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					                                               CHAPTER 6

                                  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. Calculator solutions are included in the margins next to all illustrative
examples.
          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.
       Financial calculators are introduced and instruction is provided on their use in solving time
       value problems.
    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.


                                                 128
                                        The Time Value of Money                                            129

    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
       Problem solving technique, the sinking fund concept, using a financial calculator for annuity
       problems.
    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.
       Computer spreadsheet techniques for solving time value problems are introduced and explained.
    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.


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?
130                                               Chapter 6

ANSWER:           PV = FV30 [PVF10,30]
                    = $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                                           131

          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 are we likely to be interested in that’s 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? (Hint: Think about taxes and
getting the mortgage paid off.)

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. That is, half way through the loan’s life, a great deal more than half of the original loan balance
is still unpaid.

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.
132                                              Chapter 6

         Preferred stock is a security that pays a constant dividend indefinitely. Its value is the
capitalization of the stream of its dividends.

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.


BUSINESS ANALYSIS

1. A business can be valued by capitalizing it’s earnings stream (see example 6.15). How might you use
the same idea to value securities, especially the stock of large publicly held companies? Is there a way
to calculate a value that could be compared to the stock’s market price that would tell an investor
whether it’s a good buy? (If the market price is lower than the calculated value, the stock is a bargain.)
What financial figures associated with shares of stock might be used in the calculation. Consider the per
share figures and ratios discussed in chapter 3 including EPS, dividends, book value per share etc. Does
one measure make more sense than the others? What factors would make a stock worth more or less
than your calculated value.

Answer: A privately or closely held company is valued by capitalizing a stream of earnings (net
income) because all of a firm’s earnings are available to its owners. The analogous figure for the stocks
of publicly held companies is dividends, because they represent cash received by stockholders.
Although all earnings technically belong to owners, the stockholders of larger companies generally can’t
influence how much of those earnings they receive. Hence dividends, which stockholders do receive,
are the best measure for mathematically calculating value. (However, practitioners do use EPS
regularly for less precise estimates.)
         The starting point should be the capitalized value of the current dividend, essentially assuming
it will go on forever. The rate used for capitalization should be consistent with the riskiness of the
company involved.
         This starting estimated value should be factored up or down based on expectations about future
dividends. If the stream is expected to grow or shrink, value will be higher or lower respectively.
Further, a stable stream should be worth more than one that varies substantially from year to year.
         Expectations about future performance usually come from performance in the recent past so a
stock with a record of consistent dividend (or EPS) growth should be worth more than one whose
dividends have been constant for some time or erratic.


PROBLEMS

NOTE: FOR ALL PROBLEMS INVOLVING NON-ANNUAL COMPOUNDING, SEE
CONCEPT CONNECTION EXAMPLES 6.7 AND 6.8, PAGES 263 – 264 AS WELL AS
THE PROBLEM REFERENCED.

Amount Problems
Present Value of an Amount – Example 6.2 (page 246)
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
                                      The Time Value of Money                                133

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


Future Value of an Amount – Example 6.1 (page 246)
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?

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


Finding the Interest Rate – Example 6.3, (page 250)
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.)
134                                             Chapter 6

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%


Solving for the Number of Periods – Example 6.4 (page 251)
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


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?
                                        The Time Value of Money                                           135

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.


Deferred Payment is Equivalent to a Cash Discount – Example 6.2 (page 246)
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,
and half in cash.
         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.


7. John Cleaver's grandfather died in 2007 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 2007? (Hint: [FVFk,a+b ] = [FVFk,a][FVFk,b ]
          b. How much would they have been worth if they paid interest at a rate more like that paid
         during the 1970s and 80s, say 7%?
          c. Comment on the difference between the answers to parts a and b.
136                                                Chapter 6

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,92 ] = [FVFk,60 ] [FVFk,30 ] [FVFk,2]
Hence,
                 [FVF3,92 ] = [FVF3,60 ] [FVF3,30 ] [FVF3,9]
                            = (5.8916)(2.4273)(1.0609)
                            = 15.1716
and
                 [FVF7,92 ] = [FVF7,60 ] [FVF7,30 ] [FVF7,2]
                            = (57.9464)(7.6123)(1.1449)
                            = 505.022

a.                     FV = PV [FVF3,92 ]
                          = $11.50 (15.1716)
                          = $174.47 per bond

b.                     FV = PV [FVF7,92 ]
                          = $11.50 (505.22)
                          = $5,810.03 per bond

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


Deferred Payment is Equivalent to a Cash Discount – Example 6.2 (page 246)
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.

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%
                                        The Time Value of Money                                      137


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.

9. Charlie owes Joe $8,000 on a note which is due in five years with accumulated interest at 6%. Joe
has an investment opportunity now that he thinks will earn 18%. There’s a chance, however, that it will
earn as little as 4%. A bank has offered to discount the note at 14% and give Joe cash that he can invest
today.
         a. How much ahead will Joe be if he takes the bank’s offer and the investment does turn out to
yield 18%?
         b. How much behind will he be if the investment turns out to yield only 4%?

Solution:
a.     First calculate the amount Charlie’s note will pay in five years.
                FV = PV[FVF6,5] = $8,000(1.3382) = $10,705.60
       If the bank discounts that at 14%, Joe will receive
                PV = FV[PVF14,5 ] = $10,705.60(.5194) = $5,560.49
       This amount invested at 18% for five years will grow into
                FV = PV[FVF18,5 ] = $5,560.49(2.2878) = $12,721.29.
       And Joe will be ahead by the difference
                $12,721.29 - $10,705.60 = $2,015.69

b. If the investment yie lds only 4%, the last time value calculation will be
                 FV = PV[FVF4,5] = $5,560.49(1.2167) = $6,765.45,
                 And Joe will be behind by
                 $10,705.60 - $6,765.45 = $3,940.15

Annuity Problems
Future Value of an Annuity – Example 6.5 (page 256)
10. 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

11. 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

12. What interest rate would you need to get to have an annuity of $7,500 per year accumulate to
$279,600 in 15 years?
138                                            Chapter 6

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


13. 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


Present Value of an Annuity – Example 6.9 (page 266)
14. 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


Amortized Loan Finding Payment – Example 6.11 (page 271)
15. Sam Rothstein wants borrow $15,500 to be repaid in quarterly installments over five years at 16%
compounded quarterly. How much will his payment be?

SOLUTION:
        For quarterly compounding we have
                        k = knom/12 = 16%/4 = 4%
                        n = 5 years x 12 months/year = 60 months.
Write equation 6.19 and substitute.
                        PVA = PMT[PVFAk,n ]
                        $15,500 = PMT[PVFA4,60 ]
Using Appendix A-4
                        $15,500 = PMT[22.6235]
                        PMT = $685


Amortized Loan Finding Amount Borrowed – Example 6.12 (page 271)
16. Harry Clements would like to buy a new car. He can afford payments of $650 a month. The bank
makes four-year car loans at 12% compounded monthly. How much can Harry borrow toward a new
car?

SOLUTION:
First calculate k and n for monthly compounding,
                          k = knom/12 = 12%/12 = 1%
                 n = 4 years x 12 months/year = 48 months.
Write equation 6.19 and substitute.
                                       The Time Value of Money                                      139

                         PVA = PMT[PVFAk,n ]
                         PVA = $650[PVFA1,48 ]
Using Appendix A-4
                         PVA = $650[37.9740]
                         PVA = $24,683


17. 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


18. 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%


EAR (page 261)
19.       Ralph Renner just borrowed $30,000 to pay for a new sports car. He took out a 60 month loan
          and his car payments are $761.80 per month. What is the effective annual interest rate (EAR)
          on Ralph’s loan?

SOLUTION:
     First, calculate the periodic interest rate
     n = 60; PV = 30,000; PMT = (761.80); FV = 0         CPT I/Y = 1.5%
     EAR = (1 + .015)12 – 1 = .1956 or 19.56%


Annuity Due – Example 6.17 (page 277)
20. 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?

SOLUTION: FVAd = PMT [FVFAk,n ](1+k)
               = $250 [FVFA.5,24 ](1.005)
               = $250 (25.4320) (1.005) = $6,389.79
140                                               Chapter 6

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


21. How long will it take a payment of $500 per quarter to amortize a loan of $8,000 at 16%
compounded quarterly? Approximate 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

                    PVAd = PMT [PVFAk,n ](1+k)
                   $8,000 = $500 [PVFA4,n ](1.04)
                  PVFA4,n = 15.3846
                         n = 24 1/3 quarters = 6 years 1 month
      So it will take about 5 months less time.


Mortgage Loans – Examples 6.13- 6.15 (pages 272-274)
22. Ryan and Laurie Middleton just purchased their first home with a traditional (monthly compounding
and payments) 6% 30-year mortgage loan of $178,000.
    a.     How much is their monthly payment?
    b.     How much interest will they pay the first month?
    c.     If they make all their payments on time over the 30-year period, how much interest will
           they have paid?
    d.     If Ryan and Laurie decide to move after 7 years what will the balance of their loan be at that
           time?
    e.     If they finance their home over 15 rather than 30 years at the same interest rate, how much
           less interest will they pay over the life of the loan?

SOLUTION (using a financial calculator):
  a.    PV = $178,000; n = 360, I/Y = .5; CPT PMT = $1,067.20
  b.    Interest in the first payment is the opening loan balance times the monthly rate
            $178,000 x .005 = $890
  c.    Total interest is total payments minus the amount borrowed
            Total payments are $1067.20 x 360 = $384,192
            And total interest is $384,192 - $178,000 = $206,192
  d.    The loan balance at any time is the PV of the remaining payments.
       After 7 years, the Middletons will have made 7 x 12 = 84 monthly payments which leaves
       360-84 = 276 remaining.
            n = 276, I/Y = .5, PMT = $1,067.20; CPT PV = $159,558.11
  e.    The payment on a 15 year loan is
            n = 180, I/Y = .5, PV = $178,000; CPT PMT = $1,502.07
            Total interest is total payments minus the amount borrowed
            Total payments are $1,502.07 x 180 = $270,372.60.
            And total interest is $270,372.60 - $178,000 = $92,372.60
            The difference is $113,819.40
                                       The Time Value of Money                                       141

23. 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

         Month         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


24. Construct an amortization schedule for the last six months of the loan in Problem 21. (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


25. How soon would the loan in Problem 21 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
                        Less additional payment    17,936.29
                                   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.

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

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.
142                                             Chapter 6


                 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


27.        Adam Wilson just purchased a home and took out a $250,000 mortgage for 30 years at 8%,
           compounded monthly.
a.         How much is Adam’s monthly mortgage payment?
b.         How much sooner would Adam pay off his mortgage if he made an additional $100 payment
           each month?
c.         Assume Adam makes his normal mortgage payments and at the end of five years, he refinances
           the balance of his loan at 6%. If he continues to make the same mortgage payments, how soon
           after the first five years will he pay off his mortgage?
d.         How much interest will Adam pay in the tenth year of the loan
                    i. if he does not refinance?
                    ii. if he does refinance?

SOLUTION:
a.   n = 360; I/Y = 8/12 = .666667; PV = 250,000; FV = 0 CPT PMT = $1,834.41

b.         Add $100.00 to the PMT calculated above and recompute n:
           I/Y = .666667; PV = 250,000; PMT = (1,934.41); FV = 0 CPT n = 297.62
           Adam would eliminate the last 62 months of payments by paying an additional $100/month

c.         First calculate the mortgage balance at the end of 5 years
           n = 60; I/Y = .666667; PV = 250,000; PMT = (1,834.4193) CPT FV = $237,674.75
           Then solve for n using the original PMT, the new I/Y, and the mortgage balance as the new PV
           I/Y = 6.0/12 = .50; PV = 237,674.64; PMT = (1,834.41); FV = 0 CPT n = 209.25
           The mortgage would pay off about 90 months earlier.

d. i.      FV after the 108th payment
           n = 108; I/Y = 8/12 = .666667; PV = 250,000 PMT = ($1834.41); CPT FV = $223,592.02
           FV after the 120th payment = $219,311.76
           Principal payments = $223,592.02 - $219,311.76 = $4,280.26
           Interest payments = 12 x $1834.41 = $22,012.92 - $4,280.26 = $17,732.66

     ii.   FV after the 60th payment (when the refinancing would occur)
                                       The Time Value of Money                                      143

        n = 60; I/Y = 8/12 = .666667; PV = $250,000; PMT = ($1834.41); CPT FV = $237,674.75;
        Then find the balance after 48 payments of the refinanced loan
        n = 48; I/Y = 6/12 = .50; PV = $237,674.64; PMT = ($1834.41); CPT FV = $202,725.34.
        FV after the 60th payment = $192,600.63
        Principal payments = $202,725.34 - $192,600.63 = $10,124.71
        Interest payments = 12 x $1834.41 = $22,012.92 - $10,124.71 = $11,888.21


Estimating an ARM Reset – Example 16.6 (page 275)
28. Harrison Conway is choosing between a fix rate and an adjustable rate mortgage (ARM) for
$300,000. Both are 30 year mortgages with monthly payments and compounding. The fixed rate is
offered at 8% while the initial rate on the ARM is 6%. Harrison is concerned that inflation may be a
problem within ten years and that rates may return to levels not seen since the mid 1980s, i.e., in the
neighborhood of 12%. Compare the payment on the fixed rate loan to what Harrison would have to pay
on the ARM if it reset to 12% after ten years. For simplicity assume just that one resetting occurs.
Round all calculations to the nearest dollar.

Solution:
       First calculate the initial ARM payment.

                        k = 6 / 12 = .5 n = 30 x 12 = 360
                                 PVA = PMT [PVFAk,n ]
                          $300,000 = PMT [PVFA.5,360]
                          $300,000 = PMT [166.792]
                                 PMT = $1,799

Next calculate the projected unpaid ARM balance after ten years (20 years or 240 months remaining).

                                  PVA = PMT [PVFAk,n ]
                                     = $1,799 [PVFA.5,240 ]
                                     = $1,799 [139.581]
                                     = $251,106

Next calculate the payment required to amortize $251,106 over the remaining 20 years at 12%.
                       k = 12 / 12 = 1 n = 20 x 12 = 240
                                PVA = PMT [PVFAk,n ]
                         $251,106 = PMT [PVFA1,240 ]
                         $251,106 = PMT [90.8194]
                                PMT = $2,765

The fixed rate payment at 8% is

                        k = 8 / 12 = .67 n = 30 x 12 = 360
                                 PVA = PMT [PVFAk,n ]
                          $300,000 = PMT [PVFA.67,360 ]
                          $300,000 = PMT [136.283]
                                 PMT = $2,201

Hence the ARM would be $564 per month higher
 144                                             Chapter 6

29. Lee Childs is negotiating a contract to do some work for Haas Corp. over the next five years. Haas
proposes to pay Lee $10,000 at the end of each of the third, fourth and fifth years. No payments will be
received prior to that time. If Lee discounts these payments at 8%, what is the contract worth to him
today?

 SOLUTION:
         This is an annuity problem, but the annuity doesn’t begin today. Since the payments occur at
 the end of the 3rd through 5th years, we can solve for the value of the annuity at the end of the second
 year using the following values:
                  n = 3; I/Y = 8; PMT = 10,000; FV = 0          CPT PV = $25,770.97
         This amount needs to be discounted back two periods to find the value today
                  n = 2; I/Y = 8; PMT = 0; FV = 25,770.97         CPT PV = $22,094.45


30. Referring to the previous problem, Lee wants to receive the payments for his work sooner than Haas
proposes to make them. He has counter proposed that the payments be made at the beginning of the
third, fourth and fifth years rather than at the end. What will the contract be worth to Lee if Haas accepts
his counter proposal?

SOLUTION:
 We want to retain the PV of the contract from the previous problem at $22,094.45. Since payments will
 occur at the beginning of the 3rd through 5th years under this proposal, we first need to determine the
 future value of the contract at the beginning of the 2nd period. Then we can treat the payments as an
 ordinary annuity starting at that time.
                  n = 1; I/Y = 8; PV = 22.094.45; PMT = 0          CPT FV = $23,862.01
 This amount becomes the PV of an ordinary annuity three years long
                  n = 3; I/Y = 8; PV = 23,862.01; FV = 0          CPT PMT = $9,259.26


 31. The Orion Corp. is evaluating a proposal for a new project. It will cost $50,000 to get the
 undertaking started. The project will then generate cash inflows of $20,000 in its first year and $16,000
 per year in the next five years after which it will end. Orion uses an interest rate of 15% compounded
 annually for such evaluations.
         a. Calculate the “Net Present Value” (NPV) of the project by treating the initial cost as a cash
 outflow (a negative) in the present, and adding the present value of the subsequent cash inflows as
 positives.
         b. What is the implication of a positive NPV? (Words only.)
         c. Suppose the inflows were somewhat lower, and the NPV turned out to be negative. What
 would be the implication of that result? (Words only.)
 (This problem is a preview of a technique called capital budgeting, which we’ll study in detail in
 Chapters 10, 11, and 12.)

 Solution:
        a. First take the present value of the first inflow
                  PV = FV[PVF15,1 ] = $20,000(.8696) = $17,392
        Next take the present value of the annuity in two steps
                  PVA = PMT[PVFA15,5 ] = $16,000(3.3522) = $53,635.20
        Next bring that figure backward one year into the present as an amount.
                  PV = FV[PVF15,1 ] = $53,635.20(.8696) = $46,641.17
        Hence the NPV of the project is
                  NPV = $50,000 + $17,392 + $46,641.17
                       = $14,033.17
                                        The Time Value of Money                                            145

         b. A positive NPV means that on a present value basis the project’s cash inflows exceed its
outflows. That implies it’s a good deal for the company. In essence it is expected to increase the firm’s
wealth and value to stockholders. Since that’s what management is supposed to do, they should accept
the project.
         c. A negative means just the opposite. It implies that, on balance, the project will cost the firm
money. Hence it should be rejected.


Perpetuities – Capitalization of Earnings – Example 6.19 (page 280)
32. Roper Metals Inc. is in negotiations to acquire the Hanson Sheet Metal Company. Hanson’s after
tax earnings have averaged $19 million per year for the last four years without much variation around
that average figure. So far discussions have been about the business fit of the two firms and pricing has
been conspicuously ignored. Roper’s CEO feels the venture is risky and needs to pay a price that would
yield his firm a return of about 20% if no operating improvements came out of the merger.
         a. What offering price should he put on the table to open negotiations?
         b. Hanson’s management is sure to want a higher price. Would that imply
            capitalizing earnings at a higher or lower rate. Why.
         c. What arguments is Hanson likely to use.

SOLUTION:
a. Roper’s CEO should start by capitalizing earnings as a perpetuity at the rate he feels he needs to see
   as a return. This yields

                               Earnings       $19 M
                 PHanson                           $95 M
                             Rateof Re turn    .20

b. Paying a higher price would imply a lower percentage return because Roper would be paying more
for the same earnings stream. Mathematically, a lower denominator in the equation above implies a
higher price.

c. Hanson is likely to argue that the acquisition is not very risky because the firm is stable and has a
solid ongoing business. He also may argue that there’s a good deal of growth potential which would
increase the earnings stream into the future above the value of a perpetuity.


Multi-Part Problems
Simple Multipart – Example 6.21 (page 282)
33. 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
146                                            Chapter 6

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

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


Complex Multipart – Example 6.22 (page 284)
34. 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.
      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 [FVFA k,n ] = PMT [FVFA2,6 ]
     $20,184.76 = PMT (6.3081)
          PMT = $3,199.82

35. 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?
                                        The Time Value of Money                                      147

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

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

36. Joe Trenton expects to retire in 15 years and has suddenly realized that he hasn’t saved anything
toward that goal. After giving the matter some thought, he has decided that he would like to retire with
enough money in savings to withdraw $85,000 per year for 25 years after he retires. Assume a 6%
return on investment before and after retirement and that all payments into and withdrawals from
savings are at year end.

a. How much does Joe have to save in each year for the next 15 years to reach this goal?
b. How much would Joe have needed to save each year if he had started when retirement was 25 years
away?
c. Comment on the difference between the results of parts a and b.

SOLUTION:
a. This is a 2-step solution. First, calculate the amount needed at retirement to fund 25 annual
withdrawals of $85,000. This is the amount that will be needed in savings at retirement. Then calculate
how much Joe will need to save each year until retirement to have that much.
    PVA = PMT[PVFAk,n ] = $85,000[FVA6,25 ] = $85,000 (12.7834) = $1,086,589
    This PVA becomes the FVA of the required savings over the next 15 years.

    FVA = PMT[FVFA6,15 ]
    $1,086,589 = PMT[23.2760]
    PMT = $46,683
148                                             Chapter 6

b. The first calculation is the same, $1,086,589 is required at retirement. The only change in the second
calculation is n becomes 25 rather than 15.

      FVA = PMT[FVFA6,25 ]
      $1,086,589 = PMT[54.8645]
      PMT = $19,805

c. For most people saving more than $46,000 a year is impossible, while about putting aside $20,000 or
less than half of that amount would be very difficult but might be possible. There is no substitute for
starting retirement planning early.


37.    Janet Elliott just turned 20, and received a gift of $20,000 from her rich uncle. Janet plans
ahead and would like to retire on her 55th birthday. She thinks she’ll need to have about $2 million
saved by that time in order to maintain her lavish lifestyle. She wants to make a payment at the end of
each year until she’s 50 into an account she’ll open with her uncle’s gift. After that she’d like to stop
making payments and let the money grow at interest until it reaches $2 million when she turns 55.
Assume she can invest at 7% compounded annually. Ignore the effect of taxes.

a.      How much will she have to invest each year in order to achieve her objective?
b.      What percent of the $2 million will have been contributed by Janet (including the $20,000 she
got from her uncle)?

SOLUTION:
a. First we need to know how much she will need to accumulate by the time she reaches 50 so that
amount can grow to $2 million by the time she reaches 55.
                n = 5; I/Y = 7; PMT = 0; FV = 2,000,000     CPT PV = $1,425,972.36
         That number becomes the FV in an ordinary annuity with a $20,000 PV
                n = 30; I/Y = 7; PV = 20,000; FV = (1,425,972.36) CPT PMT = $13,484.19

b.       Janet will have contributed
                          $20,000 + 30 x $13,484.19 = $424,525.70, and
                                  424,525.70/2,000,000 = 21.23%


38.     Merritt Manufacturing needs to accumulate $20 million to retire a bond issue that matures in 13
years. The firm’s manufacturing division can contribute $100,000 per quarter to an account that will
pay 8%, compounded quarterly. How much will the remaining divisions have to contribute every
month to a second account that pays 6% compounded monthly in order to reach the $20 million goal?

SOLUTION: First calculate how much the known source of funding ($100,000 per quarter) will grow
to by the end of 13 years
                 n = 13 x 4 = 52; I/Y = 8/4 = 2; PV = 0; PMT = 100,000 CPT FV = $9,001,640.93
         The remainder $10,998,359.07 will have to be raised from the other divisions and becomes the
FV of the second annuity problem
         n = 156; I/Y = .5; PV = 0; FV = 10,998,359.07 CPT PMT = $46,712.61


39.     Carol Pasca just had her fifth birthday. As a birthday present, her uncle promised to contribute
$300 per month to her education fund until she turns 18 and starts college. Carol’s parents estimate
college will cost $2,500 per month for four years, but don’t think they’ll be able to save anything toward
                                       The Time Value of Money                                         149

it for five years. How much will Carol’s parents need to contribute to the fund each month starting on
her tenth birthday to pay for her college education? Assume the fund earns 6% compounded monthly?

SOLUTION:
        First, we need to calculate how much money Carol will need when she starts college. That’s the
present value of $2,500 per month for four years.
                  n = 48; I/Y = .5; PMT = 2,500; FV = 0 CPT PV = $106,450.79
        The money from her uncle will contribute the future value of an annuity of $300 per month for
13 years:
                  n = 156; I/Y = .5; PV = 0; PMT = 300 CPT FV = $70,634.20
The shortfall is the difference of $35,816.59 which must be made up by her parents contributions over
five years
                  n = 60; I/Y = .5; FV = 35;816.59; PV=0;    CPT PMT = $513.35


40.      Joan Colby is approaching retirement and plans to purchase a condominium in Florida in three
years. She now has $40,000 saved toward the purchase in a bank account that pays 8% compounded
quarterly. She also has five $1,000 face value corporate bonds that mature in two years. She plans to
deposit the bonds’ principal repayments in the same account when they’re paid. Joan also receives
$1,200 per month alimony from her ex-husband which will continue for two more years until he retires
(24 checks including one that arrived today). She’s decided to put her remaining alimony money
toward her condo depositing it as received in a credit union account that pays 8% compounded monthly.
She’ll make the first deposit today with the check she already has. Joan anticipates buying a $200,000
property. What will her monthly payment be on a 15 year mortgage at 6%? What would the payment
be on a 30 year loan at the same interest rate?

SOLUTION: Calculate the value of money Joan will have in the bank and credit union in three years

$40,000 already on deposit
       FV = PV[FVFk,n] = $40,000[FVF2,12 ] = $40,000(1.2682) = $50,728.00

Bond principal repayment
       FV = PV[FVFk,n] = $5,000[FVF2,4 ] = $5,000(1.0824) = $5,412.00

Alimony: First calculate the future value of the stream at its end in two years recognizing that the
deposits will be an annuity due. Then project at sum forward one year to the date of purchase.
        FVA = PMT[FVFAk,n ](1+k)
              = $1,200[FVFA.67,24 ](1.0067)
              = $1,200(25.9332)(1.0067)
              = $31,119.84(1.0067)
              = $31,328.34

        FV = PV[FVFk,n]
          = $31,328.34[FVF.67,12 ]
          = $31,328.34 (1.0830)
          = $33,928.59

Funds saved
       $50,728.00 + $5,412 + $33,928.59 = $90,068.59

Next calculate the mortgage required
        $200,000 - $90,068.59 = $109,931.41
150                                             Chapter 6

Finally calculate the mortgage payment
        PVA = PMT[PVFAk,n ]
              = PMT[PVFA.5,180 ]
        $109,931.41 = PMT(118.504)
        PMT = $927.66

If Joan chooses a 30 year mortgage at the same rate, her payments will be as follows:
        PVA = PMT[PVFAk,n ]
             = PMT[PVFA.5,360 ]
        $109,931.41 = PMT(166.792)
        PMT = $659.09


Imbedded Annuity – Example 6.24 (page 283)
41. Amy’s uncle died recently and left her some money in a trust that will pay her $500 per month for
five years starting on her twenty fifth birthday. Amy is getting married soon, and would like to use this
money as a down payment on a house now. If the trust allows her to assign its future payments to a
bank, and her bank is willing to discount them at 9% compounded monthly, how much will she have
toward her down payment on home ownership? Amy just turned 23.

SOLUTION:
         This problem requires a two-step calculation. The payments are an annuity, so we need to take
its present value. That will give us a result at the beginning of the annuity, which is two years in the
future. That PVA then has to be brought to the present as an amount.
         First calculate the “present” value of the annuity at its beginning.
                           k = 9/12 = .75 n = 5  12 = 60
                  PVA = PMT[PVFA.75,60 ] = $500(48.1734) = $24,086.70
         Next bring that figure backward two years into the present as an amount.
                           k = 9/12 = .75 n = 2  12 = 24
                  PV = FV[PVF.75,24 ] = $24,086.70(.8358) = $20,131.66


COMPUTER PROBLEMS

42. 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.
                                        The Time Value of Money                                        151

                             Mortgage Payments per $100,000 Borrowed
                                        as Term Decreases

                                          MORTGAGE TERM IN YEARS

                                        30                25                20              15
                            6%
      RATES
                            8%
                           10%
                           12%


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

SOLUTION:

                                 Mortgage Payments per $100,000 Borrowed
                                          as Term Decreases

                                          MORTGAGE TERM IN YEARS

                                              30               25              20            15
        RATES                      6%           $600               $644          $836
                                   8%           $878               $909          $965        $1,075
                                  10%         $1,029             $1,053        $1,101        $1,200
                                  12%         $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 more significant. At 12%, five years of payments can be
avoided by paying only an additional $24 dollars a month, and the term can be cut in half with a 17%
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.


43. 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
152                                              Chapter 6

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.
     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 10, 11, and 12. 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 10.

SOLUTION: a.            k        NPV
                        9%       $11,405 good idea
                        10%         4,086 good idea
                        11%        (2,910) bad idea
                        12%        (9,601) bad idea
b. Indifference implies NPV = 0 which occurs at k = 10.58%


44. 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

1. 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 6-4, and
write your program to duplicate the calculation procedure described.

SOLUTION: A sample solution is in the Instructor's Material on the text website.

				
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