Week 2 Discussion 1
bfied0404
chapter04.pdf
Learning Objectives
Upon completion of Chapter 4, you will be able to:
• Express the time value of money and related mathematics, including present and future values, principal, and interest.
• Describe the significance and application of simple and compound interest.
• Explain the significance of compounding frequency in relation to future and present cash flows and effective annual percentage rates.
• Identify the values of common cash flow streams, including perpetuities, ordinary annuities, annuities due, and amortized loans.
Present and Future Value of Money
4
iStockphoto/Thinkstock
byr80656_04_c04_077-112.indd 77 3/28/13 3:31 PM
CHAPTER 4Section 4.1 The Time Value of Money
The saying “time is money” could not be more true than it is in finance. People ratio-nally prefer to collect money earlier rather than later. By delaying the receipt of cash, individuals forgo the opportunity to purchase desired goods or invest the funds to increase their wealth. The forgone interest, which could be earned if cash were received immediately, is called the opportunity cost of delaying its receipt. Individuals require compensation to reimburse them for the opportunity cost of not having the funds avail- able for immediate investment purposes. This chapter describes how such opportunity costs are calculated. Because many business activities require computing a value today for a series of future cash flows, the techniques presented in this chapter apply not only to finance but also to marketing, manufacturing, and management. Here are examples of questions that the tools introduced in this chapter can help answer:
• How much should we spend on an advertising campaign today if it will increase sales by 5% in the future?
• Which strategy should we employ, given their respective costs and estimated contributions to future earnings?
• What types of health insurance and retirement plans are best for our employees, given the amount of money we have available?
• Is it worth buying a new automated manufacturing tool for $120,000 if it reduces material waste by 15%?
Being able to give a value to cash to be received in the future, whether dividends from a share of stock, interest from a bond, or profits from a new product, is one of the primary skills needed to run a successful business. The material in this chapter provides an intro- duction to that skill.
4.1 The Time Value of Money
Suppose a friend owes you $100 and the payment is due today. You receive a phone call from this friend, who says she would like to delay paying you for 1 year. You may reasonably demand a higher future payment, but how much more should you receive? The situation is illustrated here using the timeline shown in Figure 4.1.
Figure 4.1
In this diagram “now,” the present time, is assigned t 5 0, or time zero. One year from now is assigned t 5 1. The present value of the cash payment is $100 and is denoted PV0 (and read as “present value at time zero”). Its future value at t 5 1 is denoted as FV1 (and read as “future value 1 year from now”). To find the amount that you could demand for deferring receipt of the money by 1 year, you must solve for FV1, the future value of $100
t = 0 t = 1
PV0 = $100 FV1 = ?
byr80656_04_c04_077-112.indd 78 3/28/13 3:31 PM
CHAPTER 4Section 4.1 The Time Value of Money
one year from now. The FV1 value will depend on the opportunity cost of forgoing imme- diate receipt of $100. You know, for instance, that if you had the money today, you could deposit the $100 in a bank account earning 3% interest annually. However, you know from Chapter 1 that value depends on risk. In your judgment, your friend is less likely to pay you next year than is the bank. Therefore, you will increase the rate of interest to reflect the additional risk that you think is inherent in the loan to your friend.
Suppose you decide that a 10% annual rate of interest is appropriate. The amount of the future payment, FV1, will be the original principal plus the interest that could be earned at the 10% annual rate. Algebraically, you can solve for FV1, being careful always to con- vert percentages to decimals when doing arithmetic calculations, and so
(4.1) FV1 5 $100 1 ($100)(0.10)
Factoring $100 from the right-hand side of Equation (4.1) gives
FV1 5 $100(1 1 0.10) 5 $100(1.10) 5 $110
You may demand a $110 payment at t 5 1 in lieu of an immediate $100 payment because these two amounts have equivalent value.
Let’s say that your friend agrees to this interest rate but asks to delay payment for 2 years.
Figure 4.2
Now we must find FV2, the future value of the payment 2 years from today. This situation is illustrated by the timeline in Figure 4.2. Since we know FV1 5 $110 and we know the interest rate is 10%, we can solve for FV2 by recognizing that FV2 will equal FV1 plus the interest that could be earned on FV1 during the second year. Our equation is then
(4.2) FV2 5 FV1 1 FV1(0.10) 5 $110 1 ($110)(0.10) 5 $110(1 1 0.10) 5 $110(1.10) 5 $121
You may demand a $121 payment at t 5 2 because its time value is equivalent to either $110 at t 5 1 or $100 at t 5 0, given the 10% interest rate.
t = 0
t = 0
t = 2
PV0 = $100 FV2 = ?
t = 1
FV1 = $110
FV1 = $100(1.10) ?
byr80656_04_c04_077-112.indd 79 3/28/13 3:31 PM
CHAPTER 4Section 4.2 Compound and Simple Interest
The time value of money and the mathematics associated with it provide important tools for comparing the relative values of cash flows received at different times. Just as a ham- mer may be the most useful item in a carpenter’s toolbox, time value of money mathemat- ics is indispensable to a financial manager.
For example, recall from Chapter 1 that to increase shareholder wealth, managers must make investments that have greater value than their costs. Often, such investments require an immediate cash outlay, like buying a new delivery truck. The investment (the truck) then produces cash flows for the corporation in the future (delivery fee income, increased sales, lower delivery costs, etc.). To determine whether the future cash flows have greater value than the initial cost of the truck, managers must be able to calculate the present value of the future stream of cash flows produced by this investment.
4.2 Compound and Simple Interest
The preceding section showed that, at a 10% annual interest rate, $100 today is equiv-alent to $110 a year from now and $121 in 2 years. This result may be generalized using the following formulas:
(4.3) FV1 5 PV0(1 1 r)
(4.4) FV2 5 PV0(1 1 r) 2
where FV1 and FV2 are, respectively, future values 1 year and 2 years from now, PV0 is the present value at time zero, respectively, and r is the interest rate.
Now, let’s expand Equation (4.4):
(4.5) FV2 5 PV0 5 (1 1 r)(1 1 r) 5 PV0(1 1 2r 1 r
2) 5 PV0(1 1 2r) 1 PV0(r
2)
The last line of Equation (4.5) is broken down in a special way. The first term on the right side of the equal sign, PV0(1 1 2r), would yield $120 given the information we have used in our example. The second term, PV0(r
2), yields $1. The value $120 equals your origi- nal principal ($100) plus the amount of interest earned ($20) if your friend paid simple interest. For example, if you withdraw interest earned during each year at the end of that year, you would earn simple interest. In this case, you would receive $10 interest pay- ments at the end of years 1 and 2, totaling $20. If, on the other hand, your friend credited (but did not pay) interest to you every year, then you would earn interest during year 2 on the interest credited to you at the end of year 1. Earning interest on previously earned interest is known as compounding. Thus, you would earn an extra dollar, a total of $121, over the 2-year period with interest compounded annually. In this example we assumed annual compounding since nearly all transactions are now based on compound rather than simple interest. Not all compounding is done on an annual basis, however. Some- times interest is added to an account every 6 months (semiannual compounding). Other
byr80656_04_c04_077-112.indd 80 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
contracts call for quarterly, monthly, or daily compounding. As you will see, the frequency of compounding can make a big difference when the time value of money is calculated.
4.3 The Time Value of a Single Cash Flow
Continuing our example, let us suppose that your friend who wishes to delay paying you agrees to a 10% annual rate of interest over the 2-year period and will allow you to compound interest semiannually. What will you be paid in 2 years given this agreement? Semiannual compounding means that interest will be credited to you every 6 months, based on half of the annual rate. In effect you will be earning a 5% semiannual rate of interest over four 6-month periods. In other words, the periodic interest rate will be half the annual rate because you are using semiannual compounding and you will be earning interest for four time periods (n 5 1 through 4), each period being 1/2-year long. The new situation is illustrated in Figure 4.3.
Figure 4.3
Here, FV1 is the future value of the $100 at the end of period 1 (the first 6 months). As before, FV1 equals the $100 beginning principal plus interest earned over the 6 months at the 5% semiannual interest rate. Therefore we set this up using the following equation:
(4.6) FV1 5 $100 1 $100(0.05) 5 $100(1.05) 5 $105
Therefore, at the end of period 1 (at n 5 1) the principal balance you are owed will be $105. FV2 will be equal to the principal at the beginning of period 2 plus interest earned during period 2:
(4.7) FV2 5 $105 1 $105(0.05) 5 $105(1.05) 5 $110.25
Note that we could substitute [$100(1.05)] for $105 in the second line of Equation (4.7). By doing so, FV2 could be expressed as follows:
(4.8) FV2 5 $105(1.05) 5 [$100(1.05)](1.05)
5 $100(1.05)2
n = 0
2 years1½ years1 year6 months
PV0 = $100 FV4
n = 2
FV2
n = 3
FV3
n = 1
FV1
n = 4
byr80656_04_c04_077-112.indd 81 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
By following this pattern, finding FV3 and FV4 is straightforward. For the future value at the end of the third period, we have
(4.9) FV3 5 $100(1.05) 3
5 $115.76
and that at the end of the fourth period is
(4.10) FV4 5 $100(1.05) 4
5 $121.55
Equation (4.10) gives the answer we seek. The future value at the end of four 6-month periods is $121.55. Changing from annual compounding to semiannual compounding has increased the future value of your friend’s obligation to you by $0.55. The additional interest earned from semiannual compounding, $0.55, doesn’t seem like much, but imag- ine a firm borrowing $100 million; then the compounding period—annual, semiannual, quarterly—can turn into tens of thousands of dollars.
The Future Value of a Single Cash Flow
The pattern established here may be generalized into the formula for the future value of a single cash flow using compound interest:
(4.11) FVn 5 PV0(1 1 r) n
where
FVn 5 the future value at the end of n time periods PV0 5 the present value of the cash flow
r 5 the periodic interest rate n 5 the number of compounding periods until maturity, or
(number of years until maturity)(compounding periods per year)
The periodic interest rate equals the annual nominal rate divided by the number of com- pounding periods per year,
r 5 annual nominal rate
number of periods per year
It is critical when using this formula to be certain that r and n agree with each other. If, for example, you are finding the future value of $100 after 6 years and the annual rate is 18%, compounded monthly, then the appropriate r is 1.5% per month (18%/12 5 1.5%), and n is 72 months (6 years times 12 months per year 5 72 months). Students often adjust the interest rate and then forget to adjust the number of periods (or vice versa)! The answer to this problem is
FV6312 5 a1 1 0.1812 b 6312
byr80656_04_c04_077-112.indd 82 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
Try It: Calculator Key Strokes and Excel Functions—Future Value
TI Business Analyst
Future Value of Single Cash Flow: If you put $400 in the bank today at 12% per year, and leave it there for 5 years, what will be the balance at the end of the time period?
400 [PV] The PV key is used to input the present value of the deposit, $400.
5 [N] The funds are invested for 5 years, so 5 is entered using the N key.
0 [PMT] PMT is the key used to input a constant periodic payment or deposit, but in this problem there are no such cash flows, so 0 is entered using the PMT key.
12 [I/Y] I/Y is the key used for entering the periodic interest rate, in this case 12% per year, so 12 is entered.
[CPT] [FV] CPT is the key that tells the calculator to calculate a value; in this case you are asked to find the future value of the deposit, so the calculator is told to compute the FV:
5 $704.9366.
Note: These may be input in any order so long as the [FV] and [CPT] are at the end. Also, the calcula- tor register will show the answer as a negative 704.9366, since you entered 400 as a positive number. Think of it like this: 400 is cash going one way (you are giving it to the bank), and the 704 is going the opposite direction (the bank is giving it back to you), so the two cash flows will have opposite signs. If you enter 400 [1/2] [PV] in this problem, then your answer will be a positive 704.9366. It does not matter which way you do this.
Excel
Use the FV function. The inputs for this function are RATE, NPER, PMT, PV, and TYPE, where
RATE is the interest rate per period as a percentage,
NPER is the number of compounding periods,
PMT is any periodic payment (for the FV of a single cash flow this would be zero),
PV is the present value,
and
TYPE is 0 if payments are made at the end of the period (the most common case) and 1 if payments are made at the beginning of the period.
If you put $400 in the bank today at 12% per year, and leave it there for 5 years, what will be the balance at the end of the time period?
Using the FV function in Excel gives
FV(12%,5,0,2400,0) 5 704.94
(continued)
FV72 5 $100(1.015) 72
5 $292.12
byr80656_04_c04_077-112.indd 83 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
For simple interest, without compounding, the future value is simply equal to the annual interest earned times the number of years, plus the original principal. The formula for the future value of a single cash flow using simple interest is
(4.12) FVsn 5 PV0 1 (n)(PV0)(r) 5 PV0(1 1 nr)
where
FVsn 5 the future value at the end of n periods using simple interest n 5 the number of periods until maturity (generally n simply equals the number of
years, because there is no adjustment for compounding periods) r 5 the periodic rate (which also usually equals the annual rate because there is no
adjustment for compounding periods)
For the previous example, the future value of $100 invested for 6 years in an account pay- ing 18% per year using simple interest is
FVs6 5 $100[11 (6)(0.18)] 5 $208.00
Thus, monthly compounding yielded a future value after 6 years of $292.12, or $84.12 more than simple interest in this example. Table 4.1 illustrates the future value of $100, bearing 18% annual interest, with different compounding assumptions. Be sure that you can replicate the solutions illustrated here using your calculator. Be sure your n and r agree (e.g., both are monthly, yearly, etc.) and always be sure you express percentages as decimals before doing any calculations. You should practice with your calculator until your answers match those given in Table 4.1. A graph of these results is shown in Figure 4.4.
Table 4.1: The future value of $100
Compounding assumption n r FVn Annual compounding 6 0.18 $269.96
Semiannual compounding 12 0.09 $281.27
Quarterly compounding 24 0.045 $287.60
Monthly compounding 72 0.015 $292.12
Weekly compounding 312 0.00346 $293.92
Daily compounding 2,190 0.000493 $294.39
Try It: Calculator Key Strokes and Excel Functions—Future Value (continued)
Note: Financial functions in Excel require that cash inflows and cash outflows have different arithmetic signs. We signed the PV (the amount you put in the bank today) negative because it is flowing away from you and into the bank. The result ($704.94) is positive because that is a cash flow to you. The inputs are separated by commas, so you cannot enter numbers with commas separating thousands (e.g., $1,000). Nor can you include dollar signs ($).
byr80656_04_c04_077-112.indd 84 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
Figure 4.4
The Present Value of a Cash Flow
We have solved for the future value of a current cash flow. Often, we must solve for the pres- ent value of a future cash flow, solving for PV rather than FV.
Suppose, for example, you are going to receive a bonus of $1,000 in 1 year. You could really use some cash today and are able to borrow from a bank that would charge you an annual interest rate of 12%, compounded monthly. You decide to borrow as much as you can now so that you will still be able to pay off the loan in 1 year using the $1,000 bonus. In essence, you wish to solve for the present value of a $1,000 future value, knowing the interest rate (12% per year, compounded monthly) and the term of the loan (1 year, or 12 monthly compounding periods). Figure 4.5 shows a timeline illustrating the problem.
Figure 4.5
250
260
270
280
290
300
Annually Semiannually
Fu tu
re V
al u
e
Quarterly Monthly
Number of compounding periods
Weekly Daily
n = 0 n = 12
PV0 = ? FV12 = $1,000r = 0.01
Try It: Calculator Key Strokes and Excel Functions—Present Value
Present Value of Single Cash Flow: How much money would you have to put in the bank today at 12% per year, to have $10,000 in exactly 4 years?
TI Business Analyst
1000 [FV]
3 [N]
0 [PMT]
12 [I/Y]
[CPT] [PV]
5 $711.78 (continued)
byr80656_04_c04_077-112.indd 85 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
In this case n 5 12, r 5 1%, and FV12 5 is known, whereas PV0 is unknown. We may still use Equation (4.11),
(4.11) FVn 5 PV0(1 1 r) n
Substituting in the known quantities gives
$1,000 5 PV0(1.01) 12
and using some algebra we have
(4.13) PV0 5 1,000 11.01 2212 5 $1,000 11.0112 5 $887.45
You could borrow $887.45 today and fully pay off the loan, given the bank’s terms, in 1 year using your $1,000 bonus. Equation (4.13) may be generalized into the formula for the present value of a single cash flow with compound interest. Solving for the present value of a future cash flow is also known as discounting. In fact, compounding and discounting are two sides of the same coin. Compounding is used to express a value at a future date given a rate of interest. Discounting involves expressing a future value as an equivalent amount at an earlier date.
This formula is also called the discounting formula for a single future cash flow:
(4.14) PV0 5 FVn 11 1 r 22n 5 FVn 111 1 r 2n
The variables PV0, FVn , n, and r are defined exactly as they are in the future value formula because both formulas are really the same; they are just solved for different unknowns.
Try It: Calculator Key Strokes and Excel Functions—Present Value (continued)
Note that the answer that your calculator produces will be negative if you follow these keystrokes. The future value was entered as a positive number (like a cash inflow) so the present value is negative (like a cash outflow).
Excel
Use the PV function with the format: PV(RATE,NPER,PMT,FV,TYPE).
The inputs for this example would be:
5 PV(12%,3,0,1000,0)
5 2$711.78
byr80656_04_c04_077-112.indd 86 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
Table 4.2: The present value of $1,000
Compounding assumption n r PV0
Annual compounding 1 0.12 $892.86
Semiannual compounding 2 0.06 $890.00
Quarterly compounding 4 0.03 $888.49
Monthly compounding 12 0.01 $887.45
Weekly compounding 52 0.00231 $887.04
Daily compounding 365 0.000329 $886.94
Table 4.2 solves for the present, or discounted, value of a $1,000 cash flow to be received in 1 year at a 12% per year discount rate using different compounding periods. You should be able to replicate these solutions on your calculator. A graph of these results is shown in Figure 4.6.
Figure 4.6
Present and future value formulas are very useful because they may be used to solve a variety of problems. Suppose you make a $500 deposit in a bank today and you want to know how long it will take your account to double in value, assuming that the bank pays 8% interest per year, compounded annually. Here, you are solving for the number of time periods. The timeline is shown in Figure 4.7.
886
887
888
889
890
891
Annually Semiannually
892
893
P re
se n
t V al
u e
Quarterly Monthly
Number of compounding periods
Weekly Daily
byr80656_04_c04_077-112.indd 87 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
Figure 4.7
You may substitute the known quantities PV0 5 $500, FVn 5 $1,000, r 5 0.08 into either formula and solve for n. Let’s use
(4.14) PV0 5 FVn(1 1 r) 2n
We can rearrange this equation into
PV0/FVn 5 (1 1 r) 2n
or
(1 1 r)n 5 FVn/PV0
Taking the logarithm of both sides gives us
n log(1 1 r) 5 log(FVn/PV0)
Finally, solving for n gives
n 5 log(FVn/PV0)/log(1 1 r)
Plugging in our numbers gives
n 5 log($1000/$500)/log(1 1 0.08) 5 9
Therefore in 9 years the balance in your account will double.
Suppose the account earned 8% per year compounded monthly. To find the time until the account’s balance doubled, you would convert the interest rate to reflect monthly compounding r 5 0.08/12 5 0.00667 and solve for the number of compounding periods. Starting again with
(4.14) PV0 5 FVn(1 1 r) 2n
we substitute in numbers to get
$500 5 $1,000(1.00667)2n
or
(1.00667)n 5 2
n = ?
PV0 = $500 FVn = $1,000
r = 0.08
byr80656_04_c04_077-112.indd 88 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
Using trial and error, you get the answer n 5 105. This should be interpreted as 105 months because you are dealing with monthly compounding periods. Thus, in 8.75 years the account will double in value when using monthly rather than annual compounding.
This example illustrates an important lesson. It takes less time to achieve a desired amount of wealth with more frequent compounding at a given nominal interest rate. It is no sur- prise that borrowers prefer less frequent compounding, while savers (or lenders) prefer compounding as frequently as possible. The difference between compounding frequen- cies offered at various banks makes shopping around worthwhile whether you are a bor- rower or a saver.
Another type of problem is solving for the interest rate. This time let’s suppose that an investment costing $200 will make a single payment of $275 in 5 years. What is the interest rate such an investment will yield? The timeline is shown in Figure 4.8.
Figure 4.8
Starting again with the formula
PV0 5 FVn(1 1 r) 2n
we have for n 5 5
PV0 5 FV5(1 1 r) 25
We want to solve for the interest rate r. Rearranging terms we get
11 1 r 2 5 5 FV5 PV0
so
1 1 r 5 (FV5 / PV0) 0.20
or
r 5 (FV5 / PV0) 0.20 21
Substituting in PV0 5 $200 and FV5 5 $275, we get
r 5 ($275/$200)0.20 21
or
r 5 0.06576
PV0 = 200 FV5 = 275
r = ?
byr80656_04_c04_077-112.indd 89 3/28/13 3:31 PM
CHAPTER 4Section 4.3 The Time Value of a Single Cash Flow
The answer, r 5 0.06576, is based on an annual compound rate, because we assumed n 5 5 years. It is also expressed as a decimal and could be re-expressed as a percentage, 6.576% per year compounded annually.
Effective Annual Percentage Rate
As you have seen, the frequency of compounding is important. Truth-in-lending laws now require that financial institutions reveal the effective annual percentage rate (EAR) to customers so that the true cost of borrowing is explicitly stated. Before this legislation, banks could quote custom- ers annual interest rates without revealing the compounding period. Such a lack of disclosure can be costly to borrowers. For example, borrow- ing at a 12% yearly rate from Bank A may be more costly than borrowing from Bank B, which charges 12.1% yearly, if Bank A compounds interest daily and Bank B compounds semiannually. Both 12% and 12.1% are nominal rates—they reveal the rate “in name only” but not in terms of the true economic cost. To find the effective annual rate, you must divide the nominal annual percentage rate (APR) by the number of compounding peri- ods per year and add 1, then raise this sum to an
exponent equal to the number of compounding periods per year, and, finally, subtract 1 from this result.
The general formula for the effective annual percentage rate is
(4.15) EAR 5 a1 1 APR CP
b CP
2 1
For our example,
EARA 5 a1 1 0.12365 b 365
2 1 5 0.1275 5 12.75%
EARB 5 a1 1 0.1212 b 2
2 1 5 0.1247 5 12.47%
Thus, if you are a borrower, you would prefer to borrow from Bank B despite its higher APR. The lower EAR translates into a lower cost over the life of the loan. The disclosure of EARs makes comparison shopping for rates much easier.
David Sipress/The New Yorker Collection/www.cartoonbank.com
byr80656_04_c04_077-112.indd 90 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
4.4 Valuing Multiple Cash Flows
Many problems in finance involve finding the time value of multiple cash flows. Consider the following problem (and see Demonstration Problem 4.1 for a detailed example). A charity has the opportunity to purchase a used mobile hot dog stand being sold at an auction. The charity would use the hot dog stand to raise money at special events held in the summer of each year (at the county fair and at baseball and soccer games). The old hot dog stand will only last 2 years and then will be worthless. The charity estimates that, after all operating expenses, the stand will produce cash flows of $1,000 in both June and July in each of the next 2 years and cash flows of $1,500 in each of the next two Augusts. The auction takes place January 1, and the charity requires that its fundraising projects return 12% on their invested funds. How much should the charity bid for the hot dog stand? The strategy for solving this problem is shown in Figure 4.9, where each of the forecasted cash flows is illustrated on a timeline. Note that the periods are carefully “counted” from time zero to ensure that the discounting is done correctly (this is a key to the problem!).
A Closer Look: The Rule of 72
The rule of 72, which is very useful for making estimates when dealing with the time value of money, says that if the periodic rate times the number of compounding periods equals 72, then the future value will equal approximately twice the present value for a lump sum. Stated differently, if the rate times the periods equals 72, then your original deposit will double. Use the rule of 72 to solve the following problems.
a. You deposit $500 in an account that paid 8% interest per year, compounded annually. If you leave the money in the account for 9 years, what would be your approximate balance at the end of the 9 years?
b. If you deposit $400 in an account that bears 12% interest per year (compounded annually) and leave it there for 12 years, what would be the approximate balance in your account at the end of that time?
c. Gas in 1969 cost about $0.40 per gallon. If inflation has averaged about 4.5% per year since then, use the rule of 72 to estimate whether gas is more expensive, less expensive, or equally expensive now compared to what it was then.
byr80656_04_c04_077-112.indd 91 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Figure 4.9
The present value of the stream of cash flows that the stand is expected to produce is found by applying Equation (4.14) to each of the six future cash flows. Note that 1% is used as the periodic rate (12% per year/12 months) because cash flows are spaced in monthly inter- vals. The charity should bid a maximum of $6,153.07 for the hot dog stand. Given the level of expected cash flows, paying more than this amount would result in the charity earning a lower return than its 12% objective.
The hot dog stand example illustrates the general formula for finding the present value of any cash flow stream,
(4.16) PV0 5 CF1
11 1 r 2 1 1 CF2
11 1 r 2 2 1 c1
CFn 11 1 r 2n 5 a
N n51
CFn 11 1 r 2n
where
n 5 the number of compounding periods from time 0 CFn5 the cash flow to be received exactly n compounding periods from time 0 (e.g.,
CF1 is the cash flow received at the end of period 1, etc.) r 5 the periodic interest rate
N 5 the number of periods until the last cash flow
The future value formula for a cash flow stream is also found by finding the future value of each individual cash flow and summing. Terms in the formula are defined as in the pres- ent value formula.
(4.17) FVn 5 CF1(1 1 r) N21 1 CF2(1 1 r)
N22 1 … CFn
You may question why in Equation (4.17) the first term is raised to the exponent N 2 1 and why the last term is not multiplied by an interest factor. This situation may be clarified by using a timeline. The last cash flow (CFn) occurs at the end of the last time period and therefore earns no interest. To see this, imagine this is a six-period payment stream, one each month for 6 months. You want to find FV6.
$942.05 = PV0 = $1,000(1.01) –6
$932.72 = PV0 = $1,000(1.01) –7
$1,385.22 = PV0 = $1,500(1.01) –8
$836.02 = PV0 = $1,000(1.01) –18
$827.74 = PV0 = $1,000(1.01) –19
$1,229.32 = PV0 = $1,500(1.01) –20
$6,153.07 = Total Present Value
n = 6n = 0
June $1,000
July $1,000
August $1,500
July $1,000
August $1,500
June $1,000
n = 7 n = 8 n = 18 n = 19 n = 20
byr80656_04_c04_077-112.indd 92 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Demonstration Problem 4.1: Future Value
Suppose $1,000 is deposited in an account that has an adjustable interest rate. During the first year, the rate is 10%, compounded annually. The second year the rate changes to 8%, compounded quar- terly. What is the account’s balance after 2 years?
Solution First, illustrate the problem with a timeline:
Algebra Solution
Set up the problem mathematically and solve:
FV2 5 FV1(1.02) 4
FV1 5 PV0(1.10) 1 (continued)
n = 0 n = 2
PV0 = 1,000
FV2 = ?
n = 1
FV1 r = 10%/year
r = 8%/year compounded quarterly
Figure 4.10
As the timeline in Figure 4.10 shows, CF1 will earn interest for N 2 1 period, but CFN earns no interest and is simply added to the other sums to find the total future value at time N. By convention, we assume that the cash flows from investments do not start immediately but occur at the end of the first period. This is not always the case, however. Practitioners must carefully analyze any problem to be certain exactly when cash flows will occur. A timeline is a useful aid in modeling when the cash flows from a project will occur.
n = 0
PV0 = ? $50
n = 2
$50
n = 3
$50
n = 1
$50
n = 4
PV0 (1 st $50)
PV0 (2 nd $50)
PV0 (3 rd $50)
+ PV0 (4 th $50)
= PV0 of the annuity
byr80656_04_c04_077-112.indd 93 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Demonstration Problem 4.1: Future Value (continued)
Substituting in numerical values gives
FV2 5 PV0(1.10)(1.02) 4
5 $1,000(1.10)(1.02)4
5 $1,000(1.10)(1.08243) 5 $1,000(1.190675) 5 $1,190.68
Table Solution
Solve the problem using the appropriate table. You can search for present and future value tables online—they’re readily available. Here’s an example of a site that links to the tables: http://account inginfo.com/study/pv/present-value-01.htm.
FV2 5 FV1(FV if n 5 4, r 5 2%) FV1 5 PV0(FV if n 5 1, r 5 10%)
Look up table values and substitute:
FV2 5 PV0(FV if n 5 1, r 5 10%)(FV if n 5 4, r 5 2%) 5 $1,000(1.10)(1.0824) 5 $1,190.68
Calculator
Finding FV1: Clear registers.
1000 [PV]
0 [PMT]
10 [I%]
1 [N]
[FV]
Answer: $1,100.
Finding FV2: Clear registers.
1100 [PV]
0 [PMT]
2 [I%]
4 [N]
[FV]
Answer: $1,190.68.
byr80656_04_c04_077-112.indd 94 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Perpetuities
Some special patterns of cash flows are frequently encountered in finance. Furthermore, the nature of these patterns allows the general formulas to be simplified to a more con- cise form. The first special case is that of perpetuities. These are cash flow streams where equal cash flow amounts are uniformly spaced in time (every year, or every month, etc.). Perpetuity means that these payments continue forever. To illustrate, suppose an invest- ment is expected to pay $50 every year forever. Investors require a return of 10% on this investment. What should its current price be? Recognizing that today’s price should equal the present value of the investment’s future cash flows, we can illustrate the problem using the timeline shown in Figure 4.11.
Figure 4.11
The arrow indicates that these cash flows continue into the future indefinitely. The three dots indicate that terms in between are not shown to simplify the drawing. This poses a problem: If there are an infinite number of cash flows, how can we find all of their present values? Let’s consider the algebraic expression of this problem:
(4.18) PV0 5 $50
11.10 2 1 1 $50
11.10 2 2 1 c1
$50 11.10 2 100 1
c
5 $50 c 111.10 2 1 1 1
11.10 2 2 1 c1
1 11.10 2 100 1
cd
Summing this geometric series and using some algebra yields the following formula for the present value of a perpetuity:
(4.19) PV0 5 CF r
Note that there is no subscript attached to CF because all the cash flows are the same. Therefore, there is no need to distinguish CF1 from CF2, and so on. Let’s apply the formula to the example. CF 5 $50, r 5 0.10, and
PV0 5 $50 0.10
5 $500
0
PV0 CF100 = $50
2
CF2 = $50
3
CF3 = $50
1
CF1 = $50
100
byr80656_04_c04_077-112.indd 95 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Annuities
Of all the special patterns of cash flow streams, annuities are the most common. As we shall see, millions of fixed-rate home mortgages are annuities. Retirement payments, bond interest payments, automobile loan payments, and lottery jackpot payoffs all often fit the annuity pattern.
An annuity is a stream of equally sized cash flows, equally spaced in time, which end after a fixed number of payments. Thus, annuities are like perpetuities, except they do not go on forever. The present value of an annuity can be found by summing the present values of all the individual cash flows.
(4.20) PV0 5 a N n51
CF 11 1 r 2n 5
CF 11 1 r 2 1 1
CF 11 1 r 2 2 1
c1 CF
11 1 r 2N
Here N is the number of cash flows being paid, and CF is the uniform amount of each cash flow. Solving for CF0 using Equation (4.20) would be a time-consuming problem if n were large. However, because the right-hand side of the equation is yet another geometric series, it can be simplified to yield the formula for finding the present value of an annuity:
(4.21) PV0 5 1CF 2 a 1 2 31/ 11 1 r 2N 4
r b
To convince you that Equations (4.20) and (4.21) are equivalent, let’s work an example using both approaches. Suppose you wished to know the present value of a stream of $50 payments made semiannually over the next 2 years. The first payment is scheduled to begin 6 months from today. The annual rate of interest is 10%. The problem is illustrated with the timeline shown in Figure 4.12.
Figure 4.12
Using Equation (4.20), and recognizing that r 5 5% 5 0.05 semiannually, this problem may be solved as follows:
n = 0
PV0 = ? $50
n = 2
$50
n = 3
$50
n = 1
$50
n = 4
PV0 (1 st $50)
PV0 (2 nd $50)
PV0 (3 rd $50)
+ PV0 (4 th $50)
= PV0 of the annuity
byr80656_04_c04_077-112.indd 96 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
(4.20) PV0 5 a N n51
CF 11 1 r 2n 5
CF 11 1 r 2 1 1
CF 11 1 r 2 2 1
c1 CF
11 1 r 2N
5 $50
11.05 2 1 1 $50
11 1 0.05 2 2 1 $50
11 1 0.05 2 3 1 $50
11 1 0.05 2 4 5 $47.619 1 $45.351 1 $43.192 1 41.135
5 $177.30
Alternatively, Equation (4.21) could be used to solve the same problem. Starting with
(4.21) PV0 5 1CF 2 a 1 2 31/ 11 1 r 2N 4
r b
we get
PV0 5 1$50 2 1 2 1/ 11.05 2 4
0.05
5 1$50 2 1 2 1/ 11.21550625 2 0.05
5 1$50 2 11 2 0.82270247 2
0.05
5 1$50 2 10.17729753 2
0.05
5 $8.86487626
0.05 5 $177.30
All of the steps are shown to aid you in following the calculations. It may appear that using Equation (4.21) is just as time consuming as using Equation (4.20), but consider the work involved had there been 300 payments rather than 4.
The problem just solved is an example of an ordinary annuity because cash flows com- mence at the end of the first period. Most loans require interest payments at the end of each period. Rent, on the other hand, is usually payable in advance. Annuities in which cash flows are made at the beginning of each period are called annuities due. Let’s change the example we just worked slightly to require that the cash flows be made at the begin- ning of each period.
Figure 4.13
n = 0
$50
n = 2
$50
n = 3
$50
n = 1
$50
Ordinary
n = 4
n = 0
$50
n = 2
$50
n = 3
$50
n = 1
$50
Due
n = 4
byr80656_04_c04_077-112.indd 97 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
The timeline in Figure 4.13 shows that in a four-payment annuity due, each payment occurs one period sooner than in an otherwise similar ordinary annuity. Because of this characteristic, each cash flow is discounted for one less period when finding the PV of an annuity due. The formula for finding the present value of an annuity due,
(4.22) PV0due 5 1CF 2 a 1 2 31/ 11 1 r 2N 4
r b 11 1 r 2
is simply the formula for an ordinary annuity times 1 1 r, which adjusts for one less discounting period. Thus, it is usually easier to find the PV of an ordinary annuity and multiply times 1 1 r when solving for the PV of an annuity due:
(4.23) PV0 due 5 PV0
ord (1 1 r)
Now suppose you save $100 each month for 2 years in an account paying 12% interest annually, compounded monthly. What will be the balance in the account at the end of 2 years if you make your first deposit at the end of this month? The timeline is shown in Figure 4.14.
Figure 4.14
In this case we are trying to solve for the future value of an ordinary annuity, so we use Equation (4.17):
(4.17) FVN 5 CF1(1 1 r) N 2 1 1 CF2(1 1 r)
N 2 2 1 … CFN
to get
FV24 5 $100(1.01) 23 1 100(1.01)22 1 … 1 $100
Solving our problem in this manner would take considerable time. Fortunately, the future value of an annuity is also a geometric series, which can be simplified.
The formula for the future value of an ordinary annuity is
(4.24) FVN ord 5 1CF 2 11 1 r 2N 2 1
r
n = 0
$100
n = 2
$100
n = 23
$100
n = 1
$100
n = 24
FV24
byr80656_04_c04_077-112.indd 98 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Substituting the values for our example into Equation (4.24) yields the solution:
FV24 5 1$100 2 11.01 2 24 2 1
0.01 5 $2,697.35
If the first deposit were made immediately, our problem would be one of finding the future value of an annuity due. The timeline is shown in Figure 4.15.
Figure 4.15
Each cash flow in an annuity due earns one additional period’s interest compared to the future value of an ordinary annuity. Thus, the future value of an annuity due is equal to the future value of an ordinary annuity times 1 1 r. Therefore our equation is
(4.25) FVN due 5 FVN
ord (1 1 r)
For our example, we have
FV24 due 5 ($2,697.35)(1.01)
5 $2,724.32
The future value of the deposits would therefore increase to $2,724.32 if they were made at the beginning of each period. Notice the adjustment from an ordinary annuity to an annuity due is the same whether you are solving for PV or FV [compare Equations (4.25) and (4.23)]. Note that both the present value as well as the future value of an annuity due is always larger than an otherwise similar ordinary annuity.
Loan Amortization: An Annuity Application
Many loans, such as home mortgages, require a series of equal payments made to the lender. Each payment is for an amount large enough to cover both the interest owed for the period as well as some principal. In the early stages of the loan, most of each payment covers interest owed by the borrower and very little is used to reduce the loan balance. Later in the loan’s life, the small principal reductions have added up to a sum that has sig- nificantly reduced the amount owed. Thus, as time passes, less of each payment is applied toward interest and increasing amounts are paid on the principal. This type of loan is called an amortized loan. The final payment just covers both the remaining principal bal- ance and the interest owed on that principal. An amortized loan is a direct application of
n = 0
$100
n = 2
$100
n = 23
$100
n = 1
$100
n = 24
FV24
byr80656_04_c04_077-112.indd 99 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
the present value of an annuity. The original amount borrowed is the present value of the annuity (PV0), while loan payments are the annuity’s cash flows (CFs). See Demonstration Problem 4.2 for an example of a loan amortization problem.
Demonstration Problem 4.2: Loan Amortization
An electronics shop advertises that it will sell the top-of-the-line big-screen TV for $2,000. The shop offers an interest rate of 2% per month for 24 months. What payments must be made in this amor- tized loan?
Solution
Illustrate the problem with a timeline:
Algebra Solution
PV0 5 1PMT 2 a 1 2 31/ 11 1 r 2N 4
r b
$2,000 5 1PMT 2 a1 2 31/ 11.02 2 24 4
0.02 b
So
$2,000 5 (PMT)[18.9139]
or
PMT 5 $2,000/18.9139 5 $105.74
Table Solution
PV0 5 (PMT)(PV IFAn524, r52%) $2,000 5 PMT(18.9139)
PMT 5 $105.74 PMT 5 $2,000/18.9139
Calculator Solution
Clear registers.
2000 [PV]
0 [FV] (continued)
PV0 = $2,000
n = 1 n = 3n = 0 n = 2 n = 24n = 23
PMT PMT PMT PMT PMT = ?
2%
byr80656_04_c04_077-112.indd 100 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
Demonstration Problem 4.2: Loan Amortization (continued)
2 [I%]
24 [N]
[PMT]
PMT 5 105.74
If you borrow $100,000 to buy a house, what will your monthly payments be on a 30-year mortgage if the interest rate is 9% per year? For this problem the formula for finding the present value of an annuity is used [Equation (4.21)]. The present value is the loan amount (PV0 5 $100,000), there are 360 payments (N 5 360), and the monthly interest rate is 0.75% (9%/12 months). The payment amount (CF) is determined as follows:
(4.21) PV0 ord 5 1CF 2 a 1 2 31/ 11 1 r 2N 4
r b
so
$100,000 5 1CF 2 11 2 31/ 11.0075 2 360 4 2
0.0075
5
1CF 2 a1 2 1 14.730576
b 0.0075
5 1CF 2 10.932114 2 0.0075
or
$100,000 5 (CF)(124.2819)
and therefore CF 5 $804.62.
A stream of 360 monthly payments of $804.62 will cover the interest owed each month and will pay off the entire $100,000 loan as well. Of the first payment, $750.00 will be used to pay the interest owed the lender for the use of $100,000 during the first month at the 0.75% monthly rate. $54.62 of the first payment will be applied toward the principal. Thus, for the second month of the loan only $99,945.38 is owed. This reduces the amount of inter- est owed during the second month to $749.59 and increases the second month’s principal reduction to $55.03. This pattern continues until the last payment when only a $798.63 principal balance is remaining. The last month’s interest on this balance is $5.99. There- fore, the last $804.62 payment will just pay off the loan and pay the last month’s interest too. Figure 4.16 in A Closer Look: Loan Amortization illustrates how the amount of each payment applied toward principal increases over time, with a corresponding decrease in interest expense.
byr80656_04_c04_077-112.indd 101 3/28/13 3:31 PM
CHAPTER 4Section 4.4 Valuing Multiple Cash Flows
The following feature box also presents an amortization table showing principal and inter- est payments on a 5-year, $10,000 loan, amortized using a 10% rate compounded annually.
A Closer Look: Loan Amortization
Figure 4.16: Amortized Loan
Loan amortization table: $10,000 loan, 5-year amortization, 10% interest, compounded annually
Each period’s beginning balance equals the prior period’s ending balance. Thus, the loan balance at the beginning of the first year of this loan is obviously equal to the total loan amount. (continued)
$54.62
$750.00
Total loan payment
Interest paid
$798.63
$5.99
3601
$804.62
Payment number
Principal reduction
I II III IV V
Period
Beginning principal balance
Total payment
Interest
Principal reduction
Ending principal balance
1 $10,000 $2,637.97 $1,000 $1,637.97 $8,362.03
2 $8,362.03 $2,637.97 $836.20 $1,801.77 $6,560.26
3 $6,560.26 $2,637.97 $656.03 $1,981.94 $4,578.32
4 $4,578.32 $2,637.97 $457.83 $2,180.14 $2,398.18
5 $2,398.18 $2,637.97 $239.79 $2,398.18 0
byr80656_04_c04_077-112.indd 102 3/28/13 3:31 PM
CHAPTER 4Key Terms
A Closer Look: Loan Amortization (continued)
The amount of the total payment is found by using Equation (4.21).
Interest paid each period equals the rate times the period’s beginning principal balance: Column III 5 (Column I)(r).
Principal reduction each period equals the total payment less the amount applied toward interest: Column IV 5 (Column II) 2 (Column III).
Ending principal balance equals the beginning balance minus the period’s principal reduction: Column V 5 (Column I) 2 (Column IV).
Summary
Chapter 4 has covered much of the topic of the time value of money. Next, the con-cepts and techniques covered here will be applied to finding the value of stocks, bonds, and other securities. Before that, however, it is best to practice the newly acquired skills. The authors cannot overemphasize the importance of mastering time value mathematics. Therefore, as you do your homework, make sure you feel confident in your ability. If not, now is a good time to seek out a quantitatively oriented friend or to ask your instructor for assistance.
Key Terms
amortized loan A loan that is paid off in equal periodic payments. Automobile loans and home mortgages are often amor- tized loans.
annuities due A finite stream of cash flows of a fixed amount, equally spaced in time where payments are made at the beginning of each period.
annuity A finite stream of cash flows of a fixed amount, equally spaced in time.
compounding Earning interest on previ- ously earned interest.
discount rate The interest rate used to find the present value of a future payment or series of payments. For many investments, investors’ required return is the discount rate used to find the present value.
discounting Solving for the present value of a future cash flow.
effective annual percentage rate (EAR) The annualized compound rate of interest.
future value A cash flow, or stream of cash flows, re-expressed as an equivalent amount at some future date.
interest The amount of money paid by a borrower to a lender for the use of the bor- rowed principal. The rate is expressed as a percentage of the principal owed.
nominal annual percentage rate (APR) The interest rate without adjustment for the full effect of compounding.
byr80656_04_c04_077-112.indd 103 3/28/13 3:31 PM
CHAPTER 4Key Formulas
nominal rates The stated rate or yield which reflects expectations about inflation.
opportunity cost The amount of the high- est valued forgone alternative.
ordinary annuity A finite stream of cash flows of a fixed amount, equally spaced in time, in which payment are made at the end of each period.
periodic interest rate The interest rate charged on a loan or realized on an invest- ment over a specific period of time.
perpetuities An infinite stream of equal cash flows, each equally spaced in time.
present value A future cash flow, or stream of cash flows, re-expressed as an equivalent current amount of money.
principal The amount of money borrowed.
rule of 72 A rule stating that if the periodic rate times the number of compounding periods equals 72, then the future value will equal approximately twice the present value for a lump sum.
simple interest A quick method of cal- culating the interest charge on a loan, in which one multiplies the interest rate by the principal and by the number of periods.
time value of money The idea that money available at the present time is worth more than the same amount in the future because of its potential earning capacity.
Key Formulas
Future value of a single cash flow with compound interest:
(4.11) FVn 5 PV0(1 1 r) n
Future value of a single cash flow with simple interest:
(4.12) FVsn 5 PV0(1 1 nr)
Present value of a single cash flow with compound interest:
(4.14) PV0 5 FVn(1 1 r) 2n
EAR formula:
(4.15) EAR 5 a1 1 APR CP
b CP
2 1
General formula for finding the present value of a cash flow stream:
(4.16) PV0 5 CF1
11 1 r 2 1 1 CF2
11 1 r 2 2 1 c1
CFn 11 1 r 2n
Future value formula for a cash flow stream:
(4.17) FVN 5 CF1 11 1 r 2N21 1 CF2 11 1 r 2N22 1 c CFN
byr80656_04_c04_077-112.indd 104 3/28/13 3:31 PM
CHAPTER 4Calculator Keystrokes for the TI Business Analyst Calculator
Formula for the present value of a perpetuity:
(4.19) PV0 5 CF r
Present value of an ordinary annuity:
(4.21) PVord0 5 1CF 2 a 1 2 31/ 11 1 r 2N 4
r b
Present value of an annuity due:
(4.23) PV0due 5 PV0ord 11 1 r 2
Future value of an ordinary annuity:
(4.24) FVNord 5 1CF 2 11 1 r 2N 2 1
r
Future value of an annuity due:
(4.25) FVNdue 5 FVNord 11 1 r 2
Web Resources
For a news story about a giant lottery jackpot of hundreds of millions of dollars in March 2012, see http://abcnews.go.com/Business/mega-millions-lottery-jackpot-lump-sum- annuity/story?id=16029468. The winner could choose between a lump sum payout or an annuity. After reading the story, which would you take? Do you recognize this as a time value of money problem? Are there other things to consider like psychology if you were making the choice?
Here is a link to an amortization calculator: http://www.amortization-calc.com/. It is a ready-made tool for practicing your skills before the next exam. Make up a problem and see if you and the website get the same solution.
Calculator Keystrokes for the TI Business Analyst Calculator
Be sure to also read your instruction manual! Here are a couple of general things you should do with your calculator.
First, set your “register” so that it displays more than two decimal places (we need bet- ter accuracy than that!). Press the [2nd] key, then press the [format] key, then press [6] (or the amount of decimal places you want to display), then press [enter], and your calcula- tor should now display solutions showing 6 decimal places. Never leave your calculator showing only two places—you will miss points, and maybe your job!
Second, some of the calculators come with a preset mode that tries to force everything into monthly compounding. This set of keystrokes turns off that feature (which is a recommend
byr80656_04_c04_077-112.indd 105 3/28/13 3:31 PM
CHAPTER 4Calculator Keystrokes for the TI Business Analyst Calculator
practice). Press the [2nd] key, then press the [P/Y] key. If the display says [1] (regardless of the number of zeros behind the decimal), then you are finished, and your calculator is in the correct mode. However, if your calculator says [12], then you need to change its mode by pressing [1], then pressing [enter]. Now, to ensure that you are in the correct mode, turn your calculator off, then turn it back on and press [2nd] and [P/Y] to check that it is in the [1] and not the [12] mode.
Now you can try these problems, which cover the types of problems in Chapter 4. It is a good strategy to try to do the problems first without looking at the keystrokes, and then try them with the keystrokes. For the homework problems, you can look for a similar kind of problem here and follow the pattern.
1. Future Value of Single Cash Flow: If you put $500 in the bank today at 11% per year, leave it there for 9 years, what will be the balance at the end of the time period?
500 [PV] 9 [N] 0 [PMT] 11 [I/Y] [CPT] [FV] 5 $1,279.018
Note: These may be input in any order so long as the [FV] and [CPT] are at the end. Also, the calculator register will show the answer as a negative 1,279.018, since you entered 500 as a positive number. Think of it like this: 500 is cash going one way (you are giving it to the bank), and the 1,279 is going the opposite direction (the bank is giving it back to you), so the two cash flows will have opposite signs. If you enter 500 [1/2] [PV] in this problem, then your answer will be a positive 1,279.018. It does not matter which way you do this.
2. Frequency of Compounding: Use the same problem as Question 1 but make it quar- terly compounding.
500 [PV] 36 [N]
9 years is 36 quarters.
0 [PMT] 2.75 [I/Y]
11% per year, compounded quarterly (11/4 5 2.75)
[CPT] [FV] 5 $1,327.749
3. Solving for the interest rate: If you put $400 in the bank today, and the account has grown to equal $800 in 9 years, what rate of interest have you earned per year?
400 [PV] 9 [N] 0 [PMT] 800 [1/2] [FV]
Either PV or FV must be negative (think of cash flow direction).
byr80656_04_c04_077-112.indd 106 3/28/13 3:31 PM
CHAPTER 4Calculator Keystrokes for the TI Business Analyst Calculator
[CPT] [I/Y] I/Y 5 8.0059%
Note: Be sure you understand how your computer handles cash flows, which will make the requirement of having either the 400 or 800 on this problem be a negative (it does not matter which one is negative).
4. Solving for N: If you invest $400 in an account at 12%, compounded monthly, and the balance grows to equal $1000, for how many years did you have the funds invested?
400 [PV] 0 [PMT] 1000 [1/2] [FV]
Either PV or FV must be negative (think about the cash flow direction).
1 [I/Y]
1% per month because of monthly compounding
[CPT] [N]
Your answer will be in months (since monthly data input will yield monthly output!).
N 5 92.086 months; Answer 5 7.67 years.
5. Finding the EAR: What is the effective annual rate of interest if the nominal rate is 9% per year, compounded daily?
9% per year 5 9%/365 per day 5 0.02465753% per day 5 0.0002465753 per day [1.00024657453]365 2 1 5 EAR
So the keystrokes are: 1.00024657453 [Yx] 365 [5] [2] 1 [5].
Answer: 0.094162 5 9.4162% per year.
6. PV of an annuity: What is the present value of a stream of $300 payments each month for 20 years if the appropriate discount rate is 8% per year?
0 [FV] 240 [N]
20 years is 240 months.
300 [PMT] 0.6666 [I/Y]
8% per year is 8%/12 5 0.6666% per month.
[CPT] [PV] PV 5 235,866.2875 5 $35,866.29
Note that the answer is negative (again think of cash flow directions).
7. Amortized loan payment: What is the quarterly payment on a loan of $100,000 that is amortized over a 5-year period, at an interest rate of 8% per year, compounded quarterly?
0 [FV] 20 [N]
byr80656_04_c04_077-112.indd 107 3/28/13 3:31 PM
CHAPTER 4Critical Thinking and Discussion Questions
5 years is 20 quarterly periods.
100000 [PV] 2 [I/Y]
8% per year is 2% quarterly.
[CPT] [PMT]
PMT 5 $6,115.6718 every three months.
8. Loan payoff for an amortized loan: For the loan in problem 7, what is the loan payoff (the balance owed) after 5 payments have been made?
6115.6718 [1/2] [PMT]
Note negative sign for payment; it’s outflow.
5 [N] 100000 [PV]
Note positive sign for 100,000; it’s inflow.
2 [I/Y]
8% per year is 2% quarterly.
[CPT] [FV] FV 5 $78,581.8786 would be required to pay off the loan immediately after the
fifth payment is made.
Critical Thinking and Discussion Questions
1. Suppose you own some land, purchased by your father 20 years ago for $5,000. You are able to trade this land for a brand new Porsche sports car. What economic opportunity might you forgo if you proceed with the trade? How would you esti- mate the opportunity cost of proceeding with the trade?
2. The Porsche dealership from Question 1 is also willing to trade the car for an IOU that you own that promises to pay you $2,000 at the end of each year for the next 10 years and $20,000 when it matures at the end of the 10-year period. Inves- tors are currently valuing such IOUs using a 6% discount rate. What economic opportunity might you lose if you make the trade? How would you calculate the opportunity cost of the trade?
3. If the market for new automobiles and the real estate and bond markets are all efficient, what do you think you would discover about the opportunity costs of the trade in Questions 1 and 2?
4. If you won the lottery, would you prefer a $1 million cash settlement or $100,000 a year for 10 years? Why? Assume that taxes would be the same for either choice.
5. Would a borrower prefer that interest be compounded annually or semiannually, all else the same?
6. When interest rates rise, what happens to the present value of a fixed stream of cash flows?
7. Describe the essential differences between the following cash flow stream patterns. a. annuity versus perpetuity b. annuity due versus ordinary annuity
byr80656_04_c04_077-112.indd 108 3/28/13 3:31 PM
CHAPTER 4Practice Problems
8. Why have many mortgage companies preferred to make home loans that have adjustable interest rates?
9. If you purchase a home and have the choice of making either an $800 payment at the end of each month on the mortgage or $400 payments on the first and fifteenth of every month, which option will allow you to pay off the loan more rapidly, assuming the same APR applies to both options?
10. XYZ Inc. is signing a 1-year agreement with Acme Health Services to provide XYZ’s employees with weekly aerobics classes at the firm’s factory. Acme charges a flat fee for their services, with three payment options. XYZ can choose to pay Acme either $12,000 immediately, $1,100 per month for 12 months, or $13,500 at the end of the year. How would you decide which payment option XYZ should choose?
Practice Problems
1. If $2,000 is deposited in a certificate of deposit for 3 years and when the CD matures it pays its holder $2,382.00, what annual rate of interest does the CD pay?
2. A bank promises that you can double your money in 8 years by depositing $5,000 in the Superior Savings Plan today. What rate of interest does the Superior Sav- ings Plan pay?
3. A home mortgage has exactly 14.5 years remaining until it is paid off. Monthly payments are $380 on this fully amortized loan. The interest rate is 6% per year, compounded monthly. What is the principal balance of the loan?
4. You hope to be able to withdraw $5,000 at the beginning of each month when you retire. Actuarial tables say that your life expectancy at retirement is 120 months. How much money should you have at retirement to fund these with- drawals if you expect to earn 1% per month on the account’s balance?
5. What amount of money will be in an account at the end of 10 years if the account earns 6% annually and the following deposits are made into the account?
Beginning of years Annual deposit
1–5 $2,000
6–10 $4,000
6. How much should you deposit today in an account that pays 10% per year if you wish to withdraw the following cash flows in the future?
End of years Annual withdrawal
1–5 $2,000
6–11 $4,000
12–19 $6,000
7. How much should you deposit today so that beginning 6 years from now you can withdraw $5,000 annually for the next 3 years (periods 6, 7, and 8)? Assume the interest rate is 9%.
byr80656_04_c04_077-112.indd 109 3/28/13 3:31 PM
CHAPTER 4Practice Problems
8. Given a 15% discount rate, find the present value of each of the following investments.
End of year A B C
1 $1,000 0 $7,000
2 $2,000 0 0
3 $2,000 0 0
4 $2,000 0 0
5 $2,000 $4,000 $10,000
6 $2,000 $4,000 $2,000
7 $2,000 $4,000 0
8 $2,000 $4,000 0
9 $2,000 $4,000 $1,000
10 $2,000 $4,000 0
9. Suppose you are going to purchase an automobile. The car costs $20,000 and you can afford to make an $8,000 down payment. You can borrow the balance of $12,000 from the dealership’s finance company at a 2% APR, with monthly pay- ments over 3 years, or you can borrow at an 8% APR, with monthly payments over 3 years and receive a $2,000 rebate on the purchase price. Assume that if you take the rebate option, you will apply it toward the purchase. Which alternative seems best?
10. When you retire, you hope to receive $3,000 a month for 20 years. You expect to retire in exactly 30 years. a. If you believe that you will earn 8% annually on your savings during retire-
ment, how much should you have in your retirement fund at the beginning of the retirement period?
b. Given your answer to part a, how much should you save each month during your working years to accumulate the necessary retirement fund if you expect to earn 8% per year?
11. Harland Hardcourt is debating whether or not to enter the NBA (National Bas- ketball Association) draft this year or wait 1 more year. If he enters the draft this year, he believes he will immediately be paid a $2,000,000 signing bonus followed by a salary of $1,000,000 per year for 15 years. He believes that if he waits 1 year, the signing bonus will stay the same but his salary will increase to $1,200,000 per year. In either case, his career in the pro league will last 15 years. It will cost Harland $25,000 (payable immediately) to remain in school another year while waiting to enter the draft. Assuming that salaries are paid at the end of the year in which they are earned and that savings can earn 6.75% per year, what is today’s present value of each of Harland’s alternatives? (Ignore taxes.)
byr80656_04_c04_077-112.indd 110 3/28/13 3:31 PM
CHAPTER 4Practice Problems
12. An investment promises to pay you 72 monthly payments of $450, beginning at the end of the current month. If you require a 9.5% effective annual rate of inter- est (EAR), what will you be willing to pay for this investment?
13. Suppose you are purchasing an automobile. You will borrow $7,000 and make quarterly payments on a 2-year amortized loan. The interest rate is 8% per year. a. What is your quarterly payment? b. Construct an amortization table for this loan.
14. A marina on Lake Michigan is considering purchasing a boat to rent to water skiers during the summer months. The marina requires a 15% annual return on such investments. The boat is expected to generate cash flows of $2,500 a month during the months of July and August, and $1,000 a month for June and September. The boat will last 3 years and will then be worthless. If the marina plans to purchase the boat on June 1 of this year, what is the present value of the boat investment to the marina? If the cost of the boat is $17,500, do you think the marina should make the investment? (By convention, the cash flows generated by the boat are assumed to occur at the end of each month.)
15. A lease-to-own store offered a 50-inch flat-screen TV for $99.99 per month for 24 months. An identical TV was priced at $1,689.99 at a local home electronics store. What is the implied annual interest rate of the lease-to-own offer? (Hint: Typically, a lease is structured as an annuity due.)
byr80656_04_c04_077-112.indd 111 3/28/13 3:31 PM
byr80656_04_c04_077-112.indd 112 3/28/13 3:31 PM
- _GoBack
