FIN CH-9

Chapter 9 Discussion Questions 9-1.

How How is is the the future va value (A (Appendi ndix A) A) re related to to th the pr present val value of of a single gle 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.

PV

 1 = FV  (1 + i )

n

  lue  Present va lue 

How How is is the the prese esent value of of a si singl ngle su sum (A (Appe ppendix B) B) related to th the pr 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 future payments of equal amount.

9-3.

Why does money have a time value? Money has a time value because funds received today can be invested 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.

Doe Does in inflation ha have any anytthing to to do do wi with ma makin king a dol dollar tod todaay wo worth mo more th 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.

S9-1

9-5.

Adj Adjust the an annual ual fo formula ula for for a fu future va value of of a singl ngle am amount at 12 12 pe percent ent for 10 years to a semiannual compounding compoun ding formula. What are the interest factors (FVIF) before and after? Why are they different? FV = PV × FVIF ( AppendixA ) i = 12%, n i = 6%, n

10

3.106 Annual

20

3.207 Semiannual

=

=

The more frequent compounding under the semiannual compounding assumption increases the future value so that semiannual compounding is worth .101 more per dollar. 9-6.

If, aass an an in investor, yo you ha had a choice of of da daily, mo monthly, or or qu 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 compou nding 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 di different fi finan nancial app applicati ations of the ti time va value of of mone oney. Different financial applications of the time value of money: Equipment purchase or new product decision, Present value of a contract providing future payments, Future value of an investment, Regular payment necessary to provide a future sum, Regular payment necessary to amortize a loan, Determination of return on an investment, Determination of the value of a bond.

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Chapter 9 Problems 1.

You You inve invest st $3,0 $3,000 00 a year year for for thr three ee yea years rs at 12 12 perc percen ent. t. a. b. c. d .

9-1. 9-1.

What What is the value value of you yourr inves investme tment nt after after one year? year? Mult Multipl iply y $3,000 $3,000 × 1.12. 1.12. What What is the the value value of of your your inves investme tment nt after after two two years years?? Multi Multiply ply your your answ answer er to to part part a by 1.12. What What is the the value value of of your your inves investme tment nt after after thre threee years? years? Multip Multiply ly your your answ answer er to to part part b by 1.12. This gives your final answer. Confir Confirm m that that your your final final answ answer er is correc correctt by goin going g to Appe Appendi ndix x A (futu (future re value value of of $1), and looking up the future value for n = 3, 3 , and i = 12 percent. Multiply this tabular value by $3,000 and compare your answer to the answer in part c. There may be a slight difference due to rounding.

Solu Soluti tion on::

a. $3,000 × 1.12 b. $3,360 × 1.12 c. $3,763.20 × 1.12 d. $3,000 × 1.405 2.

= = = =

$3,360.00 $3,763.20 $4,214.78 $4,215.00 (Appendix A)

What hat is is the pres presen entt val value of: of: a. b. c. d .

$9,0 $9,000 00 in 7 years ears at 8 perc percen ent? t? $20, $20,00 000 0 in in 5 years ears at 10 per percent cent?? $10, $10,00 000 0 in in 25 25 yea yearrs at at 6 per perce cent nt?? $1,0 $1,000 00 in 50 50 yea years rs at 16 per percent cent??

Solution:

Appendix B PV = FV × PVIF a. $ 9,000 × .583 b. $20,000 × .621 c. $10,000 × .233 d. $ 1, 1,000 × .001

= = = =

$5,247 $12,420 $2,330 $1

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3.

You will will recei receive ve $5,00 $5,000 0 three three year yearss from from now. now. The The disco discount unt rate rate is 8 percen percent. t. a. b. c. d .

9-3. 9-3.

What What is the the value value of of your your inves investme tment nt two two years years from from now? now? Multi Multiply ply $5,00 $5,000 0 × .926 .926 (one year’s discount rate at 8 percent). What What is the the value value of of your your inves investme tment nt one one year year from from now? now? Multi Multiply ply your your answ answer er to to part a by .926 (one year’s discount rate at 8 percent). What What is the value value of you yourr inves investme tment nt toda today? y? Mult Multipl iply y your your answ answer er to to part part b by .926 (one year’s discount rate at 8 percent). Conf Confiirm tha thatt you yourr ans answe werr to to par partt c is correct by going to Appendix B (present value of $1) for n = 3 and i = 8%. Multiply this tabular value by $5,000 and compare your answer to part c. There may be a slight difference due to rounding.


Appendix B a. $5,000 × .926 b. 4,630 × .926 c. 4,287 × .926 d. 5,000 × .794 4.

$4,630 $4,287 $3,968 $3,970

If you you inv inves estt $9,0 $9,000 00 toda today, y, how how muc much h will will you you have have:: a. b. c. d .

9-4. 9-4.

= = = =

In 2 years at 9 percent? In 7 years at 12 12 pe percent? In 25 years at 14 per percent? nt? In 25 25 yea years rs at at 14 perce percent nt (co (comp mpoun ounde ded d semi semian annua nuall lly) y)??


Appendix A FV = PV × FVIF a. $9,000 × 1.188 = b. $9,000 × 2.211 = c. $9,000 × 26.462 = d. $9,000 × 29.457 =

$ 10,692 $ 19,899 $238,158 $265,113 (7%, 50 periods)

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5.

Your Your uncle uncle offer offerss you you a choice choice of of $30,000 $30,000 in 50 years years or $95 today today.. If mone money y is disc discount ounted ed at 12 percent, which should you choose?

9-5. 9-5.


Appendix B PV = FV × PVIF (12%, 50 periods) PV = $30,000 × .003 = $90 Choose $95 today. 6.

Your Your aunt aunt offer offerss you a choic choicee of $60,000 $60,000 in in 40 years years or $850 $850 toda today. y. If money money is disco discount unted ed at 11 percent, which should you choose?

9-6. 9-6.


Appendix B PV = FV × PVIF (11%, 40 periods) PV = $60,000 × .015 = $900 Choose $60,000 in 40 years. The PV of $900 is greater than $850 today. 7.

You are are goin going g to recei receive ve $100,0 $100,000 00 in 50 50 years. years. What What is the the diffe differen rence ce in pres present ent valu valuee between using a discount rate of 14 percent versus four percent?

9-7. 9-7.


Appendix B $100, 000 .001

$100, 000 (14%, 50)

.141

$100

$14,100 The difference is $14,000 14,10 ,100

(4% (4%, 50)

100 0 (14 14% %, 50 50)) −10 $14,000

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(4%, 50)

8.

How How muc much h woul would d you you have have to to inv inves estt toda today y to to rec recei eive ve:: a. b. c. d .

9-8. 9-8.

$15, $15,00 000 0 in in 8 years ears at 10 per percent cent?? $20, $20,00 000 0 in in 12 12 yea yearrs at at 13 13 per perce cent nt?? $6,0 $6,000 00 each each year year for for 10 10 yea years rs at 9 per perce cent nt?? $50, $50,00 000 0 eac each h yea yearr for for 50 year yearss at at 7 per perce cent nt??


Appendix B (a and b) PV = FV × PVIF a. $15,000 × .467 = $7,005 b. b. $2 $20, 0,00 000 0 × .23 .231 1 = $4 $4,6 ,620 20 Appendix D (c and d) c. $ 6,000 × 6.418 = d. $50,000 × 13.801 = 9.

If you you inve invest st $2,00 $2,000 0 a year year in a retire retirement ment account account,, how much much will will you you have: have: a. b. c.

9-9. 9-9.

$38,508 $690,050

In 5 years at 6 percent? In 20 years at 10 per percent? nt? In 40 years at 12 per percent? nt?


Appendix C FVA = A × FV IFA a. $2,000 × 5.637 = $ 11,274 b. $2,000 × 57.275 = $ 114,550 c. $2,000 × 76 767.09 = $1,534,180

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10.

You invest invest a single single amount amount of $10,000 for for 5 years years at 10 10 percent. percent. At the 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?

9-10. Solution: Solution:

Appendix A FV = PV × FVIF $10,000 × 1.611 = $16,110 Appendix A FV = PV × FVIF $16,110 × 5.350 = $86,188 11.

Jean Splicing Splicing will will receive receive $8,500 $8,500 a year year for the next 15 years years from her trust. trust. If If a 7 percent percent interest rate is applied, what is the current value of the future payments?


Appendix D PVA = A × PVIFA (7%, 15 periods) = $8,500 × 9.108 = $77,418 12.

Phil Goode Goode will will receive receive $175,000 $175,000 in 50 50 years. years. His friend friendss are very very jealous jealous of him. him. If the funds are discounted back at a rate of 14 percent, what wh at is the present value of his future “pot of gold”?


Appendix B PV = FV × PVIF (14%, 50 periods) = $175,000 × .001 = $175

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13.

Polly Graham Graham will receive receive $12,000 a year for for the next next 15 years years as a result result of her her patent. patent. If a 9 percent rate is applied, should she be willing to sell out her he r future rights now for $100,000?


Appendix D PVA = A × PVIFA (9%, 20 periods) = $12,000 × 8.061 = $96,732 Yes, the present value of the annuity is worth less than $100,000. 14.

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


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. 15.

The Clearin Clearinghouse ghouse Sweepst Sweepstakes akes has has just just informed informed you you that that you have won won $1 milli million. on. The amount is to be paid out at the rate of $20,000 a year for the next 50 years. With a discount rate of 10 percent, what is the present value of your winnings.


Appendix D PVA = A × PVIFA (10%, 50 periods) PVA = $20,000 × 9.915 = $198,300

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16.

Joan Lucky Lucky won the $80 milli million on lottery. lottery. She is to receive receive $1 million million a year year for the next 50 years plus an additional lump sum payment of $30 million after 50 years. The discount rate is 12 percent. What is the current value of her winnings?


Appendix D PVA = A × FVIFA (12%, 50 periods) PVA = $1,000,000 × 8.304 = $8,304,000 Appendix B PV = FV × PVIF (12%, 50 periods) PV = $30,000,000 × .003 = $90,000 $8,304,000 90,000 $8,394,000

17.

Al Rosen Rosen invests invests $25,000 in a mint mint conditi condition on 1952 Mickey Mantle Mantle Topps baseball baseball card. card. He expects the card to increase in value 12 percent per year for the next 10 years. How much will his card be worth after 10 years?


Appendix A FV = PV × FVIF (12%, 10 periods) = $25,000 × 3.106 = $77,650

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18.

Dr. Ruth Ruth has been secretl secretly y depositing depositing $2,500 in her savings savings account account every every December December starting in 1999. Her account earns 5 percent compounded annually. How much will she have in December 2008? (Assume that a deposit is made in the year 2008.) Make sure to carefully count the years.


Appendix C FVA = A × FVIFA (5%, 10 periods) FVA = $2,500 × 12.578 = $31,445 19.

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


Appendix A If the sum is doubling, then the interest factor must equal 2. * In Appendix A, looking looking down the 9% column, column, we find find the factor closest to 2 (1.993) on the 8-year row. The factor closest to 3 (3.066) is on the 13-year row.

20.

If you you owe $40,000 payable payable at the the end of of seven seven years, years, what what amount amount should should your your credito creditor r accept in payment immediately if she could earn 12 percent on her money?


Appendix B PV = FV × PVIF (12%, 7 periods) PV = $40,000 × .452 = $18,080

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21.

Jack Hammer Hammer invest investss in a stock stock that that will pay dividend dividendss of $2.00 $2.00 at the end end of the the first first year; year; $2.20 at the end of the second year; and $2.40 at the end of the third year. Also, he believes that at the end of the third year he will be able to sell the stock for $33. What is the present p resent value of all future benefits if a discount rate of 11 percent is applied? (Round all values to two places to the right of the decimal point.)


Appendix B PV = FV × PVIF Discount rate = 11% $ 2.00 × .901 2.20 × .802 2.40 × .731 33.00 × .731

= $ 1.80 = 1.79 = 1.75 = 24.12 $29.46

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22.

Les Moore Moore retir retired ed as preside president nt of Goodman Goodman Snack Foods Foods Company Company but is is currently currently on a consulting contract for $35,000 per year for the next 10 years. a. b.

If Mr. Mr. Moore’s Moore’s oppo opportu rtunit nity y cost cost (poten (potentia tiall return return)) is 10 perc percent ent,, what is is the pres present ent value of his consulting contract? Assumi Assuming ng Mr. Mr. Moore Moore will will not not reti retire re for for two two more more years years and will will not not start start to recei receive ve his 10 payments until the end of the third year, what would be the value of his deferred annuity?


Appendix D a. PVA = A × PVIFA (10%, 10 periods) PVA = $35,000 × 6.145 = $215,075 b. b.

Defe Deferr rred ed ann annui uity ty—A —App ppen endi dix xD PVA = A × PVIFA (i = 10%, 10 periods) PVA = $35,000 × 6.145 = $215,075

Now, discount back this value for 2 periods PV = FV × PVIF (i = 10%, 2 periods) Appendix B = $215,075 × .826 = $177,652 OR

Appendix D PVA = $35,000 (6.814 – 1.7360 where n = 12, n = 2 and i = 10%) = $35,000(5.078) = $177,730 The answer is slightly different from the answer above due to rounding in the tables.

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23.

Juan Garza Garza investe invested d $20,000 $20,000 10 years years ago ago at 12 percent, percent, compounde compounded d quarterly quarterly.. How much much has he accumulated?


Appendix A FV = PV × FV IF (3%, 40 periods) FV = $20,000 × 3.262 = $65,240 24.

Determine Determine the the amount amount of money in in a savings savings account at the end of five years, years, given given an initial deposit of $5,000 and a 12 percent annual interest rate when interest is compounded (a) annually, (b) semiannually, and (c) quarterly.


Appendix A FV = PV × FV IF a. $5 $5,0 ,000 00 × 1.7 1.762 62 = $8, $8,81 810 0 (12 (12%, %, 5 per perio iods ds)) b. b. $5 $5,0 ,000 00 × 1.79 1.791 1 = $8 $8,9 ,955 55 (6%, (6%, 10 per perio iods ds)) c. $5 $5,0 ,000 00 × 1.8 1.806 06 = $9, $9,03 030 0 (3% (3%,, 20 20 per perio iods ds)) 25.

As stated stated in the the chapter, chapter, annuity annuity payments payments are assumed assumed to come at the the end of each payment 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, subtract 1 from n and add 1 to the tabular value. To find the future value of an annuity, 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, go to Appendix C for n = 6 and i = 10 percent. Look up the 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 $4,000 per period where payments come at the beginning of each period? The interest rate is 12 percent.


Appendix C FVA = A × FVIFA n = 11, i = 12% 20.655 – 1 = 19.655 FVA = $4,000 × 19.655 = $78,620

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26.

Related Related to the discus discussion sion in problem problem 25, what what is the the present present value value of a 10-year 10-year annuity annuity of of $5,000 per period in which payments come at the beginning of each period? The interest rate is 12 percent.


Appendix D PVA = A × PVIFA n = 9, i = 12% 5.328 + 1 = 6.328 PVA = $5,000 × 6.328 = $31,640 27.

Your rich rich godfather godfather has offered offered you you a choice choice of one of of the three three followin following g alternative alternatives: s: $10,000 now; $2,000 a year for eight years; or $24,000 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?


(first alternative) alternative) Present value of $10,000 received now: $10,000 (second alternative) Present value of annuity of $2,000 for eight years: Appendix D PVA = A × PVIFA = $2, 00 000× PVIFA (11%, 8 ye years) = $2,00 ,000× 5.146 = $10,292 (third alternative) Present value of $24,000 received in eight years:

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9-27. (Continued)

Appendix B PV = FV × PVIF = $2 $24,00 4,000×PV 0×PVIF (11 11% %,8 years) = $24,000×.434 = $10,416 Select $24,000 to be received in eight years. Revised answers based on 12%.

(first alternative) Present value of $10,000 received today: $10,000 (second alternative) Present value of annuity of $2,000 for 8 years: Appendix D PVA = A × PVIFA =

$2,00 ,000× PVIFA (12%, 8y 8yeears)

=

$2,00 ,000× 4.968

$9,936 (third alternative) Present value of $24,000 received in 8 years: =

Appendix B PV = FV × PVIF = $24,000×P ,000×PV VIF (12%,8 years) = $2 $24,00 4,000× 0×..40 404 4 = $9,69 ,696 Select $10,000 now.

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28.

You need need $28,974 $28,974 at the the end of of 10 years, years, and your your only only investme investment nt outlet outlet is is an 8 percent 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. b.

What What singl singlee paymen paymentt could could be made made at the begi beginni nning ng of the the firs firstt year year to to achiev achievee this this objective? What What amount amount could could you you pay pay at the end of of each each year year annu annuall ally y for for 10 years years to to achie achieve ve this same objective?


29.

a.

Appendix B PV = FV × PVIF (8%, 10 periods) = $28,974 × .463 = $13,415

b.

Appendix C A = FVA FVA/F /FVI VIFA FA (8%, (8%, 10 peri period ods) s) = $28,974/14.487 = $2,000

Sue Sussman Sussman start started ed a paper paper route route on January January 1, 2002. 2002. Every Every three three months, months, she deposit depositss $500 in her bank account, which earns 4 percent annually but is compounded quarterly. On December 31, 2005 she used the entire balance in her bank account to invest in a contract that pays 9 percent annually. How much will she have on December 31, 2008?


Appendix C FVA = A × FVIFA (1%, 16 periods) FVA = $500 × 17.258 = $8,629 December 31, 2005 amount Appendix A FV = PV × FVIF (9%, 3 periods) FV = $8 $8,629 × 1.295 FV = $11 $11,17 ,174.5 4.56 6 Decem December ber 31, 200 2008 8 amoun amountt

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30.

On January January 1, 2002, Mike Irwin, Irwin, Jr., Jr., bought 100 shares shares of of stock stock at $14 $14 per share. On December 31, 2008, he sold the stock for $21 per share. What is his annual rate of return? Interpolate to find the exact answer.


Appendix B PV

PVIF = FV (7 periods) PVIF

=

$14 $21

=

.667

Return is between 5% and 6% for 7 periods (between .711 and .665 in the table) PVIF at 6% PVIF at 5%

.711 -.665 .046

PVIF at 6% PVIF computed

.711 -.667 .044

5% + (.044/.046) (1%) 5% + .96 (1%) 5.96%

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31.

Dr. I. N. Stein Stein has just just invested invested $6,250 $6,250 for for his son (age (age one). The The money will be be used for for his son’s education 17 years from now. He calculates that he will need $50,000 $50,00 0 for his son’s education by the time the boy goes go es to school. What rate of return will Dr. Stein need to achieve this goal?


Appendix A FV FVIF = (17 pe periods) PV $50,000 = $6,250 = 8.0 Rate of Ret Return =13% OR

Appendix B PV PVIF = (17 pe periods) FV $6,250 =

=

$50,000 .125 .125 Rate Rate of Retu Return rn

=

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13% 13 %

32.

Ester Seals has has just just given given an insurance insurance company company $41,625. $41,625. In In return, return, she will will receive receive an annuity of $5,000 for 15 years. At what wh at rate of return must the insurance company invest this $41,625 to make the annual payments? Interpolate.


Appendix D PVIFA = PVA / A (1 ( 15 pe periods) =

$41,625/$5,000

=

8.325 is between between 8% and and 9% for for 15 periods periods

PVIFA at 8% PVIFA at 9%

8.559 −

8.061 .498

PVIFA at 8% PVIFA computed

8.559 −

8.325 .234

8% + (.234/.498) (1%) 8% + .470 (1%) = 8.47% 33.

Betty Bronson Bronson has just just retire retired d after 25 years years with with the electri electricc company. company. Her total total pension pension funds have an accumulated value of $180,000, and her life expectancy is 15 more years. Her pension fund manager assumes he can earn a 9 percent p ercent return on her assets. What will be her yearly annuity for the next 15 years?


Appendix D A = PVA / PV PVIFA (9%,15 periods) =

$180,000/8.061

=

$22,329.74

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34.

Morgan Jennings, Jennings, a geography geography profess professor, or, invest investss $50,000 $50,000 in a parcel of land land that is expected to increase in value by 12 percent per year for the next five years. He will take the proceeds and provide himself with a 10-year 1 0-year annuity. Assuming a 12 percent interest rate, how much will this annuity be?


Appendix A FV = PV × FV IF (12%, 5 periods) FV = $50,000 × 1.762 = $88,100 Appendix D A = PVA/PVIFA (12%, 10 periods) A = $88,100/5.650 = $15,593

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35.

You wish wish to retire retire after after 18 years, years, at which which time time you want to to have accumul accumulated ated enough enough money to receive an annuity of $14,000 a year for 20 years of retirement. During the period before retirement you can earn 11 percent annually, while after retirement you can earn 8 percent on your money. What annual contributions to the retirement fund will allow you to receive the $14,000 annually?


Determine Determine the present value of a 20-year annuity during retirement: Appendix D PVA = A × PVIFA (8%, 20 20 years) =

$14,000 × 9.818

=

$137,452

To determine the annual deposit into an account earning 11% that is necessary to accumulate $137,452 after 18 years, use the Future Value of an Annuity table: Appendix C A = FVA /FVIFA (11%, 18 18 ye years) $137,452 = $2 = $2,727. ,727.44 44 annu annual al con contr trib ibuti ution on 50.396

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36.

Del Monty Monty will will receive receive the following following payment paymentss at the the end of the the next three three years: years: $2,000, $2,000, $3,500, and $4,500. Then from the end of the fourth through the end of the tenth year, he will receive an annuity of $5,000 per year. At a discount rate of 9 percent, what is the present value of all three future benefits?


First find the present value of the first three payments. PV = FV × PVIF (Appendix B) i = 9% 1) $2, $2,00 000 0 × .9 .917 = $1 $1,,83 834 4 2) 3,500 × .842 = 2,947 3) 4,500 × .772 = 3,474 $8,255 Then find the present value of the deferred annuity. Appendix D will give a factor for a seven period annuity (fourth year through the tenth year) at a discount rate of 9 percent. The value of the annuity at the beginning of the fourth year is: PVA

=

A × PVIFA (9%, 7 periods)

=

$5,000

×

5.033 = $25,165

This value at the beginning of year four (end of year three) must now be discounted back for three years to get the present value of the deferred annuity. Use Appendix B. PV

=

FV × PVIF (9%, 3 periods)

=

$25,165 × .772 = $19.427.38

Finally, find the total present value of all future payments. Present value of first three payments Present value of the deferred annuity

S9-22

$ 8,225.00 19,427.38 $27,682.38

37.

Bridget Bridget Jones Jones has a contra contract ct in which which she will will receive receive the the following following payment paymentss for the the next five years: $1,000, $2,000, $3,000, $4,000, and $5,000. She will then receive an annuity of $8,500 a year from the end of the 6th through the end of the 15th year. The appropriate discount rate is 14 percent. If she is offered $30,000 to cancel the contract, should she do it?


First find the present value of the first five payments. PV = FV × PVIF (Appendix B) i = 14% 1) $1,000 × .877 2) 2,000 × .769 3) 3,000 × .675 4) 4,000 × .592 5) 5,000 × .519

= $ 877 = 1,538 = 2,025 = 2,368 = 2,595 $9,403

Then find the present value of the deferred annuity. Appendix D will give a factor for a ten period annuity (sixth year through the fifteenth year) at a discount rate of 14 percent. The value of the annuity at the beginning of the sixth year is: PVA

=

A

=

$8,500

×

PVIFA (14%, 10 periods) ×

5.216 = $44,336

This value at the beginning of year six (end of year five) must now be discounted back for five years to get the present value of the deferred annuity. Use Appendix B. PV = FV × PVIF (14%, 5 pe periods) = $44,33 ,336 × .516 = $23,01 ,010.38 Next, find the total present value of all future payments. Present value of first five payments $ 9,403.00 Present value of the deferred annuity 23,010.38 $32,413.38 Because this amount is greater than $30,000, Bridget should not cancel her contract. S9-23

38.

Mark Ventura Ventura has has just just purchased purchased an annuit annuity y to begin begin payment payment at the the end of 2011 2011 (that (that is the date of the first payment). Assume it is now the beginning of the year 2009. The annuity is for $8,000 per year and is designed to last 10 years. If the interest rate for this problem calculation is 13 percent, what is the most he should have paid for the annuity?


Appendix D will give a factor for a 10-year annuity when the appropriate discount rate is 13 percent (5.426). The value of the annuity at the beginning of the year it starts (2011) is:

PVA

=

A × PVIFA (13%, 10 pe p erio d s)

=

$8,0 $8,000 00 × 5.42 5.426 6

=

$43,408

The present value at the beginning of 2009 is found using Appendix B (2 years at 13%). The factor is .783. Note we are discounting from the beginning of 2011 to the beginning of 2009. PV = FV × PVIF (13%, 2 pe periods) = $43,408 ,408 × .783 = $33,98 ,988 The maximum that should be paid for the annuity is $33,988.

S9-24

39.

If you you borrow borrow $15,618 $15,618 and and are require required d to pay pay back back the loan loan in seven equal equal annual annual installments of $3,000, what is the interest rate associated with the loan?


Appendix D PVIFA

=

PVA / A (7 periods)

=

$15,618/$3,000

=

5.206

Interest rate = 8 percent 40.

Cal Lury Lury owes $10,000 $10,000 now. now. A lender lender will will carry carry the debt debt for five more more years years at 10 10 percent percent interest. That is, in this particular case, the amount owed will go up by 10 percent per year for five years. The lender then will require that Cal pay off the loan over the next 12 years at 11 percent interest. What will his annual payment be?


Appendix A FV = PV × FVIF (10%, 5 pe periods) = $10,000 ,000 × 1.611 = $16,110

Amount owed after 5 years

Appendix D A = PVA / PVIFA (11%, 12 12 pe periods) = $16,110 / 6.492 = $2, 48 482

Annual payments to retire the loan

S9-25

41.

If your your uncle uncle borrows borrows $60,000 $60,000 from from the bank bank at 10 percent percent interes interestt over the the seven-year seven-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 his first payment will be applied to interest? To principal? H ow much of his second payment will be applied to each?


Appendix D A = PVA /PVIFA (10%, 7p 7 periods) = $6 $60,0 0,000 00/4 /4.8 .868 68 = $12 $12,325 ,325 annu annual al payme payment nt First payment: $60,000 × .10 = $6,000 interest $12,32 $12 ,325 5 – $6,000 $6,000 = $6,325 $6,325 applie applied d to princi principal pal Second payment: First determine remaining remaining principal and then the interest and principal payment. $60,000 $60,00 0 – $6,325 $6,325 = $53 $53,67 ,675 5 rema remaini ining ng princi principal pal $53,675 × .10 = $ 5,368 interest $12,32 $12 ,325 5 – $5,368 $5,368 = $ 6,957 6,957 applie applied d to princi principal pal

S9-26

42.

Larry Davis borrows borrows $80,000 $80,000 at 14 percent percent intere interest st toward toward the the purchase purchase of a home. His mortgage is for 25 years. a.

b. c.

How much much will will his his annu annual al payme payments nts be? (Altho (Although ugh home home paym payment entss are are usual usually ly on on a monthly basis, we shall do our analysis on an annual basis for ease of computation. We will get a reasonably accurate answer.) How How much much int inter eres estt will will he he pay pay over over the the lif lifee of the the loan loan?? How much much shoul should d he be be willi willing ng to pay to get get out out of a 14 perc percent ent mortga mortgage ge and and into into a 10 percent mortgage with 25 years remaining on the mortgage? Assume current interest rates are 10 percent. Carefully consider the time value of money. Disregard taxes.


Appendix D a. A = PVA /PVIFA (14%, 25 pe periods) = $8 $80,00 0,000/ 0/6. 6.87 873 3 = $11,639 ,639.75 b.

$11,63 ,639.75 ×

25

$290,993 ,993.75 −

80,000 80 ,000..00

$210 $2 10,99 ,993. 3.75 75

annual payments years total tal payment repay payment of pri princ ncip ipaal tota totall inte interrest paid paid

Appendix D c.

New payments at 10% A = PVA /PVIFA (10%, 25 pe periods) = $8 $80,00 0,000/ 0/9. 9.07 077 7 = $8,81 ,813.48

S9-27

9-42. (Continued)

Difference Difference between old and new payments $11, 639.75 old 8,81 ,813.48 new $ 2,826.27 difference P.V. of difference – Appendix D PVA

43.

=

A × PVIFA (assumes 10% discount rate, 25 periods)

=

$ 2,826.27 × 9.077

=

$25,654.05

Amount that could be paid to refinance

You are chairperson chairperson of of the investm investment ent fund fund for the the Eastern Eastern Football Football League. League. You You are asked asked to set up a fund of semiannual payments to be compounded semiannually to accumulate a sum of $100,000 after 10 years at an 8 percent annual rate (20 payments). The first payment into the fund is to occur six months from today, and the last payment is to take place at the end of the 10th year. a.

Determ Determine ine how how much much the the semi semiann annual ual paym payment ent shoul should d be. (Round (Round to to whole whole numb numbers ers.) .)

On the day after the fourth payment is made (the beginning of the third year) the interest rate will go 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. Determ Determine ine how how much much the revi revised sed semi semiannu annual al payme payments nts shoul should d be after after this this rate rate b. change (there are 16 payments and compounding dates). The next payment will be in the middle of the third year. (Round all values to whole numbers.)


Appendix C a. A

=

FVA / FVIFA (4%, n

=

$100,0 $10 0,000 00 / 29.778 29.778

=

$3,358 semi-annual semi-annual payment payment

=

20)

S9-28

9-43. (Continued)

b. First First deter determi mine ne how how much much the the old old payme payments nts are are equa equall to aft after er 4 periods at 4%. Appendix C. FVA = A × FVIFA (4%, 4 periods) = $3,35 ,358 × 4.246 = $14,25 ,258 Then determine how much this value will grow to after 16 periods at 5%. Appendix A. FV = PV × FVIF (5%, 16 16 periods) = $14,258 ,258 × 2.183 = $31,125 Subtract this value from $100,000 to determine how much you need to accumulate on the next 16 payments. $100,000 −

31,125

$ 68,87 ,875 Determine the revised semi-annual payment necessary to accumulate this sum after 16 periods at 5%. Appendix C A = FVA/FVIFA (5%, 16 periods) A = $68,875/23.657 A = $2,911 revised semi-annual payment

S9-29

44.

Your younger younger sister sister,, Linda, Linda, will start start college college in in five years. years. She She has just just inform informed ed your parents that she wants to go to Hampton University, which will cost $17,000 per year for four years (cost assumed to come at the end en d of each year). Anticipating Linda’s ambitions, your parents started investing $2,000 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 Linda’s education? ed ucation? Use 10 percent as the appropriate interest rate throughout this problem (for discounting or compounding).


Present value of college costs Appendix D PVA = A × PVIFA (10%, 4 periods) = $17,00 ,000 × 3.170 = $53,89 ,890 Accumulation Accumulation based on investing $2,000 per year for 10 years. Appendix C FVA = A × FVIFA (10%, 10 10 periods) = $2,000 ,000 × 15.937 = $31,87 ,874 Additional funds required 5 years from now. $53,890 31,874 31,874 $22, $2 2,01 016 6

PV of college costs Accu Accumu mula lati tion on base based d on $2, $2,00 000 0 per per year year inv inves estm tmen entt Add ddit itio iona nall fun funds ds requ requiired

Added contribution for the next 5 years

S9-30

9-44. (Continued)

Appendix C A = FVA /FVIFA (10%, 5 periods) = $2 $22,0 2,016 16/6 /6.1 .105 05 = $3,606 ,606.22 45.

Linda (from (from problem problem 44) is now 18 years years old old (five (five years years have passed), passed), and and she wants wants to to get married instead of going to school. Your parents have accumulated the necessary funds for her education. Instead of her schooling, your parents are paying $8,000 for her upcoming wedding and plan to take year-end vacations costing $5,000 $ 5,000 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 and perhaps a PhD. If graduate school costs $14,045 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.


Funds available after the wedding $53,890 – 8,000

Wedding

$45, $4 5,89 890 0

Fund Fundss ava avail ilab able le afte afterr the the wedd weddin ing g

Less present value of vacation Appendix D PVA = A × PVIFA (10%, 3 p peeriods) =

$5, 00 000 × 2.487

S9-31

9-45. (Continued)

PV of vacation = $12,435 $45,890 – 12,435 $33, $3 3,45 455 5

Rema Remain inin ing g fund fundss for for gra gradu duat atee scho school ol

Funds available 3 years later for graduate school: Appendix A FV = PV × FVIF (10%, 3 periods) =

$33,45 ,455 × 1.331

=

$44,528.61 $44, 528.61 Funds availabl availablee for graduate school

Number of years of graduate education Appendix D PVIFA = =

PVA

(10%)

A $44,528.61 $14,045.00

=

3.17 3.170 0 (roun (rounde ded) d)

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

S9-32

COMPREHENSIVE COMPREHENSIVE PROBLEM Dr. Harold Wolf of Medical Research Corporation (MRC) was thrilled with the response he had received from drug companies for his latest discovery, a unique electronic stimulator that reduces the pain from arthritis. The process had yet to pass rigorous Federal Drug Administration (FDA) testing and was still in the early stages of development, but the interest was intense. He received the three offers described below this paragraph. (A 10 percent interest rate should be used throughout this analysis unless otherwise specified.) Offer I $1,000,000 now plus $200,000 from year 6 through 15. Also if the product did over $100 million in cumulative sales by the end of year 15, he would receive an additional $3,000,000. Dr. Wolf thought there was a 70 percent probability this would happen. Offer II Thirty percent of the buyer’s gross profit on the product for the next four years. The buyer in this case was Zbay Pharmaceutical. Zbay’s gross profit margin was 60 percent. Sales in year one were projected to be $2 million and then expected to grow by 40 percent per year. Offer III A trust fund would be set up for the next 8 years. At the end of that period, Dr. Wolf would receive the proceeds (and discount them back to the present at 10 percent). The trust fund called for semiannual payments for the next 8 years of $200,000 (a total of $400,000 per year).

The payments would start immediately. Since the payments are coming at the beginning of each period instead instead of the end, this is an annuity due. To look up the future value of an annuity due in the tables, add 1 to n (16 + 1) and subtract 1 from the value in the table. Assume the annual interest interest rate on this annuity is 10 percent percent annually annually (5 percent semiannually). semiannually). Determine Determine the present value of the trust fund’s final value. Required: Required: Find the present value of each of the three offers and indicate which one has the highest present value .

CP 9-1. 9-1. Soluti Solution: on: Medical Research Corporation Offer I

$1,000,000 now plus: + $200,000 from year 6 through 15 (deferred annuity) Appendix D PVA = A × PVIFA (10%, 10 10 years) = $200,00 ,000 × 6.145 = $1,229 $1, 229,, 000 (perc (percent ent val value ue at the the beg beginn inning ing of year 6, i.e. the end of year 5)

S9-33

CP 9-1. (Continued)

Appendix B

PV

=

FV

=

$1,2 $1,229 29,0 ,000 00 × .621 .621

=

$763,209

×

PVIF (10%, 5 years)

+ .70 × $3,000,000 = $2,100,000 Appendix B

PV

=

FV

=

$2,1 $2,100 00,,000 000 × .239 .239

×

PVIF (10%, 15 years) =

$501 $501,9 ,900 00

Total value of Offer I $1,000,000 763, 76 3,20 209 9 501,900 $2,265,109

Payment today Presen esentt val value of def deferr erred annu annuit ity y Present va value of of $3 $3 mi million bo bonus

Offer II

Gross Profit

Payment 30%

Year

Sales

(60% of Sales) of Gross Profit

1 2 3

$2,000,000 2,800,000 3,920,000

$1,200,000 1,680,000 2,352,000

$360,000 504,000 705,600

4

5,488,000

3,292,800

987,600

S9-34

CP 9-1. (Continued)

Year

Payment

Appendix B PV Factor

1 2 3 4

$360,000 504,000 705,600 987,600

.909 .826 .751 .683

Total value of Offer II

Future value of an annuity due (Appendix C) 8 years – semiannually n = 16 + 1 = 17 i = 10%/2 = 5% FVIFA = 25.840 – 1 = 24.840 (Appendix C) =

A × FVIFA

=

$200,00 ,000 × 24.840

=

$4,968 $4, 968,, 000 value value of trust fund fund after after 8 year yearss

Present value of trust fund (Appendix B) PV

=

A × PVIF (10%, 8 years)

=

$4,968,000 × .467

=

$2,320,056 $2,320,056 Total value value of offer offer III III

S9-35

$327,240 416,304 529,906 674,531 $1,947,981

Offer III

FVA

PV

CP 9-1. (Continued) Summary

Value of Offer I

$2,265,109

Value of Offer II

$1,947,981

Value of Offer III

$2,320,056

Select Offer III

S9-36

FIN CH-9

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