Assignment Due Friday 9pm

Edward1111

Brigham_2020_Chapter_4_Edition_16th1.pdf

Home >Business & Finance homework help >Management homework help >Assignment Due Friday 9pm

Document Type: Book Chapter

Title of Book: Financial Management Theory and Practice (16th Edition)

Author(s) of Book: Eugene F. Brigham, Michael C. Ehrhardt

Chapter Title: Chapter 4 Time Value of Money

Author(s) of Chapter: Eugene F. Brigham, Michael C. Ehrhardt

Year: 2020

Publisher: Cengage Learning

Place of Publishing: the United States of America

The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted materials. Under certain conditions specifies in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these conditions is that the photocopy or reproduction is not to be used for any purpose other than private study, scholarship, or research. If a user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of fair use, that user may be liable for copyright infringement.

(

Time Value of Money

Before you shop fora newcar,you will do a lotofresearchto identify the make and model

you want. This will give you a pretty good idea regarding the car's features, ratings,

safety record, and various option bundles. If you plan to trade in your current car, you

know to leave that car at home until you and the salesperson have agreed on new car's

sales price-bundling two transactions into one will reduce transparency, often at the

buyer's expense. You know that a factory rebate is paid by the manufacturer, not the

dealer, which means that the dealer's profit is not affected by the size of the rebate.

Knowing this, you won't allow the salesperson to use the rebate as a negotiating point.

You put on your best poker face, walk into the dealership, confident that you are ready

to negotiate the best possible deal.

After you and the salesperson agree on a price, you will go to the dealer's finance

department to conduct the transaction. And this is where it gets really tricky! Most

dealers offer to finance the purchase themselves and have multiple financing packages

involving different interest rates, rebates, down payments, and length of the loan.

Sometimes a zero-interest loan is offered by the manufacturer, but it might require

that you give up some or all of the factory rebate. Maybe it would be better for you to

borrow from your bank rather than the dealer. To make the best financing choice, you

need to understand the time value of money, so read on!

139

140

Part 2 Fixed Income Securities •

Corporate Valuation and the Time Value of Money

In Chapter 1 we explained (1) that managers should strive to

make their firms more valuable, and (2) that the value of a

firm is determined by the size, timing, and risk of its free cash

flows (FCFs). Recall from Chapter 2 that free cash flows are the

cash flows available for distribution to all of a firm's investors

(stockholders and creditors). We explain how to calculate the

weighted average cost of capital (WACC) in Chapter 9, but it is

enough for now to think of the WACC as the average rate of re

turn required by all of the firm's investors. The intrinsic value

of a company is given by the following diagram. Note that

central to this value is discounting the free cash flows at the

WACC in order to find the value of the firm. This discounting is

one aspect of the time value of money. We discuss time value

of money techniques in this chapter.

•

Net operating profit after taxes

Required investments in operating capital

Free cash flow (FCF)

�� Value"'-----+-----+ ... +----- (1 + WACC)l (1 + WACC)2 (1 + WACC)

Weighted average cost of capital (WACC)

•

Market interest rates Cost of debt

Cost of equity

Firm's debt/equity mix

Market risk aversion Firm's business risk

In Chapter 1, we saw that the primary objective of a corporation is to maximize the in trinsic value of its stock. We also saw that stock values depend on the timing of the cash flows investors expect from an investment-a dollar expected sooner is worth more than a dollar expected further in the future. Therefore, it is essential for financial managers to understand the time value of money and its impact on stock prices. In this chapter, we will explain exactly how the timing of cash flows affects asset values and rates of return.

The principles of time value analysis have many applications, including retirement plan ning, loan payment schedules, and decisions to invest (or not) in new equipment. In fact, of all the concepts used in finance, none is more important than the time value of money (TVM), also called discounted cash flow (DCF) analysis. Time value concepts are used throughout the remainder of the book, so it is vital that you understand the material in this chapter and be able to work the chapter's problems before you move on to other topics.

4-1 Time Lines The first step in a time value analysis is to set up a time line to help you visual ize what's happening in the particular problem. To illustrate, consider the follow ing diagram, where PV represents $100 that is in a bank account today and FV

The textbook's Web site

contains an Excel file that

will guide you through the

chapter's calculations.

The file far this chapter is

Ch04 Taal Kit.xlsx, and we

encourage you to open the

file and fol/aw along as

you read the chapter.

Chapter 4 Time Value of Money 141

is the value that will be in the account at some future time (3 years from now in this example):

Periods 0 5% 1 2 3

Cash PV = $100 FV=?

The intervals from 0 to 1, 1 to 2, and 2 to 3 are time periods such as years or months. Time 0 is today, and it is the beginning of Period l; Time 1 is one period from today, and it is both the end of Period 1 and the beginning of Period 2; and so on. In our example, the periods are years, but they could also be quar ters or months or even days. Note again that each tick mark corresponds to both the end of one period and the beginning of the next one. Thus, if the periods are years, the tick mark at Time 2 represents both the end of Year 2 and the beginning of Year 3.

Cash flows are shown directly below the tick marks, and the relevant interest rate is shown just above the time line. Unknown cash flows, which you are trying to find, are indicated by question marks. Here the interest rate is 5%; a single cash outflow, $100, is invested at Time 0; and the Time-3 value is unknown and must be found. In this example, cash flows occur only at Times 0 and 3, with no flows at Times 1 or 2. We will, of course, deal with situations where multiple cash flows occur. Note also that in our example the interest rate is constant for all 3 years. The interest rate is gen erally held constant, but if it varies, then in the diagram we show different rates for the different periods.

Time lines are especially important when you are first learning time value concepts, but even experts use them to analyze complex problems. Throughout the book, our procedure is to set up a time line to show what's happening, pro vide an equation that must be solved to find the answer, and then explain how to solve the equation with a regular calculator, a financial calculator, and a computer spreadsheet.

SELF-TEST

Do time lines deal only with years, or could other periods be used?

Set up a time line to illustrate the following situation: You currently have $2,000 in a 3-year certificate of deposit (CD) that pays a guaranteed 4% annually. You want to know the value of the CD after 3 years.

4-2 Future Values

A dollar in hand today is worth more than a dollar to be received in the future; if you had the dollar now, you could invest it, earn interest, and end up with more than one dollar in the future. The process of going forward, from present values (PVs) to future values (FVs), is called compounding. To illustrate, refer back to our 3-year time line and assume that you have $100 in a bank account that pays a guaranteed 5% interest each year. How much would you have at the end of Year 3? We first define some terms, and then we set up a time line and show how the future value is calculated.

142 Part 2 Fixed Income Securities

PV = Present value, or beginning amount. In our example, PV = $100.

FV N = Future value, or ending amount, in the account after N periods. Whereas PV is the value now, or the present value, FV N is the value N periods into the future, after interest earned has been added to the account.

CF t

= Cash flow. Cash flows can be positive or negative. For a borrower, the first cash flow is positive and the subsequent cash flows are negative, and the reverse holds for a lender. The cash flow for a particular period is often given a subscript, CF,, where t is the period. Thus, CF

0 = PV = the cash

flow at Time 0, whereas CF 3

would be the cash flow at the end of Period 3. In this example the cash flows occur at the ends of the periods, but in some problems they occur at the beginning.

I = Interest rate earned per year. (Sometimes a lowercase i is used.) Interest earned is based on the balance at the beginning of each year, and we assume that interest is paid at the end of the year. Here I = 5% or, expressed as a decimal, 0.05. Throughout this chapter, we designate the interest rate as I (or I/YR, for interest rate per year) because that symbol is used on most financial calculators. Note, though, that in later chapters we use the symbol "r" to denote the rate because r (for rate of return) is used more often in the finance literature. Also, in this chapter we generally assume that interest payments are guaranteed by the U.S. government and hence are riskless (i.e., certain). In later chapters we will deal with risky investments, where the rate actually earned might be different from its expected level.

INT = Dollars of interest earned during the year = (Beginning amount) X I. In our example, INT = $100(0.05) = $5 for Year 1, but it rises in subsequent years as the amount at the beginning of each year increases.

N = Number of periods involved in the analysis. In our example, N = 3. Sometimes the number of periods is designated with a lowercase n, so both N and n indicate number of periods.

We can use four different procedures to solve time value problems.1 These methods are described next.

4-2a Step-by-Step Approach

The time line itself can be modified and used to find the FV of $100 compounded for 3 years at 5%, as shown here:

Time 0 5% 2 3

Amount at beginning of period $100.00 ___.. $105.00 - $110.25 - $115.76

'A fifth procedure is called the tabular approach, which uses tables that provide "interest factors"; this pro cedure was used before financial calculators and computers became available. Now, though, calculators and

spreadsheets such as Excel are programmed to calculate the specific factor needed for a given problem, which is then used to find the FV. This is much more efficient than using the tables. Also, calculators and spreadsheets can handle fractional periods and fractional interest rates. For these reasons, tables are not used in business today; hence we do not discuss them.

(

Chapter 4 Time Value of Money 143

We start with $100 in the account, which is shown at t = 0. We then multiply the initial amount, and each succeeding beginning-of-year amount, by (1 + I) = (1.05).

• You earn $100(0.05) = $5 of interest during the first year, so the amount at the end of Year 1 (or at t = 1) is:

FV I = PV + INT = PV + PV(I) = PV(l + I) = $100(1 + 0.05) = $100(1.05) = $105

• We begin the second year with $105, earn 0.05($105) = $5.25 on the now larger beginning-of-period amount, and end the year with $110.25. Interest during Year 2 is $5.25, and it is higher than the first year's interest, $5, because we earned $5(0.05) = $0.25 interest on the first year's interest. This is called "compounding," and interest earned on interest is called "compound interest."

• This process continues, and because the beginning balance is higher in each succes sive year, the interest earned each year increases.

• The total interest earned, $15.76, is reflected in the final balance, $115.76.

The step-by-step approach is useful because it shows exactly what is happening. However, this approach is time-consuming, especially if the number of years is large and you are using a calculator rather than Excel, so streamlined procedures have been developed.

4-2b Formula Approach

In the step-by-step approach, we multiplied the amount at the beginning of each period by (1 + I) = (1.05). Notice that the value at the end of Year 2 is:

FV 2

= FVp + I) = PV(l + 1)(1 + I) = PV(l + 1)2

= 100(1.05)2 = $110.25

If N = 3, then we multiply PV by (1 + I) three different times, which is the same as multiplying the beginning amount by (1 + I)3. This concept can be extended, and the re sult is this key equation:

FV N

= PV(l + I)N ■ We can apply Equation 4-1 to find the FV in our example:

FV 3

= $100(1.05)3 = $115.76

Equation 4-1 can be used with any calculator, even a nonfinancial calculator that has an exponential function, making it easy to find FVs no matter how many years are involved.

4-2c Financial Calculators

Financial calculators were designed specifically to solve time value problems. First, use the calculator's main menu and choose the time value of money sub-menu. (Many calcu lators use TVM to denote this sub-menu.). There will be five keys that correspond to the

144 Part 2 Fixed Income Securities

five variables in the basic time value equations. Equation 4-1 has only four variables, but we show the full equation here because it is important for you to understand the negative and positive signs used in calculators. In the following equation, PMT denotes a constant cash flow for all periods:

[(l + I)N - l] PV(l + I)N + PMT I + FV = 0

If we delete the term with PMT, we get: PV(l + I)N + FV = 0 ■

We show the inputs for our example above their keys in the following diagram, and the output, which is the FV, below its key. Because there are no periodic payments in this example, we enter 0 for PMT. Assuming that I is less than 100%, either PV or FV must be negative if the left side of Equation 4-2a is to be equal to zero. Think about it like this: If you deposit money today (PV), that is a cash flow out of your pocket; when you withdraw it in the future (FV), that is a cash flow into your pocket. Conversely, if you borrow money today (PV), that is cash into your pocket; when you pay it back in the future (FV), that is cash out of your pocket. We describe the keys in more detail below the diagram. Inputs 3

Output

5 -100 0 ( !f YR J ( PV J ( PMT j FV

115.76

N = Number of periods = 3. Some calculators use n rather than N. I/YR = Interest rate per period = 5. Some calculators use i or I rather than I/YR. Calculators are programmed to automatically convert the 5 to the decimal 0.05 before doing the arithmetic.

PV = Present value = 100. In our example we begin by making a deposit, which is an outflow of 100, so the PV is entered with a negative sign. On most calculators you must enter the 100, then press the +/- key to switch from + 100 to -100. If you enter -100 directly, this will subtract 100 from thelast number in the calculator, which will give you an incorrect answerunless the last number was zero.

PMT = Payment. This key is used if we have a series of equal, or constant, payments. Because there are no such payments in our current problem, we enter PMT = 0. We will use the PMT key later in this chapter. FV = Future value. In our example, the calculator automatically shows the FV as a positive number because we entered the PV as a negative number. If we had entered the 100 as a positive number, then the FV would have been negative. Calculators automatically assume that either the PV or the FV must be negative.

Chapter 4 Time Value of Money

Hints on Using Financial Calculators

When using a financial calculator, make sure it is set up as in

dicated below. Refer to your calculator manual or to our cal

culator tutorial on the textbook's Web site for information on

setting up your calculator.

♦ One payment per period. Many calculators "come out of

the box" assuming that 12 payments are made per year;

that is, they assume monthly payments. However, in this

book we generally deal with problems in which only one

payment is made each year. Therefore, you should set your

calculator at one payment per year and leave it there. See

our tutorial or your calculator manual if you need assis

tance. We will show you how to solve problems with more

than 1 payment per year in Section 4-15.

♦ End mode. With most contracts, payments are made

at the end of each period. However, some contracts

call for payments at the beginning of each period. You

can switch between "End Mode" and "Begin Mode" de

pending on the problem you are solving. Because most

of the problems in this book call for end-of-period pay

ments, you should return your calculator to End Mode

after you work a problem in which payments are made

at the beginning of periods.

♦ Negative sign for outflows. There are three cash flows

in Equation 4-2: PV, PMT, or FV. Don't forget that at least

one of the three cash flows must be negative and at least

one must be positive. There are two possible situations:

(1) one negative cash flow is out of your pocket and two

positive cash flows are into your pocket, or (2) two nega

tive cash flows are out of your pocket and one positive

cash flow is into your pocket. This generally means typing

outflows as positive numbers and then pressing the +/

key to convert from + to - before hitting the enter key.

♦ Decimal places. When doing arithmetic, calculators al

low you to show from Oto 11 decimal places on the dis

play. When working with dollars, we generally specify

two decimal places. When dealing with interest rates, we

generally specify two places if the rate is expressed as a

percentage, like 5.25%, but we specify four places if the

rate is expressed as a decimal, like 0.0525.

♦ Interest rates. For arithmetic operations with a nonfinan

cial calculator, the rate 5.25% must be stated as a decimal,

.0525. However, with a financial calculator you must en

ter 5.25, not .0525, because financial calculators are pro

grammed to assume that rates are stated as percentages.

145

As noted in our example, you first enter the four known values (N, I/YR, PV, and PMT) and then press the FV key to get the answer, FV = 115.76.

1,j,,-.r.J ce

See Ch04 Tool Kit.xlsx for

all calculations.

4-2d Spreadsheets

Spreadsheets are ideally suited for solving many financial problems, including those deal ing with the time value of money.2 Spreadsheets are obviously useful for calculations, but they can also be used like a word processor to create exhibits like our Figure 4-1, which includes text, drawings, and calculations. We use this figure to show that four methods can be used to find the FV of $100 after 3 years at an interest rate of 5%. The time line on Rows 36 to 37 is useful for visualizing the problem, after which the spreadsheet calculates the required answer. Note that the letters across the top designate columns, the numbers down the left column designate rows, and the rows and columns jointly designate cells.

'The file Ch04 Taal Kit.xlsx on the book's Web site does the calculations in the chapter using Excel. We highly recommend that you study the models in this Tool Kit. Doing so will give you practice with Excel, and that will help you tremendously in later courses, in the job market, and in the workplace. Also, going through the mod els will improve your understanding of financial concepts.

146 Part 2 Fixed Income Securities

FIGURE4-1

Alternative Procedures for Calculating Future Values

A I B C D E I F G 31 INPUTS:

32 Invesbnent = CF0 = PV = -$100.00

33 Interest rate = I = 5% 34 No. of periods = N = 3 35

36 Time Line Periods: 0 1 I 2 I 37 Cash flow: -$100.00 0 0 FV=?

39 1. Step-by-Step: Multiply by (1 + I) each step $100.00➔ $105.00➔ $110.25 ➔ $115.76 40

41 2. Formula: FVN = PV(l+lt FV3 = $100(1.05) 3 = $115.76

43 Inputs: 3 5 -100 0

I 44 3. Financial Calculator: N I/YR PV PMT FV 45 Output: $115.76 46

47 4. Excel Spreadsheet: FV function: FVN = =FV(l,N,0,PV) -

48 Fixed inputs: FVN= =FV(0.05,3,0,-100) = $115.76 -

49 Cell references: FVN = =FV(C33,C34,0,C32) = $115.76

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision. Thus, inter

mediate calculations using the figure's rounded values will be inexact.

Thus, cell C32 shows the amount of the investment, $100, and it is given a minus sign be cause it is an outflow.

It is useful to put all inputs in a section of the spreadsheet designated "INPUTS." In Figure 4-1, we put the inputs in the aqua-colored range of cells. Rather than enter fixed numbers into the model's formulas, we enter the cell references for the inputs. This makes it easy to modify the problem by changing the inputs and then automatically use the new data in the calculations.

Time lines are important for solving finance problems because they help us visualize what's happening. When we work a problem by hand, we usually draw a time line, and when we work a problem with Excel, we set the model up as a time line. For example, in Figure 4-1 Rows 36 to 37 are indeed a time line. It's easy to construct time lines with Excel, with each column designating a different period on the time line.

On Row 39, we use Excel to go through the step-by-step calculations, multiplying the beginning-of-year values by (1 + I) to find the compounded value at the end of each pe riod. Cell G39 shows the final result of the step-by-step approach.

We illustrate the formula approach in Row 41, using Excel to solve Equation 4-1 to find the FV. Cell G41 shows the formula result, $115.76. As it must, it equals the step-by-step result.

Rows 43 to 45 illustrate the financial calculator approach, which again produces the same answer, $115.76.

Chapter 4 Time Value of Money 147

The last section, in Rows 47 to 49, illustrates Excel's future value (FV) function. You can access the function wizard by clicking the fx symbol in Excel's formula bar. Then select the category for Financial functions, and then the FV function, which is =FV(I,N,O,PV), as shown in Cell E47.3 Cell E48 shows how the formula would look with numbers as inputs; the actual function itself is entered in Cell G48, but it shows up in the table as the answer, $115.76. If you access the model and put the pointer on Cell G48, you will see the full for mula. Finally, Cell E49 shows how the formula would look with cell references rather than fixed values as inputs, with the actual function again in Cell G49. We generally use cell references as function inputs because this makes it easy to change inputs and see how those changes affect the output. This is called "sensitivity analysis." Many real-world financial applications use sensitivity analysis, so it is useful to form the habit of setting up an input data section and then using cell references rather than fixed numbers in the functions.

When entering interest rates in Excel, you can use either actual numbers or percent ages, depending on how the cell is formatted. For example, we first formatted Cell C33 to Percentage, and then typed in 5, which showed up as 5%. However, Excel uses 0.05 for the arithmetic. Alternatively, we could have formatted C33 as a Number, in which case we would have typed 0.05. If a cell is formatted to Number and you enter 5, then Excel would think you meant 500%. Thus, Excel's procedure is different from the convention used in financial calculators.

Sometimes students are confused about the sign of the initial $100. We used +$100 in Rows 39 and 41 as the initial investment when calculating the future value using the step-by-step method and the future value formula, but we used -$100 with a financial calculator and the spreadsheet function in Rows 43 and 48. When must you use a posi tive value, and when must you use a negative value? The answer is that whenever you set up a time line and use either a financial calculator's time value functions or Excel's time value functions, you must enter the signs that correspond to the "direction" of the cash flows. Cash flows that go out of your pocket (outflows) are negative, but cash flows that come into your pocket (inflows) are positive. In the case of the FV function in our example, if you invest $100 (an outflow and therefore negative) at Time 0, then the bank will make available to you $115. 76 (an inflow and therefore positive) at Time 3. In essence, the FV function on a financial calculator or Excel answers the question, "If I invest this much now, how much will be available to me at a time in the future?" The investment is an outflow and negative, and the amount available to you is an in flow and positive. If you use algebraic formulas, then you must keep track of whether the value is an outflow or an inflow yourself. When in doubt, refer back to a correctly constructed time line.

4-2e Comparing the Procedures

The first step in solving any time value problem is to understand what is happening and then to diagram it on a time line. Woody Allen said that 90% of success is just showing up. With time value problems, 90% of success is correctly setting up the time line.

After you diagram the problem on a time line, your next step is to pick one of the four approaches shown in Figure 4-1 to solve the problem. Any one approach may be used, but your choice will depend on the particular situation.

3 All functions begin with an equal sign. The third entry is zero in this example, which indicates that there are no periodic payments. Later in this chapter we will use the FV function in situations where we have nonzero pe riodic payments. Also, for inputs we use our own notation, which is similar but not identical to Exce/'s notation.

148

Q..urce

See Ch04 Tool Kit.xlsx for

all calculations.

Part 2 Fixed Income Securities

All business students should know Equation 4-1 by heart and should also know how to use a financial calculator. So, for simple problems such as finding the future value of a single payment, it is generally easiest and quickest to use either the formula approach or a financial calculator. However, for problems that involve several cash flows, the formula approach usually is time-consuming, so either the calculator or spreadsheet approach would generally be used. Calculators are portable and quick to set up, but if many calcula tions of the same type must be done, or if you want to see how changes in an input such as the interest rate affect the future value, then the spreadsheet approach is generally more efficient. If the problem has many irregular cash flows, or if you want to analyze alterna tive scenarios using different cash flows or interest rates, then the spreadsheet approach definitely is the most efficient procedure.

Spreadsheets have two additional advantages over calculators. First, it is easier to check the inputs with a spreadsheet because they are visible; with a calculator the inputs are bur ied somewhere in the machine. Thus, you are less likely to make a mistake in a complex problem when you use the spreadsheet approach. Second, with a spreadsheet, you can make your analysis much more transparent than you can when using a calculator. This is not necessarily important when all you want is the answer, but if you need to present your calculations to others, like your boss, it helps to be able to show intermediate steps, which enables someone to go through your exhibit and see exactly what you did. Transparency is also important when you must go back, sometime later, and reconstruct what you did.

You should understand the various approaches well enough to make a rational choice, given the nature of the problem and the equipment you have available. In any event, you must understand the concepts behind the calculations, and you also must know how to set up time lines in order to work complex problems. This is true for stock and bond valu ation, capital budgeting, lease analysis, and many other types of financial problems.

4-2f Graphic View of the Compounding Process

Figure 4-2 shows how a $100 investment grows (or declines) over time at different inter est rates. Interest rates are normally positive, but the "growth" concept is broad enough to include negative rates. We developed the curves by solving Equation 4-1 with different values for N and I. The interest rate is a growth rate: If money is deposited and earns 5% per year, then your funds will grow by 5% per year. Note also that time value con cepts can be applied to anything that grows-sales, population, earnings per share, or your future salary. Also, as noted before, the "growth rate" can be negative. For example, Campbell Soup Company had negative revenue growth in 2017.

4-2g Simple Interest versus Compound Interest

As explained earlier, when interest is earned on the interest earned in prior periods, we call it compound interest. If interest is earned only on the principal, we call it simple interest; it is also called regular interest. The total interest earned with simple inter est is equal to the principal multiplied by the interest rate times the number of periods: PV(I)(N). The future value is equal to the principal plus the interest: FV = PV + PV(I)(N). For example, suppose you deposit $100 for 3 years and earn simple interest at an annual rate of 5%. Your balance at the end of 3 years would be:

FV = PV + PV(I)(N) = $100 + $100(5%)(3) = $100 + $15 = $115

Chapter 4 Time Value of Money

FIGURE4-2

Growth of$100 at Various Interest Rates and Time Periods

FV of $100 After N Years

$600

$500

$400

$300

$200

$0 l _ _J __ =====:=:::=::;:==:i::'.=:� I

� =

�-� 20

� %

�=:::::=i l 2 3 4 5 6 7 8 9 10

Years

149

Notice that this is less than the $115.76 we calculated earlier using compound interest. Most applications in finance are based on compound interest, but you should be aware that simple interest is still specified in some legal documents.

SELF-TEST

Explain why this statement is true: '\I\ dollar in hand today is worth more than a dollar to be received next year, assuming interest rates are positive."

What is compounding? What would the future value of $100 be after 5 years at 10% compound interest? ($161.05)

Suppose you currently have $2,000 and plan to purchase a 3-year certificate of deposit (CD) that pays 4% interest, compounded annually. How much will you have when the CD matures? ($2,249.73) How would your answer change if the interest rate were 5%, or 6%, or 20%? (Hint: With a calculator, enter N = 3, I/YR = 4, PV = -2000, and PMT = 0; then press FV to get 2,249.73. Then, enter I/YR = 5 to override the 4% and press FV again to get the second answer. In general, you can change one input at a time to see how the output changes.) ($2,315.25; $2,382.03; $3,456.00)

A company's sales in 2012 were $100 million. If sales grow by 8% annually, what will they be 10 years later? ($215.89 million) What would they be if they decline by 8% per year for 10 years? ($43.44 million)

How much would $1, growing at 5% per year, be worth after 100 years? ($131.50) What would FV be if the growth rate were 10%? ($13,780.61)

150

It's a Matter of Trust

One of our Founding Fathers, Benjamin Franklin, wanted to

make a donation to the cities of Boston and Philadelphia,

but he didn't want the cities to squander the money before

it had grown enough to make a big impact. His solution was

the "Methuselah Trust." When Franklin died in 1790, his will

left £1,000, at that time about $4,550, to Philadelphia and

to Boston, but on the condition that it would be invested for

100 years, after which some of the proceeds were to be used

for the public projects (primarily trade schools and water

works) and the rest invested for another 100 years. Depend

ing on interest rates, this strategy could generate quite a bit of

money! For example, if half of his bequest, $2,275, remained

invested at 5% compound interest for 200 years, the value in

1990 would be $39.3 million! The ultimate payout, however,

was only about $7 million, because substantial amounts were

eaten up by trustee fees, taxes, and legal battles. Franklin

certainly would have been disappointed!

In 1936 an eccentric investor and New York lawyer,

Jonathan Holden, decided to expand on Franklin's idea by do

nating a series of 500-year and 1,000-year trusts to Hartwick

4-3 Present Values

Part 2 Fixed Income Securities

College and several other recipients. By 2008 Hartwick College's

trust had grown in value to about $9 million; if invested at 5%

for the remaining 928 years _of its planned life, its value would

grow to ($9 million)(l.05)928 = $4.15 x 1026• That is a lot of

dollars by any measure! For example, that amount in million

dollar bills (if they existed by then) would paper the earth

10,000 times over, or laid end to end would reach the nearest

star, Alpha Centauri, more than 1,000 times. In a move that

surely would have disappointed Holden, Hartwick College

was able to convert the trust into annual cash flows of about

$450,000 a year.

The trusts of Franklin and Holden didn't turn out exactly

as they had planned-Franklin's trust didn't grow adequately,

and Holden's trust was converted into annual cash flows. This

goes to show that you can't always trust a trust!

Sources: Jake Palmateer, "On the Bright Side: Hartwick College Receives

$9 million Trust," The Daily Star, Oneonta, NY, January 22, 2008, http://

thedallystar.com/local/x112892349/0n-The-Brlght-Side-Hartwlck

-College-recelves-9M-trust/prlnt; Lewis H. Lapham, "Trust Issues,"

Lapham's Quarterly, Friday, December 2, 2011, www.laphamsquarterly

,org/essays/trust-lssues.php?page=l.

Suppose you have some extra money and want to make an investment. A broker offers to sell you a bond that will pay a guaranteed $115.76 in 3 years. Banks are currently offering ct guaranteed 5% interest on 3-year certificates of deposit (CDs), and if you don't buy the bond, you will buy a CD. The 5% rate paid on the CD is defined as your opportunity cost, or the rate of return you would earn on an alternative investment of similar risk if you don't invest in the security under consideration. Given these conditions, what's the most you should pay for the bond?

4-3a Discounting a Future Value to Find the Present Value

First, recall from the future value example in the last section that if you invested $100 at 5% in a CD, it would grow to $115.76 in 3 years. You would also have $115.76 af ter 3 years if you bought the bond. Therefore, the most you should pay for the bond is $100-this is its "fair price," which is also its intrinsic, or fundamental, value. If you could buy the bond for less than $100, then you should buy it rather than in vest in the CD. Conversely, if its price were more than $100, you should buy the CD. If the bond's price were exactly $100, you should be indifferent between the bond and the CD.

The $100 is defined as the present value, or PV, of $115.76 due in 3 years when the ap propriate interest rate is 5%. In general, the present value of a cash JI.ow due N years in the future is the amount that, if it were on hand today, would grow to equal the given future

,:.e.s..o u rce

Chapter 4 Time Value of Money 151

amount. Because $100 would grow to $115.76 in 3 years at a 5% interest rate, $100 is the present value of $115.76 due in 3 years at a 5% rate.

Finding present values is called discounting, and as previously noted, it is the re verse of compounding: If you know the PV, you can compound it to find the FV; if instead you know the FV, you can discount it to find the PV. Indeed, we simply solve Equation 4-1, the formula for the future value, for the PV to produce the present value equation. We repeat Equation 4-1 here to see the difference between compounding and discounting:

Compounding to find future values: Future value = FV N

= PV(l + J)N

FV N

Discounting to find present values: Present value = PV = ( 1 + I)N

■

■ See Ch04 Tool Kit.xlsx for

oil colculotions.

The top section of Figure 4-3 shows inputs and a time line for finding the present value of $115.76 discounted back for 3 years. We first calculate the PV using the step by-step approach. When we found the FV in the previous section, we worked from left to right, multiplying the initial amount and each subsequent amount by (1 + I).

FIGURE4-3

Alternative Procedures for Calculating Present Values

A I B C D E I F G 97 INPUTS:

98 Future payment= CFN = FV = $115.76

99 Interest rate = I = 5.00% 100 No. of periods= N = 3 101

102 Time Line Periods: 0 1 I 2 3 -

I 103 Cash flow: PV=? 0 0 $115.76 104

105 1. Step-by-Step: $100.00 +-$105.00 +-$110.25 +-$115.76 106

107 2. Formula: PV = FVN/(1+1t PV = $115.76/(1.05) 3 = $100.00

108

109 Inputs: 3 5 0 115.76

I 110 3. Financial Calculator: N I/YR PV PMT FV -

I 111 Output: -$100.00 112

113 4. Excel Spreadsheet: PV function: PV= =PV(l,N,0,FV)

114 Fixed inputs: PV= =PV(0.05,3,0,115.76) = -$100.00

115 Cell references: PV= =PV(C99,C100,0,C98) = -$100.00

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision.

Thus, intermediate calculations using the figure's rounded values will be inexact.

152 Part 2 Fixed Income Securities

To find present values, we work backwards, or from right to left, dividing the future value and each subsequent amount by (1 + I), with the present value of $100 shown in Cell D105. The step-by-step procedure shows exactly what's happening, and that can be quite useful when you are working complex problems or trying to explain a model to others. However, it's inefficient, especially if you are dealing with more than a year or two.

A more efficient procedure is to use the formula approach in Equation 4-3, simply dividing the future value by (1 + I)N. This gives the same result, as we see in Figure 4-3, Cell Gl07.

Equation 4-2 is actually programmed into financial calculators. As shown in Figure 4-3, Rows 109 to 111, we can find the PV by entering values for N = 3, I/YR = 5, PMT = 0, and FV = 115.76, and then pressing the PV key to get -100.

Inputs 3 5 0 115.76

N I/YR PV PMT FV

Output -100

Excel also has a function that solves Equation 4-3-this is the PV function, and it is written as =PV(I,N,O,FV}.4 Cell E113 shows the inputs to this function. Next, Cell E114 shows the Excel function with fixed numbers as inputs, with the actual function and the resulting -$100 in Cell G114. Cell EllS shows the Excel function using cell references, with the actual function and the resulting -$100 in Cell GllS.

As with the future value calculation, students often wonder why the result of the present value calculation is sometimes positive and sometimes negative. In the alge braic calculations in Rows 105 and 107 the result is +$100, while the result of the cal culation using a financial calculator or Excel's function in Rows 111 and 114 is -$100. Again, the answer is in the signs of a correctly constructed time line. Outflows are negative and inflows are positive. The PV function for Excel and a financial calcula tor answer the question, "How much must I invest today in order to have available to me a certain amount of money in the future?" If you want to have $115.76 available in 3 years (an inflow to you and therefore positive), then you must invest $100 today (an outflow and therefore negative). If you use the algebraic functions as in Rows 105 and 107, you must keep track of whether the results of your calculations are inflows or outflows.

The fundamental goal of financial management is to maximize the firm's intrinsic value, and the intrinsic value of a business (or any asset, including stocks and bonds) is the present value of its expected future cash flows. Because present value lies at the heart of the valuation process, we will have much more to say about it in the remainder of this chapter and throughout the book.

4-3b Graphic View of the Discounting Process

Figure 4-4 shows that the present value of a sum to be received in the future decreases and approaches zero as the payment date is extended further and further into the future; it also shows that the higher the interest rate, the faster the present value falls. At relatively high rates, funds due far in the future are worth very little today, and even at relatively low rates present values of sums due in the very distant future are quite small. For example, at a 20%

'The third entry in the PV function is zero to indicate that there are no intermediate payments in this example.

s..Qurce

See Ch04 Tool Kit.xlsx for

oil co/culotions.

Chapter 4 Time Value of Money

FIGURE4-4

Present Value of $100 at Various Interest Rates and Time Periods

Present Value of$100

$100 .---------------------

$80

$60

$40

$20

10 20 30 40 Years

153

discount rate, $100 due in 40 years would be worth less than 7 cents today. (However, 1 cent would grow to almost $1 million in 100 years at 20%.)

SELF-TEST

What is "discounting," and how is it related to compounding? How is the future value Equation (4-1) related to the present value Equation {4-3)?

How does the present value of a future payment change as the time to receipt is lengthened? As the interest rate increases?

Suppose a risk-free bond promises to pay $2,249.73 in 3 years. If the going risk-free interest rate is 4%, how much is the bond worth today? ($2,000) How much is the bond worth if it matures in 5 rather than 3 years? ($1,849.11) If the risk-free interest rate is 6% rather than 4%, how much is the 5-year bond worth today? ($1,681.13)

How much would $1 million due in 100 years be worth today if the discount rate were 5%? ($7,604.49) What if the discount rate were 20%? ($0.0121)

4-4 Finding the Interest Rate, I / We have used Equations 4-1, 4-2, and 4-3 to find future and present values. Those equa

tions have four variables, and if we know three of them, then we (or our calculator or Excel) can solve for the fourth. Thus, if we know PV, I, and N, we can solve Equation 4-1 for FV, or if we know FV, I, and N, we can solve Equation 4-3 to find PV. That's what we did in the preceding two sections.

Now suppose we know PV, FV, and N, and we want to find I. For example, suppose we know that a given security has a cost of$100 and that it will return $150 after 10 years. Thus, we know PV, FV, and N, and we want to find the rate of return we will earn if we buy the security. Starting with Equation 4-1 and solving for I gives Equation 4-4:

154

IJ.L_(Ce

See Ch04 Tool Kit.xlsx for

all calculations.

Part 2 Fixed Income Securities

Using Equation 4-4 with the input values, I is 4.14%: [$150](1/lO)I = $l50 - 1 = 0.0414 = 4.14%

Finding the interest rate by solving the formula takes a little time and thought, but financial calculators and spreadsheets find the answer almost instantly. Here's the calculator setup: Inputs

Output

N I/YR 4.14

-100

0 PMT

150

Enter N = 10, PV = -100, PMT = 0 ( because there are no payments until the security matures), and FV = 150. Then, when you press the I/YR key, the calculator gives the answer, 4.14%. Notice that the PV is a negative value because it is a cash outflow (an investment) and the FV is positive because it is a cash inflow (a return of the investment). If you enter both PV and FV as positive numbers (or both as negative numbers), you will get an error message rather than the answer. In Excel, the RATE function can be used to find the interest rate: =RATE{N,PMT,PV,FV).For this example, the interest rate is found as =RATE(I0,0,-100,150) = 0.0414 = 4.14%. See the file Ch04 Tool Kit.xlsx on the textbook's Web site for an example. SELF-TEST

Suppose you can buy a U.S. Treasury bond that makes no payments until the bond matures 10 years from now, at which time it will pay you $1,000. What interest rate would you earn if you bought this bond for $585.43? (5.5%) What rate would you earn if you could buy the bond for $550? (6.16%) For $600? (5.24%)

Microsoft earned $0.33 per share in 1997. Fourteen years later, in 2011, it earned $2. 75. What was the growth rate in Microsoft's earnings per share (EPS) over the 14-year period? (16.35%) If EPS in 2011 had been $2.00 rather than $2. 75, what would the growth rate have been? (13.73%)

4-5 Finding the Number of Years, N We sometimes need to know how long it will take to accumulate a specific sum of money, given our beginning funds and the rate we will earn. For example, suppose we now have $500,000 and the interest rate is 4.5%. How long will it be before we have $1 million? Starting with Equation 4-1 and solving for N gives Equation 4-5:

N = [LN(FVN/PV)]LN(l + I) Using Equation 4-5 with the input values, N is:

[LN ($1,000,000/$500,000)] N = -------- = 15.7473 LN(l + 0.045)

s.c.urce

See Ch04 Tool Kit.xlsx for

oil calculations.

Chapter 4 Time Value of Money 155

As you might expect, financial calculator and spreadsheet approaches are easier. Here's the calculator setup:

Inputs 4.15

N I/YR

Output 15.7473

-500000

PMT

1000000

Enter I/YR = 4.5, PV = -500000, PMT = 0, and FV = 1000000. We press the N key to get the answer, 15.7473 years. In Excel, we would use the NPER function: =NPER(I,PMT,PV,FV). Inserting data, we have =NPER(0.045,0,-500000,1000000) = 15.7473. The chapter's tool kit, Ch04 Tool Kit.xlsx, shows this example.

SELF-TEST

How long would it take $1,000 to double if it were invested in a bank that pays 6% per year? (11.9 years) How long would it take if the rate were 10%? {7.27 years)

A company's 2013 earnings per share were $2.75, and its growth rate during the prior 14 years was 16.35% per year. If that growth rate were maintained, how long would it take for EPS to double? (4.58 years)

4-6 Perpetuities Section 4-3 showed how to find the present value of a single cash flow occurring N years in the future. However, some types of stocks and bonds pay a fixed amount each period but don't have an ending date. 5 Because the payments are perpetual, this pattern of cash flows is called a perpetuity. You can't apply the step-by-step approach because the cash flows never end, but it's easy to find the PV of a perpetuity with the following formula:6

PMT PV of a perpetuity = -

1 - I

For example, suppose the payment is $25 each period and the interest rate is 2.5%. Us ing Equation 4-6, the PV is:

PV = $25/0.025 = $1,000

What would happen to the present value if interest rates rises to 3%? The present value would fall to $833.33:

Present value if rates increase to 2% = PV = $25/0.03 = $833.33

'Chapters 5 and 7 describe types of bonds and stocks that don't have ending dates for payments.

6Here is an intuitive explanation for Equation 4-6. Suppose you deposit an amount equal to PV in a bank that pays an interest rate of I. Each year you would be able to withdraw an amount equal to I X PV. If you left your deposit in forever, you could withdraw a payment of I X PV forever: PMT = I X PV. Rearranging, we get Equation 4-6. This is only an intuitive explanation, so see Web Extension 4A on the textbook's Web site for a mathematical derivation of the perpetuity formula.

156 Part 2 Fixed Income Securities

Note, though, that if interest rates fell to 2%, then the present value would rise to $1,250.00:

Present value if rates decline to 2% = $25/0.02 = $1,250.00

These examples demonstrate an important point: When interest rates change, the pres ent value also changes but inversely to the change in rates. Thus, the present value declines if rates rise and increases if rates fall.

SELF-TEST

What is the present value of a perpetuity that pays $1,000 per year, beginning 1 year from now, if the appropriate interest rate is 5%? ($20,000)

Do bond prices move directly or inversely with interest rates-that is, what happens to the value of a bond if interest rates increase or decrease?

4-7 Annuities

Thus far, we have dealt with a single payment (i.e., a "lump sum") or infinite payments. However, assets such as bonds provide a finite series of cash inflows over time, and obliga tions such as auto loans, student loans, and mortgages call for a finite series of payments. If the payments are equal and are made at fixed intervals, then we have an annuity. For exarpple, $100 paid at the end of each of the next 3 years is a 3-year annuity.

If payments occur at the end of each period, then we have an ordinary annuity, which is also called a deferred annuity. Payments on mortgages, car loans, and student loans are generally made at the ends of the periods and thus are ordinary annuities. If the payments are made at the beginning of each period, then we have an annuity due. Rental lease pay ments, life insurance premiums, and lottery payoffs (if you are lucky enough to win one!) are examples of annuities due. Ordinary annuities are more common in finance, so when we use the term "annuity" in this book, you may assume that the payments occur at the ends of the periods unless we state otherwise.

Next we show the time lines for a $100, 3-year, 5%, ordinary annuity and for the same annuity on an annuity due basis·. With the annuity due, each payment is shifted back (to the left) by 1 year. In our example, we assume that a $100 payment will be made each year, so we show the payments with minus signs.

Ordinary Annuity:

Periods 0 5% 1 2 3

Payments -$100 -$100 -$100

Annuity Due:

Periods 0 5% 1 2 3

Payments -$100 -$100 -$100

As we demonstrate in the following sections, we can find an annuity's future value, present value, the interest rate built into the contracts, how long it takes to reach a finan cial goal using the annuity, and, if we know all of those values, the size of the annuity

(

FIGURE4-5

Chapter 4 Time Value of Money 157

payment. Keep in mind that annuities must have constant payments and a fixed number of periods. If these conditions don't hold, then the series is not an annuity.

SELF-TEST

What's the difference between an ordinary annuity and an annuity due?

Why should you prefer to receive an annuity due with payments of $10,000 per year for 10 years than an otherwise similar ordinary annuity?

4-8 Future Value of an Ordinary Annuity Consider the ordinary annuity whose time line was shown previously, where you deposit $100 at the end of each year for 3 years and earn 5% per year. Figure 4-5 shows how to calculate the future value of the annuity (FV �), using the same approaches we used for single cash flows.

As shown in the step-by-step section of Figure 4-5, we compound each payment out to Time 3, and then sum those compounded values in Cell F254 to find the annuity's FV, FVA

3 = $315.25 (we ignore the negative signs in the time line to be consistent with the

Summary: Future Value of an Ordinary Annuity

A I B C D E I F G 243 INPUTS: 244 Payment amount= PMT = -$100 245 Interest rate = I = 5.00% 246 No. of periods= N = 3 247

248 1. Step-by- Periods: 0 1 2 I 3 - 249 Cash flow: -$100 -$100 I -$100 - 250 J. J. J. - 251 J. J. $100.00 -

L➔➔ 252 Multiply each payment by J. $105.00 -

253 (1+1t·t and sum these FVs to L➔·➔➔➔➔➔➔ $110.25 -

254 find FVAN: I $315.25 255

256 2. Formula:- 257

(C1�I) N

- ½) = PMT x ( (1+1r-1) = - =PMT x I I258 FVAN $315.25

259

260

261 Inputs: 3 5 0 -100 -

I 262 3. Financial Calculator: N I/YR PV PMT FV - 263 Output: $315.25 264

265 4. Excel Spreadsheet: FV function: FVAN= =FV(l,N,PMT,PV) -

266 Fixed inputs: FVAN= =FV(0.05,3,-100,0) = $315.25 -

267 Cell references: FVAN= =FV(C245,C246,C244,0) = $315.25

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision.

Thus, intermediate calculations using the figure's rounded values will be inexact.

158 Part 2 Fixed Income Securities

results from the other methods). The first payment earns interest for two periods, the second for one period, and the third earns no interest because it is made at the end of the annuity's life. This approach is straightforward, but if the annuity extends out for many years, it is cumbersome and time-consuming. As you can see from the time line diagram, with the step-by-step approach we apply the following equation with N = 3 and I = 5%: FVA

N = PMT(l + I)N-i + PMT(l + I)N-2 + PMT(l + I)N 3 = $100(1.05)2 + $100(1.05)1 + $100(1.05}° = $315.25

For the general case, the future value of an annuity is: FVA

N = PMT(l + I)N-1 + PMT(l + I)N-2 + PMT(l + I)N-3 + · · · + PMT(l +I)0

As shown in Web Extension 4A on the textbook's Web site, the future value of an annuity can be written as follows:7

[(1 + I)N 1] [<1 + I)N - 1]FV A N

= PMT I - I = PMT I

Using Equation 4-7, the future value of the annuity is $315.25: [(l + 0.05)3 - 1] FVA

3 = $100 ----- = $315.25 0.05

As you might expect, annuity problems can be solved easily using a financial calculator or a spreadsheet. The procedure when dealing with annuities is similar to what we have done thus far for single payments, but the presence of recurring payments means that we must use the PMT key. Here's the calculator setup for our illustrative annuity: Inputs 3 5

N I/YR Output

-100

PMT FV

315.25

We enter PV = 0 because we start off with nothing, and we enter PMT = -100 because we will deposit this amount in the account at the end of each of the 3 years. The interest rate is 5%, and when we press the FV key we get the answer, FVA 3

= 315.25. Because this is an ordinary annuity, with payments coming at the end of each year, we must set the calculator appropriately. As noted earlier, most calculators "come out of the box" set to assume that payments occur at the end of each period-that is, to deal with ordinary annuities. However, there is a key that enables us to switch between ordinary annuities and annuities due. For ordinary annuities, the designation "End Mode" or something 7Section 4-6 shows that the present value of an infinitely long annuity, called a "perpetuity," is equal to PMT/I. The cash flows of an ordinary annuity of N periods are equal to the cash flows of a perpetuity minus the cash flows of a perpetuity that begins at year N + l. Therefore, the future value of an N-period annuity is equal to the future value (as of year N) of a perpetuity minus the value (as of year N) of a perpetuity that begins at year N + l. See Web Extension 4A on the textbook's Web site for details regarding derivations of Equation 4-7.

Chapter 4 Time Value of Money 159

The Power of Compound Interest

Assume that you are 26 and just received your MBA. After read

ing the introduction to this chapter, you decide to start invest

ing in the stock market for your retirement. Your goal is to

have $1 million when you retire at age 65. Assuming you earn

10% annually on your stock investments, how much must you

invest at the end of each year in order to reach your goal?

age 40, you will need to save $10,168 per year to reach

your $1 million goal, assuming you can earn 10%, but

$13,679 per year if you earn only 8%. If you wait until

age 50 and then earn 8%, the required amount will be

$36,830 per year!

Although $1 million may seem like a lot of money, it won't

be when you get ready to retire. If inflation averages 5% a year

over the next39 years, then your $1 million nest egg would be

worth only $149,148 in today's dollars. If you live for 20 years

after retirement and earn a real 3% rate of return, your annual

retirement income in today's dollars would be only $9,733

before taxes. So, after celebrating your graduation and new

job, start saving!

The answer is $2,491, but this amount depends critically

on the return your investments earn. If your return drops to

8%, the required annual contribution would rise to $4,185. On

the other hand, if the return rises to 12%, you would need to

put away only $1,462 per year.

What if you are like most 26-year-olds and wait un

til later to worry about retirement? If you wait until

esource

See Ch04 Tool Kit.xlsx for

all calculations.

similar is used, while for annuities due, the designator is "Begin," "Begin Mode," "Due," or something similar. If you make a mistake and set your calculator on Begin Mode when working with an ordinary annuity, then each payment will earn interest for 1 extra year, which will cause the compounded amounts, and thus the FVA, to be too large.

The spreadsheet approach uses Excel's FV function, =FV(I,N,PMT,PV). In our example, we have =FV(0.05,3,-100,0}, and the result is again $315.25.

SELF-TEST

For an ordinary annuity with 5 annual payments of $100 and a 10% interest rate, for how many years will the first payment earn interest, and what is the compounded value of this payment at the end? (4 years, $146.41) Answer these questions for the fifth payment. (0 years, $100)

Assume that you plan to buy a condo 5 years from now, and you estimate that you can save $2,500 per year toward a down payment. You plan to deposit the money in a bank that pays 4% interest, and you will make the first deposit at the end of this year. How much will you have after 5 years? ($13,540.81) How would your answer change if the bank's interest rate were increased to 6%, or decreased to 3%? ($14,092.73; $13,272.84)

4-9 Future Value of an Annuity Due Because each payment occurs one period earlier with an annuity due, the payments will all earn interest for one additional period. Therefore, the FV of an annuity due will be greater than that of a similar ordinary annuity.

If you followed the step-by-step procedure, you would see that our illustrative annuity due has a FV of$331.0l versus $315.25 for the ordinary annuity. See Ch04 Tool Kit.xlsx on the textbook's Web site for a summary of future value calculations.

With the formula approach, we first use Equation 4-7, but because each payment occurs one period earlier, we multiply the Equation 4-7 result by (1 + I):

FVA due =

FVAo,dinory(l + I} ■ Thus, for the annuity due, FVA

du e = $315.25(1.05) = $331.01, which is the same result

as found with the step-by-step approach.

160

eso.JJrce

See Ch04 Tool Kit.xlsx for

o/1 co/culotions.

�so..urce

See Ch04 Tool Kit.xlsx for

all aolculotions.

Part 2 Fixed Income Securities

With a calculator, we input the variables just as we did with the ordinary annuity, but we now set the calculator to Begin Mode to get the answer, $331.01: Inputs 3 5

N I/YR

Output 0

-100

PMT

Begin Mode

FV J 331.01

In Excel, we still use the FV function, but we must indicate that we have an annuity due . The function is =FV(I,N,PMT,PV,Type), where "Type" indicates the type of annuity. If Type is omitted, then Excel assumes that it is 0, which indicates an ordinary annuity. For an annuity due, Type = 1. As shown in Ch04 Tool Kit.xlsx, the function is =FV(0.05,3,-100,0,1) = $331.01.

SELF-TEST

Why does an annuity due always have a higher future value than an ordinary annuity?

If you know the value of an ordinary annuity, explain why you could find the value of the corre sponding annuity due by multiplying by (1 + /).

Assume that you plan to buy a condo 5 years from now and that you need to save for a down payment. You pion to save $2,500 per year, with the first payment being made immediately and deposited in a bank that pays 4%. How much will you have after 5 years? ($14,082.44) How much would you have if you made the deposits at the end of each year? ($13,540.81)

4-10 Present Value of Ordinary Annuities and Annuities Due

The present value of any annuity (PVA N ) can be found using the step-by-step, formula, calculator, or spreadsheet method. We begin with ordinary annuities.

4-10a Present Value of an Ordinary Annuity

See Figure 4-6 for a summary of the different approaches for calculating the present value of an ordinary annuity. As shown in the step-by-step section of Figure 4-6, we discount each payment back to Time 0, and then sum those discounted values to find the annuity's present value, PVA 3

= $272.32. (We ignore the negative signs in the time line to be consistent with the results from the other methods.) This approach is straightforward, but if the annuity extends out for many years, it is cumbersome and time-consuming. The time line diagram shows that with the step-by-step approach we apply the following equation with N = 3 and I = 5%: PVA

N = PMT/(1 + 1) 1 + PMT/(1 + 1) 2 + · .. + PMT/(1 + I) N

The present value of an annuity can be written as:8 PV � = PMT[ ½- -1(-l -:-1)-N]

'See Web Extension 4A on the textbook's Web site for details of this derivation.

Chapter 4 Time Value of Money 161

FIGURE4·6

Summary: Present Value of an Ordinary Annuity

A I B C D E I F G II 313 INPUTS: II 314 Payment amount = PMT = -$100 315 Interest rate = I = 5.00% 316 No. of periods= N = 3 317

318 1. Step-by• Periods: 0 1 2 I 3 - 319 Cash flow: -$100 -$100 I -$100 -

320 ,1. ,1. ,1. -

+- +- J 321 $95.24 ,1. ,1. -

+- +- J 322 Divide each payment by $90.70 +-+-+-+-+-+- ,1. -

323 (1+It and sum these PVs to $86.38 4-+-4-+-+-+- +-+-+-+-+-+- +- +- J -

324 find PVAN: $272.32

325

326 2. Formula: -

327 -

= PMT x (½ -�) = I I328 PVAN $272.32 - I (1+1) 329 -

330

331

332 Inputs: 3 s -100 0 333 3. Financial Calculator: N I -

PV I PMT 334 Output: 272.32 I 335

336 4. Excel Spreadsheet: PV function: PVAN= =PV(I,N,PMT,FV) -

337 Fixed inputs: PVAN = =PV(0.05,3,-100,0) = $272.32 -

338 Cell references: PVAN= =PV(C31S,C316,C314,0) = $272.32

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision. Thus,

intermediate calculations using the figure's rounded values will be inexact.

s_purce

See Ch04 Tool Kit.x/sx for

all calculations.

For our example annuity, the present value is: PVA

3 = $100[-1- - ( l )3] = $272.320.05 0.05 1 + 0.05

Financial calculators are programmed to S<?lve Equation 4-2, so we input the variablesand press the PV key, first making sure the calculator is set to End Mode. The calculatorsetup is shown here: Inputs 3 5

N I/YR Output

272.32

-100

PMT

0 FV

End Mode {Ordinary Annuity)

The last section of Figure 4-6 shows the spreadsheet solution using Excel's built-inPV function: =PV(I,N,PMT,FV). In our example, we have =PV(0.05,3,-100,0) with aresulting value of $272.32.

162

............... ~,_c. e

See Ch04 Tool Kit.xlsx for

all calculations.

Part 2 Fixed Income Securities

4-lOb Present Value of Annuities Due

Because each payment for an annuity due occurs one period earlier, the payments will all be discounted for one less period. Therefore, the PV of an annuity due must be greater than that of a similar ordinary annuity.

If you went through the step-by-step procedure, you would see that our example an nuity due has a PV of $285.94 versus $272.32 for the ordinary annuity. See Ch04 Tool Kit. xlsx for this and the other calculations.

With the formula approach, we first use Equation 4-9 to find the value of the ordinary annuity and then, because each payment now occurs one period earlier, we multiply the value given by Equation 4-9 by (1 + I):

PVA due =

PVAo,dinor./1 + I)

PVA d ue = $272.32(1.05) = $285.94

■ With a financial calculator, the inputs are the same as for an ordinary annuity, except

you must set the calculator to Begin M ode:

Inputs 3 5

N I/YR

Output

285.94

-100

PMT)

Begin Mode (Annuity

Due)

In Excel, we again use the PV function, but now we must indicate that we have an annuity due. The function is now =PV(I,N,PMT,FV,Type), where "Type" is the type of annuity. IfType is omitted then Excel assumes that it is 0, which indicates an ordinary an nuity; for an annuity due, Type = 1. As shown in Ch04 Tool Kit.xlsx, the function for this example is =PV(0.05,3,-100,0,1) = $285.94.

SELF-TEST

Why does on annuity due have a higher present value than an ordinary annuity?

If you know the present value of an ordinary annuity, what's an easy way to find the PV af the cor responding annuity due?

What is the PVA of an ordinary annuity with 10 payments of $100 if the appropriate interest rate is 10%? ($614.46) What would the PVA be if the interest rate were 4%? ($811.09) What if the inter est rate were 0%? ($1,000.00) What would the PVAs be if we were dealing with annuities due? ($675.90, $843.53, and $1,000.00)

Assume that you are offered an annuity that pays $100 at the end of each year for 10 years. You could earn 8% on your money in other equally risky investments. What is the most you should pay for the annuity? ($671.01) If the payments began immediately, then how much would the annu ity be worth? ($724.69)

4-11 Finding Annuity Payments, Periods, and Interest Rates

In the three preceding sections, we discussed how to find the FV and PV of ordinary annuities and annuities due, using these four methods: step-by-step, formula, financial calculator, and Excel. Five variables are involved-N, I, PMT, FV, and PV-and if you

()

Chapter 4 Time Value of Money 163

Variable Annuities: Good or Bad?

Retirees appreciate stable, predictable income, so they often

buy annuities. Insurance companies have been the traditional

suppliers, using the payments they receive to buy high-grade

bonds, whose interest is then used to make the promised pay

ments. Such annuities were quite safe and stable and provided

returns of around 7.5%. However, returns on stocks (dividends

plus capital gains) have historically exceeded bonds' returns (in

terest). Therefore, some insurance companies in the 1990s be

gan to offer variable annuities, which were backed by stocks

instead of bonds. If stocks earned in the future as much as they

had in the past, then variable annuities could offer returns of

about 9%; this is better than the return on fixed-rate annuities. If

stock returns turned out to be lower in the future than they had

been in the past (or even had negative returns), then the variable

annuities promised a guaranteed minimum payment of about

6.5%. Variable annuities appealed to many retirees, so compa

nies that offered them had a significant competitive advantage.

The insurance company that pioneered variable annui

ties, The Hartford Financial Services Group, tried to hedge

its position with derivatives that paid off if stocks went

down. But like so many other derivatives-based risk man

agement programs, this one went awry in 2008 because

stock losses exceeded the assumed worst-case scenario.

The Hartford, which was founded in 1810 and was one of the

oldest and largest U.S. insurance companies at the begin

ning of 2008, saw its stock price fall from $85.54 per share to

$4.16. Because of the general stock market crash, investors

feared that The Hartford would be unable to make good on

its variable annuity promises, and this would lead to bank

ruptcy. The company was bailed out by the economic stim

ulus package, but this venerable old firm will never be the

same again.

Source: Leslie Scism and Liam Pleven, "Hartford Aims to Take Risk Out of

Annuities," Online Wall Street Journal, January 13, 2009.

know any four, you can find the fifth by solving Equation 4-2 for ordinary annuities (for annuities due, substitute PMT(l+I) for PMT in Equation 4-2 and solve for the unknown variable). However, a trial-and-error procedure is generally required to find N or I, and that can be quite tedious if performed manually. Therefore, we discuss only the financial calculator and spreadsheet approaches for finding N and I.

4-11a Finding Annuity Payments, PMT

We need to accumulate $10,000 and have it available 5 years from now. We can earn 6% on our money. Thus, we know that FV = 10,000, PV = 0, N = 5, and I/YR = 6. We can enter these values in a financial calculator and then press the PMT key to find our required de posits. However, the answer depends on whether we make deposits at the end of each year (ordinary annuity) or at the beginning (annuity due), so the mode must be set properly. Here are the results for each type of annuity:

Inputs 5 6 0 10000 End Mode

N I/YR PV (Ordinary

PMT FV Annuity)

Output -1,773.96

Inputs 5 6 0 10000 Begin Mode

N I/YR PV PMT FV (Annuity

Due)

Output -1,673.55

164

See Ch04 Tool Kit.xlsx for

oil colcu/ot/ons.

Part 2 Fixed Income Securities

Thus, you must put away $1,773.96 per year if you make payments at the end of each year, but only $1,673.55 if the payments begin immediately. Finally, note that the re quired payment for the annuity due is the ordinary annuity payment divided by (1 + I): $1,773.96/l.06 = $1,673.55.

Excel can also be used to find annuity payments, as shown here for the two types of annuities. For end-of-year (ordinary) annuities, "Type" can be left blank or a O can be inserted. For beginning-of-year annuities (annuities due), the same function is used but now Type is designated as 1. Here is the setup for the two types of annuities.

Function: = PMT(I,N,PV,FV,Type) Ordinary annuity: = PMT(0.06,5,0,10000) = -$1,773.96 Annuity due: = PMT(0.06,5,0,10000,1) = - $1,673.55

4-llb Finding the Number of Periods, N

Suppose you decide to make end-of-year deposits, but you can save only $1,200 per year. Again assuming that you would earn 6%, how long would it take you to reach your $10,000 goal? Here is the calculator setup:

Inputs

Output 6.96

I/YR

-1200

PMT

10000

End Mode

With these smaller deposits, it would take 6.96 years, not 5 years, to reach the $10,000 target. If you began the deposits immediately, then you would have an annuity due and N would be slightly less, 6.63 years.

With Excel, you can use the NPER function: =NPER(I,PMT,PV,FV, Type). For our ordinary annuity example, Type is left blank (or O is inserted) and the func tion is =NPER(0.06,-1200,0,10000) = 6.96. If we put in 1 for Type, we would find N = 6.63.

4-llc Finding the Interest Rate, I

Now suppose you can save only $1,200 annually, but you still need to have the $10,000 in 5 years. What rate of return would you have to earn to reach your goal? Here is the calcu lator setup:

Inputs

Output

N I/YR

25.78

-1200

PMT

10000 End Mode

Thus, you would need to earn a whopping 25.78%! About the only way to earn such a high return would be either to invest in speculative stocks or head to a Las Vegas casino. Of course, speculative stocks and gambling aren't like making deposits in a bank with a guaranteed rate of return, so there would be a high probability that you'd end up with nothing. So, you should probably save more, lower your $10,000 target, or extend your time horizon. It might be appropriate to seek a somewhat higher return, but trying to earn 25.78% in a 6% market would involve speculation, not investing.

Chapter 4 Time Value of Money 165

Using the Internet for Personal Financial Planning

How good are your financial planning skills? For example, your financial plan. In addition to the online data sources

should you buy or lease a car? How much and how soon described in Chapter 2, excellent sources of information are

should you begin to save for your children's education? How available through the National Endowment for Financial

expensive a house can you afford? Should you refinance Education (NEFE), a not-for-profit foundation whose mission

your home mortgage? How much must you save each year is to provide "educated financial decision making for individ-

if you want to retire comfortably? The answers to these uals and families through every stage of life." If you go to their

questions are often complicated and depend on a number Web site, www.nefe.org and scroll to the bottom of the page,

of factors, such as projected housing and education costs, look on the right for "Other NEFE Resources." These include interest rates, inflation, expected family income, and stock helpful information, especially Smart About Money (SAM),

market returns. which has a helpful financial calculator to help you run the

Fortunately, you should be able to use time value of numbers for a variety of topics. You can go to this Web site

money concepts and on line resources to begin developing directly at www.smartaboutmoney.org.

In Excel, you can use the RATE function: =RATE(N,PMT,PV,FV,Type). For our ex ample, the function is =RATE(S,-1200,0,10000) = 0.2578 = 25.78%. If you decide to make the payments beginning immediately then the required rate of return would decline sharply, to 17.54%.

SELF-TEST

You just inherited $100,000 and invested it at 7% per year. How large a withdrawal could you make at the end of each of the next 10 years and end up with zero? ($14,237.75) How would your answer change if you made withdrawals at the beginning of each year? ($13,306.31)

If you have $100,000 that is invested at 7% and you wanted to withdraw $10,000 at the end of each year, how long will your funds last? (17.8 years) How long would they last if you earned 0%? (10 years) How long would they last if you earned the 7% but limited your withdrawals to $7,000 per year? (forever)

Your uncle named you as the beneficiary of his life insurance policy. The insurance company gives you a choice of $100,000 today or a 12-year annuity of $12,000 at the end of each year. What rate of return is the insurance company offering? (6.11%)

You just inherited an annuity that will pay you $10,000 per year for 10 years, and you receive the first payment today. A professional investor offers to give you $60,000 for the annuity. If you sell it to him, what rate of return will he earn on the investment? (13.70%) If you think a "fair" rate of return would be 6%, how much should you ask for the annuity? ($78,016.92)

4-12 Uneven, or Irregular, Cash Flows The definition of an annuity includes the term constant payment, which suggests that an nuities involve a set of identical payments over a given number of periods. Although many financial decisions do involve constant payments, many others involve cash flows that dif fer from year to year. These are called uneven cash flow streams or irregular cash flow streams. For example, the dividends on common stocks are typically expected to increase over time, and the investments that companies make in new products, expanded produc tion capacity, and replacement machinery almost always generate cash flows that vary from year to year. Throughout the book, we use the term payment (PMT) in situations

166 Part 2 Fixed Income Securities

where the cash flows are constant and thus an annuity is involved; if different cash flows occur in different time periods, t, then we use the term CF, to designate the cash flow in period t.

There are two important classes of uneven cash flows: (1) those in which the cash flow stream consists of a series of annuity payments plus an additional final lump sum in Year N and (2) all other uneven streams. Bonds are an instance of the first type, while stocks and capital investments illustrate the second type. Here's an example of each type.

Stream 1. Annuity plus an additional final payment:

Periods 0 I= 12% 1 2 3 4 5

Cash flows $0 $100 $100 $100 $100 $ 100 $1,000 $1,100

Stream 2. Irregular cash flow stream:

Periods 0 I= 12% 1 2 3 4 5

Cash flows $0 $100 $300 $300 $300 $500

To find the PV of either stream, you could use the step-by-step procedure of Equation 4-11:

CF 1

CF 2

PV=---+---+ (1 + 1)1 (1 + 1)2

CF N

N CF, +----""--

(1 + I)N ::-: (1 + I)' I However, as we shall see, the solution process differs significantly for the two types.

4-12a Annuity Plus Additional Final Payment

First, consider Stream 1 and notice that it is a 5-year, 12%, ordinary annuity plus a final payment of $1,000. We can find the PV of the annuity, find the PV of the final payment, and then sum them to get the PV of the stream. Financial calculators are programmed to do this for us-we use all five time value of money (TVM) keys, en tering the data for the four known values as shown below, and then pressing the PV key to get the answer:

Inputs 5 12

N I/YR

Output

-927.90

100

PMT

1000

In Excel, use the PV function: =PV(I,N,PMT,FV). For this example, use =PV(0.12,5,100,1000) =-$927.90.

,:.esource

Chapter 4 Time Value of Money 167

Whether you use a financial calculator or Excel to find the present value of the cash flow stream, notice that each approach is similar to the ones used for annuities. The only difference is that there is a nonzero value for FV.

4-12b Irregular Cash Flow Stream

Now consider the irregular stream shown in Figure 4-7. Panel A shows the basic time line, which contains the inputs, and we first use the step-by-step approach to (ind PV = $1,016.35. Note that we show the PV of each cash flow directly below the cash flow, and then we sum these PVs to find the PV of the stream.

See Ch04 Tool Kit.xlsx for

all calculations.

With a financial calculator, you can't use the TVM sub-menu because the periodic cash flows aren't constant. Instead, go to the calculator's main menu and select the sub menu for cash flows, which may be labeled "CFLO" or something similar. This sub-menu has a feature-the cash flow register-that allows you to input the individual cas� flows in chronological order.9 Cash flows are designated CF

0 , CF

1 , CF

2 , CF

3 , and so on, up to the

last cash flow, CF N

" Next, you enter the interest rate, I/YR. At this point, you have substi tuted in all the known values of Equation 4-11, so when you press the NPV key, you get the PV of the irregular cash flow stream. To input the cash flows for this problem, enter 0 (because CF

0 = 0), 100, 300, 300, 300, and 500 in that order into the cash flow register,

enter I/YR = 12, and then press NPV to obtain the answer, $1,016.35.

FIGURE4•7

Present Value of an Irregular Cash Flow Stream

A B C D E F G

466 INPUTS:

467 Interest rate = I = 12%

468

469 1. Step-by-Step:

470 Periods: 0 1 2 3 4 5

471 Cash flow: $0.00 $100.00 $300.00 $300.00 $300.00 $500.00 - 472 of the CFs: $89.29 $239.16 $213.53 $190.66 $283.71

473 m ofPVs = $1,016.35 - 474

475

476 2. Financial Calculator: -

Enter the cash flows into the cash flow register of a financial calculator, $1,016.35

477 enter I/YR, and then press the NPV key to find the answer.

478

479 3. Excel Spreadsheet:

480 Excel NPV function: -

=NPV(l,Cash flows) $1,016.35

481 Fixed inputs: =NPV(0.12,100,300,300,300,500) $1,016.35 - 482 Cell references: =NPV(C467,C471:G471) $1,016.35

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Excel's full precision. Thus,

intermediate calculations using the figure's rounded values will be inexact.

'These instructions are for the HP JObII +, but most other financial calculators work in a similar manner.

168 Part 2 Fixed Income Securities

Two points should be noted. First, when dealing with the cash flow register, the calculator uses the term "NPV" rather than "PV." The N stands for "net," so NPV is the abbreviation for "net present value," which is simply the net present value of a series of positive and negative cash flows, including any cash flow at time zero. The NPV func tion will be used extensively when we get to capital budgeting, where CF

O is generally

the cost of the project. The second point to note is that repeated cash flows with identical values can be

entered into the cash flow register more efficiently on some calculators by using the Nj key. In this illustration, you would enter CF

0 = 0, CF

1 = 100, CF

2 = 300, Nj = 3

(which tells the calculator that the 300 occurs 3 times), and CF 5

= 500.10 Then enter I = 12, press the NPV key, and 1,016.35 will appear in the display. Also, note that num bers entered into the cash flow register remain in the register until they are cleared. Thus, if you previously worked a problem with eight cash flows, and then moved to one with only four cash flows, the calculator would simply add the cash flows from the second problem to those of the first problem, and you would get an incorrect answer. Therefore, you must be sure to clear the cash flow register before starting a new problem.

Spreadsheets are especially useful for solving problems with uneven cash flows. You enter the cash flows in the spreadsheet as shown in Figure 4-7 on Row 471. To find the PV of these cash flows without going through the step-by-step process, you would use the Excel NPV function. First put the cursor on the cell where you want the answer to appear, Cell G482, click the function wizard, choose Financial, scroll down to NPV, and click OK to get the dialog box. Then enter C467 (or 0.12) for Rate and enter either the individual cash flows or the range of cells containing the cash flows, C471:G471, for Value 1. Be very careful when entering the range of cash flows. With a financial calculator, you begin by entering the Time-0 cash flow. With Excel, you do not include the Time-0 cash flow; instead, you begin with the Time-1 cash flow. Now, when you click OK, you get the PV of the stream, $1,016.35. Note that you can use the PV function if the payments are constant, but you must use the NPV function if the cash flows are not constant. Finally, note that Excel has a major advantage over financial calculators in that you can see the cash flows, which makes it easy to spot data-entry errors. With a calculator, the numbers are buried in the machine, making it harder to check your work.

SELF-TEST

Could you use Equation 4-3, once for each cash flow, to find the PV of an uneven stream of cash flows?

What is the present value of a 5-year ordinary annuity of $100 plus an additional $500 at the end of Year 5 if the interest rate is 6%? ($794.87) What would the PV be if the $100 payments occurred in Years 1 through 10 and the $500 came at the end of Year 10? ($1,015.21)

What is the present value of the following uneven cash flow stream: $0 at Time 0, $100 at the end of Year 1 (or at Time 1), $200 at the end of Year 2, $0 at the end of Year 3, and $400 at the end of Year 4-assuming the interest rate is 8%? ($558.07)

Would a "typical" common stock provide cash flows more like an annuity or more like an uneven cash flow stream?

10On some calculators, instead of entering CF, = 500, you enter CF 3 = 500 because this is the next cash flow

different from 300.

FIGURE4-8

Chapter 4 Time Value of Money 169

4-13 Future Value of an Uneven Cash Flow Stream

The future value of an uneven cash flow stream is found by compounding each payment to the end of the stream and then summing the future values:

= � CFp + I)N-t t=O

The future value of this uneven cash flow stream is $1,791.15, as shown in Figure 4-8. Most financial calculators have a net future value (NFV) key, which, after the cash flows

and interest rate have been entered, can be used to obtain the future value of an uneven cash flow stream. If your calculator doesn't have the NFV feature, you can first find the net pres ent value of the stream, and then find its net future value as NFV = NPV(l + I)N. In the illustrative problem, we find PV = 1,016.35 using the cash flow register and I = 12. Then we use the TVM register, entering N = 5, I= 12, PV =-1016.35, and PMT = 0. When we press FV, we find FV = 1,791.15, which is the same as the value shown on the time line in Figure 4-8. As Figure 4-8 also shows, the same procedure can be used with Excel.

SELF-TEST

What is the future value of this cash flow stream: $100 at the end of 1 year, $150 after 2 years, and $300 after 3 years, assuming the appropriate interest rate is 15%? ($604.75)

Future Value of an Irregular Cash Flow Stream

A B C D E F G II 11

498 INPUTS:

499 Interest rate = I = 12%

500

501 1. Step-by-Step: -

502 Periods: 0 1 2 3 4 5 -

503 Cash flow: $0.00 $100.00 $300.00 $300.00 $300.00 $500.00 -

504 of the CFs: $157.35 $421.48 $376.32 $336.00 $500.00 -

sos Sum ofFV's = $1,791.15

506

507

508 2. Financial Calculator:

Enter the cash flows into the cash flow register of a financial calculator, enter $1,791.15

509 I/YR, and then press the NFV key to find the answer.

510

511 3. Excel Spreadsheet: -

512 Step 1. Find NPV: =NPV(C499,C503:G503) $1,016.35 -

513 Step 2. Compound NPV to find NFV: =FV(C499,G502,0,-G512) $1,791.15

Source: See the file Ch04 Tool Klt.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision. Thus, intermediate calculations using the figure's rounded values will be inexact.

170 Part 2 Fixed Income Securities

4-14 Solving for I with Irregular Cash Flows Before financial calculators and spreadsheets existed, it was extremely difficult to find I if the cash flows were uneven. However, with spreadsheets and financial calculators it's easy to find I. If you have an annuity plus a final lump sum, you can input values for N, PV, PMT, and FV into the calculator's TVM registers and then press the I/YR key. Here's the setup for Stream 1 from Section 4-12, assuming we must pay $927.90 to buy the asset:

Inputs 5

Output

I/YR

12.00

-927.90

The rate of return on the $927.90 investment is 12%.

100

PMT

1000

Finding the interest rate for an irregular cash flow stream with a calculator is a bit more complicated. Figure 4-9 shows Stream 2 from Section 4-12, assuming a required investment of CF

O =-$1,000. First, note that there is no simple step-by-step method for

finding the rate of return; finding the rate for this investment requires a trial-and-error process, which is terribly time-consuming. Therefore, we always use a financial calcula tor or a spreadsheet. With a calculator, we would enter the CFs into the cash flow regis ter and then press the IRR key to get the answer. IRR stands for "internal rate of return," and it is the rate of return the investment provides. The investment is the cash flow at Time 0, and it must be entered as a negative number. When we enter those cash flows in the calculator's cash flow register and press the IRR key, we get the rate of return on the $1,000 investment, 12.55%. Finally, note that once you have entered the cash flows in the calculator's register, you can find both the investment's net present value (NPV) and its internal rate of return. For investment decisions, we typically want both of these numbers. Therefore, we generally enter the data once and then find both the NPV and the IRR.

You would get the same answer using Excel's IRR function, as shown in Figure 4-9. Notice that when using the IRR-unlike using the NPV function-you must include all cash flows, including the Time-0 cash flow.

FIGURE4-9

IRR of an Uneven Cash Flow Stream

A B C D E F G

545 Periods: 0 1 2 3 4 s

546 Cash flows: -$1,000 $100 $300 $300 $300 $500

547

1. Calculator: You could enter the cash flows Into the cash flow register of a

12.55% 548

financial calculator and then press the IRR key to find the answer.

549

550 2. Excel IRR Function: Cell references: IRR= =IRR(BS46:GS46) 12.55%

Sourc�: See the file Ch04 Tool Klt.xlsx. Numbers are reported as rounded values for clarity but are calculated using Excel's full precision. Thus,

intermediate calculations using the figure's rounded values will be inexact.

Chapter 4 Time Value of Money 171

SELF-TEST

An investment costs $465 now and is expected to produce cash flows of $100 at the end of each of the next 4 years, plus an extra lump-sum payment of $200 at the end of the fourth year. What is the expected rate of return on this investment? (9.05%)

An investment costs $465 and is expected to produce cash flows of $100 at the end of Year 1, $200 at the end of Year 2, and $300 at the end of Year 3. What is the expected rate of return on this investment? (11.71%)

4-15 Semiannual and Other Compounding Periods In most of our examples thus far, we assumed that interest is compounded once a year, or annually. This is annual compounding. Suppose, however, that you put $1,000 into a bank that pays a 6% annual interest rate but credits interest every 6 months. This is semiannual compounding. If you leave your funds in the account, how much would you have at the end of 1 year under semiannual compounding? Note that you will receive $60 of interest for the year with annual compounding. With semiannual com pounding, you will receive $30 of it after only 6 months, and you will earn interest on the first $30 during the second 6 months, so you will end the year with more than the $60 you would have had under annual compounding. You would be even better off under quarterly, monthly, weekly, or daily compounding. Note also that virtually all bonds pay interest semiannually; most stocks pay dividends quarterly; most mortgages, student loans, and auto loans involve monthly payments; and most money fund ac counts pay interest daily. Therefore, it is essential that you understand how to deal with nonannual compounding.

4-lSa Types of Interest Rates

When we move beyond annual compounding, we must deal with the following four types of interest rates:

1. Nominal annual rates, given the symbol INoM 2. Periodic rates, denoted as IPER 3. Effective annual rates, given the symbol EAR or EFF% 4. Annual percentage rates, termed APR rates

NOMINAL (OR QUOTED) RATE, INoM 11

The nominal annual interest rate (I N0M), or just the nominal rate, is the rate quoted by

banks, brokers, and other financial institutions. So, if you talk with a banker, broker, mortgage lender, auto finance company, or student loan officer about rates, the nominal rate is the one he or she will normally quote you. However, to be meaningful, the quoted nominal rate must also include the number of compounding periods per year. For ex ample, a bank might offer you a CD at 6% compounded daily, while a credit union might offer 6.1% compounded monthly.

''The term nominal rate as used here does not have the same meaning as it did in Chapter l. There, nomi nal interest rates referred to stated market rates as opposed to real (zero-inflation) rates. In this chapter, the term nominal rate means the stated, or quoted, annual rate as opposed to the effective annual rate, which we explain later. In both cases, though, nominal means stated, or quoted, as opposed to some sort of adjusted rate.

172 Part 2 Fixed Income Securities

Note that we never show the nominal rate on a time line, and we never use it as an input to a financial calculator (except when compounding occurs only once a year). 12 If more frequent compounding occurs, we use periodic rates as explained next. PERIODIC RATE, I

PER The periodic rate (I PER

) is the rate charged by a lender or paid by a borrower each period. It can be a rate per year, per 6 months (semiannually), per quarter, per month,per day, or per any other time interval. For example, a bank might charge 1.5% permonth on its credit card loans, or a finance company might charge 3% per quarter oninstallment loans. We find the periodic rate as follows: Periodic rate IPER = INOM/M ■

where INOM is the nominal annual rate and M is the number of compounding periodsper year. Thus, a 6% nominal rate with semiannual payments results in a periodicrate of Periodic rate IPER = 6%/2 = 3.00%

If only one payment is made per year, then M = l, in which case the periodic rate wouldequal the nominal rate: 6%/1 = 6%. The periodic rate is the rate shown on time lines and used in calculations. 13 To illustrate,suppose you invest $100 in an account that pays a nominal rate of 12%, compounded quarterly, or 3% per period. How much would you have after 2 years if you leave the funds ondeposit? First, here is the time line for the problem:

Quarters 0 I -$100

3% 2 I 3 I 4 I 5I 6 I

To find the FV, we would use this modified version of Equation 4-1:

( 0,12)4X2= $100 1 + 4 = $100(1 + 0.03)8 = $126.68

7 8 I I FV=?

12S ome calculators have a feature so that you can input the nominal rate and the number of periods per

year. It is easy to make mistakes with this approach, which is why we do not use nominal rates in our

calculations.

"The only exception is in cases where (I) annuities are involved and (2) the payment periods do not correspond

to the compounding periods. In such cases-for example, if you are making quarterly payments into a bank

account to build up a specified future sum but the bank pays interest on a daily basis-then the calculations are

more complicated. For such problems, the simplest procedure is to determine the periodic (daily) interest rate

by dividing the nominal rate by 365 (or by 360 if the bank uses a 360-day year), then compound each payment over the exact number of days from the payment date to the terminal point, and then sum the compounded

payments to find the future value of the annuity. This is a simple process with a computer.

Chapter 4 Time Value of Money 173

With a financial calculator, we find the FV using these inputs: N = 4 X 2 = 8, I = 12/4 = 3, PV =-100, and PMT = 0. The result is again FV = $126.68:14

Inputs 8 3

N I/YR

Output

-100

PV j 0

( PMTJ FV

126.68

EFFECTIVE {OR EQUIVALENT) ANNUAL RATE {EAR OR EFF%)

The effective (or equivalent) annual rate (EAR or EFF%) is the annual (interest oncea year) rate that produces the same final result as compounding at the periodic rate forM times per year. The EAR, also called EFF% (for effective percentage rate), is foundas follows:15

EAR = EFF% = (1 + IPl!R)M - 1.0 = ( 1 + \:r r - 1.0

Here I N0M

/M is the periodic rate, and M is the number of periods per year. If a bank wouldlend you money at a nominal rate of 12%, compounded quarterly, then the EFF% ratewould be 12.5509%: Rate on bank loan: EFF% = (1 + 0.03)4 - 1.0 = (1.03)4 - 1.0 = 1.125509 - 1.0 = 0.125509 = 12.5509%

It is even easier to use the Excel function = EFF(I N oM' M). To see the importance of the EFF%, suppose that-as an alternative to the bank loan-you could borrow on a credit card that charges 1 % per month. Would you be better off using the bank loan or credit card loan? To answer this question, the cost of each

alternative must be expressed as an EPP%. The cost of the credit card loan, with monthly payments, is: Credit card loan: EFF% = (1 + 0.01)12 - 1.0 = (1.01)12 - 1.0 = 1.126825 - 1.0 = 0.126825 = 12.6825%

"Most financial calculators have a feature that allows you to set the number of payments per year and then use the nominal annual interest rate. However, students tend to make fewer errors when using the periodic rate with their calculators set for one payment per year (i.e., per period), so this is what we recommend. Note also that you cannot use a normal time unless you use the periodic rate.

"You could also use the "interest conversion feature" of a financial calculator. Most financial calculators

are programmed to find the EPP% or, given the EPP%, to find the nominal rate; this is called "interest rate conversion." You enter the nominal rate and the number of compounding periods per year, and then press the EPP% key to find the effective annual rate. However, we generally use Equation 4-15 because it's easy and because using the equation reminds us of what we are really doing. If you do use the interest rate conversion feature on your calculator, don't forget to reset your settings afterward. Interest conversion is discussed in our calculator tutorials.

174 Part 2 Fixed Income Securities

Notice that the effective rate is higher for monthly payments than for quarterly payments: 12.6825% versus 12.5509%. This makes sense: Both loans have the same 12% nominal rate, yet you would have to make the first payment after only 1 month on the credit card versus 3 months under the bank loan.

The EFF% rate is rarely used in calculations. However, it must be used to compare the effective costs of different loans or rates of return on different investments when payment periods differ, as in our example of the credit card versus a bank loan.

ANNUAL PERCENTAGE RATES {APR)

The annual percentage rate (APR) is the nominal annual interest rate actually charged on loans. The APR is especially important for loans that use add-on interest, which calcu lates the total interest to be paid by multiplying the amount borrowed by the quoted rate, dividing this product by the number of times per year that a payment is due, and requiring the first payment due at the time the loan is made.

For example, suppose you buy a refrigerator, stove, and dishwasher for $3,000. The store might offer you one year of financing using an add-on quoted rate of 8%. The total interest is equal to $240: $3,000(0.08) = $240. Add this total interest charge to the $3,000 cost of the kitchen appliances for a total loan of $3,240. Then divide the total loan by 12 to get the monthly payments: $3,240/12 = $270 per month, with the first payment made at the time of purchase. Therefore, you have a 12-month annuity due with payments of $270. Is your cost really the 8% that you were quoted?

To find the APR, set your calculator to Begin Mode, then enter N = 12, PV = 3000, PMT = -270, and FV = 0. When you press the I/YR key, you get the periodic rate, 1.4313%. Multiply this by 12 to get the APR, 17.1758%, which is substantially higher than the 8% quoted rate. The APR might be so high that you decide to decline the in-store fi nancing and ask your bank for a loan.

Before 1968, lenders were not required to tell borrowers the APR. This changed when Congress passed the Consumer Credit Protection Act in 1968. The Act's Truth in Lending provisions require banks and other lenders to disclose the APR. The APR is helpful but not as revealing as the EFF%, which is 18.5945% for the loan with the 17.1758% APR. (We show these calculations using both the calculator and Excel, along with a time line that helps visualize what's happening, in the chapter's Excel Tool Kit.)

Your best bet is to be wary of almost all loan offers and do the math yourself to deter mine the rate that you really would pay.

4-lSb The Result of Frequent Compounding

What would happen to the future value of an investment if interest were compounded annually, semiannually, quarterly, or some other less-than-annual period? Because in terest will be earned on interest more often, you should expect higher future values the more frequently compounding occurs. Similarly, you should expect the effective annual rate to increase with more frequent compounding. As Figure 4-10 shows, these results do occur-the future value and the EFF% do increase as the frequency of compound ing increases. Notice that the biggest increase in FV (and in EFF%) occurs when com pounding goes from annual to semiannual, and notice also that moving from monthly to daily compounding has a relatively small impact. Although Figure 4-10 shows daily compounding as the smallest interval, it is possible to compound even more frequently. At the limit, compounding can occur continuously. This is explained in Web Extension 4B on the textbook's Web site.

Chapter 4 Time Value of Money 175

FIGURE4•10

Effect on $100 of Compounding More Frequently Than Once a Year

A I B I C I D I E I F I G Number of Periodic Percentage

Frequency of Nominal Periods Interest Effective Annual Increase in 568 Compounding Annual Rate per Year (M)" Rate (lpEIJ Rate (EFF%t Future Value

c FV

569 Annual 12% 1 12.0000% 12.0000% $112.00

570 Semiannual 12% 2 6.0000% 12.3600% $112.36 0.32%

571 Quarterly 12% 4 3.0000% 12.5509% $112.55 0.17%

572 Monthly 12% 12 1.0000% 12.6825% $112.68 0.12%

573 Daily 12% 365 0.0329% 12.7475% $112.75 0.06%

Source: See the file Ch04 Tool Kit.xlsx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision.

Thus, intermediate calculations using the figure's rounded values will be inexact.

Notes:

• We used 365 days per year in the calculations.

�The EFF% is calculated as (I + 1,.,)., - 1. It can also be calculated using an Excel function: =EFF(I N m,•M).

•The future value is calculated as $100(1 + EFF%).

SELF-TEST

Would you rather invest in an account that pays a 7% nominal rate with annual compounding or with monthly compounding? If you borrowed at a nominal rate of 7%, would you rather make an nual or monthly payments? Why?

What is the future value of $100 after 3 years if the appropriate interest rate is 8%, compounded annually? ($125.97) Compounded monthly? ($127.02)

What is the present value of $100 due in 3 years if the appropriate interest rate is 8%, compounded annually? ($79.38) Compounded monthly? ($78.73)

Define the following terms: annual percentage rate (APR), effective annual rate (EFF%), and nomi nal interest rate (I Nau).

A bank pays 5% with daily compounding on its savings accounts. Should it advertise the nominal or effective rate if it is seeking to attract new deposits?

Credit card issuers must by law print their annual percentage rate on their monthly statements. A common APR is 18%, with interest paid monthly. What is the EFF% on such a loan? (19.56%)

Some years ago banks weren't required to reveal the rate they charged on credit cards. Then Congress passed a "truth in lending" law that required them to publish their APR rate. Is the APR rate really the most truthful rate, or would the EFF% be even more truthful?

4-16 Fractional Time Periods16

So far we have assumed that payments occur at either the beginning or the end of periods , but not within periods. However, we occasionally encounter situations that require com pounding or discounting over fractional periods. For example, suppose you deposited $100

"This section is interesting and useful but relatively technical. It can be omitted, at the option of the instructor, without loss of continuity.

176 Part 2 Fixed Income Securities

in a bank that pays a nominal rate of 10%, compounded daily, based on a 365-day year. How much would you have after 9 months? The answer of $107.79 is found as follows:17

Periodic rate = I PER

= 0.10/365 = 0.000273973 per day Number of days = (9/12)(365) = 0.75(365) = 273.75 days, rounded to 274 Ending amount = $100(1.000273973)274 = $107.79

Now suppose that instead you borrow $100 at a nominal rate of 10% per year and are charged simple interest, which means that interest is not charged on interest. If the loan is outstanding for 274 days (or 9 months), how much interest would you have to pay? The interest owed is equal to the principal multiplied by the interest rate times the number of periods. In this case, the number of periods is equal to a fraction of a year: N = 274/365 = 0.7506849.

Interest owed = $100(10%)(0.7506849) = $7.51

Another approach would be to use the daily rate rather than the annual rate and thus to use the exact number of days rather than the fraction of the year:

Interest owed = $100(0.000273973)(274) = $7.51

You would owe the bank a total of$107.51 after 274 days. This is the procedure most banks use to calculate interest on loans, except that they generally require borrowers to pay the interest on a monthly basis rather than after 274 days; this more frequent compounding raises the EFF% and thus the total amount of interest paid.

SELF -TEST

Suppose a company borrowed $1 mi/lion at a rate of 9%, using simple interest, with interest paid at the end of each month. The bank uses a 360-day year. How much interest would the firm have to pay in a 30-day month? ($7,500.00) What would the interest be if the bank used a 365-day year? ($7,397.26)

Suppose you deposited $1,000 in a credit union that pays 7% with daily compounding and a 365-day year. What is the EFF%? (7.250098%) How much could you withdraw after 7 months, assuming this is 7/12 of a year? ($1,041.67)

4-17 Amortized Loans

An extremely important application of compound interest involves loans that are paid off in installments over time. Included are automobile loans, home mortgage loans, student loans, and many business loans. A loan that is to be repaid in equal amounts on a monthly, quarterly, or annual basis is called an amortized loan.18

"We assume that these 9 months constitute 9/12 of a year. Also, bank deposit and loan contracts specifically state whether they are based on a 360-day or a 365-day year. If a 360-day year is used, then the daily rate is higher, so the effective rate is also higher. Here we assumed a 365-day year. Finally, note that banks use software with built-in calendars, so they calculate the exact number of days.

Note also that banks often treat such loans as follows: (1) They require monthly payments, and they calcu late the interest for the month by multiplying the periodic rate by the beginning-of-month balance times the

number of days in the month. This is called "simple interest." (2) The interest for the month is either added to the next beginning of month balance or actually paid by the borrower. In this case, the EFF% is based on 12 compounding periods, not 365 as is assumed in our example.

"The word amortized comes from the Latin mors, meaning "death," so an amortized loan is one that is "killed off' over time.

Chapter 4 Time Value of Money

What You Know Is What You Get: Not in Payday Lending

When money runs low toward the end of a month, many in

dividuals turn to payday lenders. If a borrower's application

is approved, the payday lender makes a short-term loan,

which will be repaid with the next paycheck. In fact, on the

next payday the lender actually transfers the repayment from

the borrower's bank account. This repayment consists of the

amount borrowed plus a fee.

How costly are payday loans? The lender charges a fee

of about $15 to $17 per $100 borrowed. A typical loan is for

about $350, so the typical fee is about $56. A typical borrower

gets paid about every 2 weeks, so the loan is for a very short

amount of time. With a big fee and a short time until repay

ment, the typical payday loan has an APR of over 400%.

How informed are borrowers? Two professors at the Uni

versity of Chicago set out to answer this question. When loans

are approved, borrowers receive a form to sign that shows

the APR. However, subsequent telephone surveys of borrow

ers show that over 40% of borrowers thought their APR was

around 15%; perhaps not coincidentally, these are similar nu

merals to the fee schedules that are posted prominently in

the lender's office.

4-17a Payments

The professors then did an experiment (with the agree

ment of 77 payday loan stores) in which they provided more

information than just the APR. One group of borrowers re

ceived information about the APR of the payday loan as com

pared to the APRs of other loans, such as car loans. A second

group received information about the dollar cost of the pay

day loan as compared to the dollar cost of other loans, such as

car loans. A third group received information about how long

it takes most payday borrowers to repay their loans (which is

longer than the next payday; borrowers tend to extend the

loan for additional pay periods, accruing additional fees).

Compared to a control group with no additional informa

tion, the results show that some borrowers with additional

information decided not to take the loan; other borrowers re

duced the amount that they borrow. These findings suggest

that better information helps borrowers make less costly de

cisions. The more you know, the less you get, at least when it

comes to costly payday loans.

Source: Marianne Bertrand and Adair Morse, "Information Disclosure,

Cognitive Biases, and Payday Borrowing," Journal of Finance, Vol. 66, No. 6,

December 2011, pp. 1865-1893.

177

Suppose a company borrows $100,000, with the loan to be repaid in 5 equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year.

Here's a picture of the situation:

0 1=6% 1 2 3 4 5

$100,000 PMT PMT PMT PMT PMT

Our task is to find the amount of the payment, PMT, such that the sum of their PVs equals the amount of the loan, $100,000:

PMT PMT PMT PMT PMT 5 PMT $100 000=--+--+--+--+--= �--,

(1.06) 1 (1.06)2 (1.06)3 (1.06)4 (1.06)5 � (1.06)1

It is possible to solve the annuity formula, Equation 4-9, for PMT, but it is much easier to use a financial calculator or spreadsheet. With a financial calculator, we insert values as shown below to get the required payments, $23,739.64:

Inputs 5 6

N I/YR

Output

100000

PV PMT

-23,739.64

178 Part 2 Fixed Income Securities

With Excel, you would use the PMT function: = PMT(I,N,PV,FV) = PMT(0.06,5,100000,0) = -$23,739.64. Thus, we see that the borrower must pay the lender $23,739.64 per year for the next 5 years.

4-17b Amortization Schedules

Each payment will consist of two parts-part interest and part repayment of principal. This breakdown is shown in the amortization schedule given in Figure 4-11. The interest component is relatively high in the first year, but it declines as the loan balance decreases. For tax purposes, the borrower would deduct the interest component while the lender would report the same amount as taxable income. Over the 5 years, the lender will earn 6% on its investment and also recover the amount of its investment.

4-17c Mortgage Payments, Remaining Balances, and Interest Paid

Suppose you buy a house and take out a 30-year home mortgage of $250,000 at an annual rate of 6%. What is the monthly payment? What is the remaining balance after a year? What are the total principal payments and interest payments during the first year? What are the total payments and interest payments over the life of the loan?

FIGURE4-11

Loan Amortization Schedule, $100,000 at 6% for 5 Years

A B C D I E I F 675 INPUTS:

676 Amount borrowed: $100,000

677 Years: 5

678 Rate: 6%

679 Intermediate calculation:

680 PMT: $23,739.64 =PMT(C678,C677,-C676) Repayment Ending

Beginning of Principalb Balance Amount Payment Interest" (2) - (3) = (1) - (4) =

681 Year (1) (2) (3) (4) (5)

682 1 $100,000.00 $23,739.64 $6,000.00 $17,739.64 $82,260.36

683 2 $82,260.36 $23,739.64 $4,935.62 $18,804.02 $63,456.34 -

684 3 $63,456.34 $23,739.64 $3,807.38 $19,932.26 $43,524.08 -

685 4 $43,524.08 $23,739.64 $2,611.44 $21,128.20 $22,395.89

686 5 $22,395.89 $23,739.64 $1,343.75 $22,395.89 $0.00

Source: See the file Ch04 Tool Kit,x/sx. Numbers are reported as rounded values for clarity but are calculated using Exce/'s full precision. Thus,

intermediate calculations using the figure's rounded values will be inexact.

Notes:

• Interest in each period is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefore, interest in Year I is $100,000(0.06) = $6,000; in Year 2 it is $82,260.36(0.06) = $4,935.62; and so on.

•Repayment of principal is the $23,739.64 annual payment minus the interest charge for the year, $17,739.64 for Year I.

Chapter 4 Time Value of Money 179

MONTHLY PAYMENTS

To find the payment, first adjust N and I/YR to reflect monthly payments. The financial calculator inputs are shown below:

Inputs

Output

360

0.5%

I/YR

250,000

( PV J ( PMT J -1,498.8763

After rounding to cents, the monthly payment is $1,498.88. We will use this rounded value to determine how much you will owe at the end of the first year. In other words, what is your remaining balance after the 12th payment?

REMAINING BALANCE

Imagine you sell the house after the 12th payment. How much would you still owe? In other words, what is the future value of the initial loan amount and the subsequent 12 loan payments at t = 12? Considering this is from the perspective of the lender making a loan of $250,000 and receiving 12 payments of $1,498.88, the financial calculator solu tion is $264,929.93:

Inputs 348

0.5%

I/YR

-250,000

-1,498.88

PMT FV

Output -246,929.93

TOTAL PRINCIPAL PAYMENTS AND INTEREST PAYMENTS

DURING THE FIRST YEAR

The total amount of principal repaid in the first year is the difference between the starting principal and the end-of-year principal: $250,000 - $246,929.93 = $3,070.07. The total payments during the year are the product of the 12 months and the monthly payment: 12($1,498.88) = $17,986.56. Therefore, the total interest paid in the year is:

Total interest in first year = Total payments - Total principal payments = $17,986.56 - $3,070.07 = $14,916.49.

Dividing the total first-year interest ($14,917) by the total first-year payments shows that about 83% of the payments go to interest!

TOTAL PAYMENTS AND INTEREST PAYMENTS DURING

THE MORTGAGE'S LIFE

Suppose you don't sell the house. The total amount of your payments is 360($1,498.88) = $539,597. The borrower pays back the borrowed $250,000 over the life of the loan, so the total interest paid is $539,597 - $250,000 = $289,597.

Now consider a 15-year mortgage. To compare apples to apples, assume the interest rate stays at 6%, although it probably would be a bit lower. Changing N to 180, the new payment is $2,109.6421. The total amount of payments is 180($2,109.6421) = $379,736 and the total interest paid is $379,736 - $250,000 = $129,736, a big decrease from the $289,597 paid on the 30-year mortgage. As this example shows, increasing the monthly payment can dramatically reduce the total interest paid and the time required to pay off the mortgage.

180 Part 2 Fixed Income Securities

An Accident Waiting to Happen: Option Reset Adjustable Rate Mortgages

Option reset adjustable rate mortgages (ARMs) give the

borrower some choices regarding the initial monthly pay

ment. One popular option ARM allowed borrowers to make

a monthly payment equal to only half of the interest due in

the first month. Because the monthly payment was less than

the interest charge, the loan balance grew each month. When

the loan balance exceeded 110% of the original principal, the

monthly payment was reset to fully amortize the now larger

loan at the prevailing market interest rates.

Here's an example. Someone borrows $325,000 for

30 years at an initial rate of 7%. The interest accruing in the

first month is ( 7%/12)($325,000) = $1,895.83. Therefore,

the initial monthly payment is 50%($1,895.83) = $947.92.

Another $947.92 of deferred interest is added to the loan

balance, taking it up to $325,000 + $947.92 = $325,947.82.

Because the loan is now larger, interest in the second month

is higher, and both interest and the loan balance will continue

to rise each month. The first month after the loan balance

exceeds 110%($325,000) = $357,500, the contract calls for

the payment to be reset so as to fully amortize the loan at the

then prevailing interest rate.

First, how long would it take for the balance to exceed

$357,500? Consider this from the lender's perspective: The

lender initially pays out$325,000, receives $947.92 each month,

and then would receive a payment of $357,500 if the loan were

payable when the balance hit that amount, with interest ac

cruing at a 7% annual rate and with monthly compounding.

We enter these values into a financial calculator: I = 7%/12, PV = -325000, PMT = 947.92, and FV = 357500. We solve for N = 31.3 months, rounded up to 32 months. Thus, the bor rower will make 32 payments of $947.92 before the ARM resets.

The payment after the reset depends upon the terms of

the original loan and the market interest rate at the time of the

reset. For many borrowers, the initial rate was a lower-than

market "teaser" rate, so a higher-than-market rate would be

applied to the remaining balance. For this example, we will as

sume that the original rate wasn't a teaser and that the rate re

mains at 7%. Keep in mind, though, that for many borrowers the

reset rate was higher than the initial rate. The balance after the

32nd payment can be found as the future value of the original

loan and the 32 monthly payments, so we enter these values

in the financial calculator: N = 32, I = 7%/12, PMT = 947.92, PV = -325000, and then solve for FV = $358,242.84. The num

ber of remaining payments to amortize the $358,242.84 loan

balance is 360 - 32 = 328, so the amount of each payment is

found by setting up the calculator as N = 328, I = 7%/12, PV = 358242.84, and FV = 0. Solving, we find that PMT = $2,453.94.

Even if interest rates don't change, the monthly payment

jumps from $947.92 to $2,453.94 and would increase even

more if interest rates were higher at the reset. This is exactly

what happened to millions of American homeowners who took

out option reset ARMS in the early 2000s. When large num

bers of resets began in 2007, defaults ballooned. The accident

caused by option reset AR Ms didn't wait very long to happen!

4-17d Auto Loans and the Remaining Balance

Auto dealers and manufacturers often offer loans for cars purchased at the dealership. Even if the nominal interest rate, number of payments, and monthly payment amount are the same as those of a regular amortizing loan, the monthly interest charges often differ. Recall that a regular amortizing loan calculates a monthly interest charge by multiplying the beginning loan balance by the periodic rate. An auto loan instead cal culates monthly interest charges by multiplying the total interest charges by a different predefined percentage each month. 19 These percentages sum to 100%, so the total inter est paid is the same for both types of loans. However, the interest portions are larger at the beginning of the loan's life and smaller toward the end. Because the payment is constant, the corresponding principal payments are lower at the beginning months and higher at the later months. However, in every month the beginning balance will be higher than it would have been with a regular amortizing loan. This has two nega tive consequences. First, if you sell the car before you pay off the loan, you will owe more than you would have with a regular amortizing loan. Second, if you itemize your deductions, your interest rate deductions in the early months will be smaller than they would have been with a regular amortizing loan.

esource

See Ch04 Tool Kit.xlsx for

oil colcu/ations.

Chapter 4 Time Value of Money 181

SELF-TEST

Consider again the example in Figure 4-11. If the loan were amortized over 5 years with 60 equal monthly payments, how much would each payment be, and how would the first payment be divided between interest and principal? (Each payment would be $1,933.28; the first payment would have $500 of interest and $1,433.28 of principal repayment.)

Suppose you borrowed $30,000 on a student loan at a rate of 8% and now must repay it in three equal installments at the end of each of the next 3 years. How large would your payments be, how much of the first payment would represent interest and how much would be principal, and what would your ending balance be after the first year? (PMT = $11,641.01; interest = $2,400; principal= $9,241.01; balance at end of Vear 1 = $20,758.99)

4-18 Growing Annuities20

Notice that the term "growing annuity" isn't logical because an annuity is a series of con stant amounts to be received or paid over a specified number of periods. Despite its name being an oxymoron, a growing annuity is a series of amounts either received or paid that grow at a constant rate.

4-18a Example 1: Finding a Constant Real Income

Growing annuities are often used in the area of financial planning, where a prospective retiree wants to determine the maximum constant real, or inflation-adjusted, withdrawals that he or she can make over a specified number of years. For example, suppose a 65-year old is contemplating retirement. The individual expects to live for another 20 years, has a $1 million nest egg, expects the investments to earn a nominal annual rate of 6%, expects inflation to average 3% per year, and wants to withdraw a constant real amount annually over the next 20 years so as to maintain a constant standard of living. If the first with drawal is to be made today, what is the amount of that initial withdrawal?

This problem can be solved in three ways: (1) Set up a spreadsheet model that is similar to an amortization table, where the account earns 6% per year, withdrawals rise at the 3% inflation rate, and Excel's Goal Seek feature is used to find the initial infla tion-adjusted withdrawal. A zero balance will be shown at the end of the 20th year. (2) Use a financial calculator, where we first calculate the real rate of return, adjusted for inflation, and use it for I/YR when finding the payment for an annuity due. (3) Use a relatively complicated and obtuse formula to find this same amount.21 We will focus on the first two approaches.

We illustrate the spreadsheet approach in the chapter model, Ch04 Tool Kit.xlsx. The spreadsheet model provides the most transparent picture because it shows the value of the retirement portfolio, the portfolio's annual earnings, and each withdrawal over the 20-year planning horizon-especially if you include a graph. A picture is worth a thou sand numbers, and graphs make it easy to explain the situation to people who are plan ning their financial futures.

"This section is interesting and useful but relatively technical. It can be omitted, at the option of the instructor, without loss of continuity. Also, although an annuity is constant, the expression "growing annuity" is widely used. Therefore, we will use it, too, even though it is a self-contradictory term.

"For example, the formula used to find the payment of a growing annuity due is shown here. If g = annuity growth rate and r = nominal rate of return on investment, then:

PVIF of a growing annuity due = PVIFGA 0

., = {l - [(I + g)/(1 + r)]NJ [(I + r)/(r - g)]

PMT :c PV/PVIFGA 0

where PVIF denotes "present value interest factor." Similar formulas are available for growing ordinary annuities.

182 Part 2 Fixed Income Securities

To implement the calculator approach, we first find the expected real rate of return, where r, is the real rate of return and r

NOM the nominal rate of return. The real rate of re

turn is the return that we would see if there were no inflation. We calculate the real rate as:

Real rate = r, = [(l + r N0M

)/(l + Inflation)] - 1.0

= [l.06/1.03] - 1.0 = 0.029126214 = 2.9126214%

■ Using this real rate of return, we solve the annuity due problem exactly as we did

earlier in the chapter. We set the calculator to Begin Mode, after which we input N = 20, I/YR= real rate = 2.9126214, PV = -1000000, and FV = O; then we press PMT to get $64,786.88. This is the amount of the initial withdrawal at Time O (today), and future withdrawals will increase at the inflation rate of 3%. These withdrawals, growing at the inflation rate, will provide the retiree with a constant real income over the next 20 years provided the inflation rate and the rate of return do not change.

In our example, we assumed that the first withdrawal would be made immediately. The procedure would be slightly different if we wanted to make end-of-year withdrawals. First, we would set the calculator to End Mode. Second, we would enter the same inputs into the calculator as just listed, including the real interest rate for I/YR. The calculated PMT would be $66,673.87. However, that value is in beginning-of-year terms, and because inflation of 3% will occur during the year, we must make the following adjustment to find the inflation-adjusted initial withdrawal:

Initial end-of-year withdrawal = $66,673.87(1 + Inflation) = $66,673.87(1.03) = $68,674.09

Thus, the first withdrawal at the end of the year would be $68,674.09; it would grow by 3% per year; and after the 20th withdrawal (at the end of the 20th year), the balance in the retirement fund would be zero.

We also demonstrate the solution for this end-of-year payment example in Ch04 Tool Kit.xlsx. There we set up a table showing the beginning balance, the annual withdrawals, the annual earnings, and the ending balance for each of the 20 years. This analysis con firms the $68,674.09 initial end-of-year withdrawal derived previously.

4-18b Example 2: Initial Deposit to Accumulate a Future Sum

As another example of growing annuities, suppose you need to accumulate $100,000 in 10 years. You plan to make a deposit in a bank now, at Time 0, and then make 9 more deposits at the beginning of each of the following 9 years, for a total of 10 deposits. The bank pays 6% interest, you expect inflation to be 2% per year, and you plan to increase your annual deposits at the inflation rate. How much must you deposit initially? First, we calculate the real rate:

Real rate = r, = [l.06/1.02] - 1.0 = 0.0392157 = 3.9215686%

Next, because inflation is expected to be 2% per year, in 10 years the target $100,000 will have a real value of:

$100,000/(1 + 0.02)10 = $82,034.83

Chapter 4 Time Value of Money 183

Now we can find the size of the required initial payment by setting a financial calculator to the Begin Mode and then inputting N = 10, I/YR = 3.9215686, PV = 0, and FV = 82034.83. Then, when we press the PMT key, we get PMT = -6598.87. Thus, a deposit of $6,598.87 made at time 0 and growing by 2% per year will accumulate to $100,000 by Year 10 if the interest rate is 6%. Again, this result is confirmed in the chapter's Tool Kit.The key to this analysis is to express I/YR, FV, and PMT in real, not nominal, terms. SELF-TEST

Differentiate between a "regular" and a "growing" annuity.

What three methods can be used to deal with growing annuities?

If the nominal interest rote is 10% and the expected inflation rote is 5%, what is the expected real rate of return? (4.7619%)

Most financial decisions involve situations in which someone makes a payment at one point in time and receives money later. Dollars paid or received at two different points in time are different, and this difference is dealt with using time value of money (TVM) analysis.

• Compounding is the process of determining the future value (FV) of a cash flow or aseries of cash flows. The compounded amount, or future value, is equal to the beginning amount plus interest earned.• Future value of a single payment = FV N

= PV(l + I)N.• Discounting is the process of finding the present value (PV) of a future cash flow or aseries of cash flows; discounting is the reciprocal, or reverse, of compounding. FV

N Present value of a payment received at the end of Time N = PV = ( N '1 + I)

• An annuity is defined as a series of equal periodic payments (PMT) for a specifiednumber of periods.• An annuity whose payments occur at the end of each period is called an ordinary annuity.

[(1 + I)N 1] Future value of an (ordinary) annuity = FV A N

= PMT I - 1 Present value of an ( ordinary) annuity = PV A

N = PMT[.!. - (

1 N J I I 1 + I)

• If payments occur at the beginning of the periods rather than at the end, then wehave an annuity due. The PV of each payment is larger, because each payment isdiscounted back 1 year less, so the PV of the annuity is also larger. Similarly, the FVof the annuity due is larger because each payment is compounded for an extra year.The following formulas can be used to convert the PV and FV of an ordinary annuity to an annuity due: PVA

due = PVA

ordinary(l + I) FVA due = FVAo,dinary(l + I)

Part 2 Fixed Income Securities

• A perpetuity is an annuity with an infinite number of payments: PMT Value ofa perpetuity = -1 -

• To find the PV or FV of an uneven series, find the PV or FV of each individual cashflow and then sum them. • If you know the cash flows and the PV (or FV) of a cash flow stream, you can determine its interest rate. • When compounding occurs more frequently than once a year, the nominal rate must be converted to a periodic rate, and the number of years must be converted to periods: Periodic rate (IPER) = Nominal annual rate + Periods per yearNumber of periods = Years X Periods per year

The periodic rate and number of periods are used for calculations and are shown ontime lines. • If you are comparing the costs of alternative loans that require payments more than once a year, or the rates of return on investments that pay interest more than once ayear, then the comparisons should be based on effective (or equivalent) rates of return. Here is the formula: EAR= EFF% = (1 + IPER)M _ 1.0 = (I+ 1;;1r _ 1.0

• The general equation for finding the future value of a current cash flow (PV) for anynumber of compounding periods per year is:

where FV = PV( 1 + I )Numberofperiods = PV 1 + NOM ( l )MN N PER M

INoM = Nominal quoted interest rate M = Number of compounding periods per yearN = Number of years • An amortized loan is paid off with equal payments over a specified period. An amor

tization schedule shows how much of each payment constitutes interest, how much is used to reduce the principal, and the unpaid balance at the end of each period. Theunpaid balance at Time N must be zero. • A "growing annuity" is a stream of cash flows that grows at a constant rate for a specified number of years. The present and future values of growing annuities can befound with relatively complicated formulas or, more easily, with an Excel model. • Web Extension 4A provides derivations of the annuity formulas. • Web Extension 4B explains continuous compounding.

Define each of the following terms: a. PV; I; INT; FV N; PVAN; FVAN; PMT; M; INoM b. Opportunity cost rate c. Annuity; lump-sum payment; cash flow; uneven cash flow streamd. Ordinary (or deferred) annuity; annuity due

(4-2)

(4-3)

(4-4)

(4-5)

Chapter 4 Time Value of Money

e. Perpetuity f. Outflow; inflow; time line; terminal value g. Compounding; discounting h. Annual, semiannual, quarterly, monthly, and daily compounding i. Effective annual rate (EAR or EFF%); nominal (quoted) interest rate; APR;

periodic rate j. Amortization schedule; principal versus interest component of a payment; amor-

tized loan

185

What is an opportunity cost rate? How is this rate used in discounted cash flow analysis, and where is it shown on a time line? Is the opportunity rate a single number that is used to evaluate all potential investments?

An annuity is defined as a series of payments of a fixed amount for a specific number of periods. Thus, $100 a year for 10 years is an annuity, but $100 in Year 1, $200 in Year 2, and $400 in Years 3 through 10 does not constitute an annuity. However, the entire series does contain an annuity. Is this statement true or false?

Ifa firm's earnings per share grew from $1 to $2 over a 10-year period, the total growth would be 100%, but the annual growth rate would be less than 10%. True or false? Explain.

Would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% interest compounded daily? Explain.

S E L F - T E S T P R O B L E M S S O L U T I O I\J S S H O W 1\1 I N A P P E N D I X A

(ST-1) Future Value

(ST-2) Time Value

of Money

Assume that 1 year from now you plan to deposit $1,000 in a savings account that pays a nominal rate of 8%.

a. If the bank compounds interest annually, how much will you have in your account 4 years from now?

b. What would your balance be 4 years from now if the bank used quarterly com pounding rather than annual compounding?

c. Suppose you deposited the $1,000 in 4 payments of $250 each at the end of Years 1, 2, 3, and 4. How much would you have in your account at the end of Year 4, based on 8% annual compounding?

d. Suppose you deposited 4 equal payments in your account at the end of Years 1, 2, 3, and 4. Assuming an 8% interest rate, how large would each of your payments have to be for you to obtain the same ending balance as you calculated in part a?

Assume that 4 years from now you will need $1,000. Your bank compounds interest at an 8% annual rate.

a. How much must you deposit 1 year from now to have a balance of $1,000 at Year 4? b. If you want to make equal payments at the end of Years 1 through 4 to accumulate

the $1,000, how large must each of the 4 payments be? c. If your father were to offer either to make the payments calculated in part b

($221.92) or to give you a lump sum of $750 one year from now, which would you choose?

d. If you will have only $750 at the end of Year l, what interest rate, compounded an nually, would you have to earn to have the necessary $1,000 at Year 4?

e. Suppose you can deposit only $186.29 each at the end of Years 1 through 4, but you still need $1,000 at the end of Year 4. What interest rate, with annual compounding, is required to achieve your goal?

186

(ST-3) Effective

Annual Rates

(4-1) Future Value of a

Single Payment

(4-2) Present Value of a

Single Payment

(4-3) Interest Rate on a

Single Payment

(4-4) Number of Periods

of a Single Payment

(4-5) Number of Periods

for an Annuity

(4-6) Future Value:

Ordinary Annuity versus Annuity Due

(4-7) Present and Future Value of an Uneven

Cash Flow Stream

(4-8) Annuity Payment

and EAR

(4-9) Present and Future

Values of Single Cash Flows for

Different Periods

Part 2 Fixed Income Securities

f. To help you reach your $1,000 goal, your father offers to give you $400 one year from now. You will get a part-time job and make 6 additional deposits of equal amounts each 6 months thereafter. If all of this money is deposited in a bank that pays 8%, compounded semiannually, how large must each of the 6 deposits be?

g. What is the effective annual rate being paid by the bank in part f?

Bank A pays 8% interest, compounded quarterly, on its money market account. The managers of Bank B want its money market account's effective annual rate to equal that of Bank A, but Bank B will compound interest on a monthly basis. What nominal, or quoted, rate must Bank B set?

EASY PROBLEMS 1-8

If you deposit $10,000 in a bank account that pays 10% interest annually, how much will be in your account after 5 years?

What is the present value of a security that will pay $5,000 in 20 years if securities of equal risk pay 7% annually?

Your parents will retire in 18 years. They currently have $250,000, and they think they will need $1 million at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds?

If you deposit money today in an account that pays 6.5% annual interest, how long will it take to double your money?

You have $42,180.53 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $250,000. You expect to earn 12% annually on the account. How many years will it take to reach your goal?

What is the future value of a 7%, 5-year ordinary annuity that pays $300 each year? If this were an annuity due, what would its future value be?

An investment will pay $100 at the end of each of the next 3 years, $200 at the end of Year 4, $300 at the end of Year 5, and $500 at the end of Year 6. If other investments of equal risk earn 8% annually, what is this investment's present value? Its future value?

You want to buy a car, and a local bank will lend you $20,000. The loan would be fully amor tized over 5 years (60 months), and the nominal interest rate would be 12%, with interest paid monthly. What is the monthly loan payment? What is the loan's EFF%?

INTERMEDIATE PROBLEMS 9-29

Find the following values, using the equations, and then work the problems using a financial calculator to check your answers. Disregard rounding differences. (Hint: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key

(4-10) Present and

Future Values of Single Cash

Flows for Different Interest Rates

(4-11) Time for a Lump

Sum to Double

(4-12) Future Value of

an Annuity

(4-13) Present Value of an Annuity

(4-14) Uneven Cash Flow Stream

Chapter 4 Time Value of Money 187

for the unknown variable to obtain the second answer. This procedure can be used in parts b and d, and in many other situations, to see how changes in input variables affect the output variable.)

a. An initial $500 compounded for 1 year at 6% b. An initial $500 compounded for 2 years at 6% c. The present value of $500 due in 1 year at a discount rate of 6% d. The present value of $500 due in 2 years at a discount rate of 6%

Use both the TVM equations and a financial calculator to find the following values. See the Hint for Problem 4-9.

a. An initial $500 compounded for 10 years at 6% b. An initial $500 compounded for 10 years at 12% c. The present value of $500 due in 10 years at a 6% discount rate d. The present value of $500 due in 10 years at a 12% discount rate

To the closest year, how long will it take $200 to double if it is deposited and earns the fol lowing rates? [Notes: (1) See the Hint for Problem 4-9. (2) This problem cannot be solved exactly with some financial calculators. For example, if you enter PV = -200, PMT = 0, FV = 400, and I = 7 in an HP-12C and then press the N key, you will get 11 years for part a. The correct answer is 10.2448 years, which rounds to 10, but the calculator rounds up. However, the HPlOBII gives the exact answer.]

a. 7%

b. 10% c. 18% d. 100%

Find the future value of the following annuities. The first payment in these annuities is made at the end of Year l, so they are ordinary annuities. (Notes: See the Hint to Problem 4-9. Also, note that you can leave values in the TVM register, switch to Begin Mode, press FV, and find the FV of the annuity due.)

a. $400 per year for 10 years at 10% b. $200 per year for 5 years at 5% c. $400 per year for 5 years at 0% d. Now rework parts a, b, and c assuming that payments are made at the beginning of

each year; that is, they are annuities due.

Find the present value of the following ordinary annuities (see the Notes to Problem 4-12).

a. $400 per year for 10 years at 10% b. $200 per year for 5 years at 5% c. $400 per year for 5 years at 0% d. Now rework parts a, b, and c assuming that payments are made at the beginning of

each year; that is, they are annuities due.

a. Find the present values of the following cash flow streams. The appropriate interest rate is 8%. (Hint: It is fairly easy to work this problem dealing with the individual cash flows. However, if you have a financial calculator, read the section of the man ual that describes how to enter cash flows such as the ones in this problem. This will take a little time, but the investment will pay huge dividends throughout the course. Note that, when working with the calculator's cash flow register, you must enter

188

(4-15) Effective Rate

of Interest

(4-16) Future Value

for Various Compounding

Periods

(4-17) Present Value

for Various Compounding

Periods

(4-18) Future Value of an

Annuity for Various Compounding

Periods

(4-19) Effective versus

Nominal Interest Rates

(4-20) Amortization

Schedule

Part 2 Fixed Income Securities

CF 0

= 0. Note also that it is quite easy to work the problem with Excel, using proce dures described in the file Ch04 Tool Kit.xlsx.)

Year Cash Stream A Cash Stream B

1 $100 $300 2 400 400 3 400 400 4 400 400 5 300 100

b. What is the value of each cash flow stream at a 0% interest rate?

Find the interest rate (or rates of return) in each of the following situations.

a. You borrow $700 and promise to pay back $749 at the end of 1 year. b. You lend $700 and receive a promise to be paid $749 at the end of 1 year. c. You borrow $85,000 and promise to pay back $201,229 at the end of 10 years. d. You borrow $9,000 and promise to make payments of $2,684.80 at the end of each of

the next 5 years.

Find the amount to which $500 will grow under each of the following conditions.

a. 12% compounded annually for 5 years b. 12% compounded semiannually for 5 years c. 12% compounded quarterly for 5 years d. 12% compounded monthly for 5 years

Find the present value of $500 due in the future under each of the following conditions.

a. 12% nominal rate, semiannual compounding, discounted back 5 years b. 12% nominal rate, quarterly compounding, discounted back 5 years c. 12% nominal rate, monthly compounding, discounted back 1 year

Find the future values of the following ordinary annuities.

a. FV of $400 each 6 months for 5 years at a nominal rate of 12%, compounded semiannually

b. FV of $200 each 3 months for 5 years at a nominal rate of 12%, compounded quarterly c. The annuities described in parts a and b have the same total amount of money paid

into them during the 5-year period, and both earn interest at the same nominal rate, yet the annuity in part b earns $101.75 more than the one in part a over the 5 years. Why does this occur?

Universal Bank pays 7% interest, compounded annually, on time deposits. Regional Bank pays 6% interest, compounded quarterly.

a. Based on effective interest rates, in which bank would you prefer to deposit your money?

b. Could your choice of banks be influenced by the fact that you might want to with draw your funds during the year as opposed to at the end of the year? In answering this question, assume that funds must be left on deposit during an entire com pounding period in order for you to receive any interest.

Consider a $25,000 loan to be repaid in equal installments at the end of each of the next 5 years. The interest rate is 10%.

a. Set up an amortization schedule for the loan. b. How large must each annual payment be if the loan is for $50,000? Assume that the

interest rate remains at 10% and that the loan is still paid off over 5 years.

(4-21) Growth Rates

(4-22) Expected Rate

of Return

(4-23) Effective Rate

of Interest

(4-24) Required Lump

Sum Payment

(4-25)

Repaying a Loan

(4-26) Reaching a

Financial Goal

(4-27) Present Value of

a Perpetuity

(4-28) PV and Effective

Annual Rate

Chapter 4 Time Value of Money 189

c. How large must each payment be if the loan is for $50,000, the interest rate is 10%, and the loan is paid off in equal installments at the end of each of the next 10 years? This loan is for the same amount as the loan in part b, but the payments are spread out over twice as many periods. Why are these payments not half as large as the pay ments on the loan in part b?

Sales for Hanebury Corporation's just-ended year were $12 million. Sales were $6 million 5 years earlier.

a. At what rate did sales grow? b. Suppose someone calculated the sales growth for Hanebury in part a as follows:

"Sales doubled in 5 years. This represents a growth of 100% in 5 years; dividing 100% by 5 results in an estimated growth rate of 20% per year." Explain what is wrong with this calculation.

Washington-Pacific (W-P) invested $4 million to buy a tract ofland and plant some young pine trees. The trees can be harvested in 10 years, at which time W-P plans to sell the forest at an expected price of$8 million. What is W-P's expected rate of return?

A mortgage company offers to lend you $85,000; the loan calls for payments of $8,273.59 at the end of each year for 30 years. What interest rate is the mortgage company charging you?

To complete your last year in business school and then go through law school, you will need $10,000 per year for 4 years, starting next year (that is, you will need to withdraw the first $10,000 one year from today). Your uncle offers to put you through school, and he will deposit in a bank paying 7% interest a sum of money that is sufficient to provide the four payments of $10,000 each. His deposit will be made today.

a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal?

After the last withdrawal?

While Mary Corens was a student at the University of Tennessee, she borrowed $12,000 in student loans at an annual interest rate of 9%. If Mary repays $1,500 per year, then how long (to the nearest year) will it take her to repay the loan?

You need to accumulate $10,000. To do so, you plan to make deposits of $1,250 per year with the first payment being made a year from today-into a bank account that pays 12% annual interest. Your last deposit will be less than $1,250 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal, and how large will the last deposit be?

What is the present value of a perpetuity of $100 per year if the appropriate discount rate is 7%? If interest rates in general were to double and the appropriate discount rate rose to 14%, what would happen to the present value of the perpetuity?

Assume that you inherited some money. A friend of yours is working as an unpaid intern at a local brokerage firm, and her boss is selling securities that call for 4 payments of $50 (1 payment at the end of each of the next 4 years) plus an extra payment of $1,000 at the end of Year 4. Your friend says she can get you some of these securities at a cost of $900 each. Your money is now invested in a bank that pays an 8% nominal (quoted) inter est rate but with quarterly compounding. You regard the securities as being just as safe, and as liquid, as your bank deposit, so your required effective annual rate of return on the securities is the same as that on your bank deposit. You must calculate the value of the se curities to decide whether they are a good investment. What is their present value to you?

190

(4-29) Loan Amortization

(4-30) Loan Amortization

(4-31) Nonannual

Compounding

(4-32) Nominal Rate

of Return

(4-33) Required Annuity

Payments

(4-34) Growing Annuity

Payments

Part 2 Fixed Income Securities

Assume that your aunt sold her house on December 31, and to help close the sale she took a second mortgage in the amount of $10,000 as part of the payment. The mortgage has a quoted (or nominal) interest rate of 10%; it calls for payments every 6 months, beginning on June 30, and is to be amortized over 10 years. Now, 1 year later, your aunt must in form the IRS and the person who bought the house about the interest that was included in the two payments made during the year. (This interest will be income to your aunt and a deduction to the buyer of the house.) To the closest dollar, what is the total amount of interest that was paid during the first year?

CHALLENGING PROBLEMS 30-34

Your company is planning to borrow $1 million on a 5-year, 15%, annual payment, fully am ortized term loan. What fraction of the payment made at the end of the second year will represent repayment of principal?

It is now January 1. You plan to make a total of 5 deposits of $100 each, one every 6 months, with the first payment being made today. The bank pays a nominal inter est rate of 12% but uses semiannual compounding. You plan to leave the money in the bank for 10 years.

a. How much will be in your account after 10 years? b. You must make a payment of $1,432.02 in 10 years. To get the money for this pay

ment, you will make five equal deposits, beginning today and for the following 4 quarters, in a bank that pays a nominal interest rate of 12% with quarterly com pounding. How large must each of the five payments be?

Anne Lockwood, manager of Oaks Mall Jewelry, wants to sell on credit, giving customers 3 months to pay. However, Anne will have to borrow from her bank to carry the accounts receivable. The bank will charge a nominal rate of 15% and will compound monthly. Anne wants to quote a nominal rate to her customers (all of whom are expected to pay on time) that will exactly offset her financing costs. What nominal annual rate should she quote to her credit customers?

Assume that your father is now 50 years old, plans to retire in 10 years, and expects to live for 25 years after he retires-that is, until age 85. He wants his first retirement pay ment to have the same purchasing power at the time he retires as $40,000 has today. He wants all of his subsequent retirement payments to be equal to his first retirement payment. (Do not let the retirement payments grow with inflation: Your father realizes that if inflation occurs the real value of his retirement income will decline year by year after he retires.) His retirement income will begin the day he retires, 10 years from today, and he will then receive 24 additional annual payments. Inflation is expected to be 5% per year from today forward. He currently has $100,000 saved and expects to earn a return on his savings of 8% per year with annual compounding. To the nearest dollar, how much must he save during each of the next 10 years (with equal deposits being made at the end of each year, beginning a year from today) to meet his retire ment goal? (Note: Neither the amount he saves nor the amount he withdraws upon retirement is a growing annuity.)

You want to accumulate $1 million by your retirement date, which is 25 years from now. You will make 25 deposits in your bank, with the first occurring today. The bank

(4-35) Build a Model:

The Time Value of Money

resource

Chapter 4 Time Value of Money 191

pays 8% interest, compounded annually. You expect to receive annual raises of 3%, which will offset inflation, and you will let the amount you deposit each year also grow by 3% (i.e., your second deposit will be 3% greater than your first, the third will be 3% greater than the second, etc.). How much must your first deposit be if you are to meet your goal?

Start with the partial model in the file Ch04 P35 Build a Model.xlsx from the textbook's Web site. Answer the following questions, using the spreadsheet model to do the calculations.

a. Find the FV of $1,000 invested to earn 10% annually 5 years from now. Answer this question first by using a math formula and then by using the Excel function wizard.

b. Now create a table that shows the FV at 0%, 5%, and 20% for 0, 1, 2, 3, 4, and 5 years. Then create a graph with years on the horizontal axis and FV on the vertical axis to display your results.

c. Find the PV of $1,000 due in 5 years if the discount rate is 10% per year. Again, first work the problem with a formula and then by using the function wizard.

d. A security has a cost of $1,000 and will return $2,000 after 5 years. What rate of return does the security provide?

e. Suppose California's population is 30 million people and its population is expected to grow by 2% per year. How long would it take for the population to double?

f. Find the PV of an ordinary annuity that pays $1,000 at the end of each of the next 5 years if the interest rate is 15%. Then find the FV of that same annuity.

g. How would the PV and FV of the above annuity change if it were an annuity due rather than an ordinary annuity?

h. What would the FV and PV for parts a and c be if the interest rate were 10% with semiannual compounding rather than 10% with annual compounding?

i. Find the PV and FV of an investment that makes the following end-of-year pay ments. The interest rate is 8%.

Year

1 2

Payment

$100 200 400

j. Suppose you bought a house and took out a mortgage for $50,000. The interest rate is 8%, and you must amortize the loan over 10 years with equal end-of-year pay ments. Set up an amortization schedule that shows the annual payments and the amount of each payment that repays the principal and the amount that constitutes interest expense to the borrower and interest income to the lender.

(1) Create a graph that shows how the payments are divided between interest and principal repayment over time.

(2) Suppose the loan called for 10 years of monthly payments, 120 payments in all, with the same original amount and the same nominal interest rate. What would the amortization schedule show now?

192 Part 2 Fixed Income Securities

Assume that you are nearing graduation and have applied for a job with a local bank. The bank's evaluation process requires you to take an examination that covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.

a. Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2, (2) an or dinary annuity of $100 per year for 3 years, and (3) an uneven cash flow stream of -$50, $100, $75, and $50 at the end of Years 0 through 3.

b. (1) What's the future value of an initial $100 after 3 years if it is invested in an ac count paying 10% annual interest?

(2) What's the present value of $100 to be received in 3 years if the appropriate inter est rate is 10%?

c. We sometimes need to find out how long it will take a sum of money (or something else, such as earnings, population, or prices) to grow to some specified amount. For example, if a company's sales are growing at a rate of 20% per year, how long will it take sales to double?

d. If you want an investment to double in 3 years, what interest rate must it earn? e. What's the difference between an ordinary annuity and an annuity due? What type

of annuity is shown below? How would you change the time line to show the other type of annuity?

0 1 2 3

100 100 100

f. (1) What's the future value of a 3-year ordinary annuity of $100 if the appropriate interest rate is 10%?

(2) What's the present value of the annuity? (3) What would the future and present values be if the annuity were an

annuity due? g. What is the present value of the following uneven cash flow stream? The appropriate

interest rate is 10%, compounded annually.

0 1 2 3 4

0 100 300 300 -50

h. (1) Define the stated (quoted) or nominal rate INOM as well as the periodic rate IPER" (2) Will the future value be larger or smaller if we compound an initial amount

more often than annually-for example, every 6 months, or semiannually holding the stated interest rate constant? Why?

(3) What is the future value of $100 after 5 years under 12% annual compounding? Semiannual compounding? Quarterly compounding? Monthly compounding? Daily compounding?

(4) What is the effective annual rate (EAR or EFF%)? What is the EFF% for a nominal rate of 12%, compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily?

i. Will the effective annual rate ever be equal to the nominal (quoted) rate?

Chapter 4 Time Value of Money

j. (1) Construct an amortization schedule for a $1,000, 10% annual rate loan with three equal installments.

193

(2) During Year 2, what is the annual interest expense for the borrower, and what is the annual interest income for the lender?

k. Suppose that on January 1 you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later?

I. (1) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%, compounded semiannually?

0 1

100

(2) What is the PV of the same stream? (3) Is the stream an annuity?

100

3 Years

100

(4) An important rule is that you should never show a nominal rate on a time line or use it in calculations unless what condition holds? (Hint: Think of annual com pounding, when I

NOM = EFF% = I

PER ') What would be wrong with your answers

to parts (1) and (2) if you used the nominal rate of 10% rather than the periodic rate, I

N0M /2 = 10%/2 = 5%?

m. Suppose someone offered to sell you a note calling for the payment of $1,000 in 15 months. They offer to sell it to you for $850. You have $850 in a bank time deposit that pays a 6.76649% nominal rate with daily compounding, which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you buy the note. The note is not risky-you are sure it will be paid on schedule. Should you buy the note? Check the decision in three ways: (1) by comparing your future value if you buy the note versus leaving your money in the bank; (2) by comparing the PV of the note with your current bank account; and (3) by comparing the EFF% on the note with that of the bank account.