ANSWERS TO
END-OF-CHAPTER QUESTIONS
5-1.The concept of time value of money is a recognition that a dollar received today is worth more than a dollar received a year from now or at any future date. It exists because there are investment opportunities on money, that is, we can place our dollar received today in a savings account and one year from now have more than a dollar.
5-2.Compounding and discounting are inverse processes of each other. In compounding, money is moved forward in time, while in discounting money is moved back in time. This can be shown mathematically in the
compounding equation:
FVn=PV (1 + i)n
We can derive the discounting equation by multiplying each side of
this equation by and we get:
PV=FVn
5-3.We know that
FVn=PV(1 + i)n
Thus, an increase in i will increase FVn and a decrease innwill
decrease FVn.
5-4.Bank C which compounds continuously pays the highest interest. This occurs because, while all banks pay the same interest, 5 percent, bank C compounds the 5 percent continuously. Continuous compounding allows interest to be earned more frequently than any other compounding period.
5-5.The values in the present value of an annuity table (Table 5-8) are actually derived from the values in the present value table (Table 5-4). This can be seen by examining the value represented in each table. The present value table gives values of
for various values of i and n, while the present value of an annuity table gives values of
for various values of i and n. Thus the value in the present value of annuity for an n-year annuity for any discount rate i is merely the sum of the first n value in the present value table.
5-6.An annuity is a series of equal dollar payments for a specified number of years. Examples of annuities include mortgage payments, interest payments on bonds, fixed lease payments, and any fixed contractual payment. A perpetuity is an annuity that continues forever, that is, every year from now on this investment pays the same dollar amount. The difference between an annuity and a perpetuity is that a perpetuity has no termination date whereas an annuity does.
SOLUTIONS TO
END-OF-CHAPTER PROBLEMS
Solutions to Problem Set A
5-1A.(a)FVn=PV (1 + i)n
FV10=$5,000(1 + 0.10)10
FV10=$5,000 (2.594)
FV10=$12,970
(b)FVn=PV (1 + i)n
FV7=$8,000 (1 + 0.08)7
FV7=$8,000 (1.714)
FV7=$13,712
(c)FV12=PV (1 + i)n
FV12=$775 (1 + 0.12)12
FV12=$775 (3.896)
FV12=$3,019.40
(d)FVn=PV (1 + i)n
FV5=$21,000 (1 + 0.05)5
FV5=$21,000 (1.276)
FV5=$26,796.00
5-2A.(a)FVn=PV (1 + i)n
$1,039.50=$500 (1 + 0.05)n
2.079=FVIF 5%, n yr.
Thus n=15 years (because the value of 2.079 occurs in the 15 year row of the 5 percent column of Appendix B).
(b)FVn=PV (1 + i)n
$53.87=$35 (1 + .09)n
1.539=FVIF 9%, n yr.
Thus, n=5 years
(c)FVn=PV (1 + i)n
$298.60=$100 (1 + 0.2)n
2.986=FVIF 20%, n yr.
Thus, n=6 years
(d)FVn=PV (1 + i)n
$78.76=$53 (1 + 0.02)n
1.486=FVIF 2%, n yr.
Thus, n=20 years
5-3A.(a)FVn=PV (1 + i)n
$1,948=$500 (1 + i)12
3.896=FVIF i%, 12 yr.
Thus, i=12% (because the Appendix B value of 3.896 occurs in the 12 year row in the 12 percent column)
(b)FVn=PV (1 + i)n
$422.10=$300 (1 + i)7
1.407=FVIF i%, 7 yr.
Thus, i =5%
(c)FVn=PV (1 + i)n
$280.20=$50 ( 1 + i)20
5.604=FVIF i%, 20 yr.
Thus, I=9%
(d)FVn=PV ( 1 + i)n
$497.60=$200 (1 + i)5
=FVIF i%, 5 yr.
Thus, i =20%
5-4A.(a)PV=FVn
PV=$800
PV=$800 (0.386)
PV=$308.80
(b)PV=FVn
PV=$300
PV=$300 (0.784)
PV=$235.20
(c)PV=FVn
PV=$1,000
PV=$1,000 (0.789)
PV=$789
(d)PV=FVn
PV=$1,000
PV=$1,000 (0.233)
PV=$233
5-5A.(a)FVn=PMT
FV=$500
FV10=$500 (12.578)
FV10=$6,289
(b)FVn=PMT
FV5=$100
FV5=$100 (6.105)
FV5=610.50
(c)FVn=PMT
FV7=$35
FV7=$35 (8.654)
FV7=$302.89
(d)FVn=PMT
FV3=$25
FV3=$25 (3.060)
FV3=$76.50
5-6A.(a)PV=PMT
PV=$2,500
PV=$2,500 (7.024)
PV=$17,560
(b)PV=PMT
PV=$70
PV=$70 (2.829)
PV=$198.03
(c)PV=PMT
PV=$280
PV=$280 (5.582)
PV=$1,562.96
(d)PV=PMT
PV=$500
PV=$500 (6.145)
PV=$3,072.50
5-7A.(a)FVn=PV (1 + i)n
compounded for 1 year
FV1=$10,000 (1 + 0.06)1
FV1=$10,000 (1.06)
FV1=$10,600
compounded for 5 years
FV5= $10,000 (1 + 0.06)5
FV5=$10,000 (1.338)
FV5=$13,380
compounded for 15 years
FV15=$10,000 (1 + 0.06)15
FV15=$10,000 (2.397)
FV15=$23,970
(b)FVn=PV (1 + i)n
compounded for 1 year at 8%
FV1=$10,000 (1 + 0.08)1
FV1=$10,000 (1.080)
FV1=$10,800
compounded for 5 years at 8%
FV5=$10,000 (1 + 0.08)5
FV5=$10,000 (1.469)
FV5=$14,690
compounded for 15 years at 8%
FV15=$10,000 (1 + 0.08)15
FV15=$10,000 (3.172)
FV15=$31,720
compounded for 1 year at 10%
FV1=$10,000 (1 + 0.1)1
FV1=$10,000 (1 + 1.100)
FV1=$11,000
compounded for 5 years at 10%
FV5=$10,000 (1 + 0.1)5
FV5=$10,000 (1.611)
FV5=$16,110
compounded for 15 years at 10%
FV15=$10,000 (1 + 0.1)15
FV15=$10,000 (4.177)
FV15=$41,770
(c)There is a positive relationship between both the interest rate used to compound a present sum and the number of years for which the compounding continues and the future value of that sum.
5-8A.FVn=PV (1 + )mn
AccountPVimn(1 + )mnPV(1 + )mn
Theodore Logan III$ 1,00010%110 2.594$ 2,594
Vernell Coles 95,00012%121 1.127107,065
Thomas Elliott8,00012%62 1.26810,144
Wayne Robinson120,0008%42 1.172140,640
Eugene Chung30,00010%24 1.47744,310
Kelly Cravens15,00012%33 1.42321,345
5-9A.(a)FVn=PV (1 + i)n
FV5=$5,000 (1 + 0.06)5
FV5=$5,000 (1.338)
FV5=$6,690
(b)FVn=PV (1 + )mn
FV5=$5,000 (1 + )2X5
FV5=$5,000 (1 + 0.03)10
FV5=$5,000 (1.344)
FV5=$6,720
FVn=PV (1 + )mn
FV5=5,000 (1 + )6X5
FV5=$5,000 (1 + 0.01)30
FV5=$5,000 (1.348)
FV5=$6,740
(c) FVn=PV (1 + i)n
FV5=$5,000 (1 + 0.12)5
FV5=$5,000 (1.762)
FV5=$8,810
FV5=PV mn
FV5=$5,000 2X5
FV5=$5,000 (1 + 0.06)10
FV5=$5,000 (1.791)
FV5=$8,955
FV5=PV mn
FV5=$5,000 6X5
FV5=$5,000 (1 + 0.02)30
FV5=$5,000 (1.811)
FV5=$9,055
(d) FVn=PV (1 + i)n
FV12=$5,000 (1 + 0.06)12
FV12=5,000 (2.012)
FV12= $10,060
(e)An increase in the stated interest rate will increase the future value of a given sum. Likewise, an increase in the length of the holding period will increase the future value of a given sum.
5-10A.Annuity A:PV=PMT
PV=$8,500
PV=$8,500 (6.492)
PV=$55,182
Since the cost of this annuity is $50,000 and its present value is $55,182, given an 11 percent opportunity cost, this annuity has value and should be accepted.
Annuity B: PV=PMT
PV=$7,000
PV=$7,000 (8.422)
PV=$58,952
Since the cost of this annuity is $60,000 and its present value is only $59,094, given an 11 percent opportunity cost, this annuity should not be accepted.
Annuity C:PV=PMT
PV=$8,000
PV=$8,000 (7.963)
PV=$63,704
Since the cost of this annuity is $70,000 and its present value is only $63,704, given an 11 percent opportunity cost, this annuity should not be accepted.
5-11A.Year 1:FVn=PV (1 + i)n
FV1=15,000(1 + 0.2)1
FV1= 15,000(1.200)
FV1=18,000 books
Year 2:FVn=PV (1 + i)n
FV 2=15,000(1 + 0.2)2
FV 2=15,000(1.440)
FV2 =21,600 books
Year 3: FVn=PV (1 + i)n
FV3=15,000(1.20)3
FV3=15,000(1.728)
FV3=25,920 books
The sales trend graph is not linear because this is a compound growth trend. Just as compound interest occurs when interest paid on the investment during the first period is added to the principal of the second period, interest is earned on the new sum. Book sales growth was compounded; thus, the first year the growth was 20 percent of 15,000 books, the second year 20 percent of 18,000 books, and the third year 20 percent of 21,600 books.
5-12A.FVn=PV (1 + i)n
FV1=41(1 + 0.10)1
FV1= 41(1.10)
FV1=45.1 Home Runs in 1981 (in spite of the baseball strike).
FV2=41(1 + 0.10)2
FV2=41(1.21)
FV2=49.61 Home Runs in 1982
FV3=41(1 + 0.10)3
FV3=41(1.331)
FV3=54.571 Home Runs in 1983.
FV3=41(1 + 0.10)4
FV4=41(1.464)
FV4=60.024 Home Runs in 1984.
FV5=41(1 + 0.10)5
FV5=41(1.611)
FV5=66.051 Home Runs in 1985 (for a new major league record).
5-13A.PV=PMT
$60,000=PMT
$60,000=PMT (9.823)
Thus, PMT=$6,108.11 per year for 25 years.
5-14A.FVn=PMT
$15,000=PMT
$15,000=PMT (23.276)
Thus, PMT=$644.44
5-15A. FVn=PV (1 + i)n
$1,079.50=$500 (FVIF i%, 10 yr.)
2.159=FVIF i%, 10 yr.
Thus, i=8%
5-16A.The value of the home in 10 years
FV10=PV (1 + .05)10
=$100,000(1.629)
=$162,900
How much must be invested annually to accumulate $162,900?
$162,900=PMT
$162,900=PMT(15.937)
PMT=$10,221.50
5-17A.FVn=PMT
$10,000,000=PMT
$10,000,000=PMT(15.193)
Thus, PMT=$658,197.85
5-18A.One dollar at 12.0% compounded monthly for one year
FVn=PV nm
FV12=$1(1 + .01)12
=$1(1.127)
=$1.127
One dollar at 13.0% compounded annually for one year
FVn=PV (1 + i)n
FV1=$1(1 + .13)1
=$1(1.13)
=$1.13
The loan at 12% compounded monthly is more attractive.
5-19A.Investment A
PV=PMT
=$10,000
=$10,000(2.991)
=$29,910
Investment B
First, discount the annuity back to the beginning of year 5, which is the end of year 4, Then discount this equivalent sum to present.
PV=PMT
=$10,000
=$10,000(3.326)
=$33,260--then discount the equivalent sum back to present.
PV=FVn
=$33,260
=$33,260(.482)
=$16,031.32
Investment C
PV=FVn
=$10,000 + $50,000
+ $10,000
=$10,000(.833) + $50,000(.335) + $10,000(.162)
=$8,330 + $16,750 + $1,620
=$26,700
5-20A.PV=FVn
PV=$1,000
PV=$1,000(.513)
PV=$513
5-21A.(a)PV=
PV=
PV=$3,750
(b)PV=
PV=
PV=$8,333.33
(c)PV=
PV=
PV=$1,111.11
(d)PV=
PV=
PV=$1,900
5-22A.PV(annuity due)=PMT(PVIFAi,n)(l+i)
=$1,000(6.145)(1+.10)
=$6145(1.10)
=$6759.50
5-23A.FVn=PV (1 + )m . n
4=1(1 + )2 . n
4=(1 + 0.08)2 . n
4=FVIF 8%, 2n yr.
A value of 3.996 occurs in the 8 percent column and 18-year row of the table in Appendix B. Therefore, 2n = 18 years and n = approximately 9 years.
5-24A.Investment A:
PV=FVn (PVIFi,n)
PV=$2,000(PVIF10%, year 1) + $3,000(PVIF10%, year 2) + $4,000(PVIF10%, year 3) - $5,000(PVIF10%, year 4) + $5,000(PVIF10%, year 5)
=$2,000(.909) + $3,000(.826) + $4,000(.751) - $5,000(.683) + $5,000(.621)
=$1,818 + $2,478 + $3,004 - $3,415 + $3,105
=$6,990.
Investment B:
PV=FVn (PVIFi,n)
PV=$2,000(PVIF10%, year 1) + $2,000(PVIF10%, year 2) + $2,000(PVIF10%, year 3) + $2,000(PVIF10%, year 4) + $5,000(PVIF10%, year 5)
=$2,000(.909) + $2,000(.826) + $2,000(.751) + $2,000(.683) + $5,000(.621)
=$1,818 + $1,652 + $1,502 + $1,366 + $3,105
=$9,443.
Investment C:
PV=FVn (PVIFi,n)
PV=$5,000(PVIF10%, year 1) + $5,000(PVIF10%, year 2) - $5,000(PVIF10%, year 3) - $5,000(PVIF10%, year 4) + $15,000(PVIF10%, year 5)
=$5,000(.909) + $5,000(.826) - $5,000(.751) - $5,000(.683) + $15,000(.621)
=$4,545 + $4,130 - $3,755 - $3,415 + $9,315
=$10,820.
5-25A.The Present value of the $10,000 annuity over years 11-15.
PV=PMT
=$10,000(9.712 - 7.360)
=$10,000(2.352)
=$23,520
The present value of the $20,000 withdrawal at the end of year 15:
PV=FV15
=$20,000(.417)
=$8,340
Thus, you would have to deposit $23,520 + $8,340 or $31,860 today.
5-26A.PV=PMT
$40,000=PMT (6.145)
PMT=$6,509
5-27A.PV=PMT
$30,000=$10,000 (PVIFAi%, 5 yr.)
3.0=PVIFAi%, 5 yr.
i =20%
5-28A.PV=FVn
$10,000=$27,027 (PVIFi%, 5 yr.)
.370=PVIF22%, 5 yr.
Thus, i =22%
5-29A.PV=PMT
$25,000=PMT
$25,000=PMT (3.605)
PMT=$6,934.81
5-30A.The present value of $10,000 in 12 years at 11 percent is:
PV=FVn ()
PV=$10,000 ()
PV=$10,000 (.286)
PV=$2,860
The present value of $25,000 in 25 years at 11 percent is:
PV=$25,000 ()
=$25,000 (.074)
=$1,850
Thus take the $10,000 in 12 years.
5-31A. FVn=PMT
$20,000=PMT
$20,000=PMT(6.353)
PMT=$3,148.12
5-32A.
(a)FV=
$50,000=
$50,000=PMT (FVIFA7%, 15 yr.)
$50,000=PMT(25.129)
A=$1,989.73. per year
(b)PV=FVn
PV=$50,000 (PVIF7%, 15 yr.)
PV=$50,000(.362)
PV=$18,100 deposited today
(c)The contribution of the $10,000 deposit toward the $50,000 goal is
FVn=PV(1 + i)n
FVn=$10,000 (FVIF7%, 10 yr.)
FV10=$10,000(1.967)
=$19,670
Thus only $30,330 need be accumulated by annual deposit.
FV=PMT
$30,330=PMT (FVIFA7%, 15 yr.)
$30,330=PMT [25.129]
PMT=$1,206.97 per year
5-33A.(a)This problem can be subdivided into (1) the compound value of the $100,000 in the savings account (2) the compound value of the $300,000 in stocks, (3) the additional savings due to depositing $10,000 per year in the savings account for 10 years, and (4) the additional saving due to depositing $10,000 per year in the savings account at the end of years 6-10. (Note the $20,000 deposited in years 6-10 is covered in parts (3) and (4).)
(1)Future value of $100,000
FV10=$100,000 (1 + .07)10
FV10=$100,000 (1.967)
FV10=$196,700
(2)Future value of $300,000
FV10=$300,000 (1 + .12)10
FV10=$300,000 (3.106)
FV10=$931,800
(3)Compound annuity of $10,000, 10 years
FV10=PMT
=$10,000
=$10,000 (13.816)
=$138,160
(4)Compound annuity of $10,000 (years 6 - 10)
FV5=$10,000
=$10,000 (5.751)
=$57,510
At the end of ten years you will have $196,700 + $931,800 + $138,160 + $57,510 = $1,324,170.
(b)PV=PMT
$1,324,170=PMT (8.514)
PMT=$155,528
5-34A. PV=PMT (PVIFAi%, n yr.)
$100,000=PMT (PVIFA15%, 20 yr.)
$100,000=PMT(6.259)
PMT=$15,977
5-35A.PV=PMT (PVIFAi%, n yr.)
$150,000=PMT (PVIFA10%, 30 yr.)
$150,000=PMT(9.427)
PMT=$15,912
5-36A.At 10%:
PV=$50,000 + $50,000 (PVIFA10%, 19 yr.)
PV=$50,000 + $50,000 (8.365)
PV=$50,000 + $418,250
PV=$468,250
At 20%:
PV=$50,000 + $50,000 (PVIFA20%, 19 yr.)
PV=$50,000 + $50,000 (4.843)
PV=$50,000 + $242,150
PV=$292,150
5-37A.FVn(annuity due)=PMT(FVIFAi,n)(l+i)
=$1000(FVIFA10%,10 years)(1+.10)
=$1000(15.937)(1.1)
=$17,530.70
FVn(annuity due)=PMT(FVIFAi,n)(l+i)
=$1,000(FVIFA15%,10 years)(1+.15)
=$1,000(20.304)(1.15)
=$23,349.60
5-38A.PVn(annuity due)=PMT(PVIFAi,n)(l+i)
=$1,000(PVIFA10%,10 years)(1+.10)
=$1,000(6.145)(1.10)
=$6,759.50
PVn(annuity due)=PMT(PVIFAi,n)(l+i)
=$1,000(PVIFA15%,10 years)(l+.15)
=$1,000(5.019)(1.15)
=$5,771.85
5-39A.PV=PMT(PVIFAi,n)(PVIFi,n)
=PMT(PVIFA10%,10 years)(PVIF10%,7 years)
=$1,000(6.145)(.513)
=$3,152.39
5-40A.FV=PV (FVIFi%, n yr.)
$6,500=.12(FVIFi%, 37 yr.)
solving using a financial calculator:
i=34.2575%
5-41A. (a)
There are a number of equivalent ways to discount these cash flows back to present, one of which is as follows (in equation form):
PV=$50,000(PVIFA10%, 19 yr. - PVIFA10%, 4 yr.)
+ $250,000(PVIF10%, 20 yr.)
+ $50,000(PVIF10%, 23 yr. + PVIF10%, 24 yr.)
+ $100,000 (PVIF10%, 25 yr.)
=$50,000 (8.365-3.170) + $250,000 (.149)
+ $50,000 (0.112 + .102) + $100,000 (.092)
=$259,750 + $37,250 + $10,700 + $9,200
=$316,900
(b)If you live longer than expected you could end up with no money later on in life.
5-42A.
rate (i)=8%
number of periods (n)=7
payment (PMT)=$0
present value (PV)=$900
type (0 = at end of period)=0
Future value=$1,542.44
Excel formula: =FV(rate,number of periods,payment,present value,type)
Notice that present value ($500) took on a negative value.
5-43A.
In 20 years you'd like to have $250,000 to buy a home, but you only have $30,000. At what rate must your $30,000 be compounded annually for it to grow to $250,000 in 20 years?
number of periods (n)=20
payment (PMT)=$0
present value (PV) =$30,000
future value (FV)=$250,000
type (0 = at end of period)=0
guess=
i =11.18%
Excel formula: =RATE(number of periods,payment,present value,future value,type,guess)
Notice that present value ($30,000) took on a negative value.
5-44A.
To buy a new house you take out a 25 year mortgage for $300,000. What will your monthly interest rate payments be if the interest rate on your mortgage is 8 percent?
Two things to keep in mind when you're working this problem: first, you'll have to convert the annual rate of 8 percent into a monthly rate by dividing it by 12, and second, you'll have to convert the number of periods into months by multiplying 25 times 12 for a total of 300 months.
Excel formula: =PMT(rate,number of periods,present value,future value,type)
rate (i) = 8%/12
number of periods (n)=300
present value (PV)=$300,000
future value (FV) =$0
type (0 = at end of period)=0
monthly mortgage payment=($2,315.45)
Notice that monthly payments take on a negative value because you pay them.
You can also use Excel to calculate the interest and principal portion of any loan amortization payment. You can do this using the following Excel functions:
Calculation:Formula:
Interest portion of payment=IPMT(rate,period,number of periods,present value,future value,type)
Principal portion of payment=PPMT(rate,period,number of periods,present value,future value,type)
where period refers to the number of an individual periodic payment.
Thus, if you would like to determine how much of the 48th monthly payment went toward interest and principle you would solve as follows:
Interest portion of payment 48: ($1,884.37)
The principle portion of payment 48: ($431.08)
Solution to Integrative Problem
1.Discounting is the inverse of compounding. We really only have one formula to move a single cash flow through time. In some instances we are interested in bringing that cash flow back to the present (finding its present value) when we already know the future value. In other cases we are merely solving for the future value where we know the present value.
2.The values in the present value of an annuity table (Table 5-8) are actually derived from the values in the present value table (Table 5-4). This can be seen by examining the value represented in each table. The present value table gives values of
for various values of i and n, while the present value of an annuity table gives values of
for various values of i and n. Thus the value in the present value of annuity for an n-year annuity for any discount rate i is merely the sum of the first n value in the present value table.
3.(1)FVn=PV (1 + i)n
FV11=$5,000(1 + 0.08)10
FV11=$5,000 (2.159)
FV11=$10,795
(2)FVn =PV (1 + i)n
$1,671=$400 (1 + 0.10)n
4.1775=FVIF 10%, n yr.
Thus n=15 years (because the value of 4.177 occurs in the 15 year row of the 10 percent column of Appendix B).
(3)FVn=PV (1 + i)n
$4,046=$1,000 (1 + i)10
4.046=FVIF i%, 10 yr.
Thus, i=15% (because the Appendix B value of 4.046 occurs in the 10 year row in the 15 percent column)
4.FVn=PVmn
=$1,0002•5
=$1,000(1+.05)10
=$1,629
5.An annuity due is an annuity in which the payments occur at the beginning of each period as opposed to occurring at the end of each period, which is when the payment occurs in an ordinary annuity.
6.PV=PMT(PVIFAi,n)
=$1,000(PVIFA10%,7 years)
=$1,000(4.868)
=$4,868
PV(annuity due)=PMT(PVIFAi,n)(l+i)
=$1000(4.868)(l+.10)
=$5,354.80
7.FV=PMT(FVIFAi,n)
=$1,000(9.487)
=$9,487
FVn(annuity due)=PMT(FVIFAi,n)(l+i)
=$1000(9.487)(l+.10)
=$10,435.70
8.PV=PMT(PVIFAi,n)
$100,000=PMT(PVIFA10%, 25 years)
$100,000=PMT(9.077)
$11,016.86=PMT
9.PV=
=
=$12,500
10.PV=PMT(PVIFAi,n)(PVIFi,n)
=$1,000(PVIFA10%,10 years)(PVIF10%, 9 years)
=$1,000(6.145)(.424)
=$2,605.48
11.PV=
=PVIF10%, 9 years)
=
=$4,240.00
12.APY=m - 1
=4 - 1
=[1 + .02]4 - 1
=1.0824 - 1
=.0824 or 8.24%
Solutions to Problem Set B
5-1B.
(a)FVn=PV (1 + i)n
FV11=$4,000(1 + 0.09)11
FV11=$4,000 (2.580)
FV11=$10,320
(b)FVn=PV (1 + i)n
FV10=$8,000 (1 + 0.08)10
FV10=$8,000 (2.159)
FV10=$17,272
(c)FVn=PV (1 + i)n
FV12=$800 (1 + 0.12)12
FV12=$800 (3.896)
FV12=$3,117
(d) FVn=PV (1 + i)n
FV6=$21,000 (1 + 0.05)6
FV6=$21,000 (1.340)
FV6 =$28,140
5-2B.
(a)FVn=PV (1 + i)n
$1,043.90=$550 (1 + 0.06)n
1.898=FVIF6%, n yr.
Thus n=11 years (because the value of 1.898 occurs in the 11 year row of the 6 percent column of Appendix B).
(b)FVn=PV (1 + i)n
$88.44=$40 (1 + .12)n
2.211 =FVIF12%, n yr.
Thus, n =7 years
(c)FVn =PV (1 + i)n
$614.79 =$110 (1 + 0.24)n
5.589 =FVIF24%, n yr.
Thus, n =8 years
(d)FVn =PV (1 + i)n
$78.30 =$60 (1 + 0.03)n
1.305 =FVIF3%, n yr.
Thus, n =9 years
5-3B.(a)FVn=PV (1 + i)n
$1,898.60=$550 (1 + i)13
3.452=FVIFi%, 13 yr.
Thus, i=10% (because the Appendix B value of 3.452 occurs in the 12 year row in the 10 percent column)
(b)FVn=PV (1 + i)n
$406.18=$275 (1 + i)8
1.477=FVIFi%, 8 yr.
Thus, i =5%
(c)FVn=PV (1 + i)n
$279.66 =$60 ( 1 + i)20
4.661 =FVIFi%, 20 yr.
Thus, i =8%
(d)FVn =PV ( 1 + i)n
$486.00 =$180 (1 + i)6
2.700 =FVIFi%, 6 yr.
Thus, i =18%
5-4B.
(a)PV =FVn
PV =$800
PV =$800 (0.386)
PV =$308.80
(b)PV =FVn
PV =$400
PV =$400 (0.705)
PV =$282.00
(c)PV =FVn
PV =$1,000
PV =$1,000 (0.677)
PV =$677
(d)PV =FVn
PV =$900
PV =$900 (0.194)
PV =$174.60
5-5B.
(a)FVn =PMT
FV =$500
FV10=$500 (13.181)
FV10 =$6,590.50
(b)FVn =PMT
FV5 =$150
FV5=$150 (6.228)
FV5 =$934.20
(c)FVn=PMT
FV7 =$35
FV7=$35 (10.260)
FV7 =$359.10
(d)FVn=PMT
FV3=$25
FV3=$25 (3.060)
FV3=$76.50
5-6B.
(a)PV=PMT
PV=$3,000
PV=$3,000 (6.710)
PV=$20,130
(b)PV=PMT
PV=$50
PV =$50 (2.829)
PV =$141.45
(c)PV =PMT
PV =$280
PV=$280 (5.971)
PV=$1,671.88
(d)PV =PMT
PV = $600
PV =$600 (6.145)
PV =$3,687.00
5-7B.
(a)FVn=PV (1 + i)n
compounded for 1 year
FV1=$20,000 (1 + 0.07)1
FV1=$20,000 (1.07)
FV1=$21,400
compounded for 5 years
FV5=$20,000 (1 + 0.07)5
FV5=$20,000 (1.403)
FV5=$28,060
compounded for 15 years
FV15=$20,000 (1 + 0.07)15
FV15=$20,000 (2.759)
FV15=$55,180
(b)FVn=PV (1 + i)n
compounded for 1 year at 9%
FV1=$20,000 (1 + 0.09)1
FV1=$20,000 (1.090)
FV1=$21,800
compounded for 5 years at 9%
FV5=$20,000 (1 + 0.09)5
FV5=$20,000 (1.539)
FV5=$30,780
compounded for 15 years at 9%
FV15 =$20,000 (1 + 0.09)15
FV15=$20,000 (3.642)
FV15=$72,840
compounded for 1 year at 11%
FV1=$20,000 (1 + 0.11)1
FV1=$20,000 (1.11)
FV1=$22,200
compounded for 5 years at 11%
FV5=$20,000 (1 + 0.11)5
FV5=$20,000 (1.685)
FV5= $33,700
compounded for 15 years at 11%
FV15=$20,000 (1 + 0.11)15
FV15=$20,000 (4.785)
FV15=$95,700
(c) There is a positive relationship between both the interest rate used to compound a present sum and the number of years for which the compounding continues and the future value of that sum.
5-8B.FVn =PV (1 + )mn
AccountPVimn(1 + )mnPV(1 + )mn
Korey Stringer2,00012%62 1.268$2,536
Eric Moss50,00012%121 1.12756,350
Ty Howard7,00018%62 1.4269,982
Rob Kelly130,00012%42 1.267164,710
Matt Christopher20,00014%24 1.71834,360
Juan Porter15,00015%33 1.55123,265
5-9B.
(a)FVn =PV (1 + i)n
FV5=$6,000 (1 + 0.06)5
FV5=$6,000 (1.338)
FV5=$8,028
(b)FVn=PV (1 + )mn
FV5=$6,000 (1 + )2 x 5
FV5=$6,000 (1 + 0.03)10
FV5=$6,000 (1.344)
FV5=$8,064
FVn=PV (1 + )mn
FV5=$6,000 (1 + )6X5
FV5=$6,000 (1 + 0.01)30
FV5=$6,000 (1.348)
FV5=$8,088
(c) FVn=PV (1 + i)n
FV5=$6,000 (1 + 0.12)5
FV5=$6,000 (1.762)
FV5=$10,572
FV5=PV mn
FV5=$6,000 2X5
FV5=$6,000 (1 + 0.06)10
FV5=6,000 (1.791)
FV5=$10,746
FV5=PV mn
FV5=$6,000 6X5
FV5=$6,000 (1 + 0.02)30
FV5=$6,000 (1.811)
FV5=$10,866
(d) FVn=PV (1 + i)n
FV12=$6,000 (1 + 0.06)12
FV12=$6,000 (2.012)
FV12=$12,072
(e)An increase in the stated interest rate will increase the future value of a given sum. Likewise, an increase in the length of the holding period will increase the future value of a given sum.
5-10B.Annuity A:PV=PMT
PV=$8,500
PV=$8,500 (6.194)
PV =$52,649
Since the cost of this annuity is $50,000 and its present value is $52,649, given a 12 percent opportunity cost, this annuity has value and should be accepted.
Annuity B: PV=PMT
PV =$7,000
PV=$7,000 (7.843)
PV =$54,901
Since the cost of this annuity is $60,000 and its present value is only $54,901 given a 12 percent opportunity cost, this annuity should not be accepted.
Annuity C:PV =PMT
PV =$8,000
PV =$8,000 (7.469)
PV =$59,752
Since the cost of this annuity is $70,000 and its present value is only $59,752, given a 12 percent opportunity cost, this annuity should not be accepted.
5-11B.Year 1:FVn=PV (1 + i)n
FV1=10,000(1 + 0.15)1
FV1=10,000(1.15)
FV1=11,500 books
Year 2:FVn=PV (1 + i)n
FV 2=10,000(1 + 0.15)2
FV 2=10,000(1.322)
FV2=13,220 books
Year 3: FVn=PV (1 + i)n
FV3=10,000(1 + 0.15)3
FV3= 10,000(1.521)
FV3=15,210 books
The sales trend graph is not linear because this is a compound growth trend. Just as compound interest occurs when interest paid on the investment during the first period is added to the principal of the second period, interest is earned on the new sum. Book sales growth was compounded; thus, the first year the growth was 15 percent of 10,000 books, the second year 15 percent of 11,500 books, and the third year 15 percent of 13,220 books.
5-12B.FVn =PV (1 + i)n
FV1=41(1 + 0.12)1
FV1=41(1.12)
FV1 =45.92 Home Runs in 1981 (in spite of the baseball strike).
FV2=41(1 + 0.12)2
FV2=41(1.254)
FV2=51.414 Home Runs in 1982
FV3=41(1 + 0.12)3
FV3=41(1.405)
FV3=57.605 Home Runs in 1983.
FV3=41(1 + 0.12)4
FV4=41(1.574)
FV4 =64.534 Home Runs in 1984 (for a new major league record).
FV5 =41(1 + 0.12)5
FV5=41(1.762)
FV5=72.242 Home Runs in 1985 (again for a new major league record).
Actually, Reggie never hit more than 41 home runs in a year. In 1982, he only hit 15, in1983 he hit 39, in 1984 he hit 14, in 1985 25 and 26 in 1986. He retired at the end of 1987 with 563 career home runs.
5-13B.PV=PMT
$120,000=PMT
$120,000=PMT(9.077)
Thus, PMT=$13,220.23 per year for 25 years
5-14B.FVn=PMT
$25,000 =PMT
$25,000=PMT(25.129)
Thus, PMT =$994.87
5-15B. FVn=PV (1 + i)n
$2,376.50=$700 (FVIFi%, 10 yr.)
3.395 =FVIFi%, 10 yr.
Thus, i =13%
5-16B.The value of the home in 10 years
FV10=PV (1 + .05)10
=$125,000(1.629)
=$203,625
How much must be invested annually to accumulate $203.625?
$203,625=PMT
$203,625=PMT(15.937)
PMT =$12,776.87
5-17B.FVn=PMT
$15,000,000 = PMT
$15,000,000 =PMT(15.937)
Thus, PMT =$941,206
5-18B.One dollar at 24.0% compounded monthly for one year
FVn= PV (1 + )nm
FV12=$1(1 + .02)12
=$1(1.268)
=$1.268
One dollar at 26.0% compounded annually for one year
FVn=PV (1 + i)n
FV1= $1(1 + .26)1
= $1(1.26)
= $1.26
The loan at 26% compounded annual is more attractive.
5-19B.Investment A
PV= PMT
= $15,000
= $15,000(2.991)
= $44,865
Investment B
First, discount the annuity back to the beginning of year 5, which is the end of year 4. Then discount this equivalent sum to present.
PV=PMT
=$15,000
=$15,000(3.326)
=$49,890--then discount the equivalent sum back to present.
PV=FVn
=$49,890
=$49,890(.482)
=$24,046.98
Investment C
PV=FVn
=$20,000 + $60,000
+ $20,000
= $20,000(.833) + $60,000(.335) + $20,000(.162)
=$16,660 + $20,100 + $3,240
=$40,000
5-20B.PV=FVn
PV= $1,000
= $1,000(.502)
= $502
5-21B. (a)PV=
PV=
PV =$4,444
(b)PV =
PV =
PV =$11,538
(c)PV =
PV=
PV =$1,500
(d)PV =
PV =
PV = $1,667
5-22B.PV(annuity due)=PMT(PVIFAi,n)(l + i)
=$1000(3.791)(1 + .10)
=$3791(1.1)
=$4,170.10
5-23B.FVn=PV (1 + )m . n
7=1(1 + )2 . n
7 =(1 + 0.05)2 . n
7=FVIF5%, 2n yr.
A value of 7.040 occurs in the 5 percent column and 40-year row of the table in Appendix B. Therefore, 2n = 40 years and n = approximately 20 years.
5-24A.Investment A:
PV =FVn (PVIFi,n)
PV =$5,000(PVIF10%, year 1) + $5,000(PVIF10%, year 2) + $5,000(PVIF10%, year 3) - $15,000(PVIF10%, year 4) + $15,000(PVIF10%, year 5)
=$5,000(.909) + $5,000(.826) + $5,000(.751) - $15,000(.683) + $15,000(.621)
=$4,545 + $4,130 + $3,755 - $10,245 + $9,315
=$11,500.
Investment B:
PV=FVn (PVIFi,n)
PV =$1,000(PVIF10%, year 1) + $3,000(PVIF10%, year 2) + $5,000(PVIF10%, year 3) + $10,000(PVIF10%, year 4) - $10,000(PVIF10%, year 5)
=$1,000(.909) + $3,000(.826) + $5,000(.751) + $10,000(.683) - $10,000(.621)
=$909 + $2,478 + $3,755 + $6,830 - $6,210
=$7,762.
Investment C:
PV =FVn (PVIFi,n)
PV =$10,000(PVIF10%, year 1) + $10,000(PVIF10%, year 2) + $10,000(PVIF10%, year 3) + $10,000(PVIF10%, year 4) - $40,000(PVIF10%, year 5)
=$10,000(.909) + $10,000(.826) + $10,000(.751) + $10,000(.683) - $40,000(.621)
=$9,090 + $8,260 + $7,510 + $6,830 - $24,840
=$6,850.
5-25B.
The Present value of the $10,000 annuity over years 11-15.
PV=PMT
=$10,000(9.108 - 7.024)
=$10,000(2.084)
=$20,840
The present value of the $15,000 withdrawal at the end of year 15:
PV=FV15
=$15,000(.362)
=$5,430
Thus, you would have to deposit $20,840 + $5,430 or $26,270 today.
5-26B. PV=PMT
$45,000=PMT(6.418)
PMT=$7,012
5-27B. PV =PMT
$45,000 =$9,000 (PVIFAi%, 5 yr.)
5.0=PVIFAi%, 5 yr.
i =0%
5-28B. PV =FVn
$15,000 =$37,313 (PVIFi%, 5 yr.)
.402=PVIF20%, 5 yr.
Thus, i =20%
5-29B. PV =PMT
$30,000=PMT
$30,000 =PMT(2.974)
PMT =$10,087
5-30B. The present value of $10,000 in 12 years at 11 percent is:
PV = FVn ()
PV =$10,000 ()
PV=$10,000 (.286)
PV=$2,860
The present value of $25,000 in 25 years at 11 percent is:
PV =$25,000 ()
=$25,000 (.074)
=$1,850
Thus take the $10,000 in 12 years.
5-31B. FVn =PMT
$30,000 =PMT
$30,000 =PMT(6.105)
PMT =$4,914
5-32B.(a)FV =
$75,000=
$75,000 =PMT (FVIFA8%, 15 yr.)
$75,000 =PMT(27.152)
PMT =$2,762.23. per year
(b)PV =FVn
PV =$75,000 (PVIF8%, 15 yr.)
PV =$75,000(.315)
PV =$23,625 deposited today
(c)The contribution of the $20,000 deposit toward the $75,000 goal is
FVn=PV (1 + i)n
FVn =$20,000 (FVIF8%, 10 yr.)
FV10=$20,000(2.159)
= $43,180
Thus only $31,820 need be accumulated by annual deposit.
FV =PMT
$31,820=PMT (FVIFA8%, 15 yr.)
$31,820 =PMT [27.152]
PMT =$1,171.92 per year
5-33B.(a)This problem can be subdivided into (1) the compound value of the $150,000 in the savings account, (2) the compound value of the $250,000 in stocks, (3) the additional savings due to depositing $8,000 per year in the savings account for 10 years, and (4) the additional saving due to depositing $2,000 per year in the savings account at the end of years 6-10. (Note the $10,000 deposited in years 6-10 is covered in parts (3) and (4).)
(1)Future value of $150,000
FV10=$150,000 (1 + .08)10
FV10=$150,000 (2.159)
FV10=$323,850
(2)Future value of $250,000
FV10=$250,000 (1 + .12)10
FV10=$250,000 (3.106)
FV10=$776,500
(3)Compound annuity of $8,000, 10 years
FV10=PMT
=$8,000
= $8,000 (14.487)
=$115,896
(4)Compound annuity of $2,000 (years 6-10)
FV5=$2,000
= $2,000 (5.867)
=$11,734
At the end of ten years you will have $323,850 + $776,500 + $115,896
+ $11,734 = $1,227,980.
(b)PV=PMT
$1,227,980= PMT (7.963)
PMT=$154,210.72
5-34B. PV=PMT (PVIFAi%, n yr.)
$200,000 =PMT (PVIFA10%, 20 yr.)
$200,000 =PMT(8.514)
PMT =$23,491
5-35B.PV=PMT (PVIFAi%, n yr.)
$250,000 =PMT (PVIFA9%, 30 yr.)
$250,000=PMT(10.274)
PMT=$24,333
5-36B.At 10%:
PV=$40,000 + $40,000 (PVIFA10%, 24 yr.)
PV =$40,000 + $40,000 (8.985)
PV=$40,000 + $359,400
PV=$399,400
At 20%:
PV=$40,000 + $40,000 (PVIFA20%, 24 yr.)
PV=$40,000 + $40,000 (4.938)
PV=$40,000 + $197,520
PV=$237,520
5-37BFVn(annuity due)=PMT(FVIFAi,n)(l + i)
=$1000(FVIFA5%, 5 years)(l + .05)
=$1000(5.526)(1.05)
=$5802.30
FVn(annuity due)=PMT(FVIFAi,n)(l + i)
=$1,000(FVIFA8%, 5 years)(1 + .08)
=$1,000(5.867)(1.08)
=$6,336.36
5-38B.PVn(annuity due)=PMT(PVIFAi,n)(l + i)
=$1000 (PVIFA12%, 15 years)(1 + .12)
=$1000(6.811)(1.12)
=$7,628.32
PVn(annuity due)=PMT(PVIFAi,n)(l + i)
=$1000(PVIFA15%, 15 years)(l + .15)
=$1000(5.847)(1.15)
=$6,724.05
5-39B.PV=PMT(PVIFAi,n)(PVIFi,n)
=$1000(PVIFA15%,10 years)(PVIF15%, 7 years)
=$1000(5.019)(.376)
=$1,887.14
5-40B.FV = PMT (FVIFi%, n yr.)
$3,500=.12(FVIFi%, 38 yr.)
solving using a financial calculator:
i =31.0681%
5-41B.(a)
There are a number of equivalent ways to discount these cash flows back to present, one of which is as follows (in equation form):
PV =$60,000 (PVIFA10%, 19 yr. - PVIFA10%, 4 yr.)
+ $300,000 (PVIF10%, 20 yr.)
+ $60,000 (PVIF10%, 23 yr. + PVIF10%, 24 yr.)
+ $100,000 (PVIF10%, 25 yr.)
=$60,000 (8.365-3.170) + $300,000 (.149)
+ $60,000 (0.112 + .102) + $100,000 (.092)
=$311,700 + $44,700 + $12,840 + $9,200
=$378,440
(b) If you live longer than expected you could end up with no money later on in life.
1