Finance Chapter 9 Questions and Solutions

Chapter 1

Chapter 9

Discussion Questions

9-1.How is the future value (Appendix A) related to the present value of a single sum (Appendix B)?

The future value represents the expected worth of a single amount, whereas the present value represents the current worth.

FV = PV (1 + I)n future value

9-2.How is the present value of a single sum (Appendix B) related to the present value of an annuity (Appendix D)?

The present value of a single amount is the discounted value for one future payment, whereas the present value of an annuity represents the discounted value of a series of consecutive payments of equal amount.

9-3.Why does money have a time value?

Money has a time value because funds received today can be reinvested to reach a greater value in the future. A person would rather receive $1 today than $1 in ten years, because a dollar received today, invested at 6 percent, is worth $1.791 after ten years.

9-4.Does inflation have anything to do with making a dollar today worth more than a dollar tomorrow?

Inflation makes a dollar today worth more than a dollar in the future. Because inflation tends to erode the purchasing power of money, funds received today will be worth more than the same amount received in the future.

9-5.Adjust the annual formula for a future value of a single amount at 12 percent for 10 years to a semiannual compounding formula. What are the interest factors (FVIF) before and after? Why are they different?

The more frequent compounding under the semiannual compounding assumption increases the future value.

9-6.If, as an investor, you had a choice of daily, monthly, or quarterly compounding, which would you choose? Why?

The greater the number of compounding periods, the larger the future value. The investor should choose daily compounding over monthly or quarterly.

9-7.What is a deferred annuity?

A deferred annuity is an annuity in which the equal payments will begin at some future point in time.

9-8.List five different financial applications of the time value of money.

Different financial applications of the time value of money:

Equipment purchase or new product decision,

Present value of a contract providing future payments,

Future worth of an investment,

Regular payment necessary to provide a future sum,

Regular payment necessary to amortize a lone,

Determination of return on an investment,

Determination of the value of a bond.

Problems

9-1.You invest $2,500 a year for 3 years at 8%.

a.What is the value of your investment after 1 year? Multiply $2,500 * 1.08.

b.What is the value of your investment after 2 years? Multiply your answer to part a by 1.08.

c.What is the value of your investment after 3 years? Multiply your answer to part b by 1.08. This gives you your final answer.

d.Confirm that your final answer is correct by going to Appendix A (future value of a $1), and looking up the future value for n = 3, and I = 8%. Multiply this tabular value by $2,500 and compare your answer to the answer in part c. There may be a slight difference due to rounding.

Solution:

Future Value

a.$2,500 * 1.08 = $2,700

b.$2,700 * 1.08 = $2,916

c.$2,916 * 1.08 = $3,149.28

d.Appendix A (8%, 3 periods)

FV = PV * FVIF

$2,500 * 1.260 = $3,150

9-2.What is the present value of:

a.$8,000 in 10 years at 6 percent?

b.$16,000 in 5 years at 12 percent?

c.$25,000 in 15 years at 8 percent?

d.$1,000 in 40 periods at 20 percent?

Solution:

Appendix B

PV = FV * PVIF

a.$ 8,000 * .558 = $4,464

b.$16,000 * .567 = $9,072

c.$25,000 * .315 = $7,875

d.$ 1,000 * .001 = $1

9-3.If you invest $12,000 today, how much will you have:

a.In 6 years at 7 percent?

b.In 15 years at 12 percent?

c.In 25 years at 10 percent?

d.In 25 years at 10 percent (compounded semiannually)?

Solution:

Appendix A

FV = PV * FVIFa.$12,000 * 1.501 = $ 18,012

b.$12,000 * 5.474 = $ 65,688

c.$12,000 * 10.835 = $130,020

d.$12,000 * 11.467 = $137,604 (5%, 50 periods)

9-4.How much would you have to invest today to receive:

a.$12,000 in 6 years at 12 percent?

b.$15,000 in 15 years at 8 percent?

c.$5,000 each year for 10 years at 8 percent?

d.$40,000 each year for 40 years at 5 percent?

Solution:

Appendix B (a and b)

PV = FV * PVIFa.$12,000 * .507 = $ 6,084

b.$15,000 * .315 = $ 4,725

Appendix D (c and d)

c.$ 5,000 * 6.710 = $ 33,550

d.$40,000 * 17.159 = $686,360

9-5.If you invest $8,000 per period for the following number of periods, how much would you have?

a.7 years at 9 percent

b.40 years at 11 percent

Solution:

Appendix C

FVA = A * FVIFAa.$8,000 * 9.20 = $ 73,600

b.$8,000 * 581.83 = $4,654,640

9-6.You invest a single amount of $12,000 for 5 years at 10 percent. At the end of 5 years you take the proceeds and invest them for 12 years at 15 percent. How much will you have after 17 years?

Solution:

Appendix A

FV = PV * FVIF$12,000 * 1.611 = $ 19,332

Appendix A

FV = PV * FVIF$19,332 * 5.350 = $103,426

9-7.Mrs. Crawford will receive $6,500 a year for the next 14 years from her trust. If an 8 percent interest rate is applied, what is the current value of the future payments?

Solution:

Appendix D

PVA = A * PVIFA (8%, 14 periods)

= $6,500 * 8.244 = $53,586

9-8.John Longwaite will receive $100,000 in 50 years. His friends are very jealous of him. If the funds are discounted back at a rate of 14 percent, what is the present value of his future "pot of gold"?

Solution:

Appendix B

PV = FV * PVIF (14%, 50 periods)

= $100,000 * .001 = $100

9-9.Carrie Tune will receive $19,500 a year for the next 20 years as a result of the new song she has written. If a 10 percent rate is applied, should she be willing to sell out her future rights now for $160,000?

Solution:

Appendix D

PVA = A * PVIFA (10%, 20 periods)

PVA = $19,500 * 8.514 = $166,023

No, the present value of the annuity is worth more than $160,000.

9-10.General Mills will receive $27,500 for the next 10 years as a payment for a weapon he invented. If a 12 percent rate is applied, should he be willing to sell out his future rights now for $160,000.

Solution:

Appendix D

PVA = A * PVIFA (12%, 10 periods)

PVA = $27,500 * 5.650 = $155,375

Yes, the present value of the annuity is worth less than $160,000.

9-11.Al Lopez invests $2,000 in a mint condition Nolan Ryan baseball card. He expects the card to increase in value by 20 percent a years for the next five years. After that, he anticipates a 15 percent annual increase for the next three years. What is the projected value of the card after eight years?

Solution:

Appendix A

FV = PV * FVIF (20%, 5 periods)

= $2,000 * 2,488 = $4,976

FV = PV * FVIF (15%, 3 periods)

= $4,976 * 1.521 = $7,568.50

9-12.Martha Reed has been depositing $1,500 in her savings account every December since 1992. Her account earns 6 percent compounded annually. How much will she have in December of 2001? (Assume that a deposit is made in 2001. Make sure to count the years carefully.)

Solution:

Appendix C

FVA = A * FVIFA (6%, n = 10)

FVA = $1,500 * 13.181 = $19,771.50

9-13.At a growth (interest) rate of 8 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer.

Solution:

Appendix A

If the sum is doubling, then the tabular value must equal 2.

In Appendix A, looking down the 8% column, we find the factor closest to 2 (1.999) on the 9-year row. The factor closest to 3 (2.937) is on the 14-year row.

9-14.If you owe $30,000 payable at the end of five years, what amount should your creditor accept in payment immediately if she could earn 11 percent on her money?

Solution:

Appendix B

PV = FV * PVIF (11%, 5 periods)

PV = $30,000 * .593 = $17,790

9-15.Mr. Flint retired as president of the Color Tile Company but is current on a consulting contract for $45,000 per year for the next 10 years.

a.If Mr. Flint's opportunity cost (potential return) is 10 percent, what is the present value of his consulting contract?

b.Assuming that Mr. Flint will not retire for two more years and will not start to receive his 10 payments until the end of the third year, what would be the value of his deferred annuity?

Solution:

Using a Two Step Procedure

Appendix D

a.PVA = A * PVIFA (i = 10%, 10 periods)

= $45,000 * 6.145 = $276,525

Appendix B

b.PV = FV * PVIF (i = 10%, 2 periods)

$276,525 * .826 = $228,410

Alternative Solution

Appendix D

a.PVA = A * PVIFA (10%, 10 periods)

PVA = $45,000 * 6.145 = $276,525

b.Deferred annuity-Appendix D

PVA = $45,000 (6.814 1.736) where n = 12; n = 2 andi = 10%

= $45,000 (5.078)

= $228,510 (or use a two step solution)

The answer is slightly different from the answer above due to rounding in the tables.

9-16.Cousin Bertha invested $100,000 10 years ago at 12 percent, compounded quarterly. How much has she accumulated?

Solution:

Appendix A

FV = PV * FVIF (3%, 40 periods)

FV = $100,000 * 3.262 = $326,200

9-17.Determine the amount of money in a savings account at the end of 5 years, given an initial deposit of $3,000 and an 8 percent annual interest rate when interest is compounded (a) annually, (b) semiannually, and (c) quarterly.

Solution:

Appendix A

FV = PV * FVIFa.$3,000 * 1.469 = $4,407

b.$3,000 * 1.480 = $4,440

c.$3,000 * 1.486 = $4,458

9-18.As stated in the chapter, the annuity payments are assumed to come at the end of each payment period (termed an ordinary annuity). However, an exception occurs when the annuity payments come at the beginning of each period (termed an annuity due). To find the present value of an annuity due, you subtract one from n an add 1 to the tabular value. To find the future value of an annuity, you add 1 to n and subtract 1 from the tabular value. For example, to find the future value of a $100 payment at the beginning of each period for five periods at 10 percent, you would go to Appendix C for n = 6 and I = 10%. You look up a value of 7.716 and subtract 1 from it for an answer of 6.716 or $671.60 ($100 * 6.716).

What is the future value of a 10-year annuity of $2,000 per period where payments come at the beginning of each period. The interest rate is 8 percent.

Solution:

Appendix C

FVA = A * FVIFAn = 11, i = 8% 16.645 1 = 15.645

FVA = $2,000 * 15.645 = $31,290

9-19.Your grandfather has offered you a choice of one of the three following alternatives: $5,000 now; $1,000 a year for each years; or $12,000 at the end of eight years. Assuming you could earn 11 percent annually, which alternative should you choose? If you could earn 12 percent annually, would you still choose the same alternative?

Solution:

(first alternative) Present value of $5,000 received now:

$5,000

(second alternative) Present value of annuity of $1,000 for eight years: Appendix D

PVA= A * PVIFA

= $1,000 * PVIFA (11%, 8 years)

= $1,000 * 5.146

= $5,146

(third alternative) Present value of $12,000 received in eight years: Appendix B

PV= FV * PVIF

= $12,000 * PVIF (11%, 8 years)

= $12,000 * .434

= $5,208

Select $12,000 to be received in eight years.

Revised answers based on 12%.

(first alternative) Present value of $5,000 received today: $5,000

(second alternative) Present value of annuity of $1,000 at 12% for 8 years: Appendix D

PVA= A * PVIFA

= $1,000 * PVIFA (12%, 8 years)

= $1,000 * 4.968

= $4,968

(third alternative) Present value of $12,000 received in 8 years at 12%: Appendix B

PV= FV * PVIF

= $12,000 * PVIF (12%, 8 years)

= $12,000 * .404

= $4,848

Select $5,000 now.

9-20.You need $23,956 at the end of nine years, and your only investment outlet is a 7 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year.

a.What single payment could be made at the beginning of the first year to achieve this objective?

b.What amount could you pay at the end of each year annually for nine years to achieve this same objective?

Solution:

a.Appendix B

PV= FV * PVIF (7%, 9 periods)

PV= $23,956 * .544 = $13,032.06

b.Appendix C

A= FVA/FVIFA

A= $23,956/11.978 = $2,000

9-21.Beverly Hills started a paper route on January 1, 1995. Every three months, she deposits $300 in her bank account, which earns 8 percent annually but is compounded quarterly. On December 31, 1998, she used the entire balance in her bank account to invest in a certificate of deposit at 12 percent annually. How much will she have on December 31, 2001?

Solution:

Appendix C

FVA= A * FVIFA (2%, 16 periods)

FVA= $300 * 18.639 = $5,591.70 after four years

Appendix A

FV= PV * FVIF (12%, 3 periods)

FV= $5,591.70 * 1.405

FV= $7,856.34 after three more years

9-22.On January 1, 1999, Mr. Dow bought 100 shares of stock at $12 per share. On December 31, 2001, he sold the stock for $18 per share.

What is his annual rate of return? Interpolate to find the answer.

Solution:

Appendix B

PVIF=

PVIF=

PVIF at 14% .675

PVIF at 15%.658

.017

PFIF at 14% .675

PVIF computed.667

.008

14% + (.008/.017) (1%)

14% + .471 (1%)

14.47%

9-23.Tom Phillips has just invested $8,760 for his son (age one). This money will be used for his son's education 17 years from now. He calculates that he will need $60,000 by the time the boy goes to school.

What rate of return will Mr. Phillips need in order to achieve this goal?

Solution:

Appendix B

PVIF=

PVIF=

9-24.C. D. Rom has just given an insurance company $30,000. In return, he will receive an annuity of $3,200 for 20 years.

At what rate of return must the insurance company invest this $30,000 in order to make the annual payments? Interpolate.

Solution:

Appendix D

PVIFA= PVA/A (20 periods)

= $30,000/$3,200

= 9.375 is between 8% and 9% for 20 periods

PVIFA at 8% 9.818

PVIFA at 9%9.129

.689

PVIFA at 8% 9.818

PVIFA computed9.375

.443

8% + (.443/.689) (1%)

8% + .643 (1%) = 8.64%

9-25.Frank Bell has just retired from the telephone company. His total pension funds have an accumulated value of $200,000, and his life expectancy is 16 more years. His pension fund manager assumes he can earn a 12 percent return on his assets.

What will be his yearly annuity for the next 16 years?

Solution:

Appendix D

A= PVA/PVIFA (12%, 16 periods)

= $200,000/6.974

= $28,677.95

9-26.Dr. Oats, a nutrition professor, invests $80,000 in a piece of land that is expected to increase in value by 14 percent per year for the next five years. She will then take the proceeds and provide herself with a 10-year annuity. Assuming a 14 percent interest rate for the annuity, how much will this annuity be?

Solution:

Appendix A

FV = PV * FVIF (14%, 5 periods)

FV = $80,000 * 1.925 = $154,000

Appendix D

A= PVA/PVIFA (14%, 10 periods)

A= $154,000/5.216 = $29,524.54

9-27.You wish to retire in 20 years, at which time you want to have accumulated enough money to receive an annuity of $12,000 for 25 years after retirement. During the period before retirement you can earn 8 percent annually, while after retirement you can earn 10 percent on your money.

What annual contributions to the retirement fund will allow you to receive the $12,000 annuity?

Solution:

Determine the present value of an annuity during retirement: Appendix D

PVA= A * PVIFA (10%, 25 years)

= $12,000 * 9.077 = $108,924

To determine the annual deposit into an account earning 8% that is necessary to accumulate $108,924 after 20 years, use the Future Value of an Annuity table: Appendix C

A= FVA/FVIFA (8%, 20 years)

=

9-28.Judy Green has purchased an annuity to begin payment at the end of 2003 (the date of the first payment). Assume it is now the beginning of 2001. The annuity is for $12,000 per year and is designed to last eight years.

If the discount rate for the calculation is 11 percent, what is the most she should have paid for the annuity?

Solution:

Appendix D will give a factor for a 8 years annuity when the appropriate discount rate is 11 percent (5.146). The value of the annuity at the beginning of the year it starts (2003) is:

PVA= A* PVIFA (11%, 8 periods)

= $12,000 * 5.146

= $61,752

The present value at the beginning of 2001 is found using Appendix B (2 years at 11%). The factor is .812. Note we are discounting from the beginning of 2003 to the beginning of 2001.

PV= PV * PVIF (11%, 2 periods)

= $61,752 * .812

= $50,142.62

The maximum that should be paid for the annuity is $50,142.62.

9-29.If you borrow $9,725 and are required to pay back the loan in five equal annual installments of $2,500, what is the interest rate associated with the loan?

Solution:

Appendix D

PVIFA= PVA/A (5 periods)

= $9,725/$2,500

= 3.890

Interest rate = 9 percent

9-30.Tom Busby owes $20,000 now. A lender will carry the debt for four more years at 8 percent interest. That is, in this particular case, the amount owed will go up 8 percent per year for four years. The lender then will require Busby to pay off the loan over 12 years at 11 percent interest. What will his annual payments be?

Solution:

Appendix A

FV= PV * FVIFA (8%, 4 periods)

FV= $20,000 * 1.360

= $27,200

Appendix D

A= PVA/PVIFA (11%, 12 periods)

= $27,200/6.492

= $4,189.77

9-31.If your aunt borrows $50,000 from the bank at 10 percent interest over the eight-year life of the loan, what equal annual payments must be made to discharge the loan, plus pay the bank its required rate of interest (round to the nearest dollar)? How much of her first payment will be applied to interest? To principal? How much of her second payment will be applied to each?

Solution:

Appendix D

A= PVA/PVIFA (10%, 8 periods)

= $50,000/5.335

= $9,372.07

First payment:

$50,000 * .10= $5,000 interest

$9,372.07 $5,000= $4,372.07 applied to principal

Second payment: First determine remaining principal

$50,000 $4,372.07= $45,627.93

$45,627.93 * .10= $4,562.79 interest

$9,372.07 $4,562.79= $4,809.28 applied to principal

9-32.Jim Thomas borrows $70,000 at 12 percent interest toward the purchase of a home. His mortgage is for 30 years.

a.How much will his annual payments be? (Although home payments are usually on a monthly basis, we shall do our analysis on an annual basis for ease of computation. We get a reasonably accurate answer.)

b.How much interest will he pay over the life of the loan?

c.How much should he be willing to pay to get out of a 12 percent mortgage and into a 10 percent mortgage with 30 years remaining on the mortgage? Suggestion: Find the annual savings and then discount them back to the present at the current interest rate (10 percent).

Solution:

Appendix D

a.A= PVA/PVIFA (12%, 30 periods)

= $70,000/8.055

= $8,690.25

b.$ 8,690.25annual payments

* 30years

$260,707.50total payment

70,000.00repayment of principal

$190,707.50

Appendix D

c.New payments at 10%

A= PVA/PVIFA (10%, 30 periods)

= $70,000/9.427

= $7,425.48

Difference between old and new payments

$8,690.25old

7,425.48new

$1,264.77difference

P.V. of difference-Appendix D

PVA= A * PVIFA (assumes 10% discount rate, 30 periods)

= $1,264.77 * 9.427

= $11,922.99 Amount that could be paid to refinance

9-33.You are chairperson of the investment fund for the Continental Soccer League. You are asked to set up a fund of semiannual payments to be compounded semiannually to accumulate a sum of $200,000 after ten years at an 8 percent annual rate (20 payments). The first payment into the fund is to take place six months from today, and the last payment is to take place at the end of the tenth year.

a.Determine how much the semiannual payment should be. (Round to whole numbers.)

On the day after the sixth payment is made (the beginning of the fourth year) the interest rate goes up to a 10 percent annual rate, and you can earn a 10 percent annual rate on funds that have been accumulated as well as all future payments into the fund. Interest is to be compounded semiannually on all funds.

b.Determine how much the revised semiannual payments should be after this rate change (there are 14 payments and compounding dates). The next payment will be in the middle of the fourth year. (Round all values to whole numbers.)

Solution:

Appendix C

a.A= FVA/FVIFA

= $200,000/29.778 (4%, 20 periods)

= $6,716

b.First determine how much the old payments are equal to after 6 periods at 4%. Appendix C.

FVA= A * FVIFA (4%, 6 periods)

= $6,716 * 6.633

= $44,547

Then determine how much this value will grow to after 14 periods at 5%.

Appendix A

FV= PV * FVIF (5%, 14 periods)

= $44,547 * 1.980

= 88,203

Subtract this value from $200,000 to determine how much you need to accumulate on the next 14 payments.

$200,000

88,203

$111,797

Determine the revised semi-annual payment necessary to accumulate this sum after 14 periods at 5%.

Appendix C

A= FVA/FVIFA

A= $111,797/19.599

A= $5,704

9-34.Your younger sister, Jennifer, will start college in five years. She has just informed your parents that she wants to go to Eastern State U., which will cost $18,000 per year for four years (cost assumed to come at the end of each year). Anticipating Jennifer's ambitions, your parents started investing $3,000 per year five years ago and will continue to do so for five more years.

How much more will your parents have to invest each year for the next five years to have the necessary funds for Jennifer's education? Use 10 percent as the appropriate interest rate throughout this problem (for discounting or compounding). Round all values to whole numbers.

Solution:

Present value of college costs

Appendix D

PVA= A * PVIFA (10%, 4 periods)

= $18,000 * 3.170

= $57,060

Accumulation based on investing $3,000 per year for 10 years.

Appendix C

FVA= A * FVIFA (10%, 10 periods)

= $3,000 * 15.937

= $47,811

Additional funds required 5 years from now.

$57,060PV of college costs

47,811Accumulation based on $3,000 per year

$ 9,249Additional funds required

Added contribution for the next 5 years

Appendix C

A= FVA/FVIFA (10%, 5 periods)

= $9,249/6.105

= $1,515

9-35.Jennifer (from problem 34) is now 18 years old (five years have passed), and she wants to get married instead of going to college. Your parents have accumulated the necessary funds for her education.

Instead of her schooling, your parents are paying $7,000 for her current wedding and plan to take year-end vacations costing $2,000 per year for the next three years.

How much money will your parents have at the end of three years to help you with graduate school, which you will start then? You plan to work on a master's perhaps a Ph.D. If graduate school costs $18,930 per year, approximately how long will you be able to stay in school based on these funds? Use 10 percent as the appropriate interest rate throughout this problem. (Round all values to whole numbers.)

Solution:

Funds available after the wedding

$57,060Funding available before the wedding

7,000Wedding

$50,060Funds available after the wedding

Less present value of vacation

Appendix D

PVA= A* PVIFA (10%, 3 periods)

= $2,000 * 2.487 = $4,974

$50,060

4,974$45,086Remaining funds for graduate school

Appendix A

FV= PV * FVIF (10%, 3 periods)

= $45,086 * 1.331

= $60,009 Funds available for starting graduate school

Number of years of graduate education

Appendix D

PVIFA=

=

with i = 10%, n = 4 for 3.170, the answer is 4 years.

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