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5 years. Because Gina doesnâ€™t really need the money today, she plans to let it

accumulate in an account that earns 7% annual interest. Given her desire to buy

a house at the end of 5 years after closing on the sale of the lot, she decides to

choose the payment alternativeâ€”$24,000 single amount or the mixed stream of

payments in the following tableâ€”that provides the higher future value at the end

of 5 years.

Mixed stream

Beginning of year Cash flow

1 $ 2,000

2 4,000

3 6,000

4 8,000

5 10,000

a. What is the future value of the single amount at the end of year 5?

b. What is the future value of the mixed stream at the end of year 5?

c. On the basis of your findings in parts a and b, which alternative should Gina

take?

d. If Gina could earn 10% rather than 7 percent on the funds, would your rec-

ommendation in part c change? Explain.

4â€“27 Present valueâ€”Mixed streams Find the present value of the streams of cash

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flows shown in the following table. Assume that the firmâ€™s opportunity cost

is 12%.

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CHAPTER 4 Time Value of Money

A B C

Year Cash flow Year Cash flow Year Cash flow

1 $2,000 1 $10,000 1â€“5 $10,000/yr

2 3,000 2â€“5 5,000/yr 6â€“10 8,000/yr

3 4,000 6 7,000

4 6,000

5 8,000

4â€“28 Present valueâ€”Mixed streams Consider the mixed streams of cash flows

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shown in the following table.

Cash flow stream

Year A B

1 $ 50,000 $ 10,000

2 40,000 20,000

3 30,000 30,000

4 20,000 40,000

5 10,000 50,000

Totals $150,000 $150,000

a. Find the present value of each stream using a 15% discount rate.

b. Compare the calculated present values and discuss them in light of the fact

that the undiscounted cash flows total $150,000 in each case.

4â€“29 Present value of a mixed stream Harte Systems, Inc., a maker of electronic sur-

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veillance equipment, is considering selling to a well-known hardware chain the

rights to market its home security system. The proposed deal calls for payments

of $30,000 and $25,000 at the end of years 1 and 2 and for annual year-end

payments of $15,000 in years 3 through 9. A final payment of $10,000 would

be due at the end of year 10.

a. Lay out the cash flows involved in the offer on a time line.

b. If Harte applies a required rate of return of 12% to them, what is the present

value of this series of payments?

c. A second company has offered Harte a one-time payment of $100,000 for the

rights to market the home security system. Which offer should Harte accept?

4â€“30 Funding budget shortfalls As part of your personal budgeting process, you

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have determined that in each of the next 5 years you will have budget shortfalls.

In other words, you will need the amounts shown in the following table at the

end of the given year to balance your budgetâ€”that is, to make inflows equal

outflows. You expect to be able to earn 8% on your investments during the next

5 years and wish to fund the budget shortfalls over the next 5 years with a single

amount.

180 PART 2 Important Financial Concepts

End of year Budget shortfall

1 $ 5,000

2 4,000

3 6,000

4 10,000

5 3,000

a. How large must the single deposit today into an account paying 8% annual

interest be to provide for full coverage of the anticipated budget shortfalls?

b. What effect would an increase in your earnings rate have on the amount cal-

culated in part a? Explain.

4â€“31 Relationship between future value and present valueâ€”Mixed stream Using only

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the information in the accompanying table, answer the questions that follow.

Future value interest factor

Year (t) Cash flow at 5% (FVIF5%,t)

1 $ 800 1.050

2 900 1.102

3 1,000 1.158

4 1,500 1.216

5 2,000 1.276

a. Determine the present value of the mixed stream of cash flows using a 5%

discount rate.

b. How much would you be willing to pay for an opportunity to buy this

stream, assuming that you can at best earn 5% on your investments?

c. What effect, if any, would a 7% rather than a 5% opportunity cost have on

your analysis? (Explain verbally.)

4â€“32 Changing compounding frequency Using annual, semiannual, and quarterly

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compounding periods, for each of the following: (1) Calculate the future value if

$5,000 is initially deposited, and (2) determine the effective annual rate (EAR).

a. At 12% annual interest for 5 years.

b. At 16% annual interest for 6 years.

c. At 20% annual interest for 10 years.

4â€“33 Compounding frequency, future value, and effective annual rates For each of

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the cases in the following table:

a. Calculate the future value at the end of the specified deposit period.

b. Determine the effective annual rate, EAR.

c. Compare the nominal annual rate, i, to the effective annual rate, EAR. What

relationship exists between compounding frequency and the nominal and

effective annual rates.

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CHAPTER 4 Time Value of Money

Compounding

Amount of Nominal frequency, m Deposit period

Case initial deposit annual rate, i (times/year) (years)

A $ 2,500 6% 2 5

B 50,000 12 6 3

C 1,000 5 1 10

D 20,000 16 4 6

4â€“34 Continuous compounding For each of the cases in the following table, find the

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future value at the end of the deposit period, assuming that interest is com-

pounded continuously at the given nominal annual rate.

Amount of Nominal Deposit

Case initial deposit annual rate, i period (years), n

A $1,000 9% 2

B 600 10 10

C 4,000 8 7

D 2,500 12 4

4â€“35 Compounding frequency and future value You plan to invest $2,000 in an

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individual retirement arrangement (IRA) today at a nominal annual rate of 8%,

which is expected to apply to all future years.

a. How much will you have in the account at the end of 10 years if interest is

compounded (1) annually? (2) semiannually? (3) daily (assume a 360-day

year)? (4) continuously?

b. What is the effective annual rate, EAR, for each compounding period in

part a?

c. How much greater will your IRA account balance be at the end of 10 years

if interest is compounded continuously rather than annually?

d. How does the compounding frequency affect the future value and effective

annual rate for a given deposit? Explain in terms of your findings in parts a

through c.

4â€“36 Comparing compounding periods RenĂ© Levin wishes to determine the future

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value at the end of 2 years of a $15,000 deposit made today into an account

paying a nominal annual rate of 12%.

a. Find the future value of RenĂ©â€™s deposit, assuming that interest is

compounded (1) annually, (2) quarterly, (3) monthly, and

(4) continuously.

b. Compare your findings in part a, and use them to demonstrate the relation-

ship between compounding frequency and future value.

c. What is the maximum future value obtainable given the $15,000 deposit, the

2-year time period, and the 12% nominal annual rate? Use your findings in

part a to explain.

182 PART 2 Important Financial Concepts

4â€“37 Annuities and compounding Janet Boyle intends to deposit $300 per year in a

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credit union for the next 10 years, and the credit union pays an annual interest

rate of 8%.

a. Determine the future value that Janet will have at the end of 10 years,

given that end-of-period deposits are made and no interest is withdrawn, if

(1) $300 is deposited annually and the credit union pays interest

annually.

(2) $150 is deposited semiannually and the credit union pays interest

semiannually.

(3) $75 is deposited quarterly and the credit union pays interest

quarterly.

b. Use your finding in part a to discuss the effect of more frequent deposits and

compounding of interest on the future value of an annuity.

4â€“38 Deposits to accumulate future sums For each of the cases shown in the follow-

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ing table, determine the amount of the equal annual end-of-year deposits neces-

sary to accumulate the given sum at the end of the specified period, assuming the

stated annual interest rate.

Sum to be Accumulation

Case accumulated period (years) Interest rate

A $ 5,000 3 12%

B 100,000 20 7

C 30,000 8 10

D 15,000 12 8

4â€“39 Creating a retirement fund To supplement your planned retirement in exactly

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42 years, you estimate that you need to accumulate $220,000 by the end of

42 years from today. You plan to make equal annual end-of-year deposits into

an account paying 8% annual interest.

a. How large must the annual deposits be to create the $220,000 fund by the

end of 42 years?

b. If you can afford to deposit only $600 per year into the account, how much

will you have accumulated by the end of the 42nd year?

4â€“40 Accumulating a growing future sum A retirement home at Deer Trail Estates

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now costs $85,000. Inflation is expected to cause this price to increase at 6%

per year over the 20 years before C. L. Donovan retires. How large an equal

annual end-of-year deposit must be made each year into an account paying an

annual interest rate of 10% for Donovan to have the cash to purchase a home at

retirement?

4â€“41 Deposits to create a perpetuity You have decided to endow your favorite uni-

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versity with a scholarship. It is expected to cost $6,000 per year to attend the

university into perpetuity. You expect to give the university the endowment in

183

CHAPTER 4 Time Value of Money

10 years and will accumulate it by making annual (end-of-year) deposits into an

account. The rate of interest is expected to be 10% for all future time periods.

a. How large must the endowment be?

b. How much must you deposit at the end of each of the next 10 years to accu-

mulate the required amount?

4â€“42 Loan payment Determine the equal annual end-of-year payment required each

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year, over the life of the loans shown in the following table, to repay them fully

during the stated term of the loan.

Loan Principal Interest rate Term of loan (years)

A $12,000 8% 3

B 60,000 12 10

C 75,000 10 30

D 4,000 15 5

4â€“43 Loan amortization schedule Joan Messineo borrowed $15,000 at a 14%

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annual rate of interest to be repaid over 3 years. The loan is amortized into three

equal annual end-of-year payments.

a. Calculate the annual end-of-year loan payment.

b. Prepare a loan amortization schedule showing the interest and principal

breakdown of each of the three loan payments.

c. Explain why the interest portion of each payment declines with the passage

of time.

4â€“44 Loan interest deductions Liz Rogers just closed a $10,000 business loan that is

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to be repaid in three equal annual end-of-year payments. The interest rate on the

loan is 13%. As part of her firmâ€™s detailed financial planning, Liz wishes to

determine the annual interest deduction attributable to the loan. (Because it is a

business loan, the interest portion of each loan payment is tax-deductible to the

business.)

a. Determine the firmâ€™s annual loan payment.

b. Prepare an amortization schedule for the loan.

c. How much interest expense will Lizâ€™s firm have in each of the next 3 years as

a result of this loan?

4â€“45 Monthly loan payments Tim Smith is shopping for a used car. He has found

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one priced at $4,500. The dealer has told Tim that if he can come up with a

down payment of $500, the dealer will finance the balance of the price at a 12%

annual rate over 2 years (24 months).

a. Assuming that Tim accepts the dealerâ€™s offer, what will his monthly (end-of-

month) payment amount be?

b. Use a financial calculator or Equation 4.15a (found in footnote 9) to help

you figure out what Timâ€™s monthly payment would be if the dealer were

willing to finance the balance of the car price at a 9% yearly rate.

184 PART 2 Important Financial Concepts

4â€“46 Growth rates You are given the series of cash flows shown in the following

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table.

Cash flows

Year A B C

1 $500 $1,500 $2,500

2 560 1,550 2,600

3 640 1,610 2,650

4 720 1,680 2,650

5 800 1,760 2,800

6 1,850 2,850

7 1,950 2,900

8 2,060

9 2,170

10 2,280

a. Calculate the compound annual growth rate associated with each cash flow

stream.

b. If year-1 values represent initial deposits in a savings account paying annual

interest, what is the annual rate of interest earned on each account?

c. Compare and discuss the growth rate and interest rate found in parts a and b,

respectively.

4â€“47 Rate of return Rishi Singh has $1,500 to invest. His investment counselor sug-

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gests an investment that pays no stated interest but will return $2,000 at the end

of 3 years.

a. What annual rate of return will Mr. Singh earn with this investment?

b. Mr. Singh is considering another investment, of equal risk, that earns an

annual return of 8%. Which investment should he make, and why?

4â€“48 Rate of return and investment choice Clare Jaccard has $5,000 to invest.

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Because she is only 25 years old, she is not concerned about the length of the

investmentâ€™s life. What she is sensitive to is the rate of return she will earn on the

investment. With the help of her financial advisor, Clare has isolated the four

equally risky investments, each providing a single amount at the end of its life, as

shown in the following table. All of the investments require an initial $5,000

payment.

Investment Single amount Investment life (years)

A $ 8,400 6

B 15,900 15

C 7,600 4

D 13,000 10

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CHAPTER 4 Time Value of Money

a. Calculate, to the nearest 1%, the rate of return on each of the four invest-

ments available to Clare.

b. Which investment would you recommend to Clare, given her goal of maxi-

mizing the rate of return?

4â€“49 Rate of returnâ€”Annuity What is the rate of return on an investment of $10,606

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if the company will receive $2,000 each year for the next 10 years?

4â€“50 Choosing the best annuity Raina Herzig wishes to choose the best of four

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immediate-retirement annuities available to her. In each case, in exchange for

paying a single premium today, she will receive equal annual end-of-year cash

benefits for a specified number of years. She considers the annuities to be equally

risky and is not concerned about their differing lives. Her decision will be based

solely on the rate of return she will earn on each annuity. The key terms of each

of the four annuities are shown in the following table.

Annuity Premium paid today Annual benefit Life (years)

A $30,000 $3,100 20

B 25,000 3,900 10

C 40,000 4,200 15

D 35,000 4,000 12

a. Calculate to the nearest 1% the rate of return on each of the four annuities

Raina is considering.

b. Given Rainaâ€™s stated decision criterion, which annuity would you

recommend?

4â€“51 Interest rate for an annuity Anna Waldheim was seriously injured in an

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industrial accident. She sued the responsible parties and was awarded a judg-

ment of $2,000,000. Today, she and her attorney are attending a settlement

conference with the defendants. The defendants have made an initial offer of

$156,000 per year for 25 years. Anna plans to counteroffer at $255,000

per year for 25 years. Both the offer and the counteroffer have a present value of

$2,000,000, the amount of the judgment. Both assume payments at the end of

each year.

a. What interest rate assumption have the defendants used in their offer

(rounded to the nearest whole percent)?

b. What interest rate assumption have Anna and her lawyer used in their coun-

teroffer (rounded to the nearest whole percent)?

c. Anna is willing to settle for an annuity that carries an interest rate assump-

tion of 9%. What annual payment would be acceptable to her?

4â€“52 Loan rates of interest John Flemming has been shopping for a loan to finance

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the purchase of a used car. He has found three possibilities that seem attractive

and wishes to select the one with the lowest interest rate. The information

186 PART 2 Important Financial Concepts

available with respect to each of the three $5,000 loans is shown in the follow-

ing table.

Loan Principal Annual payment Term (years)

A $5,000 $1,352.81 5

B 5,000 1,543.21 4

C 5,000 2,010.45 3

a. Determine the interest rate associated with each of the loans.

b. Which loan should Mr. Flemming take?

4â€“53 Number of yearsâ€”Single amounts For each of the following cases, determine

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the number of years it will take for the initial deposit to grow to equal the future

amount at the given interest rate.

Case Initial deposit Future amount Interest rate

A $ 300 $ 1,000 7%

B 12,000 15,000 5

C 9,000 20,000 10

D 100 500 9

E 7,500 30,000 15

4â€“54 Time to accumulate a given sum Manuel Rios wishes to determine how long it

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will take an initial deposit of $10,000 to double.

a. If Manuel earns 10% annual interest on the deposit, how long will it take for

him to double his money?

b. How long will it take if he earns only 7% annual interest?

c. How long will it take if he can earn 12% annual interest?

d. Reviewing your findings in parts a, b, and c, indicate what relationship exists

between the interest rate and the amount of time it will take Manuel to dou-

ble his money?

4â€“55 Number of yearsâ€”Annuities In each of the following cases, determine the

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number of years that the given annual end-of-year cash flow must continue in

order to provide the given rate of return on the given initial amount.

Case Initial amount Annual cash flow Rate of return

A $ 1,000 $ 250 11%

B 150,000 30,000 15

C 80,000 10,000 10

D 600 275 9

E 17,000 3,500 6

187

CHAPTER 4 Time Value of Money

4â€“56 Time to repay installment loan Mia Salto wishes to determine how long it will

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take to repay a loan with initial proceeds of $14,000 where annual end-of-year

installment payments of $2,450 are required.

a. If Mia can borrow at a 12% annual rate of interest, how long will it take for

her to repay the loan fully?

b. How long will it take if she can borrow at a 9% annual rate?

c. How long will it take if she has to pay 15% annual interest?

d. Reviewing your answers in parts a, b, and c, describe the general relationship

between the interest rate and the amount of time it will take Mia to repay the

loan fully.

CHAPTER 4 CASE Finding Jill Moranâ€™s Retirement Annuity

S unrise Industries wishes to accumulate funds to provide a retirement annuity

for its vice president of research, Jill Moran. Ms. Moran by contract will

retire at the end of exactly 12 years. Upon retirement, she is entitled to receive an

annual end-of-year payment of $42,000 for exactly 20 years. If she dies prior to

the end of the 20-year period, the annual payments will pass to her heirs. During

the 12-year â€śaccumulation periodâ€ť Sunrise wishes to fund the annuity by mak-

ing equal annual end-of-year deposits into an account earning 9% interest. Once

the 20-year â€śdistribution periodâ€ť begins, Sunrise plans to move the accumulated

monies into an account earning a guaranteed 12% per year. At the end of the

distribution period, the account balance will equal zero. Note that the first

deposit will be made at the end of year 1 and that the first distribution payment

will be received at the end of year 13.

Required

a. Draw a time line depicting all of the cash flows associated with Sunriseâ€™s view

of the retirement annuity.

b. How large a sum must Sunrise accumulate by the end of year 12 to provide

the 20-year, $42,000 annuity?

c. How large must Sunriseâ€™s equal annual end-of-year deposits into the account

be over the 12-year accumulation period to fund fully Ms. Moranâ€™s retire-

ment annuity?

d. How much would Sunrise have to deposit annually during the accumulation

period if it could earn 10% rather than 9% during the accumulation period?

e. How much would Sunrise have to deposit annually during the accumulation

period if Ms. Moranâ€™s retirement annuity were a perpetuity and all other

terms were the same as initially described?

WEB EXERCISE Go to Web site www.arachnoid.com/lutusp/finance_old.html. Page down to the

WW portion of this screen that contains the financial calculator.

W

1. To determine the FV of a fixed amount, enter the following:

Into PV, enter 1000; into np, enter 1; into pmt, enter 0; and, into ir,

enter 8.

188 PART 2 Important Financial Concepts

Now click on Calculate FV, and 1080.00 should appear in the FV window.

2. Determine FV for each of the following compounding periods by changing

only the following:

a. np to 2, and ir to 8/2

b. np to 12, and ir to 8/12

c. np to 52, and ir to 8/52

3. To determine the PV of a fixed amount, enter the following:

Into FV, 1080; into np, 1; into pmt, 0; and, into ir, 8. Now click on

Calculate PV. What is the PV?

4. To determine the FV of an annuity, enter the following:

Into PV, 0; into FV, 0; into np, 12; into pmt, 1000; and, into ir, 8. Now

click on Calculate FV. What is the FV?

5. To determine the PV of an annuity, change only the FV setting to 0; keep

the other entries the same as in question 4. Click on Calculate PV. What is

the PV?

6. Check your answers for questions 4 and 5 by using the techniques discussed

in this chapter.

Go to Web site www.homeowners.com/. Click on Calculators in the left column.

Click on Mortgage Calculator.

7. Enter the following into the mortgage calculator: Loan amount, 100000;

duration in years, 30; and interest rate, 10. Click on compute payment.

What is the monthly payment?

8. Calculate the monthly payment for $100,000 loans for 30 years at 8%, 6%,

4%, and 2%.

9. Calculate the monthly payment for $100,000 loans at 8% for 30 years,

20 years, 10 years, and 5 years.

Remember to check the bookâ€™s Web site at

www.aw.com/gitman

for additional resources, including additional Web exercises.

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