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Chapter 6

By Ana Bell,2014-06-29 08:54
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Chapter 6 ...

Chapter 6

Time Value of Money

LEARNING OBJECTIVES

? Convert time value of money (TVM) problems from words to time lines. After reading this chapter, students should be able to:

? Explain the relationship between compounding and discounting, between future and

present value.

? Calculate the future value of some beginning amount, and find the present value

of a single payment to be received in the future.

? Solve for time or interest rate, given the other three variables in the TVM

equation.

? Find the future value of a series of equal, periodic payments (an annuity) as

well as the present value of such an annuity.

? Explain the difference between an ordinary annuity and an annuity due, and

calculate the difference in their values.

? Calculate the value of a perpetuity.

? Demonstrate how to find the present and future values of an uneven series of

cash flows.

? Distinguish among the following interest rates: Nominal (or Quoted) rate,

Periodic rate, and Effective (or Equivalent) Annual Rate; and properly choose

between securities with different compounding periods.

? Solve time value of money problems that involve fractional time periods.

? Construct loan amortization schedules for both fully-amortized and partially-

amortized loans.

Learning Objectives: 6 - 1

LECTURE SUGGESTIONS

We regard Chapter 6 as the most important chapter in the book, so we spend a good bit

of time on it. We approach time value in three ways. First, we try to get students

to understand the basic concepts by use of time lines and simple logic. Second, we

explain how the basic formulas follow the logic set forth in the time lines. Third,

we show how financial calculators and spreadsheets can be used to solve various time

value problems in an efficient manner. Once we have been through the basics, we have

students work problems and become proficient with the calculations and also get an

idea about the sensitivity of output, such as present or future value, to changes in

input variables, such as the interest rate or number of payments.

Some instructors prefer to take a strictly analytical approach and have students focus on the formulas themselves. Others prefer to use the Present Value Tables,

which have for many years been supplied with the text. In both cases, the argument is

made that students treat their calculators as “black boxes,” and that they do not

understand where their answers are coming from or what they mean. We disagree. We

think that our approach shows students the logic behind the calculations as well as

alternative approaches, and because calculators are so efficient, students can

actually see the significance of what they are doing better if they use a calculator.

We also think it is important to teach students how to use the type of technology

(calculators and spreadsheets) they must use when they venture into the real world.

In the past, the biggest stumbling block to many of our students has been time value, and the biggest problem there has been that they did not know how to use their

calculator when we got into time value. Therefore, we strongly encourage students to

get a calculator early, learn to use it, and bring it to class so they can work

problems with us as we go through the lectures. Our urging, plus the fact that we can

now provide relatively brief, course-specific manuals for the leading calculators, has

reduced if not eliminated the problem.

Our research suggests that the best calculator for the money for most students is the HP-10B. Finance and accounting majors might be better off with a more powerful

calculator, such as the HP-17B. We recommend these two for people who do not already

have a calculator, but we tell them that any financial calculator that has an IRR

function will do.

We also tell students that it is essential that they work lots of problems, including the end-of-chapter problems. We emphasize that this chapter is critical, so

they should invest the time now to get the material down. We stress that they simply

cannot do well with the material that follows without having this material down cold.

Cost of capital and capital budgeting make little sense, and one certainly cannot work

problems in these areas, without understanding time value of money first.

For other suggestions about the lecture, please see the “Lecture Suggestions” in Chapter 2, where we describe how we conduct our classes.

DAYS ON CHAPTER: 4 OF 58 DAYS (50-minute periods)

Lecture Suggestions: 6 - 2

6-1 The opportunity cost rate is the rate of interest one could earn on an

alternative investment with a risk equal to the risk of the investment in

question. This is the value of i in the TVM equations, and it is shown on the

top of a time line, between the first and second tick marks. It is not a single

rate--the opportunity cost rate varies depending on the riskiness and maturity

of an investment, and it also varies from year to year depending on inflationary

expectations (see Chapter 5).

6-2 True. The second series is an uneven payment stream, but it contains an annuity

of \$400 for 8 years. The series could also be thought of as a \$100 annuity for

10 years plus an additional payment of \$100 in Year 2, plus additional payments

of \$300 in Years 3 through 10.

6-3 True, because of compounding effects--growth on growth. The following example

demonstrates the point. The annual growth rate is i in the following equation:

10\$1(1 + i) = \$2. 10The term (1 + i) is the FVIF for i percent, 10 years. We can find i as

follows:

Using a financial calculator input N = 10, PV = -1, PMT = 0, FV = 2, and

I = ? Solving for I you obtain 7.18 percent.

Viewed another way, if earnings had grown at the rate of 10 percent per year for

10 years, then EPS would have increased from \$1.00 to \$2.59, found as follows:

Using a financial calculator, input N = 10, I = 10, PV = -1, PMT = 0, and FV = ?.

Solving for FV you obtain \$2.59. This formulation recognizes the “interest on

interest” phenomenon.

6-4 For the same stated rate, daily compounding is best. You would earn more

“interest on interest.”

6-5 False. One can find the present value of an embedded annuity and add this PV to

the PVs of the other individual cash flows to determine the present value of the

cash flow stream.

6-6 The concept of a perpetuity implies that payments will be received forever. FV

(Perpetuity) = PV (Perpetuity)(1 + i)

? = ?.

Answers and Solutions: 6 - 3

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

6-1 0 1 2 3 4 5 10% | | | | | | PV = 10,000 FV = ? 5 5FV = \$10,000(1.10) 5 = \$10,000(1.61051) = \$16,105.10.

Alternatively, with a financial calculator enter the following: N = 5,

I = 10, PV = -10000, and PMT = 0. Solve for FV = \$16,105.10.

6-2 0 5 10 15 20 7% | | | | |

PV = ? FV = 5,000 20

With a financial calculator enter the following: N = 20, I = 7, PMT = 0, and FV

= 5000. Solve for PV = \$1,292.10.

6-3 0 n = ? 6.5% | |

PV = 1 FV = 2 n n2 = 1(1.065).

With a financial calculator enter the following: I = 6.5, PV = -1, PMT = 0, and

FV = 2. Solve for N = 11.01 ? 11 years.

6-4 Using your financial calculator, enter the following data: I = 12; PV =

-42180.53; PMT = -5000; FV = 250000; N = ? Solve for N = 11. It will take 11

years for John to accumulate \$250,000.

i = ? 6-5 0 18 | |

PV = 250,000 FV = 1,000,000 18

With a financial calculator enter the following: N = 18, PV = -250000, PMT = 0,

and FV = 1000000. Solve for I = 8.01% ? 8%. 7%

6-6 0 1 2 3 4 5

| | | | | |

300 300 300 300 300

FVA = ? 5 Answers and Solutions: 6 - 4 With a financial calculator enter the following: N = 5, I = 7, PV = 0, and

PMT = 300. Solve for FV = \$1,725.22.

6-7 0 1 2 3 4 5

| | | | | |

300 300 300 300 300 7%

With a financial calculator, switch to “BEG” and enter the following: N = 5, I

= 7, PV = 0, and PMT = 300. Solve for FV = \$1,845.99. Don’t forget to switch

back to “END” mode.

6-8 0 1 2 3 4 5 6

| | | | | | |

100 100 100 200 300 500 8% PV = ? FV = ?

Using a financial calculator, enter the following:

= 0 CF0CF = 100, N = 3 1jCF = 200 (Note calculator will show CF on screen.) 42CF = 300 (Note calculator will show CF on screen.) 53CF = 500 (Note calculator will show CF on screen.) 64and I = 8. Solve for NPV = \$923.98.

To solve for the FV of the cash flow stream with a calculator that doesn’t have

the NFV key, do the following: Enter N = 6, I = 8, PV = -923.98, and PMT = 0.

Solve for FV = \$1,466.24. You can check this as follows:

0 1 2 3 4 5 6

| | | | | | |

100 100 100 200 300 500 8% 324.00 ? (1.08) 2 233.28 ? (1.08) 125.97 3? (1.08) 136.05 4? (1.08) 146.93 5? (1.08) \$1,466.23

6-9 Using a financial calculator, enter the following: N = 60, I = 1, PV =

-20000, and FV = 0. Solve for PMT = \$444.89.

mi??Nom1?EAR = - 1.0 ??m??12 = (1.01) - 1.0

= 12.68%.

Alternatively, using a financial calculator, enter the following: NOM% = 12 and

P/YR = 12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on

Answers and Solutions: 6 - 5

?

6-10 a. 1997 1998 1999 2000 2001 2002

| | | | | |

-6 12 (in millions)

With a calculator, enter N = 5, PV = -6, PMT = 0, FV = 12, and then solve

for I = 14.87%. 5 = \$6,000,000(2.4883) = \$14,929,800, b. The calculation described in the quotation fails to take account of the compounding effect. It can be demonstrated to be incorrect as follows: which is greater than \$12 million. Thus, the annual growth rate is less than 20 percent; in fact, it is about 15 percent, as shown in Part a. \$6,000,000(1.20)

6-11 0 1 2 3 4 5 6 7 8 9 10 i = ? | | | | | | | | | | |

-4 8 (in millions)

With a calculator, enter N = 10, PV = -4, PMT = 0, FV = 8, and then solve for I

= 7.18%.

6-12 0 1 2 3 4 30

| | | | | i = ? ? ? ? |

85,000 -8,273.59 -8,273.59 -8,273.59 -8,273.59 -8,273.59

With a calculator, enter N = 30, PV = 85000, PMT = -8273.59, FV = 0, and then

solve for I = 9%.

6-13 a. 0 1 2 3 4 7% | | | | |

PV = ? -10,000 -10,000 -10,000 -10,000

With a calculator, enter N = 4, I = 7, PMT = -10000, and FV = 0. Then press

PV to get PV = \$33,872.11.

b. 1. At this point, we have a 3-year, 7 percent annuity whose value is

\$26,243.16. You can also think of the problem as follows:

\$33,872(1.07) - \$10,000 = \$26,243.04.

2. Zero after the last withdrawal.

6-14 0 1 2 3 4 5 6 12% | | | | | | |

1,250 1,250 1,250 1,250 1,250 ?

FV = 10,000

With a financial calculator, get a “ballpark” estimate of the years by entering

I = 12, PV = 0, PMT = -1250, and FV = 10000, and then pressing the N key to find

Answers and Solutions: 6 - 6

N = 5.94 years. This answer assumes that a payment of \$1,250 will be made

94/100th of the way through Year 5.

Now find the FV of \$1,250 for 5 years at 12 percent; it is \$7,941.06. Compound

this value for 1 year at 12 percent to obtain the value in the account after 6

years and before the last payment is made; it is \$7,941.06(1.12) = \$8,893.99.

Thus, you will have to make a payment of \$10,000 - \$8,893.99 = \$1,106.01 at Year

6, so the answer is: it will take 6 years, and \$1,106.01 is the amount of the

last payment.

\$3,000,000\$3,000,000\$3,000,000\$3,000,000 ???2341.1(1.1)(1.1)(1.1)6-15 Contract 1: PV = = \$2,727,272.73 + \$2,479,338.84 + \$2,253,944.40 + \$2,049,040.37

= \$9,509,596.34.

Using your financial calculator, enter the following data: CF = 0; CF = 01-43000000; I = 10; NPV = ? Solve for NPV = \$9,509,596.34.

\$2,000,000\$3,000,000\$4,000,000\$5,000,000Contract 2: PV = ???2341.10(1.10)(1.10)(1.10)

= \$1,818,181.82 + \$2,479,338.84 + \$3,005,259.20 + \$3,415,067.28

= \$10,717,847.14.

Alternatively, using your financial calculator, enter the following data: CF = 00; CF = 2000000; CF = 3000000; CF = 4000000; CF = 5000000; I = 10; NPV = ? 1234Solve for NPV = \$10,717,847.14.

\$7,000,000\$1,000,000\$1,000,000\$1,000,000Contract 3: PV = ???2341.10(1.10)(1.10)(1.10)

= \$6,363,636.36 + \$826,446.28 + \$751,314.80 + \$683,013.46

= \$8,624,410.90.

Alternatively, using your financial calculator, enter the following data: CF = 00; CF = 7000000; CF = 1000000; CF = 1000000; CF = 1000000; I = 10; NPV = ? 1234Solve for NPV = \$8,624,410.90.

Contract 2 gives the quarterback the highest present value; therefore, he should

accept Contract 2.

6-16 PV = \$100/0.07 = \$1,428.57. PV = \$100/0.14 = \$714.29.

When the interest rate is doubled, the PV of the perpetuity is halved.

6-17 0 4 8 12 16

| | | | | | | | | | | | | | | | | 2% PV = ? 0 0 0 50 0 0 0 50 0 0 0 50 0 0 0 1,050

i

= 8%/4 = 2%. PER Answers and Solutions: 6 - 7

= 0, CF = 0, 01-3CF = 50, CF = 0, CF = 50, CF = 0, CF = 50, = 0, CF45-789-111213-15CF = 1050; enter I = 2, and then press the NPV key to find PV = \$893.16. 16The cash flows are shown on the time line above. With a financial calcu-lator

enter the following cash flows into your cash flow register: CF

6-18 This can be done with a calculator by specifying an interest rate of

5 percent per period for 20 periods with 1 payment per period.

N = 10 ? 2 = 20.

I = 10%/2 = 5.

PV = -10000.

FV = 0.

Solve for PMT = \$802.43.

Set up an amortization table:

Beginning Payment of Ending

Period Balance Payment Interest Principal Balance

1 \$10,000.00 \$802.43 \$500.00 \$302.43 \$9,697.57

2 9,697.57 802.43 484.88

\$984.88

You can also work the problem with a calculator having an amortization function.

Find the interest in each 6-month period, sum them, and you have the answer.

Even simpler, with some calculators such as the HP-17B, just input 2 for periods

and press INT to get the interest during the first year, \$984.88. The HP-10B

does the same thing.

6-19 \$1,000,000 loan @ 15 percent, annual PMT, 5-year amortization. What is the

fraction of PMT that is principal in the second year? First, find PMT by using

your financial calculator: N = 5, I/YR = 15, PV = -1000000, and

FV = 0. Solve for PMT = \$298,315.55.

Then set up an amortization table:

Beginning Ending

Year Balance Payment Interest Principal Balance

1 \$1,000,000.00 \$298,315.55 \$150,000.00 \$148,315.55 \$851,684.45

2 851,684.45 298,315.55 127,752.67 170,562.88 681,121.57

Fraction that is principal = \$170,562.88/\$298,315.55 = 0.5718 = 57.18% ? 57.2%.

6-20 a. Begin with a time line:

0 1 2 3 4 5 6 7 8 9 10 16 17 18 19 20 6-mos.

0 1 2 3 4 5 8 9 10 Years 6% | | | | | | | | | | | ? ? ? | | | | |

100 100 100 100 100 FVA

Answers and Solutions: 6 - 8

Since the first payment is made today, we have a 5-period annuity due. The

applicable interest rate is 12%/2 = 6%. First, we find the FVA of the

annuity due in period 5 by entering the following data in the financial

calculator: N = 5, I = 12/2 = 6, PV = 0, and PMT = -100. Setting the

calculator on “BEG,” we find FVA (Annuity due) = \$597.53. Now, we must

compound out for 15 semiannual periods at 6 percent. ? ? ? | \$597.53 20 5 = 15 periods @ 6% \$1,432.02. PMT PMT PMT PMT PMT FV = 1,432.02 b. 0 1 2 3 4 5 40 quarters 3% The time line depicting the problem is shown above. Because the payments | | | | | | only occur for 5 periods throughout the 40 quarters, this problem cannot be

immediately solved as an annuity problem. The problem can be solved in two

steps:

1. Discount the \$1,432.02 back to the end of Quarter 5 to obtain the PV of

that future amount at Quarter 5.

Input the following into your calculator: N = 35, I = 3, PMT = 0, FV =

1432.02, and solve for PV at Quarter 5. PV = \$508.92.

2. Then solve for PMT using the value solved in Step 1 as the FV of the

five-period annuity due.

The PV found in step 1 is now the FV for the calculations in this step.

Change your calculator to the BEGIN mode. Input the following into your

calculator: N = 5, I = 3, PV = 0, FV = 508.92, and solve for PMT =

\$93.07.

6-21 Here we want to have the same effective annual rate on the credit extended as on

the bank loan that will be used to finance the credit extension.

First, we must find the EAR = EFF% on the bank loan. Enter NOM% = 15, P/YR = 12, and press EFF% to get EAR = 16.08%.

Now recognize that giving 3 months of credit is equivalent to quarterly compounding--interest is earned at the end of the quarter, so it is available to

earn interest during the next quarter. Therefore, enter P/YR = 4, EFF% = EAR =

16.08%, and press NOM% to find the nominal rate of 15.19 percent. (Don’t forget

to change your calculator back to P/YR = 1.)

Therefore, if you charge a 15.19 percent nominal rate and give credit for 3 months, you will cover the cost of the bank loan.

Alternative solution: We need to find the effective annual rate (EAR) the bank

is charging first. Then, we can use this EAR to calculate the nominal rate that

Bank EAR: EAR = (1 + i

m12/m) - 1 = (1 + 0.15/12) - 1 = 16.08%. Nom

Answers and Solutions: 6 - 9

4/4) - 1 Nom41.1608 = (1 + i/4) NomNominal rate you should quote customers: 1.0380 = 1 + i/4 Nom i = 0.0380(4) = 15.19%. Nom16.08% = (1 + i

6-22 Information given:

1. Will save for 10 years, then receive payments for 25 years.

2. Wants payments of \$40,000 per year in today’s dollars for first payment only.

Real income will decline. Inflation will be 5 percent. Therefore, to find

the inflated fixed payments, we have this time line:

0 5 10 5% | | |

40,000 FV = ?

Enter N = 10, I = 5, PV = -40000, PMT = 0, and press FV to get FV =

\$65,155.79.

3. He now has \$100,000 in an account that pays 8 percent, annual compounding.

We need to find the FV of the \$100,000 after 10 years. Enter N = 10, I = 8,

PV = -100000, PMT = 0, and press FV to get FV = \$215,892.50.

4. He wants to withdraw, or have payments of, \$65,155.79 per year for 25 years,

with the first payment made at the beginning of the first retirement year.

So, we have a 25-year annuity due with PMT = 65,155.79, at an interest rate

of 8 percent. (The interest rate is 8 percent annually, so no adjustment is

required.) Set the calculator to “BEG” mode, then enter N = 25, I = 8, PMT

= 65155.79, FV = 0, and press PV to get PV = \$751,165.35. This amount must

be on hand to make the 25 payments.

5. Since the original \$100,000, which grows to \$215,892.50, will be available,

we must save enough to accumulate \$751,165.35 - \$215,892.50 = \$535,272.85.

6. The \$535,272.85 is the FV of a 10-year ordinary annuity. The payments will

be deposited in the bank and earn 8 percent interest. Therefore, set the

calculator to “END” mode and enter N = 10, I = 8, PV = 0, FV = 535272.85,

and press PMT to find PMT = \$36,949.61.

6-23 a. Begin with a time line: 8%

0 1 19 20

| | ? ? ? | |

1.75 1.75 1.75 (in millions) PV = ?

It is important to recognize that this is an annuity due since payments

start immediately. Using a financial calculator input the following after

switching to BEGIN mode:

Answers and Solutions: 6 - 10

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