# 52 CHAPTER 5 THE TIME VALUE OF MONEY FOCUS by ruq19861

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```									                                              CHAPTER 5

THE TIME VALUE OF MONEY

FOCUS
This chapter develops and applies time value formulas for amounts and annuities. The focus is
on using time value concepts to solve business problems.

PEDAGOGY
Students learn and retain discounted cash flow concepts best when they become comfortable
with formulas and tables before relying on financial calculators. For that pedagogical reason the
formula approach is presented in the text even though calculators are used almost universally in
practice. Switching to a calculator after one understands what's going on is a trivial matter. A large
number of illustrative problems are included and instructors can supplement solution techniques with
calculators if they're so inclined.
The chapter's illustrative examples serve two purposes. In addition to demonstrating technique,
several are designed to teach practice in specific areas. For example, by the time readers finish with
annuity problems, they're familiar with some of the intricacies of mortgage loans. That means they're
relatively knowledgeable about real estate finance.

TEACHING OBJECTIVES
Students should develop an understanding of discounted cash flow concepts and a facility for
solving problems. Their problem solving ability should extend to relatively complex applications.

OUTLINE

I.   OUTLINE OF APPROACH
A brief explanation of the fact that money promised in the future is worth less than money in hand
today.
Four kinds of problems and using time lines.

II. AMOUNT PROBLEMS
A. The Future Value of an Amount
The expression relating future and present values of a single amount is developed in terms of
the future value factor and the associated table.
Problem solving technique is introduced.
Applications include deferred payment terms as the equivalent of cash discounts and the
opportunity cost rate.
B. The Expression for The Present Value of an Amount
An alternate formulation emphasizing the present value in terms of the future value. More on
technique.

III. ANNUITY PROBLEMS
A. Annuities
The concept of an annuity and its present and future values.
B. The Future Value of an Annuity - Developing a Formula
The FVA expression, factor and tables are developed.
C. The Future Value of an Annuity - Solving Problems

52
The Time Value of Money                                            53

Problem solving technique, the sinking fund concept.
D. Compound Interest and Non-Annual Compounding
Compound interest concepts and how to handle non-annual compounding in problems. The
EAR and APR.
E. The Present Value of an Annuity
The formula, factor, and table are developed.
F. The Present Value of an Annuity - Solving Problems
Applications include discounting a stream of payments, amortized loans and amortization
schedules, and working with mortgages.
G. The Annuity Due
The concept and formula are developed and applied.
H. Perpetuities
The idea of a never-ending stream with a finite present value is developed intuitively.
Applications include the capitalization of earnings and preferred stock.
I. Multi-Part Problems
Dealing with situations in which the solutions to problems become inputs to other problems.
Visualizing and time lining complex problems.
J. Uneven Streams and Imbedded Annuities
Recognizing and dealing with annuities imbedded in uneven payment streams.
K. Financial Calculators
The role of financial calculators and why we learn to work without them before we come to use
them regularly in practice.

DISCUSSION QUESTIONS

1. Why are time value concepts important in ordinary business dealings, especially those involving
contracts?

ANSWER: Business contracts and agreements generally specify payments that are due at future times.
If such payments are more than a few months into the future, the correct analysis of the value of the
agreement depends on a recognition of the time value of money.

2. Why are time value concepts crucial in determining what a bond or a share of stock should be
worth?

ANSWER: All securities derive their value solely from the future cash flows that come from owning
them. The only way to value a future cash flow today is through the present value concept. Therefore,
the value of a security depends entirely on time value ideas.

3. In a retail store a discount is a price reduction. What's a discount in finance? Are the two ideas
related?

ANSWER: Discounting in finance means taking the present value of a sum promised in the future.
The present value process always results in an figure that's less than the future amount, so in a sense,
present valuing reduces the price of the future payment.

4. Calculate the present value of one dollar 30 years in the future at 10% interest. What does the result
tell you about very long-term contracts?
54                                                Chapter 5

= \$1 [.0573]
= \$0.0573
= A little less than six cents.

Money promised in a very long-term contract isn't worth much today, even if its receipt is
certain. Therefore, we should be careful about what we give up today for a commitment in the distant
future.

5. Write a brief, verbal description of the logic behind the development of the time value formulas for
annuities.

ANSWER: To develop a time value formula for an annuity we take an annuity with a finite number of
payments and develop its present or future value by treating each payment as an amount. We then
examine the resulting expression looking for a pattern that can be extended to longer streams of
payments. Recognizing such a pattern allows us to generalize the expression to an arbitrary number of
annuity payments, n.

6. Deferred payment terms are equivalent to a cash discount. Discuss and explain this idea.

ANSWER: Deferred payment means the seller will accept the promise of a future payment instead of
the full price today (with no interest charged on the amount deferred). Since the present value of the
deferred amount is less than its nominal value, the transaction is actually being conducted at a price (in
present value terms) that's less than the stated price of the article. In essence this is a sale at a discount.
Looking at it another way, the buyer could put the deferred amount in the bank until it was due.
At that time she could withdraw the amount deposited, pay the bill, and keep the interest that would be
approximately the amount of the discount.

7. What's an opportunity cost interest rate?

ANSWER: An opportunity rate is the rate that would be earned on money given up for some purpose.
For example, in the previous question, if the seller finances his business at 12%, his cost of deferring
payment is 12%, because he could have paid off some 12% debt if he'd have received the money at the
time of the sale.

8. Discuss the idea of a sinking fund. How is it related to time value?

ANSWER: Lenders are often concerned that borrowers won't be able to raise the cash to pay off loan
principals even though they're able make interest payments during the lives of loans. A sinking fund is
an arrangement in which a borrower is required to put aside money periodically during the life of a loan,
so that at maturity funds are available to pay off the principal.
In one such arrangement, funds are deposited at interest to accumulate into the principal
amount by the loan's maturity. Time value is involved because it takes a future value of an annuity
problem to calculate the periodic payment that will just pay off the loan at maturity.

9. The amount formulas share a closer relationship than the annuity formulas. Explain and interpret
this statement.

ANSWER: The two amount formulas are really the same expression written differently for
convenience. Both expressions involve the present value and the future value of the same money, and
either can be used to solve any amount problem.
The Time Value of Money                                            55

The annuity formulas are two distinct, separately derived expressions. They deal with two
different things, the present and future value of the annuity in question. Annuity problems must be
classified as either present or future value, and the correct expression has to be used to solve either.

10. Describe the underlying meaning of compounding and compounding periods. How does it relate to
time value? Include the idea of an effective annual rate (EAR). What is the annual percentage rate
(APR)? Is the APR related to the EAR?

ANSWER: Compounding relates to earning interest on previously earned interest. Money initially
deposited at interest earns interest on that amount until the end of a compounding period. At that time
the interest is "credited" to the account, and future interest is earned on the sum of the original deposit
plus the interest earned in the first period. Interest earned in the second period is credited at its end, so
interest earned in the third period is based on the original deposit plus the first and second period's
interest. And so on. The more frequently (shorter compounding periods) interest is compounded, the
more interest is earned. Interest rates are generally stated in annual terms (the nominal rate) and must
be adjusted to reflect the compounding periods in use.
Time value assumes compound interest. In problems, compounding periods and rates must be
consistently specified.
The Effective Annual Rate (EAR) is the rate of annually compounded interest that is just
equivalent to a nominal rate compounded more frequently.
The APR is the nominal rate. The APR and the EAR are not directly related.

11. What information is contained in a loan amortization schedule?

ANSWER: A loan amortization schedule details the interest and principal components of every loan
payment as well as the beginning and ending loan balance for every payment period.

12. Discuss mortgage loans in terms of the time value of money and loan amortization. What important
points should every homeowner know about how mortgages work?

ANSWER: A mortgage is an amortized loan, generally of fairly long term. 30 years is common.
Payments are made monthly so there are 360 payments in a 30-year mortgage.
Amortized loan payments are generally constant in amount, but the split between interest and
principal within the payments varies during the life of the loan. Early payments contain relatively more
interest and later payments relatively more principal repayment.
This phenomenon is extreme in long term loans like mortgages. Early payments are nearly all
interest while later payments are nearly all principal repayment. This leads to two important facts for
homeowners. First, because interest is tax deductible, early mortgage payments save a lot on taxes
while later payments save only a little. Second, a loan is not reduced by much during the early years of
its life.

13. Discuss the idea of capitalizing a stream of earnings in perpetuity. Where is this idea useful? Is
there a financial asset that makes use of this idea?

ANSWER: A constant stream of earnings that can be expected to go on forever has a finite present
value which is known as the capitalized value of the stream.
The idea is useful in valuing businesses. Essentially, any firm is worth the capitalized value of
its expected future earnings. Where the best estimate of future earnings is simply a continuation of
current earnings, capitalizing a stream of that magnitude gives an estimate of value.
Preferred stock is a security that pays a constant dividend indefinitely. Its value is the
capitalization of the stream of its dividends.
56                                               Chapter 5

14. When an annuity begins several time periods into the future, how do we calculate its present value
today? Describe the procedure in a few words.

ANSWER: The formula for the present value of an annuity gives a value at the beginning of the
annuity. If that time is in the future, the "present value" of the annuity has to be brought back in time to
the true present as an amount. Hence two consecutive calculations are required. First take the present
value of the annuity, then treat that figure as an amount and take its present value.

PROBLEMS

Amount Problems

1. The Lexington Property Development Company has a \$10,000 note receivable from a customer due
in three years. How much is the note worth today if the interest rate is
a. 9%?
b. 12% compounded monthly?
c. 8% compounded quarterly?
d. 18% compounded monthly?
e. 7% compounded continuously?

SOLUTION:         PV = FV [PVFk,n]

a.                PV = \$10,000 [PVF9,3]
= \$10,000 (.7722)
= \$7,722

b.               PV = \$10,000 [PVF1,36]
= \$10,000 (.6989)
= \$6,989

c.               PV = \$10,000 [PVF2,12]
= \$10,000 (.7885)
= \$7,885

d.               PV = \$10,000 [PVF1.5,36]
= \$10,000 (.5851)
= \$5,851

e.              FV = PV (ekn)
\$10,000 = PV [e.07(3)]
\$10,000 = PV [1.2337]
PV = \$8,105.70

2. What will a deposit of \$4,500 left in the bank be worth under the following conditions:
a. Left for nine years at 7% interest?
b. Left for six years at 10% compounded semiannually?
c. Left for five years at 8% compounded quarterly?
d. Left for 10 years at 12% compounded monthly?
The Time Value of Money                                   57

SOLUTION:      FV = PV [FVFk,n)

a.            FV = \$4,500 [FVF7,9] = \$4,500 (1.8385) = \$8,273.25
b.            FV = \$4,500 [FVF5,12] = \$4,500 (1.7959) = \$8,081.55
c.            FV = \$4,500 [FVF2,20] = \$4,500 (1.4859) = \$6,686.55
d.            FV = \$4,500 [FVF1,120] = \$4,500 (3.3004) = \$14,851.80

3. What interest rates are implied by the following lending arrangements?
a. You borrow \$500 and repay \$555 in one year.
b. You lend \$1,850 and are repaid \$2,078.66 in two years.
c. You lend \$750 and are repaid \$1,114.46 in five years with quarterly compounding.
d. You borrow \$12,500 and repay \$21,364.24 in three years under monthly compounding.
(Note: In c and d, be sure to give your answer as the annual nominal rate.)

SOLUTION:      FV = PV [FVFk,n]

a.              \$555 = \$500 [FVFk,1]
FVFk,1 = 1.1100
k = 11%

b.          \$2,078.66 = \$1,850.00 [FVFk,2]
FVFk,2 = 1.1236
k = 6%

c.          \$1,114.46 = \$750.00 [FVFk,20]
FVFk,20 = 1.4859
k = 2%
knom = 8%

d.         \$21,364.24 = \$12,500.00 [FVFk,36]
FVFk,36 = 1.7091
k = 1.5%
knom = 18%

4. How long does it take for the following to happen?
a. \$856 grows into \$1,122 at 7%.
b. \$450 grows into \$725.50 at 12% compounded monthly.
c. \$5,000 grows into \$6724.44 at 10% compounded quarterly.

SOLUTION: PV = FV [PVFk,n]

a.              \$856 = \$1,122 [PVF7,n]
PVF7,n = .7629
n = 4 years

b.            \$450.00 = \$725.50 [PVF1,n]
PVF1,n = .6203
n = 48 months = 4 years
58                                               Chapter 5

c.               \$5,000 = \$6,724.44 [PVF2.5,n]
PVF2.5,n = 0.7436
n = 12 quarters = 3 years

5. Sally Guthrie is looking for an investment vehicle that will double her money in five years.
a. What interest rate, to the nearest whole percentage, does she have to receive?
b. At that rate, how long will it take the money to triple?
c. If she can't find anything that pays more than 11%, approximately how long will it take to
double her investment?
d. What kind of financial instruments do you think Sally is looking at? Are they risky? What
could happen to Sally's investment?

SOLUTION: FV = PV [FVFk,n]

a.               2 = 1 [FVFk,5]
FVFk,5 = 2
k= 15%

b.          FVF15,n = 3
n = 7.9 years (approximate with 8 years)

c.           FVF11,n = 2
n = 6.6 years (approximate with 7 years)

d. Investments with anticipated returns like these are probably growth-oriented stocks with
considerable risk. She could lose money.

6. Branson Inc. has sold product to the Brandywine Company, a major customer, for \$20,000. As a
courtesy to Brandywine, Branson has agreed to take a note due in two years for half of the amount due.
a. What is the effective price of the transaction to Branson if the interest rate is: (1) 6%, (2) 8%,
(3) 10%, (4) 12%?
b. Under what conditions might the effective price be even less as viewed by Brandywine?

SOLUTION:
a. 1)  PV = FV [PVF6,2] = \$10,000 (.8900) = \$8,900
\$8,900 + \$10,000 = \$18,900
Effective Discount = 5.5%
2)  PV = FV [PVF8,2] = \$10,000 (.8573) = \$8,573
\$8,573 + \$10,000 = \$18,573
Effective Discount = 7.1%
3)  PV = FV [PVF10,2] = \$10,000 (.8264) = \$8,264
\$8,264 + \$10,000 = \$18,264
Effective Discount = 8.7%
4)  PV = FV [PVF12,2] = \$10,000 (.7972) = \$7,972
\$7,972 + \$10,000 = \$17,972
Effective Discount = 10.1%

b. The discount from Brandywine's perspective is calculated as in part a), but using the interest rate at
which that firm borrows. If Brandywine's rate is higher than Branson's, it will perceive a greater
discount.
The Time Value of Money                                       59

7. John Cleaver's grandfather died recently and left him a trunk that had been locked in his attic for
years. At the bottom of the trunk John found a packet of 50 World War I "liberty bonds" that had never
been cashed in. The bonds were purchased for \$11.50 each in 1918, and pay 3% interest as long as
they're held. (Government savings bonds like these accumulate and compound their interest unlike
corporate bonds, which regularly pay out interest to bondholders.)
a. How much are the bonds worth in 2000?
b. How much would they have been worth if they paid interest at a rate more like that paid in
recent years, say 7%?
c. Comment on the difference between the answers to parts a and b.

SOLUTION:
First notice that
[FVFk,a+b] = [FVFk,a] [FVFk,b]
because
(1+k)a+b = (1+k)a (1+k)b,
and
FVFk,n = (1+k)n
Therefore,
[FVFk,82] = [FVFk,50] [FVFk,30] [FVFk,2].
So,
[FVF3,82] = [FVF3,50] [FVF3,30] [FVF3,2]
= (4.3839)(2.4273)(1.0609)
= 11.289
and
[FVF7,82] = [FVF7,50] [FVF7,30] [FVF7,2]
= (29.457)(7.6123)(1.1449)
= 256.73

a.                    FV = PV [FVF3,82]
= \$11.50 (11.289)
= \$129.82 per bond

b.                    FV = PV [FVF7,82]
= \$11.50 (256.73)
= \$2,952.40 per bond

c. Over a long period the interest rate makes an enormous difference in investment results!

8. Paladin Enterprises manufactures printing presses for small-town newspapers that are often short of
cash. To accommodate these customers, Paladin offers the following payment terms:
1/3 on delivery
1/3 after six months
1/3 after 18 months
The Littleton Sentinel is a typically cash-poor newspaper considering one of Paladin's presses
a. What discount is implied by the terms from Paladin's point of view if it can invest excess
funds at 8% compounded quarterly?
b. The Sentinel can borrow limited amounts of money at 12% compounded monthly. What
discount do the payment terms imply to the Sentinel?
c. Reconcile these different views of the same thing in terms of opportunity cost.
60                                             Chapter 5

SOLUTION: Assume a price of \$300.
a.          PV = \$100 + \$100 [PVF2,2] + \$100 [PVF2,6]
= \$100 + \$100 (.9612)+ \$100 (.8880)
= \$284.92
Discount = \$15.08/\$300 = 5%

b.             PV = \$100 + \$100 [PVF1,6] + \$100 [PVF1,18]
= \$100 + \$100 (.9420)+ \$100 (.8360)
= \$277.80
Discount = \$22.20/\$300 = 7.4%

c. The buyer is avoiding more financing cost than the seller is giving up because funds are available to
both of them at different opportunity rates.

Annuity Problems

9. How much will \$650 per year be worth in eight years at interest rates of
a. 12%
b. 8%
c. 6%

SOLUTION:        FVA = PMT [FVFAk,n]
a.               FVA = \$650 [FVFA12,8] = \$650 (12.2997) = \$7,994.81
b.               FVA = \$650 [FVFA8,8] = \$650 (10.6366) = \$6,913.79
c.               FVA = \$650 [FVFA6,8] = \$650 (9.8975) = \$6,433.38

10. The Wintergreens are planning ahead for their son's education. He's eight now and will start college
in 10 years. How much will they have to set aside each year to have \$65,000 when he starts if the
interest rate is 7%?

SOLUTION: FVA = PMT [FVFAk,n]
\$65,000 = PMT [FVFA7,10] = PMT (13.8164)
PMT = \$4,704.55

11. What interest rate would you need to get to have an annuity of \$7,500 per year accumulate to
\$279,600 in 15 years?

SOLUTION: FVA = PMT [FVFAk,n]
\$279,600 = \$7,500 [FVFAk,15]
FVFAk,15 = 37.28
k = 12%

12. How many years will it take for \$850 per year to amount to \$20,000 if the interest rate is 8%?
Interpolate and give an answer to the nearest month.

SOLUTION: FVA = PMT [FVFAk,n]
\$20,000 = \$850 [FVFA8,n]
FVFA8,n = 23.529
n = 13.75 yrs = 13 yrs 9 mos
The Time Value of Money                                          61

13. What would you pay for an annuity of \$2,000 paid every six months for 12 years if you could invest
your money elsewhere at 10% compounded semiannually?

SOLUTION: PVA = PMT [PVFAk,n]
PVA = \$2,000 [PVFA5,24] = \$2,000 (13.7986)
PVA = \$27,597.20

14. Construct an amortization schedule for a four-year, \$10,000 loan at 6% interest compounded
annually.

SOLUTION: PVA = PMT [PVFAk,n]
\$10,000 = PMT [PVFA6,4]
PMT = \$2,885.92

Year       Beg Bal         PMT         INT         Prin Red        End Bal

1         \$10,000.00      \$2,885.92 \$600.00       \$2,285.92       \$7,714.08
2           7,714.08       2,885.92 462.84         2,423.08        5,291.00
3           5,291.00       2,885.92  317.46        2,568.46        2,722.54
4           2,722.54       2,885.92  163.35        2,722.54            0.00

15. A \$10,000 car loan has payments of \$361.52 per month for three years. What is the interest rate?
Assume monthly compounding and give the answer in terms of an annual rate.

SOLUTION: PVA = PMT [PVFAk,n]
\$10,000 = \$361.52 [PVFAk,36]
PVFAk,36 = 27.661
k = 1.5%
knom = 18%

16. Joe Ferro's uncle is going to give him \$250 a month for the next two years starting today. If Joe
banks every payment in an account paying 6% compounded monthly, how much will he have at the end
of three years?

= \$250 [FVFA.5,24](1.005)
= \$250 (25.4320) (1.005) = \$6,389.79
which stays in the bank for another year:
FV = \$6,389.79 [FVF.5,12]
= \$6,389.79 (1.0617) = \$6,784.04

17. How long will it take a payment of \$500 per quarter to amortize a loan of \$8,000 at 16%
compounded quarterly? Interpolate and give your answer in terms of years and months. How much less
time will it take if loan payments are made at the beginning of each month rather than at the end?

SOLUTION: PVA = PMT [PVFAk,n]
\$8,000 = \$500 [PVFA4,n]
PVFA4,n = 16
n = 26 quarters = 6.5 years = 6 years 6 months
62                                                Chapter 5

\$8,000 = \$500 [PVFA4,n](1.04)
PVFA4,n = 15.3846
n = 24 1/3 quarters = 6 years 1 month

18. What are the monthly mortgage payments on a 30-year loan for \$150,000 at 12%? Construct an
amortization table for the first six months of the loan.

SOLUTION: PVA = PMT [PVFAk,n]
\$150,000 = PMT [PVFA1,360] = PMT(97.2183)
PMT = \$1,542.92

Year          Beg Bal         PMT          INT          Prin Red     End Bal

1          \$150,000.00    \$1,542.92     \$1,500.00 \$42.92         \$149,957.08
2           149,957.08     1,542.92      1,499.57  43.35          149,913.73
3           149,913.73     1,542.92      1,499.13  43.79          149,869.94
4           149,869.94     1,542.92      1,498.70  44.22          149,825.72
5           149,825.72     1,542.92      1,498.26  44.66          149,781.06
6           149,781.06     1,542.92      1,497.81  45.11          149,735.95

19. Construct an amortization schedule for the last six months of the loan in Problem 18. (Hint: What is
the unpaid balance at the end of 29 ½ years?)

SOLUTION: PVA = PMT [PVFA1,6]
= \$1,542.92 (5.7955)
= \$8,941.99

Month Beg Bal             PMT          INT        Prin Red      End Bal

355     \$8,941.99        \$1,542.92    \$89.42      \$1,453.50 \$7,488.49
356      7,488.49         1,542.92     74.88       1,468.04  6,020.45
357      6,020.45         1,542.92     60.20       1,482.72  4,537.73
358      4,537.73         1,542.92     45.38       1,497.54  3,040.19
359      3,040.19         1,542.92     30.40       1,512.52  1,527.67
360      1,527.67         1,542.92     15.25       1,527.67      0.00

20. How soon would the loan in Problem 18 pay off if the borrower made a single additional payment
of \$17,936.29 to reduce principal at the end of the fifth year?

SOLUTION:              360 − 60 = 300
PVA = PMT [PVFA1,300]
PVA = \$1,542.92 (94.9466) = \$146,495.01
New Balance = \$128,558.72
PVA = PMT [PVFA1,n]
\$128,558.72 = \$1,542.92 [PVFA1,n]
PVFA1,n = 83.3217
n = 180 months = 15 years
So the loan would pay off in a total of 20 years.
The Time Value of Money                                          63

21. What are the payments to interest and principal during the twenty-fifth year of the loan in Problem
18?

SOLUTION: Calculate the loan balance after 25 and 26 years (five and four years remaining). The
difference is the amount paid into principal during the twenty-fifth year.

PVA = PMT [PVFA1,60]
= \$1,542.92 (44.9550)
= \$69,361.97

PVA = PMT [PVFA1,48]
= \$1,542.92 (37.9740)
= \$58,590.84

Payments to principal:    \$69,361.97
− 58,590.84
\$10,771.13

Total payments: \$1,542.92 × 12 = \$18,515.04

Interest payments:       \$18,515.04
−10,771.13
\$ 7,743.91

Multi-Part Problems

22. The Tower family wants to make a home improvement that is expected to cost \$60,000. They want
to fund as much of the cost as possible with a home equity loan, but can afford payments of only \$600
per month. Their bank offers equity loans at 12% compounded monthly for a maximum term of 10
years.
a. How much cash do they need as a down payment?
b. Their bank account pays 8% compounded quarterly.

If they delay starting the project for two years, how much would they have to save each quarter to
make the required down payment if the loan rate and estimated cost remains the same?

SOLUTION:
Loan:   PVA = PMT [PVFAk,n] = \$600 [PVFA1,120]
= \$600 (69.7005) = \$41,820.30

a. Savings req: \$60,000 − \$41,820.30 = \$18,179.70

b. Save-up:      FVA = PMT [FVFAk,n] = PMT [PVFA2,8]
\$18,179.70 = PMT (8.5830)
PMT = \$2,118.10

23. The Stein family wants to buy a small vacation house in a year and a half. They expect it to cost
\$75,000 at that time. They have the following sources of money
1. They currently have \$10,000 in a bank account that pays 6% compounded monthly.
2. Uncle Murray has promised to give them \$1,000 a month for 18 months starting today.
3. At the time of purchase, they'll take out a mortgage.
64                                             Chapter 5

They anticipate being able to make payments of about \$300 a month on a 15-year, 12% loan. In
addition, they plan to make quarterly deposits to an investment account to save up any shortfall in the
amount required. How much must those additions be if the investment account pays 8% compounded
quarterly?

SOLUTION:
Current bank account:
FV = PV [FVFk,n] = \$10,000 [FVF.5,18]
= \$10,000 (1.0939) = \$10,939

Uncle Murray:
FVAd = PMT [FVFAk,n](1+k) = \$1,000 [FVFA.5,18](1+k)
= \$1,000 (18.7858) (1.005) = \$18,879.73

Loan:       PVA = PMT [PVFAk,n] = \$300 [PVFA1,180]
= \$300 (83.3217) = \$24,996.51

Sources:      \$10,939.00   Shortfall:    \$75,000.00
18,879.73                 −54,815.24
24,996.51                 \$20,184.76
\$54,815.24

Save-up: FVA = PMT [FVFAk,n] = PMT [FVFA2,6]
\$20,184.76 = PMT (6.3081)
PMT = \$3,199.82

24. Clyde Atherton wants to buy a car when he graduates college in two years. He has the following
sources of money:
1. He has \$5,000 now in the bank in an account paying 8% compounded quarterly.
2. He will receive \$2,000 in one year from a trust.
3. He'll take out a car loan at the time of purchase on which he'll make \$500 monthly payments
at 18% compounded monthly over four years.
4. Clyde's uncle is going to give him \$1,500 a quarter starting today for one year.
In addition Clyde will save up money in a credit union through monthly payroll deductions at
his part-time job. The credit union pays 12% compounded monthly. If the car is expected to cost
\$40,000 (Clyde has expensive taste!), how much must he save each month?

SOLUTION:
Current bank account:
FV = PV [FVFk,n] = \$5,000 [FVF2,8]
= \$5,000 (1.1717) = \$5,858.50

Trust:            FV = PV [FVFk,n] = \$2,000 [FVF2,4]
= \$2,000 (1.0824) = \$2,164.80

Loan:           PVA = PMT [PVFAk,n] = \$500 [PVFA1.5,48]
= \$500 (34.0426) = \$17,021.30
Uncle:
FVAd = PMT [FVFAk,n](1+k) = \$1,500 [FVFA2,4](1.02)
= \$1,500 (4.1216) (1.02) = \$6,306.05
The Time Value of Money                                            65

Bring this forward as an amount

FV = PV [FVFk,n] = \$6,306.05 [FVF2,4]
= \$6,306.05 (1.0824) = \$6,825.67

Sources:    \$ 5,858.50            Shortfall: \$40,000.00
2,164.80                        −31,870.27
17,021.30                        \$ 8,129.73
6,825.67
\$31,870.27

Save-up:       FVA = PMT [FVFAk,n] = PMT [FVFA1,24]
\$8,129.73 = PMT (26.9735)
PMT = \$301.40

COMPUTER PROBLEMS

25. At any particular time, home mortgage rates are determined by market forces and individual
borrowers can't do much about them. The length of time required to pay off a mortgage loan, however,
varies a great deal with the size of the monthly payment made, which is under the borrower's control.
You're a junior loan officer for a large metropolitan bank. The head of the mortgage
department is concerned that customers don't fully appreciate that a relatively small increase in the size
of mortgage payments can make a big difference in how long the payments have to be made. She feels
homeowners may be passing up an opportunity to make their lives better in the long run by not choosing
shorter-term mortgages that they can readily afford.
To explain the phenomenon to customers she's asked you to put together a chart that displays
the variation in term with payment size at typical interest rates. The starting point for the charts should
be the term for a typical thirty-year (360-month) loan. Use the TIMEVAL program to construct the
following chart.

Mortgage Payments per \$100,000 Borrowed
as Term Decreases

MORTGAGE TERM IN YEARS

30             25               20              15

Rates                 8%

10%

12%

14%

Write a paragraph using the chart to explain the point. What happens to the effect as interest
rates rise? Why?
66                                               Chapter 5

SOLUTION:

Mortgage Payments per \$100,000 Borrowed
as Term Decreases

MORTGAGE TERM IN YEARS

30              25              20               15

Rates                  8%          \$734             \$772           \$836            \$956

10%         \$878             \$909           \$965          \$1,075

12%       \$1,029           \$1,053          \$1,101         \$1,200

14%       \$1,185           \$1,204          \$1,244         \$1,332

Notice how much faster the homeowner's mortgage is paid off if a little extra money is devoted
to house payment each month. At 8% an extra \$38 a month pays off the typical mortgage loan a full
five years sooner, while the term is cut in half by paying \$222 a month extra. That means a 30%
increase (or 222/734) in payment brings a 50% reduction in the time mortgage payments have to be
made. At higher interest rates the effect is amazing. At 14%, five years of payments can be avoided by
paying only seven extra dollars a month and the term can be cut in half with only a 5% increase in the
monthly payment.
The term shortening effect of an extra dollar paid increases with the interest rate, because the
additional payment reduces principal ahead of schedule, and early pay-down avoids more interest at
higher rates.

26. Amitron Inc. is considering an engineering project that requires an investment of \$250,000 and is
expected to generate the following stream of payments (income) in the future. Use the TIMEVAL
program to determine if the project is a good idea in a present value sense. That is, does the present
value of expected cash inflows exceed the value of the investment that has to be made today?

Year         Payment
1            \$63,000
2            \$69,500
3            \$32,700
4            \$79,750
5            \$62,400
6            \$38,250

a. Answer the question if the relevant interest rate for taking present values is 9%, 10%, 11% and
12%. In the program, notice that period zero represents a cash flow made at the present time, which
isn't discounted. The program will do the entire calculation for if you input the initial investment as a
negative number in this cell.
b. Use trial and error in the program to find the interest rate (to the nearest hundredth of a percent) at
which Amitron would be just indifferent to the project.
The Time Value of Money                                           67

This problem is a preview of an important method of evaluating projects known as capital
budgeting. We'll study the topic in detail in chapters 9 and 10. In part a of this problem, we found the
net present value (NPV) of the project's cash flows at various interest rates, and reasoned intuitively that
the project was a good idea if that figure was positive. In part b we found the return inherent in the
project itself, which is called the internal rate of return (IRR). We'll learn how to use that in Chapter 9.

SOLUTION: a.             k         NPV
9%        \$11,405    good idea
10%         4,086    good idea

b. Indifference implies NPV = 0 which occurs at k = 10.58%

27. The Centurion Corp. is putting together a financial plan for the company covering the next three
years, and needs to forecast its interest expense and the related tax savings. The firm's most significant
liability is a fully amortized mortgage loan on its real estate. The loan was made exactly ten and one
half years ago for \$3.2 million at 11% compounded monthly for a term of thirty years. Use the
AMORTIZ program to predict the interest expense associated with the real estate mortgage over the
next three years. (Hint: Run AMORTIZ from the loan's beginning and add up the months in each of the
next three years.)

SOLUTION: In the AMORTIZ output, the next three years are months 127-138, 139-150, and 151-
162. Interest each year is:
1 \$320,213.11
2 \$314,950.35
3 \$309,078.59

DEVELOPING SOFTWARE

28. Write your own program to amortize a ten-year, \$20,000 loan at 10% compounded annually. Input
the loan amount, the payment, and the interest rate. Set up your spreadsheet just like Table 5-4, and
write your program to duplicate the calculation procedure described.

SOLUTION: A sample solution is on the Instructor's Disk.

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