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 loan,
Determination of return on an investment,
Determination of the value of a bond.
Problems
9-1.
You invest $3,000 a year for 3 years at 12 percent.
a. What is the value of your investment after one year? Multiply $3,000 x 1.12.
b. What is the value of your investment after two years? Multiply your answer to part a by 1.12.
c. What is the value of your investment after three years? Multiply your answer to part b by 1.12,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 = 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.
Solution:
a. $3,000 x 1.12 = $3,360.00
b. $3,360 x 1.12 = $3,763.20
c. $3,763.20 x 1.12 = $4,214.78
d. $3,000 x 1.405 = $4,215.00 (Appendix A)
9-2.
What is the present value of:
a. $9,000 in 7 years at 8 percent?
b. $20,000 in 5 years at 10 percent?
c. $10,000 in 25 years at 6 percent?
d. $1,000 in 50 years at 16 percent?
Solution:
Appendix B
PV = FV x PVIF
a. $ 9,000 x,583 = $5,247
b. $20,000 x,621 = $12,420
c. $10,000 x,233 = $2,330
d. $ 1,000 x,001 = $1
9-3.
If you invest $9,000 today,how much will you have:
a. In 2 years at 9 percent?
b. In 7 years at 12 percent?
c. In 25 years at 14 percent?
d. In 25 years at 14 percent (compounded semiannually)?
Solution:
Appendix A
FV = PV x FVIF
a. $9,000 x 1.188 = $ 10,692
b. $9,000 x 2.211 = $ 19,899
c. $9,000 x 26.462 = $238,158
d. $9,000 x 29.457 = $265,113 (7%,50 periods)
9-4.
Your uncle offers you a choice of $30,000 in 50 years or $95 today,If money is discounted at 12 percent,which should you choose?
Solution:
Appendix B
PV = FV x PVIF (12%,50 periods)
PV = $30,000 x,003 = $90
Choose $95 today.
9-5.
How much would you have to invest today to receive:
a. $15,000 in 8 years at 10 percent?
b. $20,000 in 12 years at 13 percent?
c. $6,000 each year for 10 years at 9 percent?
d. $50,000 each year for 50 years at 7 percent?
Solution:
Appendix B (a and b)
PV = FV x PVIF
a. $15,000 x,467 = $7,005
b. $20,000 x,231 = $4,620
Appendix D (c and d)
c. $ 6,000 x 6.418 = $ 38,508
d. $50,000 x 13.801 = $690,050
9-6.
If you invest $2,000 a year in a retirement account,how much would you have:
a. In 5 years at 6 percent?
b. In 20 years at 10 percent?
c. In 40 years at 12 percent?
Solution:
Appendix C
FVA = A x FVIFA
a. $2,000 x 5.637 = $ 11,274
b. $2,000 x 57.275 = $ 114,550
c. $2,000 x 767.09 = $1,534,180
9-7.
You invest a single amount of $10,000 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 x FVIF
$10,000 x 1.611 = $16,110
Appendix A
FV = PV x FVIF
$16,110 x 5.350 = $86,188
9-8.
Jean Splicing will receive $8,500 a year for the next 15 years from her trust,If a 7 percent interest rate is applied,what is the current value of the future payments?
Solution:
Appendix D
PVA = A x PVIFA (7%,15 periods)
= $8,500 x 9.108 = $77,418
9-9.
Phil Goode will receive $175,000 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 x PVIF (14%,50 periods)
= $175,000 x,001 = $175
9-10.
Polly Graham will receive $12,000 a year for the next 15 years as a result of her patent,If a 9 percent rate is applied,should she be willing to sell out her future rights now for $100,000?
Solution:
Appendix D
PVA = A x PVIFA (9%,15 periods)
= $12,000 x 8.061 = $96,732
Yes,the present value of the annuity is worth less than $100,000.
9-11.
Carrie Tune will receive $19,500 for the next 20 years as a payment for a 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 x PVIFA (10%,20 periods)
PVA = $19,500 x 8.514 = $166,023
No,the present value of the annuity is worth more than $160,000.
9-12.
The Clearinghouse Sweepstakes has just informed you that you have won $1 million,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?
Solution:
Appendix D
PVA = A x PVIFA (10%,50 periods)
PVA = $20,000 x 9.915 = $198,300
9-13.
Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball 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?
Solution:
Appendix A
FV = PV x FVIF (12%,10 periods)
= $25,000 x 3.106 = $77,650
9-14.
Dr,Ruth has been secretly depositing $2,500 in her savings account every December starting in 1995,Her account earns 5 percent compounded annually,How much will she have in December of 2004? (Assume that a deposit is made in 2004.) Make sure to carefully count the years.
Solution:
Appendix C
FVA = A x FVIFA (5%,10 periods)
FVA = $2,500 x 12.578 = $31,445
9-15.
At a growth (interest) rate of 9 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 interest factor must equal 2,
* In Appendix A,looking down the 9% column,we 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.
9-16.
If you owe $40,000 payable at the end of seven years,what amount should your creditor accept in payment immediately if she could earn 12 percent on her money?
Solution:
Appendix B
PV = FV x PVIF (12%,7 periods)
PV = $40,000 x,452 = $18,080
9-17.
Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first 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 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.)
Solution:
Appendix B
PV = FVIF
Discount rate = 11%
$ 2.00 x,901 = $ 1.80
2.20 x,802 = 1.79
2.40 x,731 = 1.75
33.00 x,731 = 24.12
$29.46
9-18.
Les Moore retired as president of Goodman Snack Foods Company but is currently on a consulting contract for $35,000 per year for the next 10 years.
a. If Mr,Moore's opportunity cost (potential return) is 10 percent,what is the present value of his consulting contract?
b. Assume Mr,Moore 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:
Appendix D
a. PVA = A x PVIFA (10%,10 periods)
PVA = $35,000 x 6.145 = $215,075
b. Deferred annuity—Appendix D
PVA = A x PVIFA (i = 10%,10 periods)
PVA = $35,000 x 6.145 = $215,075
Now,discount back this value for 2 periods
PV = FV x PVIF (i = 10%,2 periods) Appendix B
$215,075 x,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.
9-19.
Juan Garza invested $20,000 10 years ago at 12 percent,compounded quarterly,How much has he accumulated?
Solution:
Appendix A
FV = PV x FVIF (3%,40 periods)
FV = $20,000 x 3.262 = $65,240
9-20.
Determine the amount of money in a savings account at the end of five years,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.
Solution:
Appendix A
FV = PV x FVIF
a. $5,000 x 1.762 = $8,810 (12%,5 periods)
b. $5,000 x 1.791 = $8,955 (6%,10 periods)
c. $5,000 x 1.806 = $9,030 (3%,20 periods)
9-21.
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,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,you would 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 x 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.
Solution:
Appendix C
FVA = A x FVIFA
n = 11,i = 12% 20.655 – 1 = 19.655
FVA = $4,000 x 19.655 = $78,620
9-22.
Your rich godfather has offered you a choice of one of the three following alternatives,$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?
Solution:
(first 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 x PVIFA
= $2,000 x PVIFA (11%,8 years)
= $2,000 x 5.146
= $10,292
(third alternative) Present value of $24,000 received in eight years,
Appendix B
PV = FV x PVIF
= $24,000 x PVIF (11%,8 years)
= $24,000 x,434
= $10,416
Select $24,000 to be received in eight years.
9-22,Continued
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 x PVIFA
= $2,000 x PVIFA (12%,8 years)
= $2,000 x 4.968
= $9,936
(third alternative) Present value of $24,000 received in 8 years,
Appendix B
PV = FV x PVIF
= $24,000 x PVIF (12%,8 years)
= $24,000 x,404
= $9,696
Select $10,000 now.
9-23.
You need $28,974 at the end of nine years,and your only investment outlet is an 8 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 10 years to achieve this same objective?
Solution:
a. Appendix B
PV = FV x PVIF (8%,10 periods)
= $28,974 x,463 = $13,415
b. Appendix C
A = FVA/FVIFA (8%,10 periods)
= $28,974/14.487 = $2,000
9-24.
Sue Sussman started a paper route on January 1,1998,Every three months,she deposits $500 in her bank account,which earns 4 percent annually but is compounded quarterly,On December 31,2001,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 31,2004?
Solution:
Appendix C
FVA = A x FVIFA (1%,16 periods)
FVA = $500 x 17.258 = $8,629 December 31,2001 amount
Appendix A
FV = PV x FVIF (9%,3 periods)
FV = $8,629 x 1.295
FV = $11,174.56 December 31,2004 amount
9-25.
On January 1,2002,Mike Irwin,Jr.,bought 100 shares of stock at $14 per share,On December 31,2004,he sold the stock for $21 per share,What is his annual rate of return? Interpolate to find the exact answer.
Solution:
Appendix B
PVIF = (3 periods)
PVIF =
Return is between 14%-15% for 3 periods
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%
9-26.
Dr,I,N,Stein has just invested $6,250 for his son (age one),The money will be used for his son's education 17 years from now,He calculates that he will need $50,000 for his son’s education by the time the boy goes to school,What rate of return will Dr,Stein need to achieve this goal?
Solution:
Appendix A
Appendix B
9-27.
Ester Seals has just given an insurance company $41,625,In return,she will receive an annuity of $5,000 for 15 years,At what rate of return must the insurance company invest this $41,625 to make the annual payments? Interpolate.
Solution:
Appendix D
PVIFA = PVA/A (15 periods)
= $41,625/$5,000
= 8.325 is between 8% and 9% for 15 periods
PVIFA at 8% 8.559
PVIFA at 9% –8.061
,498
PVIFA at 8% 8.559
PVIFA computed –8.325
,234
8% + (.234/.498) (1%)
8% +,470 (1%) = 8.47%
9-28.
Betty Bronson has just retired after 25 years with the electric company,Her total 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 return on her assets,What will be her yearly annuity for the next 15 years?
Solution:
Appendix D
A = PVA/PVIFA (9%,15 periods)
= $180,000/8.061
= $22,329.74
9-29.
Morgan Jennings,a geography professor,invests $50,000 in a parcel of 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 annuity,Assuming a 12 percent interest rate,how much will this annuity be?
Solution:
Appendix A
FV = PV x FVIF (12%,5 periods)
FV = $50,000 x 1.762 = $88,100
Appendix D
A = PVA/PVIFA (12%,10 periods)
A = $88,100/5.650 = $15,593
9-30.
You wish to retire after 18 years,at which time you want to have accumulated 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 annuity?
Solution:
Determine the present value of a 20-year annuity during retirement,
Appendix D
PVA = A x PVIFA (8%,20 years)
= $14,000 x 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 years)
=
9-31.
Del Monty will receive the following payments at the end of the next three years,$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 percent value of all three future benefits?
Solution:
First find the present value of the first three payments.
PV = FV x PVIF (Appendix B) i = 9%
$2,000 x,917 = $1,834
2) 3,500 x,842 = 2,947
3) 4,500 x,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 x PVIFA (9%,7 periods)
= $5,000 x 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 x PVIF (9%,3 periods)
= $25,165 x,772 = $19,427.38
Finally,find the total present value of all future payments.
Present value of first three payments $ 8,255.00
Present value of the deferred annuity 19,427.38
$27,682.38
9-32.
Bridget Jones has a contract in which she will receive the following payments for 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?
Solution:
First find the present value of the first five payments.
PV = FV x PVIF (Appendix B) i = 14%
1) $1,000 x,877 = $ 877
2) 2,000 x,769 = 1,538
3) 3,000 x,675 = 2,025
4) 4,000 x,592 = 2,368
5) 5,000 x,519 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 x PVIFA (14%,10 periods)
= $8,500 x 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 x PVIF (14%,5 periods)
= $44,336 x,519 = $23,010.38
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
Since the present value of all future benefits under the contract is greater than $30,000,Bridget Jones should not accept this amount to cancel the contract.
9-33.
Mark Ventura has just purchased an annuity to begin payment at the end of 2007 (that is the date of the first payment),Assume it is now the beginning of the year 2005,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?
Solution:
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 (2007) is:
PVA = A x PVIFA (13%,10 periods)
= $8,000 x 5.426
= $43,408
The present value at the beginning of 2005 is found using Appendix B (2 years at 13%),The factor is,783,Note we are discounting from the beginning of 2007 to the beginning of 2005.
PV = FV x PVIF (13%,2 periods)
= $43,408 x,783
= $33,988
The maximum that should be paid for the annuity is $33,988.
9-34.
If you borrow $15,618 and are required to pay back the loan in seven equal annual installments of $3,000,what is the interest rate associated with the loan?
Solution:
Appendix D
PVIFA = PVA/A (7 periods)
= $15,618/$3,000
= 5.206
Interest rate = 8 percent
9-35.
Cal Lury owes $10,000 now,A lender will carry the debt for five more years at 10 percent interest,That is,in this particular case,the amount owed will go up 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 payments be?
Solution:
Appendix A
FV = PV x FVIF (10%,5 periods)
FV = $10,000 x 1.611
= $16,110 Amount owed after 5 years
Appendix D
A = PVA/PVIFA (11%,12 periods)
= $16,110/6.492
= $2,482 Annual payments to retire the loan
9-36.
If your uncle borrows $60,000 from the bank at 10 percent interest over the 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? How much of his second payment will be applied to each?
Solution:
Appendix D
A = PVA/PVIFA (10%,7 periods)
= $60,000/4.868
= $12,325 annual payment
First payment,
$60,000 x,10 = $6,000 interest
$12,325 – $6,000 = $6,325 applied to principal
Second payment,
First determine remaining principal and then the interest and principal payment.
$60,000 – $6,325 = $53,675 remaining principal
$53,675 x,10 = $ 5,368 interest
$12,325 – $5,368 = $ 6,957 applied to principal
9-37.
Larry Davis borrows $80,000 at 14 percent interest toward the purchase of a home,His mortgage is for 25 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 will 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 14 percent mortgage and 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.
9-37,Continued
Solution:
Appendix D
a. A = PVA/PVIFA (14%,25 periods)
= $80,000/6.873
= $11,639.75
b. $ 11,639.75 annual payments
x 25 years
$290,993.75 total payment
– 80,000.00 repayment of principal
$210,993.75 total interest paid
Appendix D
c. New payments at 10%
A = PVA/PVIFA (10%,25 periods)
= $80,000/9.077
= $8,813.48
Difference between old and new payments
$11,639.75 old
8,813.48 new
$ 2,826.27 difference
P.V,of difference-Appendix D
PVA = A x PVIFA (assumes 10% discount rate,25 periods)
= $ 2,826.27 x 9.077
= $25,654.05 Amount that could be paid to refinance
9-38.
You are chairperson of the investment fund for the Eastern Football League,You are 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. Determine how much the semiannual payment should be,(Round to whole numbers.)
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.
b. Determine how much the revised semiannual payments should be after this rate 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.)
Solution:
Appendix C
a. A = FVA/FVIFA (4%,n = 20)
= $100,000/29.778
= $3,358 semi-annual payment
b. First determine how much the old payments are equal to after 4 periods at 4%,Appendix C.
FVA = A x FVIFA (4%,4 periods)
= $3,358 x 4.246
= $14,258
Then determine how much this value will grow to after 16 periods at 5%,Appendix A.
9-38,Continued
FV = PV x FVIF (5%,16 periods)
= $14,258 x 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
$ 68,875
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
9-39.
Your younger sister,Linda,will start college in five years,She has just informed 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 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? Use 10 percent as the appropriate interest rate throughout this problem (for discounting or compounding),
9-39,Continued
Solution:
Present value of college costs
Appendix D
PVA = A x PVIFA (10%,4 periods)
= $17,000 x 3.170
= $53,890
Accumulation based on investing $2,000 per year for 10 years.
Appendix C
FVA = A x FVIFA (10%,10 periods)
= $2,000 x 15.937
= $31,874
Additional funds required 5 years from now.
$53,890 PV of college costs
31,874 Accumulation based on $2,000 per year investment
$22,016 Additional funds required
Added contribution for the next 5 years
Appendix C
A = FVA/FVIFA (10%,5 periods)
= $22,016/6.105
= $3,606.22
9-40.
Linda (from problem 39) is now 18 years old (five years have passed),and she wants 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 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 Ph.D,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,
Solution:
Funds available after the wedding
$53,890
– 8,000 Wedding
$45,890 Funds available after the wedding
Less present value of vacation
Appendix D
PVA = A x PVIFA (10%,3 periods)
= $5,000 x 2.487
PV of vacation = $12,435
$45,890
–12,435
$33,455 Remaining funds for graduate school
Funds available 3 years later for graduate school
9-40,Continued
Appendix A
FV = PV x FVIF (10%,3 periods)
= $33,455 x 1.331
= $44,528.61 Funds available for graduate school
Number of years of graduate education
Appendix D
PVIFA =
=
with i = 10%,n = 4 for 3.170 the answer is 4 years.
Comprehensive Problem
CP 9-1.
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 following three offers,(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 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 rate on this annuity is 10 percent annually (5 percent semiannually),Determine the present value of the trust fund’s final value.
Required,Find the present value of each of the three offers and indicate which one has the highest present value.
CP 9-1,Continued
Solution:
Offer I
$1,000,000 now plus:
+ $200,000 from year 6 through 15 (deferred annuity)
Appendix D
PVA = A x PVIFA (10%,10 years)
= $200,000 x 6.145
= $1,229,000 (percent value at the beginning of
year 6,i.e,the end of year 5)
Appendix B
PV = FV x PVIF (10%,5 years)
= $1,229,000 x,621
= $763,209
+,70 x $3,000,000 = $2,100,000
Appendix B
PV = FV x PVIF (10%,15 years)
= $2,100,000 x,239 = $501,900
Total value of Offer I
$1,000,000 Payment today
763,209 Present value of deferred annuity
501,900 Present value of $3 million bonus
$2,265,109
CP 9-1,Continued
Offer II
Year
Sales
Gross Profit
(60% of Sales)
Payment 30%
of Gross Profit
1
$2,000,000
$1,200,000
$360,000
2
2,800,000
1,680,000
504,000
3
3,920,000
2,352,000
705,600
4
5,488,000
3,292,800
987,600
Year
Payment
Appendix B
PV Factor
PV
1
$360,000
.909
$327,240
2
504,000
.826
416,304
3
705,600
.751
529,906
4
987,600
.683
674,531
Total value of Offer II
$1,947,981
Offer III
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)
FVA = A x FVIFA
= $200,000 x 24.840
= $4,968,000 value of trust fund after 8 years
CP 9-1,Continued
Present value of trust fund (Appendix B)
PV = A x PVIF (10%,8 years)
= $4,968,000 x,467
= $2,320,056 Total value of offer III
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
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 loan,
Determination of return on an investment,
Determination of the value of a bond.
Problems
9-1.
You invest $3,000 a year for 3 years at 12 percent.
a. What is the value of your investment after one year? Multiply $3,000 x 1.12.
b. What is the value of your investment after two years? Multiply your answer to part a by 1.12.
c. What is the value of your investment after three years? Multiply your answer to part b by 1.12,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 = 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.
Solution:
a. $3,000 x 1.12 = $3,360.00
b. $3,360 x 1.12 = $3,763.20
c. $3,763.20 x 1.12 = $4,214.78
d. $3,000 x 1.405 = $4,215.00 (Appendix A)
9-2.
What is the present value of:
a. $9,000 in 7 years at 8 percent?
b. $20,000 in 5 years at 10 percent?
c. $10,000 in 25 years at 6 percent?
d. $1,000 in 50 years at 16 percent?
Solution:
Appendix B
PV = FV x PVIF
a. $ 9,000 x,583 = $5,247
b. $20,000 x,621 = $12,420
c. $10,000 x,233 = $2,330
d. $ 1,000 x,001 = $1
9-3.
If you invest $9,000 today,how much will you have:
a. In 2 years at 9 percent?
b. In 7 years at 12 percent?
c. In 25 years at 14 percent?
d. In 25 years at 14 percent (compounded semiannually)?
Solution:
Appendix A
FV = PV x FVIF
a. $9,000 x 1.188 = $ 10,692
b. $9,000 x 2.211 = $ 19,899
c. $9,000 x 26.462 = $238,158
d. $9,000 x 29.457 = $265,113 (7%,50 periods)
9-4.
Your uncle offers you a choice of $30,000 in 50 years or $95 today,If money is discounted at 12 percent,which should you choose?
Solution:
Appendix B
PV = FV x PVIF (12%,50 periods)
PV = $30,000 x,003 = $90
Choose $95 today.
9-5.
How much would you have to invest today to receive:
a. $15,000 in 8 years at 10 percent?
b. $20,000 in 12 years at 13 percent?
c. $6,000 each year for 10 years at 9 percent?
d. $50,000 each year for 50 years at 7 percent?
Solution:
Appendix B (a and b)
PV = FV x PVIF
a. $15,000 x,467 = $7,005
b. $20,000 x,231 = $4,620
Appendix D (c and d)
c. $ 6,000 x 6.418 = $ 38,508
d. $50,000 x 13.801 = $690,050
9-6.
If you invest $2,000 a year in a retirement account,how much would you have:
a. In 5 years at 6 percent?
b. In 20 years at 10 percent?
c. In 40 years at 12 percent?
Solution:
Appendix C
FVA = A x FVIFA
a. $2,000 x 5.637 = $ 11,274
b. $2,000 x 57.275 = $ 114,550
c. $2,000 x 767.09 = $1,534,180
9-7.
You invest a single amount of $10,000 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 x FVIF
$10,000 x 1.611 = $16,110
Appendix A
FV = PV x FVIF
$16,110 x 5.350 = $86,188
9-8.
Jean Splicing will receive $8,500 a year for the next 15 years from her trust,If a 7 percent interest rate is applied,what is the current value of the future payments?
Solution:
Appendix D
PVA = A x PVIFA (7%,15 periods)
= $8,500 x 9.108 = $77,418
9-9.
Phil Goode will receive $175,000 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 x PVIF (14%,50 periods)
= $175,000 x,001 = $175
9-10.
Polly Graham will receive $12,000 a year for the next 15 years as a result of her patent,If a 9 percent rate is applied,should she be willing to sell out her future rights now for $100,000?
Solution:
Appendix D
PVA = A x PVIFA (9%,15 periods)
= $12,000 x 8.061 = $96,732
Yes,the present value of the annuity is worth less than $100,000.
9-11.
Carrie Tune will receive $19,500 for the next 20 years as a payment for a 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 x PVIFA (10%,20 periods)
PVA = $19,500 x 8.514 = $166,023
No,the present value of the annuity is worth more than $160,000.
9-12.
The Clearinghouse Sweepstakes has just informed you that you have won $1 million,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?
Solution:
Appendix D
PVA = A x PVIFA (10%,50 periods)
PVA = $20,000 x 9.915 = $198,300
9-13.
Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball 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?
Solution:
Appendix A
FV = PV x FVIF (12%,10 periods)
= $25,000 x 3.106 = $77,650
9-14.
Dr,Ruth has been secretly depositing $2,500 in her savings account every December starting in 1995,Her account earns 5 percent compounded annually,How much will she have in December of 2004? (Assume that a deposit is made in 2004.) Make sure to carefully count the years.
Solution:
Appendix C
FVA = A x FVIFA (5%,10 periods)
FVA = $2,500 x 12.578 = $31,445
9-15.
At a growth (interest) rate of 9 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 interest factor must equal 2,
* In Appendix A,looking down the 9% column,we 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.
9-16.
If you owe $40,000 payable at the end of seven years,what amount should your creditor accept in payment immediately if she could earn 12 percent on her money?
Solution:
Appendix B
PV = FV x PVIF (12%,7 periods)
PV = $40,000 x,452 = $18,080
9-17.
Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first 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 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.)
Solution:
Appendix B
PV = FVIF
Discount rate = 11%
$ 2.00 x,901 = $ 1.80
2.20 x,802 = 1.79
2.40 x,731 = 1.75
33.00 x,731 = 24.12
$29.46
9-18.
Les Moore retired as president of Goodman Snack Foods Company but is currently on a consulting contract for $35,000 per year for the next 10 years.
a. If Mr,Moore's opportunity cost (potential return) is 10 percent,what is the present value of his consulting contract?
b. Assume Mr,Moore 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:
Appendix D
a. PVA = A x PVIFA (10%,10 periods)
PVA = $35,000 x 6.145 = $215,075
b. Deferred annuity—Appendix D
PVA = A x PVIFA (i = 10%,10 periods)
PVA = $35,000 x 6.145 = $215,075
Now,discount back this value for 2 periods
PV = FV x PVIF (i = 10%,2 periods) Appendix B
$215,075 x,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.
9-19.
Juan Garza invested $20,000 10 years ago at 12 percent,compounded quarterly,How much has he accumulated?
Solution:
Appendix A
FV = PV x FVIF (3%,40 periods)
FV = $20,000 x 3.262 = $65,240
9-20.
Determine the amount of money in a savings account at the end of five years,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.
Solution:
Appendix A
FV = PV x FVIF
a. $5,000 x 1.762 = $8,810 (12%,5 periods)
b. $5,000 x 1.791 = $8,955 (6%,10 periods)
c. $5,000 x 1.806 = $9,030 (3%,20 periods)
9-21.
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,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,you would 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 x 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.
Solution:
Appendix C
FVA = A x FVIFA
n = 11,i = 12% 20.655 – 1 = 19.655
FVA = $4,000 x 19.655 = $78,620
9-22.
Your rich godfather has offered you a choice of one of the three following alternatives,$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?
Solution:
(first 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 x PVIFA
= $2,000 x PVIFA (11%,8 years)
= $2,000 x 5.146
= $10,292
(third alternative) Present value of $24,000 received in eight years,
Appendix B
PV = FV x PVIF
= $24,000 x PVIF (11%,8 years)
= $24,000 x,434
= $10,416
Select $24,000 to be received in eight years.
9-22,Continued
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 x PVIFA
= $2,000 x PVIFA (12%,8 years)
= $2,000 x 4.968
= $9,936
(third alternative) Present value of $24,000 received in 8 years,
Appendix B
PV = FV x PVIF
= $24,000 x PVIF (12%,8 years)
= $24,000 x,404
= $9,696
Select $10,000 now.
9-23.
You need $28,974 at the end of nine years,and your only investment outlet is an 8 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 10 years to achieve this same objective?
Solution:
a. Appendix B
PV = FV x PVIF (8%,10 periods)
= $28,974 x,463 = $13,415
b. Appendix C
A = FVA/FVIFA (8%,10 periods)
= $28,974/14.487 = $2,000
9-24.
Sue Sussman started a paper route on January 1,1998,Every three months,she deposits $500 in her bank account,which earns 4 percent annually but is compounded quarterly,On December 31,2001,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 31,2004?
Solution:
Appendix C
FVA = A x FVIFA (1%,16 periods)
FVA = $500 x 17.258 = $8,629 December 31,2001 amount
Appendix A
FV = PV x FVIF (9%,3 periods)
FV = $8,629 x 1.295
FV = $11,174.56 December 31,2004 amount
9-25.
On January 1,2002,Mike Irwin,Jr.,bought 100 shares of stock at $14 per share,On December 31,2004,he sold the stock for $21 per share,What is his annual rate of return? Interpolate to find the exact answer.
Solution:
Appendix B
PVIF = (3 periods)
PVIF =
Return is between 14%-15% for 3 periods
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%
9-26.
Dr,I,N,Stein has just invested $6,250 for his son (age one),The money will be used for his son's education 17 years from now,He calculates that he will need $50,000 for his son’s education by the time the boy goes to school,What rate of return will Dr,Stein need to achieve this goal?
Solution:
Appendix A
Appendix B
9-27.
Ester Seals has just given an insurance company $41,625,In return,she will receive an annuity of $5,000 for 15 years,At what rate of return must the insurance company invest this $41,625 to make the annual payments? Interpolate.
Solution:
Appendix D
PVIFA = PVA/A (15 periods)
= $41,625/$5,000
= 8.325 is between 8% and 9% for 15 periods
PVIFA at 8% 8.559
PVIFA at 9% –8.061
,498
PVIFA at 8% 8.559
PVIFA computed –8.325
,234
8% + (.234/.498) (1%)
8% +,470 (1%) = 8.47%
9-28.
Betty Bronson has just retired after 25 years with the electric company,Her total 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 return on her assets,What will be her yearly annuity for the next 15 years?
Solution:
Appendix D
A = PVA/PVIFA (9%,15 periods)
= $180,000/8.061
= $22,329.74
9-29.
Morgan Jennings,a geography professor,invests $50,000 in a parcel of 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 annuity,Assuming a 12 percent interest rate,how much will this annuity be?
Solution:
Appendix A
FV = PV x FVIF (12%,5 periods)
FV = $50,000 x 1.762 = $88,100
Appendix D
A = PVA/PVIFA (12%,10 periods)
A = $88,100/5.650 = $15,593
9-30.
You wish to retire after 18 years,at which time you want to have accumulated 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 annuity?
Solution:
Determine the present value of a 20-year annuity during retirement,
Appendix D
PVA = A x PVIFA (8%,20 years)
= $14,000 x 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 years)
=
9-31.
Del Monty will receive the following payments at the end of the next three years,$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 percent value of all three future benefits?
Solution:
First find the present value of the first three payments.
PV = FV x PVIF (Appendix B) i = 9%
$2,000 x,917 = $1,834
2) 3,500 x,842 = 2,947
3) 4,500 x,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 x PVIFA (9%,7 periods)
= $5,000 x 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 x PVIF (9%,3 periods)
= $25,165 x,772 = $19,427.38
Finally,find the total present value of all future payments.
Present value of first three payments $ 8,255.00
Present value of the deferred annuity 19,427.38
$27,682.38
9-32.
Bridget Jones has a contract in which she will receive the following payments for 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?
Solution:
First find the present value of the first five payments.
PV = FV x PVIF (Appendix B) i = 14%
1) $1,000 x,877 = $ 877
2) 2,000 x,769 = 1,538
3) 3,000 x,675 = 2,025
4) 4,000 x,592 = 2,368
5) 5,000 x,519 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 x PVIFA (14%,10 periods)
= $8,500 x 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 x PVIF (14%,5 periods)
= $44,336 x,519 = $23,010.38
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
Since the present value of all future benefits under the contract is greater than $30,000,Bridget Jones should not accept this amount to cancel the contract.
9-33.
Mark Ventura has just purchased an annuity to begin payment at the end of 2007 (that is the date of the first payment),Assume it is now the beginning of the year 2005,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?
Solution:
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 (2007) is:
PVA = A x PVIFA (13%,10 periods)
= $8,000 x 5.426
= $43,408
The present value at the beginning of 2005 is found using Appendix B (2 years at 13%),The factor is,783,Note we are discounting from the beginning of 2007 to the beginning of 2005.
PV = FV x PVIF (13%,2 periods)
= $43,408 x,783
= $33,988
The maximum that should be paid for the annuity is $33,988.
9-34.
If you borrow $15,618 and are required to pay back the loan in seven equal annual installments of $3,000,what is the interest rate associated with the loan?
Solution:
Appendix D
PVIFA = PVA/A (7 periods)
= $15,618/$3,000
= 5.206
Interest rate = 8 percent
9-35.
Cal Lury owes $10,000 now,A lender will carry the debt for five more years at 10 percent interest,That is,in this particular case,the amount owed will go up 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 payments be?
Solution:
Appendix A
FV = PV x FVIF (10%,5 periods)
FV = $10,000 x 1.611
= $16,110 Amount owed after 5 years
Appendix D
A = PVA/PVIFA (11%,12 periods)
= $16,110/6.492
= $2,482 Annual payments to retire the loan
9-36.
If your uncle borrows $60,000 from the bank at 10 percent interest over the 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? How much of his second payment will be applied to each?
Solution:
Appendix D
A = PVA/PVIFA (10%,7 periods)
= $60,000/4.868
= $12,325 annual payment
First payment,
$60,000 x,10 = $6,000 interest
$12,325 – $6,000 = $6,325 applied to principal
Second payment,
First determine remaining principal and then the interest and principal payment.
$60,000 – $6,325 = $53,675 remaining principal
$53,675 x,10 = $ 5,368 interest
$12,325 – $5,368 = $ 6,957 applied to principal
9-37.
Larry Davis borrows $80,000 at 14 percent interest toward the purchase of a home,His mortgage is for 25 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 will 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 14 percent mortgage and 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.
9-37,Continued
Solution:
Appendix D
a. A = PVA/PVIFA (14%,25 periods)
= $80,000/6.873
= $11,639.75
b. $ 11,639.75 annual payments
x 25 years
$290,993.75 total payment
– 80,000.00 repayment of principal
$210,993.75 total interest paid
Appendix D
c. New payments at 10%
A = PVA/PVIFA (10%,25 periods)
= $80,000/9.077
= $8,813.48
Difference between old and new payments
$11,639.75 old
8,813.48 new
$ 2,826.27 difference
P.V,of difference-Appendix D
PVA = A x PVIFA (assumes 10% discount rate,25 periods)
= $ 2,826.27 x 9.077
= $25,654.05 Amount that could be paid to refinance
9-38.
You are chairperson of the investment fund for the Eastern Football League,You are 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. Determine how much the semiannual payment should be,(Round to whole numbers.)
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.
b. Determine how much the revised semiannual payments should be after this rate 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.)
Solution:
Appendix C
a. A = FVA/FVIFA (4%,n = 20)
= $100,000/29.778
= $3,358 semi-annual payment
b. First determine how much the old payments are equal to after 4 periods at 4%,Appendix C.
FVA = A x FVIFA (4%,4 periods)
= $3,358 x 4.246
= $14,258
Then determine how much this value will grow to after 16 periods at 5%,Appendix A.
9-38,Continued
FV = PV x FVIF (5%,16 periods)
= $14,258 x 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
$ 68,875
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
9-39.
Your younger sister,Linda,will start college in five years,She has just informed 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 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? Use 10 percent as the appropriate interest rate throughout this problem (for discounting or compounding),
9-39,Continued
Solution:
Present value of college costs
Appendix D
PVA = A x PVIFA (10%,4 periods)
= $17,000 x 3.170
= $53,890
Accumulation based on investing $2,000 per year for 10 years.
Appendix C
FVA = A x FVIFA (10%,10 periods)
= $2,000 x 15.937
= $31,874
Additional funds required 5 years from now.
$53,890 PV of college costs
31,874 Accumulation based on $2,000 per year investment
$22,016 Additional funds required
Added contribution for the next 5 years
Appendix C
A = FVA/FVIFA (10%,5 periods)
= $22,016/6.105
= $3,606.22
9-40.
Linda (from problem 39) is now 18 years old (five years have passed),and she wants 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 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 Ph.D,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,
Solution:
Funds available after the wedding
$53,890
– 8,000 Wedding
$45,890 Funds available after the wedding
Less present value of vacation
Appendix D
PVA = A x PVIFA (10%,3 periods)
= $5,000 x 2.487
PV of vacation = $12,435
$45,890
–12,435
$33,455 Remaining funds for graduate school
Funds available 3 years later for graduate school
9-40,Continued
Appendix A
FV = PV x FVIF (10%,3 periods)
= $33,455 x 1.331
= $44,528.61 Funds available for graduate school
Number of years of graduate education
Appendix D
PVIFA =
=
with i = 10%,n = 4 for 3.170 the answer is 4 years.
Comprehensive Problem
CP 9-1.
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 following three offers,(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 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 rate on this annuity is 10 percent annually (5 percent semiannually),Determine the present value of the trust fund’s final value.
Required,Find the present value of each of the three offers and indicate which one has the highest present value.
CP 9-1,Continued
Solution:
Offer I
$1,000,000 now plus:
+ $200,000 from year 6 through 15 (deferred annuity)
Appendix D
PVA = A x PVIFA (10%,10 years)
= $200,000 x 6.145
= $1,229,000 (percent value at the beginning of
year 6,i.e,the end of year 5)
Appendix B
PV = FV x PVIF (10%,5 years)
= $1,229,000 x,621
= $763,209
+,70 x $3,000,000 = $2,100,000
Appendix B
PV = FV x PVIF (10%,15 years)
= $2,100,000 x,239 = $501,900
Total value of Offer I
$1,000,000 Payment today
763,209 Present value of deferred annuity
501,900 Present value of $3 million bonus
$2,265,109
CP 9-1,Continued
Offer II
Year
Sales
Gross Profit
(60% of Sales)
Payment 30%
of Gross Profit
1
$2,000,000
$1,200,000
$360,000
2
2,800,000
1,680,000
504,000
3
3,920,000
2,352,000
705,600
4
5,488,000
3,292,800
987,600
Year
Payment
Appendix B
PV Factor
PV
1
$360,000
.909
$327,240
2
504,000
.826
416,304
3
705,600
.751
529,906
4
987,600
.683
674,531
Total value of Offer II
$1,947,981
Offer III
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)
FVA = A x FVIFA
= $200,000 x 24.840
= $4,968,000 value of trust fund after 8 years
CP 9-1,Continued
Present value of trust fund (Appendix B)
PV = A x PVIF (10%,8 years)
= $4,968,000 x,467
= $2,320,056 Total value of offer III
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
