing money for five years. Three financial institutions have submitted interest rate quotes; all are APRs:

Lender A: 10.40 percent compound monthly

Lender B: 10.90 percent compounded annually

Lender C: 10.50 percent compounded quarterly

Although all the loans have the same maturity, the loans are not comparable because the APRs have different compounding periods. To make the adjustments for the different time periods, we apply Equation 6.7 to convert each of the APR quotes into an EAR:

As shown, Lender B offers the lowest interest cost at 10.90 percent.

Notice the shift in rankings that takes place as a result of the EAR calculations. When we initially looked at the APR quotes, it appeared that Lender A offered the lowest rate and Lender B had the highest. After computing the EAR, we find that when we account for the effect of compounding, Lender B actually offers the lowest interest rate.

Another important point is that if all the interest rates are quoted as APRs with the same annualizing period, such as monthly, the interest rates are comparable and you can select the correct rate by simply comparing the quotes. That is, the lowest APR corresponds with the lowest cost of funds. Thus, it is correct for borrowers or lenders to make economic decisions with APR data as long as interest rates have the same maturity and the same compounding period. To find the true cost of the loan, however, it is still necessary to compute the EAR.

APPLICATION 6.8 LEARNING BY DOING

What Is the True Cost of a Loan?

PROBLEM: During a period of economic expansion, Frank Smith became financially overextended and was forced to consolidate his debt with a loan from a consumer finance company. The consolidated debt provided Frank with a single loan and lower monthly payments than he had previously been making. The loan agreement quotes an APR of 20 percent, and Frank must make monthly payments. What is the true cost of the loan?

APPROACH: The true cost of the loan is the EAR, not the APR. Thus, we must convert the quoted rate into the EAR, using Equation 6.7, to get the true cost of the loan.

SOLUTION:

The true cost of the loan is 21.94 percent, not the 20 percent APR.

Consumer Protection Acts and Interest Rate Disclosure

In 1968 Congress passed the Truth-in-Lending Act to ensure that all borrowers receive meaningful information about the cost of credit so that they can make intelligent economic decisions. 9  The act applies to all lenders that extend credit to consumers, and it covers credit card loans, auto loans, home mortgage loans, home equity loans, home improvement loans, and some small-business loans. Similar legislation, the so-called Truth-in-Savings Act, applies to consumer savings vehicles such as certificates of deposit (CDs). These two pieces of legislation require by law that the APR be disclosed on all consumer loans and savings plans and that it be prominently displayed on advertising and contractual documents.

We know that the EAR, not the APR, represents the true economic interest rate. So why did the Truth-in-Lending and Truth-in-Savings Acts specify that the APR must be the disclosed rate? The APR was selected because it's easy to calculate and easy to understand. When the legislation was passed in 1969, PCs and handheld calculators did not exist. 10  Down at the auto showroom, salespeople needed an easy way to explain and annualize the monthly interest charge, and the APR provided just such a method. And most important, if all the auto lenders quoted monthly APR, consumers could use this rate to select the loan with the lowest economic interest cost.

Today, although lenders and borrowers are legally required to quote the APR, they run their businesses using interest rate calculations based on the present value and future value formulas. Consumers are bombarded with both APR and EAR rates, and confusion reigns. At the car dealership, for example, you may find that your auto loan's APR is 5 percent but the actual borrowing rate is 5.12 percent. And at the bank where your grandmother gets free coffee and doughnuts, she may be told that the bank's one-year CD has an APR of 3 percent, but it really pays 3.04 percent. Because of confusion arising from conflicting interest rates in the marketplace, some observers believe that the APR calculation has outlived its usefulness and should be replaced by the EAR.

Truth-in-Lending Act

a federal law requiring lenders to fully inform borrowers of important information related to loans, including the annual percentage rate charged

Truth-in-Savings Act

a federal law requiring institutions offering consumer savings vehicles, such as certificates of deposit (CDs), to fully inform consumers of important information about the savings vehicles, including the annual percentage rate paid

In addition to requiring that lenders report the APR on all consumer loans, the Truth-in-Lending Act provides other important protections for consumers. For example, it also limits the liability of credit card holders to $50 if a credit card is stolen or used without the cardholder's approval. Since this Act was passed in 1968, a number of subsequent acts have added to the protections of the Truth-in-Lending Act. The most recent of these, which you may be familiar with, is the Credit Card Act of 2009. This act was passed in response to criticisms of actions by credit card companies leading up to the financial crisis of 2008. Among other things, it places new limits on the ability of credit card companies to raise interest rates, limits the fees that they can charge, requires better disclosure of rate increases and how long it will take a cardholder to pay off the outstanding balance with minimum monthly payments, and makes it more difficult for credit card companies to issue new cards to people under age 21.

 You can read more about credit protection laws, including the latest laws passed after the financial crisis at the federal reserve web site  http://federalreserve.gov/creditcard/regs.html .

The Appropriate Interest Rate Factor

Here is a final question to consider: What is the appropriate interest rate to use when making future or present value calculations? The answer is simple: use the EAR. Under no circumstance should the APR or any other quoted rate be used as the interest rate in present or future value calculations. Consider an example of using the EAR in such a calculation.

Petra, an MBA student at Georgetown University, has purchased a $100 savings note with a two-year maturity from a small consumer finance company. The contract states that the note has a 20 percent APR and pays interest quarterly. The quarterly interest rate is thus = 5 percent (20 percent/4 quarters 5 percent per quarter). Petra has several questions about the note: (1) What is the note's effective annual interest rate (EAR)? (2) How much money will she have at the end of two years? (3) When making the future value calculation, should she use the quarterly interest rate or the EAR?

To answer Petra's questions, we first compute the EAR, which is the actual interest earned on the note:

Next, we calculate the future value of the note using the EAR. Because the EAR is an annual rate, for this problem we use a total of two compounding periods. The calculation is as follows:

We can also calculate the future value using the quarterly rate of interest of 5 percent with a total of eight compounding periods. In this case, the calculation is as follows:

The two calculation methods yield the same answer, $147.75.

In sum, any time you do a future value or present value calculation, you must use either the interest rate per period (quoted rate/m) or the EAR as the interest rate factor. It does not matter which of these you use. Both will properly account for the impact of compounding on the value of cash flows. Interest rate proxies such as the APR should never be used as interest rate factors for calculating future or present values. Because they do not properly account for the number of compounding periods, their use can lead to answers that are economically incorrect.

> BEFORE YOU GO ON

1. What is the APR, and why are lending institutions required to disclose this rate?

2. What is the correct way to annualize an interest rate in financial decision making?

3. Distinguish between quoted interest rate, interest rate per period, and effective annual interest rate.

SUMMARY OF Learning Objectives

 Explain why cash flows occurring at different times must be adjusted to reflect their value as of a common date before they can be compared, and compute the present value and future value for multiple cash flows.

When making decisions involving cash flows over time, we should first identify the magnitude and timing of the cash flows and then adjust each individual cash flow to reflect its value as of a common date. For example, the process of discounting (compounding) cash flows adjusts them for the time value of money because today's dollars are not equal in value to dollars in the future. Once all of the cash flows are in present (future) value terms, they can be compared to make decisions. Section 6.1 discusses the computation of present values and future values of multiple cash flows.

 Describe how to calculate the present value and the future value of an ordinary annuity and how an ordinary annuity differs from an annuity due.

An ordinary annuity is a series of equally spaced, level cash flows over time. The cash flows for an ordinary annuity are assumed to take place at the end of each period. To find the present value of an ordinary annuity, we multiply the present value of an annuity factor, which is equal to (1 − present value factor)/i, by the amount of the constant cash flow.

An annuity due is an annuity in which the cash flows occur at the beginning of each period. A lease is an example of an annuity due. In this case, we are effectively prepaying for the service. To calculate the value of an annuity due, we calculate the present value (or future value) as though the cash flows are from an ordinary annuity. We then multiply the ordinary annuity value times (1 + i). Section 6.2 discusses the calculation of the present value of an ordinary annuity and annuity due.

 Explain what a perpetuity is and where we see them in business, and calculate the value of a perpetuity.

A perpetuity is like an annuity except that the cash flows are perpetual—they never end. British Treasury Department bonds, called consols, were the first widely used securities of this kind. The most common example of a perpetuity today is preferred stock. The issuer of preferred stock promises to pay fixed-rate dividends forever. The cash flows from corporations can also look like perpetuities. To calculate the present value of a perpetuity, we simply divide the constant cash flow (CF) by the interest rate (i).

 Discuss growing annuities and perpetuities, as well as their application in business, and calculate their values.

Financial managers often need to value cash flow streams that increase at a constant rate over time. These cash flow streams are called growing annuities or growing perpetuities. An example of a growing annuity is a 10-year lease with an annual adjustment for the expected rate of inflation over the life of the contract. If the cash flows continue to grow at a constant rate indefinitely, this cash flow stream is called a growing perpetuity. Application and calculation of cash flows that grow at a constant rate are discussed in Section 6.3.

 Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest rates, and calculate the EAR.

The EAR is the annual growth rate that takes compounding into account. Thus, the EAR is the true cost of borrowing or lending money. When we need to compare interest rates, we must make sure that the rates to be compared have the same time and compounding periods. If interest rates are not comparable, they must be converted into common terms. The easiest way to convert rates to common terms is to calculate the EAR for each interest rate. The use and calculation of EAR are discussed in Section 6.4.

SUMMARY OF Key Equations

Self-Study Problems

· 6.1 Kronka, Inc., is expecting cash inflows of $13,000, $11,500, $12,750, and $9,635 over the next four years. What is the present value of these cash flows if the appropriate discount rate is 8 percent?

· 6.2 Your grandfather has agreed to deposit a certain amount of money each year into an account paying 7.25 percent annually to help you go to graduate school. Starting next year, and for the following four years, he plans to deposit $2,250, $8,150, $7,675, $6,125, and $12,345 into the account. How much will you have at the end of the five years?

· 6.3 Mike White is planning to save up for a trip to Europe in three years. He will need $7,500 when he is ready to make the trip. He plans to invest the same amount at the end of each of the next three years in an account paying 6 percent. What is the amount that he will have to save every year to reach his goal of $7,500 in three years?

· 6.4 Becky Scholes has $150,000 to invest. She wants to be able to withdraw $12,500 every year forever without using up any of her principal. What interest rate would her investment have to earn in order for her to be able to so?

· 6.5 Dynamo Corp. is expecting annual payments of $34,225 for the next seven years from a customer. What is the present value of this annuity if the discount rate is 8.5 percent?

Solutions to Self-Study Problems

· 6.1 The time line for Kronka's cash flows and their present value is as follows:

· 6.2 The time line for your cash flows and their future value is as follows:

· 6.3 Amount Mike White will need in three years = FVA3 = $7,500

Number of years = n = 3

Interest rate on investment = i = 6.0%

Amount that Mike needs to invest every year = PMT = ?

Mike will have to invest $2,355.82 every year for the next three years.

· 6.4 Present value of Becky Scholes' investment = $150,000

Amount needed annually = $12,500

This is a perpetuity!

· 6.5 The time line for Dynamo's cash flows and their present value is as follows:

Critical Thinking Questions

· 6.1 Identify the steps involved in computing the future value when you have multiple cash flows.

· 6.2 What is the key economic principle involved in calculating the present value and future value of multiple cash flows?

· 6.3 What is the difference between a perpetuity and an annuity?

· 6.4 Define annuity due. Would an investment be worth more if it were an ordinary annuity or an annuity due? Explain.

· 6.5 Raymond Bartz is trying to choose between two equally risky annuities, each paying $5,000 per year for five years. One is an ordinary annuity, the other is an annuity due. Which of the following statements is most correct?

5. The present value of the ordinary annuity must exceed the present value of the annuity due, but the future value of an ordinary annuity may be less than the future value of the annuity due.

5. The present value of the annuity due exceeds the present value of the ordinary annuity, while the future value of the annuity due is less than the future value of the ordinary annuity.

5. The present value of the annuity due exceeds the present value of the ordinary annuity, and the future value of the annuity due also exceeds the future value of the ordinary annuity.

5. If interest rates increase, the difference between the present value of the ordinary annuity and the present value of the annuity due remains the same.

1. 6.6 Which of the following investments will have the highest future value at the end of three years? Assume that the effective annual rate for all investments is the same.

6. You earn $3,000 at the end of three years (a total of one payment).

6. You earn $1,000 at the end of every year for the next three years (a total of three payments).

6. You earn $1,000 at the beginning of every year for the next three years (a total of three payments).

1. 6.7 Explain whether or not each of the following statements is correct.

7. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of the same amount and same interest rate.

7. If an investment pays 10 percent interest compounded annually, its effective rate will also be 10 percent.

1. 6.8 When will the annual percentage rate (APR) be the same as the effective annual rate (EAR)?

1. 6.9 Why is the EAR superior to the APR in measuring the true economic cost or return?

1. 6.10 Suppose three investments have equal lives and multiple cash flows. A high discount rate tends to favor:

10. The investment with large cash flows early.

10. The investment with large cash flows late.

10. The investment with even cash flows.

10. None of the investments since they have equal lives.

Questions and Problems

· 6.1 Future value with multiple cash flows: Konerko, Inc., expects to earn cash flows of $13,227, $15,611, $18,970, and $19,114 over the next four years. If the company uses an 8 percent discount rate, what is the future value of these cash flows at the end of year 4?

· 6.2 Future value with multiple cash flows: Ben Woolmer has an investment that will pay him the following cash flows over the next five years: $2,350, $2,725, $3,128, $3,366, and $3,695. If his investments typically earn 7.65 percent, what is the future value of the investment's cash flows at the end of five years?

· 6.3 Future value with multiple cash flows: You are a freshman in college and are planning a trip to Europe when you graduate from college at the end of four years. You plan to save the following amounts annually, starting today: $625, $700, $700, and $750. If the account pays 5.75 percent annually, how much will you have at the end of four years?

· 6.4 Present value with multiple cash flows: Saul Cervantes has just purchased some equipment for his landscaping business. For this equipment he must pay the following amounts at the end of each of the next five years: $10,450, $8,500, $9,675, $12,500, and $11,635. If the appropriate discount rate is 10.875 percent, what is the cost in today's dollars of the equipment Saul purchased today?

· 6.5 Present value with multiple cash flows: Jeremy Fenloch borrowed some money from his friend and promised to repay him the amounts of $1,225, $1,350, $1,500, $1,600, and $1,600 over the next five years. If the friend normally discounts investment cash flows at 8 percent annually, how much did Jeremy borrow?

· 6.6 Present value with multiple cash flows: Biogenesis Inc. management expects the following cash flow stream over the next five years. They discount all cash flows using a 23 percent discount rate. What is the present value of this cash flow stream?

· 6.7 Present value of an ordinary annuity: An investment opportunity requires a payment of $750 for 12 years, starting a year from today. If your required rate of return is 8 percent, what is the value of the investment to you today?

· 6.8 Present value of an ordinary annuity: Dynamics Telecommunications Corp. has made an investment in another company that will guarantee it a cash flow of $22,500 each year for the next five years. If the company uses a discount rate of 15 percent on its investments, what is the present value of this investment?

· 6.9 Future value of an ordinary annuity: Robert Hobbes plans to invest $25,000 a year at the end of each year for the next seven years in an investment that will pay him a rate of return of 11.4 percent. How much money will Robert have at the end of seven years?

· 6.10 Future value of an ordinary annuity: Cecelia Thomas is a sales executive at a Baltimore firm. She is 25 years old and plans to invest $3,000 every year in an IRA account, beginning at the end of this year until she turns 65 years old. If the IRA investment will earn 9.75 percent annually, how much will she have in 40 years, when she turns 65?

· 6.11 Future value of an annuity due: Refer to Problem 6.10. If Cecelia invests at the beginning of each year, how much will she have at age 65?

· 6.12 Computing annuity payment: Kevin Winthrop is saving for an Australian vacation in three years. He estimates that he will need $5,000 to cover his airfare and all other expenses for a week-long holiday in Australia. If he can invest his money in an S&P 500 equity index fund that is expected to earn an average return of 10.3 percent over the next three years, how much will he have to save every year if he starts saving at the end of this year?

· 6.13 Computing annuity payment: The Elkridge Bar & Grill has a seven-year loan of $23,500 with Bank of America. It plans to repay the loan in seven equal installments starting today. If the rate of interest is 8.4 percent, how much will each payment be?

· 6.14 Perpetuity: Your grandfather is retiring at the end of next year. He would like to ensure that his heirs receive payments of $10,000 a year forever, starting when he retires. If he can earn 6.5 percent annually, how much does your grandfather need to invest to produce the desired cash flow?

· 6.15 Perpetuity: Calculate the annual cash flows for each of the following investments:

15. $250,000 invested at 6 percent.

15. $50,000 invested at 12 percent.

15. $100,000 invested at 10 percent.

1. 6.16 Effective annual interest rate: Raj Krishnan bought a Honda Civic for $17,345. He put down $6,000 and financed the rest through the dealer at an APR of 4.9 percent for four years. What is the effective annual interest rate (EAR) if the loan payments are made monthly?

1. 6.17 Effective annual interest rate: Cyclone Rentals borrowed $15,550 from a bank for three years. If the quoted rate (APR) is 6.75 percent, and the compounding is daily, what is the effective annual interest rate (EAR)?

1. 6.18 Growing perpetuity: You are evaluating a growing perpetuity investment from a large financial services firm. The investment promises an initial payment of $20,000 at the end of this year and subsequent payments which will grow at a rate of 3.4 percent annually. If you use a 9 percent discount rate for investments like this, what is the present value of this growing perpetuity?

1. 6.19 Future value with multiple cash flows: Trigen Corp. management will invest cash flows of $331,000, $616,450, $212,775, $818,400, $1,239,644, and $1,617,848 in research and development over the next six years. If the appropriate interest rate is 6.75 percent, what is the future value of these investment cash flows six years from today?

1. 6.20 Future value with multiple cash flows: Stephanie Watson plans to make the following investments beginning next year. She will invest $3,125 in each of the next three years and will then make investments of $3,650, $3,725, $3,875, and $4,000 over the following four years. If the investments are expected to earn 11.5 percent annually, how much will Stephanie have at the end of the seven years?

1. 6.21 Present value with multiple cash flows: Carol Jenkins, a lottery winner, will receive the following payments over the next seven years. If she can invest her cash flows in a fund that will earn 10.5 percent annually, what is the present value of her winnings?

1. 6.22 Computing annuity payment: Gary Whitmore is a high school sophomore. He currently has $7,500 in a savings account that pays 5.65 percent annually. Gary plans to use his current savings plus what he can save over the next four years to buy a car. He estimates that the car will cost $12,000 in four years. How much money should Gary save each year if he wants to buy the car?

1. 6.23 Growing annuity: Modern Energy Company owns several gas stations. Management is looking to open a new station in the western suburbs of Baltimore. One possibility they are evaluating is to take over a station located at a site that has been leased from the county. The lease, originally for 99 years, currently has 73 years before expiration. The gas station generated a net cash flow of $92,500 last year, and the current owners expect an annual growth rate of 6.3 percent. If Modern Energy uses a discount rate of 14.5 percent to evaluate such businesses, what is the present value of this growing annuity?

1. 6.24 Future value of annuity due: Jeremy Denham plans to save $5,000 every year for the next eight years, starting today. At the end of eight years, Jeremy will turn 30 years old and plans to use his savings toward the down payment on a house. If his investment in a mutual fund will earn him 10.3 percent annually, how much will he have saved in eight years when he buys his house?

1. 6.25 Present value of an annuity due: Grant Productions has borrowed a large sum from the California Finance Company at a rate of 17.5 percent for a seven-year period. The loan calls for a payment of $1,540,862.19 each year beginning today. How much did Grant borrow?

1. 6.26 Present value of an annuity due: Sharon Kabana has won a state lottery and will receive a payment of $89,729.45 every year, starting today, for the next 20 years. If she invests the proceeds at a rate of 7.25 percent, what is the present value of the cash flows that she will receive? Round to the nearest dollar.

1. 6.27 Present value of an annuity due: You wrote a piece of software that does a better job of allowing computers to network than any other program designed for this purpose. A large networking company wants to incorporate your software into their systems and is offering to pay you $500,000 today, plus $500,000 at the end of each of the following six years for permission to do this. If the appropriate interest rate is 6 percent, what is the present value of the cash flow stream that the company is offering you?

1. 6.28 Present value of an annuity: Suppose that the networking company in Problem 6.27 will not start paying you until the first of the new systems that uses your software is sold in two years. What is the present value of that annuity? Assume that the appropriate interest rate is still 6 percent.

1. 6.29 Perpetuity: Calculate the present value of the following perpetuities:

29. $1,250 discounted to the present at 7 percent.

29. $7,250 discounted to the present at 6.33 percent.

29. $850 discounted to the present at 20 percent.

1. 6.30 Effective annual interest rate: Find the effective annual interest rate (EAR) for each of the following:

30. 6 percent compounded quarterly.

30. 4.99 percent compounded monthly.

30. 7.25 percent compounded semiannually.

30. 5.6 percent compounded daily.

1. 6.31 Effective annual interest rate: Which of the following investments has the highest effective annual interest rate (EAR)?

31. A bank CD that pays 8.25 percent compounded quarterly.

31. A bank CD that pays 8.25 percent compounded monthly.

31. A bank CD that pays 8.45 percent compounded annually.

31. A bank CD that pays 8.25 percent compounded semiannually.

31. A bank CD that pays 8 percent compounded daily (on a 365-day basis).

1. 6.32 Effective annual interest rate: You are considering three alternative investments: (1) a three-year bank CD paying 7.5 percent compounded quarterly; (2) a three-year bank CD paying 7.3 percent compounded monthly; and (3) a three-year bank CD paying 7.75 percent compounded annually. Which investment has the highest effective annual interest rate?

1. 6.33 You have been offered the opportunity to invest in a project which is expected to provide you with the following cash flows: $4,000 in 1 year, $12,000 in 2 years, and $8,000 in 3 years. If the appropriate interest rates are 6 percent for the first year, 8 percent for the second year, and 12 percent for the third year, what is the present value of these cash flows?

1. 6.34 Tirade Owens, a professional athlete, currently has a contract that will pay him a large amount in the first year of his contract and smaller amounts thereafter. He and his agent have asked the team to restructure the contract. The team, though reluctant, obliged. Tirade and his agent came up with a counter offer. What are the present values of each of the contracts using a 14 percent discount rate? Which of the three contacts has the highest present value?

1. 6.35 Gary Kornig will be 30 years old next year and wants to retire when he is 65. So far he has saved (1) $6,950 in an IRA account in which his money is earning 8.3 percent annually and (2) $5,000 in a money market account in which he is earning 5.25 percent annually. Gary wants to have $1 million when he retires. Starting next year, he plans to invest the same amount of money every year until he retires in a mutual fund in which he expects to earn 9 percent annually. How much will Gary have to invest every year to achieve his savings goal?

1. 6.36 The top prize for the state lottery is $100,000,000. You have decided it is time for you to take a chance and purchase a ticket. Before you purchase the ticket, you must decide whether to choose the cash option or the annual payment option. If you choose the annual payment option and win, you will receive $100,000,000 in 25 equal payments of $4,000,000—one payment today and one payment at the end of each of the next 24 years. If you choose the cash payment, you will receive a one-time lump sum payment of $59,194,567.18. If you can invest the proceeds and earn 6 percent, which option should you choose?

1. 6.37 At what interest rate would you be indifferent between the cash and annual payment options in Problem 6.36?

1. 6.38 Babu Baradwaj is saving for his son's college tuition. His son is currently 11 years old and will begin college in seven years. Babu has an index fund investment worth $7,500 that is earning 9.5 percent annually. Total expenses at the University of Maryland, where his son says he plans to go, currently total $15,000 per year, but are expected to grow at roughly 6 percent each year. Babu plans to invest in a mutual fund that will earn 11 percent annually to make up the difference between the college expenses and his current savings. In total, Babu will make seven equal investments with the first starting today and with the last being made a year before his son begins college.

38. What will be the present value of the four years of college expenses at the time that Babu's son starts college? Assume a discount rate of 5.5 percent.

38. What will be the value of the index mutual fund when his son just starts college?

38. What is the amount that Babu will have to have saved when his son turns 18 if Babu plans to cover all of his son's college expenses?

38. How much will Babu have to invest every year in order to have enough funds to cover all his son's expenses?

1. 6.39 You are now 50 years old and plan to retire at age 65. You currently have a stock portfolio worth $150,000, a 401(k) retirement plan worth $250,000, and a money market account worth $50,000. Your stock portfolio is expected to provide annual returns of 12 percent, your 401(k) investment will earn 9.5 percent annually, and the money market account earns 5.25 percent, compounded monthly.

39. If you do not save another penny, what will be the total value of your investments when you retire at age 65?

39. Assume you plan to invest $12,000 every year in your 401(k) plan for the next 15 years (starting one year from now). How much will your investments be worth when you retire at 65?

39. Assume that you expect to live 25 years after you retire (until age 90). Today, at age 50, you take all of your investments and place them in an account that pays 8 percent (use the scenario from part b in which you continue saving). If you start withdrawing funds starting at age 66, how much can you withdraw every year (e.g., an ordinary annuity) and leave nothing in your account after a 25th and final withdrawal at age 90?

39. You want your current investments, which are described in the problem statement, to support a perpetuity that starts a year from now. How much can you withdraw each year without touching your principal?

1. 6.40 Trevor Diaz is looking to purchase a Mercedes Benz SL600 Roadster, which has an invoice price of $121,737 and a total cost of $129,482. Trevor plans to put down $20,000 and will pay the rest by taking on a 5.75 percent five-year bank loan. What is the monthly payment on this auto loan? Prepare an amortization table using Excel.

1. 6.41 The Sundarams are buying a new 3,500-square-foot house in Muncie, Indiana, and will borrow $237,000 from Bank One at a rate of 6.375 percent for 15 years. What will be their monthly loan payment? Prepare an amortization schedule using Excel.

1. 6.42 Assume you will start working as soon as you graduate from college. You plan to start saving for your retirement on your 25th birthday and retire on your 65th birthday. After retirement, you expect to live at least until you are 85. You wish to be able to withdraw $40,000 (in today's dollars) every year from the time of your retirement until you are 85 years old (i.e., for 20 years). The average inflation rate is likely to be 5 percent.

42. Calculate the lump sum you need to have accumulated at age 65 to be able to draw the desired income. Assume that the annual return on your investments is likely to be 10 percent.

42. What is the dollar amount you need to invest every year, starting at age 26 and ending at age 65 (i.e., for 40 years), to reach the target lump sum at age 65?

42. Now answer questions a. and b. assuming the rate of return to be 8 percent per year, then again at 15 percent per year.

42. Now assume you start investing for your retirement when you turn 30 years old and analyze the situation under rate of return assumptions of (i) 8 percent, (ii) 10 percent, and (iii) 15 percent.

42. Repeat the analysis by assuming that you start investing when you are 35 years old.

Sample Test Problems

· 6.1 Groves Corp. is expecting annual cash flows of $225,000, $278,000, $312,500, and $410,000 over the next four years. If it uses a discount rate of 6.25 percent, what is the present value of this cash flow stream?

· 6.2 Freisinger, Inc., is expecting a new project to start paying off, beginning at the end of next year. It expects cash flows to be as follows:

If Freisinger can reinvest these cash flows to earn a return of 7.8 percent, what is the future value of this cash flow stream at the end of five years?

· 6.3 Sochi, Russia is the site of the next Winter Olympics in 2014. City officials plan to build a new multi-purpose stadium. The projected cost of the stadium in 2014 dollars is $7.5 million. Assume that it is the end of 2011 and city officials intend to invest an equal amount of money at the end of each of the next three years in an account that will pay 8.75 percent. What is the annual investment necessary to meet the projected cost of the stadium?

· 6.4 You have just won a lottery that promises an annual payment of $118,312 beginning immediately. You will receive a total of 10 payments. If you can invest the cash flows in an investment paying 7.65 percent annually, what is the present value of this annuity?

· 6.5 Which of the following investments has the highest effective annual interest rate (EAR)?

5. A bank CD that pays 5.50 percent compounded quarterly.

5. A bank CD that pays 5.45 percent compounded monthly.

5. A bank CD that pays 5.65 percent compounded annually.

5. A bank CD that pays 5.55 percent compounded semiannually.

5. A bank CD that pays 5.35 percent compounded daily (on a 365-day basis).

Appendix: Deriving the Formula for the Present Value of an Ordinary Annuity

In this chapter we showed that the formula for a perpetuity can be obtained from the formula for the present value of an ordinary annuity if n is set equal to. It is also possible to go the other way. In other words, the present value of an ordinary annuity formula can be derived from the formula for a perpetuity. In fact, this is how the annuity formula was originally obtained. To see how this was done, assume that someone has offered to pay you $1 per year forever, beginning next year, but that, in return, you will have to pay that person $1 per year forever, beginning in year n + 1.

The cash flows you will receive and the cash flows you will pay are represented in the following time line:

The first row of dollar values shows the cash flows for the perpetuity that you will receive. This perpetuity is worth:

The second row shows the cash flows for the perpetuity that you will pay. The present value of what you owe is the value of a $1 perpetuity that is discounted for n years.

Notice that if you subtract, year by year, the cash flows you would pay from the cash flows you would receive, you get the cash flows for an n-year annuity.

Therefore, the value of the offer equals the value of an n-year annuity. Solving for the difference between PVP Receive and PVP Pay  we see that this is the same as Equation 6.1.

Problem

· 6A.1 In the chapter text, you saw that the formula for a growing perpetuity can be obtained from the formula for the present value of a growing annuity if n is set equal to ∞. It is also possible to go the other way. In other words, the present value of a growing annuity formula can be derived from the formula for a growing perpetuity. In fact, this is how Equation 6.5 was actually derived. Show how Equation 6.5 can be derived from Equation 6.6.

ETHICS CASE: Buy It on Credit and Be True to Your School

At the start of every school year, major banks offer students “free” credit cards. There are good reasons for banks to solicit students' business even though most students have neither steady jobs nor credit histories. First, students have a better record of paying their bills than the general public, because if they can't pay, usually their parents will. Second, students turn into loyal customers. Studies have shown that students keep their first credit card for an average of 15 years. That enables banks to sell them services over time, such as car loans, first mortgages, and (somewhat ironically) debt consolidation loans. Third and perhaps most importantly, students are ideal customers because they do not tend to pay off their credit balances each month. A 2009 study by Sallie Mae, the largest student loan provider, found that among undergraduates who have credit cards, the credit card balances of only 18 percent are paid off each month. The other 82 percent carry a balance and pay interest charges. Sallie Mae also found that the percentage of undergraduates with at least one credit card increased from 76 percent in 2004 to 84 percent in 2009. Furthermore, students with credit cards had an average of 4.6 cards and owed an average of $3,173 in 2009. Seniors owed the most, with average debt of $4,100. Nineteen percent of students with credit cards owed over $7,000 on those cards!

© AP/Wide World Photos

Concern over Growing Student Debt

Concern has been growing that students cannot handle the debt they are taking on. In addition to credit card debt, the average graduating senior in the class of 2009 had $24,000 in student educational loan debt and 10 percent had more than $40,000 of such debt. The average student loan debt among graduating seniors in 2009 was almost twice as large as the average in 1996. Many students fail to realize that when they apply for a car loan or a mortgage, the total ratio of debt to income is usually the most important factor determining whether they get the loan. Student educational loans are added to credit card debt, and that, in turn, is added to the requested loan amount to determine eligibility. When all the debt is summed up, many do not qualify for the loan they want. In many cases, people are forced to postpone marriage or the purchase of a house because of their outstanding student loans and credit card debt.

To understand how students get into this kind of situation, consider the following hypothetical case. Suppose a student has a balance of $2,000 on a credit card. She makes the minimum payment every month but does not make any other purchases. Assuming a typical rate of interest, it would take six and one half years to pay off the credit card debt, and the student would have incurred interest charges of $2,500. As one observer noted, students like this one “will still be paying for all that pizza they bought in college when they are 30 years old.” A book published in 2000, Credit Card Nation: The Consequences of America's Addiction to Credit, was particularly critical of marketing credit cards to college students. The author, Robert Manning, identified a wide range of concerns, such as lowering of the age at which students can obtain credit cards, increasing credit limits on credit cards, students financing their education with credit card debt, and students using credit cards to conceal activities their parents might not approve of. Critics also point out that some of the advertising and marketing practices of the credit card companies are deceptive. In one case, for example, a credit card was touted as having no interest. That was true for the first month, but the annual percentage rate (APR) soared to 21 percent in the second month. Finally, many—including the students themselves—say that students do not receive sufficient education about how to manage credit card debt.

Supporters of credit card programs counter that most students do not “max out” their credit limits and that the three most common reasons for taking out a credit card are the establishment of a credit history, convenience, and emergency protection—all laudable goals.

The Credit Card Act of 2009, passed by Congress and signed by the President in 2009, includes a provision that is aimed at limiting the ability of credit card companies to market cards to students and other young adults. This provision, effective February 22, 2010, prohibits credit card companies from issuing credit cards to anyone under 21 unless that person can produce either (1) proof of a sufficiently high independent income to pay the credit card loans or (2) a willing co-signer who is over the age of 21. It remains to be seen whether this provision helps reverse the trend toward greater student credit card debt.

Affinity Credit Cards

The marketing of credit cards to students took a new twist in the 1990s. Banks began to compete fiercely to sign up students for their credit cards, and some banks entered into exclusive arrangements with universities for the right to issue an affinity card—a credit card that features the university's name and logo. The card issuer may be willing to support the university to the tune of several million dollars to gain the exclusive right to issue the affinity card and to keep other banks off campus.

The “Report to the Congress on College Credit Card Agreements,” which was required by the Credit Card Act of 2009, revealed for the first time in October 2010 exactly how pervasive this practice had become. In 2009 alone, banks paid $83 million to U.S. colleges, universities, and affiliated organizations for the right to market their credit cards to students and alumni. Universities usually receive a half percent of the purchase value when the card is used. Often, they receive a fee for each new account, and sometimes they receive a small percentage of the loans outstanding. Every time a student uses the credit card, the university benefits. The total benefits to individual universities can be substantial. For example, in 2009 alone, the University of Notre Dame du Lac received $1,860,000, the University of Southern California received $1,502,850, and the University of Tennessee received $1,428,571 from credit card agreements. In previous years, some universities received even larger direct payments from credit card issuers seeking to do business with students and alumni. Georgetown University, for example, received $2 million from MBNA for a career counseling center; Michigan State received $5.5 million from MBNA for athletic and academic scholarship programs; and the University of Tennessee received $16 million from First USA primarily for athletics and scholarships.

Universities have been facing difficult financial times, and it is easy to understand why they enter into these arrangements. However, the price the university pays is that it becomes ensnared in the ethical issue of contributing to the rising level of student credit card debt. Moreover, universities with affinity credit cards cannot escape a conflict of interest: the higher student credit card debt climbs, the greater the revenues the university earns from the bank. As a result of these issues, some universities have increased the amount of information they provide to students about handling credit card debt, both through counseling and formal courses.

Certainly, learning to responsibly manage credit card purchases and any resulting debt is a necessary part of the passage to adulthood. We can applaud the fact that universities educate students about the dangers of excessive credit card debt. However, if universities make money on that debt, we must question whether they have less incentive to educate students about the associated problems.

DISCUSSION QUESTIONS

1. Should universities enter into agreements to offer affinity credit cards to students?

2. Whether or not a university has an affinity credit card, does it have an obligation to educate students about credit card misuse and debt management?

3. Does the existence of an affinity credit card create a conflict of interest for a university if and when it adopts an education program on credit card misuse and debt management?

4. To what extent are students themselves responsible for their predicament?

Sources: “Big Cards on Campus,” Business Week, September 20, 1999, pp. 136–137; Marilyn Gardner, “A Generation Weighed Down by Debt,” Christian Science Monitor, November 24, 2004; “Survey Reveals Aggressive Marketing of Credit Cards Continues on Many Maryland College Campuses,” U.S. PIRG press release, February 19, 2004; and “Golden Eggs,” Boston Globe, June 25, 2006; “How Undergraduate Students Use Credit Cards: Sallie Mae's National Study of Usage Rates and Trends, 2009,” Sallie Mae, 2009; “Student Debt and the Class of 2009,” Project on Student Debt, 2009; “Federal Reserve Board of Governors Report to the Congress on College Credit Card Agreements,” Board of Governors of the Federal Reserve System, October 2010.

1  Recall that, when using a financial calculator, it is common practice to enter cash outflows as negative numbers and cash inflows as positive numbers. See  Chapter 5  for a complete discussion the importance of assigning the proper sign (or) to cash flows when using a financial calculator.

2  Notice that we have rounded the PV annuity factor to three decimal places (7.722). If we use a financial calculator and do not round, we get a more precise answer of $386,086.75.

3  The annuity present value factor for 10 years at 10 percent is 6.1446. Thus, PVA10 = CF × Annuity factor $3,500 × 6.1446 = $21,506.10. Using a financial calculator, PVA10 = $21,505.98. The difference is due to rounding errors.

4  Conversely, we can derive the formula for the present value of an ordinary annuity, Equation 6.1, from the formula for a perpetuity, as explained in the appendix at the end of this chapter.

5  Another easy way to calculate the present value or future value of an annuity due is by using the BEG/END switch in your financial calculator. All financial calculators have a key that switches the cash flow from the end of each period to the beginning of each period. The keys are typically labeled “BEG” for cash flows at the beginning of the period and “END” for the cash flows at the end of the period. To calculate the PV of an annuity due: (1) switch the calculator to the BEG mode, (2) enter the data, and (3) press the PV key for the answer. As an example, work the problem from  Exhibit 6.7 B using your financial calculator.

6  In Equation 6.5 we represent the present value of a growing annuity of n periods using the same notation (PVA n ) we use for a regular annuity in Equation 6.1. We do this because the regular annuity is just a special case of the growing annuity, where g = 0. Equation 6.5 is the more general form of the annuity formula.

7  For those interested, the Empire State Building has three coffee shops.

8  If you have any doubt about the total credit card debt at the end of one year, make the calculation 12 times on your calculator: the first month is $100 × 1.01 = 101.00; the second month is $101.00 × 1.01 = $102.01; the third month is $102.01 × 1.01 = $103.03; and so on for 12 months.

9  The Truth-in-Lending Act is Title I of the Consumer Credit Protection Act.

10  The first handheld calculator was the Bomar Brain, which was first sold in 1971.