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8/2/2019 Financial Management Chapter 05 IM 10th Ed
http://slidepdf.com/reader/full/financial-management-chapter-05-im-10th-ed 1/54
Prof. Rushen Chahal
CHAPTER 5
The Time Value of Money
CHAPTER ORIENTATION
In this chapter the concept of a time value of money is introduced, that is, a dollar today isworth more than a dollar received a year from now. Thus if we are to logically compare projects and financial strategies, we must either move all dollar flows back to the present or out to some common future date.
CHAPTER OUTLINE
I. Compound interest results when the interest paid on the investment during the first period is added to the principal and during the second period the interest is earned onthe original principal plus the interest earned during the first period.
A. Mathematically, the future value of an investment if compounded annually ata rate of i for n years will be
FVn = PV (l + i)n
where n = the number of years during which the compounding
occursi = the annual interest (or discount) rate
PV = the present value or original amount invested at the beginning of the first period
FVn = the future value of the investment at the end of n
years
1. The future value of an investment can be increased by either increasing the number of years we let it compound or by compoundingit at a higher rate.
2. If the compounded period is less than one year, the future value of an
investment can be determined as follows:
FVn = PVmn
where m= the number of times compounding occurs during the year
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II. Determining the present value, that is, the value in today's dollars of a sum of moneyto be received in the future, involves nothing other than inverse compounding. Thedifferences in these techniques come about merely from the investor's point of view.
A. Mathematically, the present value of a sum of money to be received in thefuture can be determined with the following equation:
PV = FVn
where: n = the number of years until payment will be received,i = the interest rate or discount ratePV = the present value of the future sum of moneyFVn = the future value of the investment at the end of n
years
1. The present value of a future sum of money is inversely related to boththe number of years until the payment will be received and the interestrate.
III. An annuity is a series of equal dollar payments for a specified number of years.Because annuities occur frequently in finance, for example, bond interest payments,we treat them specially.
A. A compound annuity involves depositing or investing an equal sum of moneyat the end of each year for a certain number of years and allowing it to grow.
1. This can be done by using our compounding equation, andcompounding each one of the individual deposits to the future or byusing the following compound annuity equation:
FVn = PMT
+
∑
−
=
1n
0t
ti)(1
where: PMT = the annuity value deposited at the end of eachyear
i = the annual interest (or discount) raten = the number of years for which the annuity will
lastFVn = the future value of the annuity at the end of the
nth year
B. Pension funds, insurance obligations, and interest received from bonds allinvolve annuities. To compare these financial instruments we would like to
know the present value of each of these annuities.
1. This can be done by using our present value equation and discountingeach one of the individual cash flows back to the present or by usingthe following present value of an annuity equation:
PV = PMT
+∑
=
n
1tti)(1
1
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where: PMT = the annuity deposited or withdrawn at the endof each year
i = the annual interest or discount ratePV = the present value of the future annuityn = the number of years for which the annuity will
lastC. This procedure of solving for PMT, the annuity value when i, n, and PV are
known, is also the procedure used to determine what payments are associatedwith paying off a loan in equal installments. Loans paid off in this way, in periodic payments, are called amortized loans. Here again we know three of the four values in the annuity equation and are solving for a value of PMT, theannual annuity.
IV. Annuities due are really just ordinary annuities where all the annuity payments have been shifted forward by one year. Compounding them and determining their presentvalue is actually quite simple. Because an annuity, due merely shifts the paymentsfrom the end of the year to the beginning of the year, we now compound the cash
flows for one additional year. Therefore, the compound sum of an annuity due is
FVn(annuity due) = PMT (FVIFAi,n) (1 + i)
A. Likewise, with the present value of an annuity due, we simply receive eachcash flow one year earlier – that is, we receive it at the beginning of each year rather than at the end of each year. Thus the present value of an annuity dueis
PV(annuity due) = PMT (PVIFAi,n) (1 + i)
V. A perpetuity is an annuity that continues forever, that is every year from now on thisinvestment pays the same dollar amount.
A. An example of a perpetuity is preferred stock which yields a constant dollar dividend infinitely.
B. The following equation can be used to determine the present value of a perpetuity:PV =where: PV = the present value of the perpetuity
pp = the constant dollar amount provided by the perpetuityi = the annual interest or discount rate
VI. To aid in the calculations of present and future values, tables are provided at the back of Financial Management (FM).
A. To aid in determining the value of FVn in the compounding formula
FVn = PV (1 + i)n = PV (FVIFi,n)
tables have been compiled for values of FVIFi,n or (i + 1)n in Appendix B,
"Compound Sum of $1," in FM.
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B. To aid in the computation of present values
PV = FVn = FVn (PVIFi,n)
tables have been compiled for values of
or PVIFi,n
and appear in Appendix C in the back of FM.
C. Because of the time-consuming nature of compounding an annuity,
FVn = PMT ∑−
=+
1n
0t
ti)(1 = PMT (FVIFAi,n)
Tables are provided in Appendix D of FM for
∑−
=
+1n
0t
ti)(1 or FVIFAi,n
for various combinations of n and i.
D. To simplify the process of determining the present value of an annuity
PV = PMT
+∑=
n
1tti)(1
1 = PMT (PVIFAi,n)
tables are provided in Appendix E of FM for various combinations of n and ifor the value
∑= +
n
1t ti)(1
1 or PVIFA
i,n
V. Spreadsheets and the Time Value of Money.
A. While there are several competing spreadsheets, the most popular one isMicrosoft Excel. Just as with the keystroke calculations on a financialcalculator, a spreadsheet can make easy work of most common financialcalculations. Listed below are some of the most common functions used withExcel when moving money through time:
Calculation: Formula:
Present Value = PV(rate, number of periods, payment, future value, type)
Future Value = FV(rate, number of periods, payment, present value, type)Payment = PMT(rate, number of periods, present value, future value,type)
Number of Periods = NPER(rate, payment, present value, future value, type)Interest Rate = RATE(number of periods, payment, present value, future
value, type, guess)
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where: rate = i, the interest rate or discount ratenumber of periods = n, the number of years or periods payment = PMT, the annuity payment deposited or received at the
end of each periodfuture value = FV, the future value of the investment at the end of n
periods or years present value = PV, the present value of the future sum of moneytype = when the payment is made, (0 if omitted)
0 = at end of period1 = at beginning of period
guess = a starting point when calculating the interest rate, if omitted, the calculations begin with a value of 0.1 or 10%
ANSWERS TO
END-OF-CHAPTER QUESTIONS
5-1. The concept of time value of money is recognition that a dollar received today isworth 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 in n will
decrease FVn.
5-4. Bank C which compounds daily pays the highest interest. This occurs because, whileall banks pay the same interest, 5 percent, bank C compounds the 5 percent daily.Daily compounding allows interest to be earned more frequently than the other compounding periods.
5-5. The values in the present value of an annuity table (Table 5-8) are actually derived
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from the values in the present value table (Table 5-4). This can be seen, byexamining the values represented in each table. The present value table gives valuesof
for various values of i and n, while the present value of an annuity table gives values
of
∑= +
n
1tti)(1
1
for various values of i and n. Thus the value in the present value of annuity table for an n-year annuity for any discount rate i is merely the sum of the first n values in the
present value table. PVIFA 10%,10yrs = 6.145. ∑=
10
1n
PVIF10%,n = 6.144 = 0.909 +
0.826 + 0.751 + 0.683 + 0.621 + 0.564 + 0.513 + 0.467 + 0.424 + 0.386
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, fixedlease payments, and any fixed contractual payment. A perpetuity is an annuity thatcontinues forever, that is, every year from now on this investment pays the samedollar amount. The difference between an annuity and a perpetuity is that a perpetuity has no termination date whereas an annuity does.
SOLUTIONS TOEND-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
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(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
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the 12 year row in the 12 percent column)
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(b) FVn = PV (1 + i)n
$422.10 = $300 (1 + i)7
1.407 = FVIFi%, 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
= FVIFi%, 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
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(d) PV = FVn
PV = $1,000
PV = $1,000 (0.233)
PV = $233
5-5A. (a) FVn = PMT
+∑
−
=
1n
0t
ti)(1
FV10 = $500
+∑
−
=
110
0t
t0.05)(1
FV10 = $500 (12.578)
FV10 = $6,289
(b) FVn = PMT
+∑−
=
t1n
0t
i)(1
FV5 = $100
+∑
−
=
t15
0t
0.1)(1
FV5 = $100 (6.105)
FV5 = 610.50
(c) FVn = PMT
+
∑
−
=
t1n
0t
i)(1
FV7 = $35
+∑
−
=
t17
0t
0.07)(1
FV7 = $35 (8.654)
FV7 = $302.89
(d) FVn = PMT
+∑
−
=
t1n
0t
i)(1
FV3 = $25
+∑−=
t 0.02)(1 13
0t
FV3 = $25 (3.060)
FV3 = $76.50
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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)15FV15 = $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
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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 compoundingcontinues and the future value of that sum.
5-8A. FVn = PV (1 + )mn
Account PV i m n (1 + )mn PV(1 + )mn
Theodore Logan III $ 1,000 10% 1 10 2.594 $ 2,594Vernell Coles 95,000 12% 12 1 1.127 107,065Thomas Elliott 8,000 12% 6 2 1.268 10,144Wayne Robinson 120,000 8% 4 2 1.172 140,640Eugene Chung 30,000 10% 2 4 1.477 44,310Kelly Cravens 15,000 12% 3 3 1.423 21,345
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
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FV5 = PVmn
FV5 = $5,0002X5
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 givensum. Likewise, an increase in the length of the holding period will increasethe future value of a given sum.
5-10A. Annuity A: PV = PMT
+∑= t
n
1t i)(1
1
PV = $8,500
+∑=
t
12
1t 0.11)(1
1
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 beaccepted.
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Annuity B: PV = PMT
+∑=
t
n
1t i)(1
1
PV = $7,000
+
∑=
t
25
1t 0.11)(1
1
PV = $7,000 (8.442)
PV = $59,094
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
+∑=
t
n
1t i)(1
1
PV = $8,000
+∑= t
20
1t 0.11)(1
1
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
FV2 = 15,000(1 + 0.2)2
FV2 = 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
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Book sales
25,000 20,000
15,000
1
2
3
years
The sales trend graph is not linear because this is acompound growth trend. Just as compound interest occurswhen interest paid on the investment during the first periodis added to the principal of the second period, interest isearned on the new sum. Book sales growth wascompounded; thus, the first year the growth was 20 percentof 15,000 books for a total of 18,000 books, the second year20 percent of 18,000 books for a total of 21,600, and thethird year 20 percent of 21,600 books for a total of 25,920.
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.
FV4 = 41(1 + 0.10)4
FV4 = 41(1.464)
FV4 = 60.024 Home Runs in 1984.
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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
+∑= t
n
1t i)(1
1
$60,000 = PMT
+∑=
t
25
1t 0.09)(1
1
$60,000 = PMT (9.823)
Thus, PMT = $6,108.11 per year for 25 years.
5-14A. FVn = PMT
+∑
−
=
t1n
0t
i)(1
$15,000 = PMT
+∑
−
=
t115
0t
0.06)(1
$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
+∑
−
=
t110
0t
.10)(1
$162,900 = PMT(15.937)
PMT = $10,221.50
5-17A. FVn = PMT
+∑
−
=
t1n
0t
i)(1
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$10,000,000 = PMT
+∑
−
=
t110
0t
.09)(1
$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
FV1 = $1(1 + .01)1
= $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
+∑=
t
n
1t i)(1
i
= $10,000
+∑= t
5
1t .20)(11
= $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
+∑=
t
n
1ti)(1
1
= $10,000
+∑=
t
6
1t .20)(1
1
= $10,000(3.326)
= $33,260--then discount the equivalent sum back to present.PV = FVn
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= $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
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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 inAppendix 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.
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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
+−
+ ∑∑ ==t
10
1tt
15
1t .06)(1
1
.06)(1
1
= $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)
= $8,340Thus, you would have to deposit $23,520 + $8,340 or $31,860 today.
5-26A. PV = PMT
+∑=
t
10
1t .10)(1
1
$40,000 = PMT (6.145)
PMT = $6,509
5-27A. PV = PMT
+∑=
t
5
1t i)(1
1
$30,000 = $10,000 (PVIFAi%, 5 yr.)
3.0 = PVIFAi%, 5 yr.
i = 20%
5-28A. PV = FVn
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$10,000 = $27,027 (PVIFi%, 5 yr.)
.370 = PVIF22%, 5 yr.
Thus, i = 22%
5-29A. PV = PMT
+∑= ti)(1
1n1t
$25,000 = PMT
+∑= t.12)(1
15
1t
$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
+ ni)(1
1
PV = $10,000
+ 12.11)(1
1
PV = $10,000 (.286)
PV = $2,860
The present value of $25,000 in 25 years at 11 percent is:
PV = $25,000
+ 25.11) (1
1
= $25,000 (.074)
= $1,850
Thus take the $10,000 in 12 years.
5-31A. FVn = PMT
+∑
−
=
t1n
0t
i)(1
$20,000 = PMT
+∑
−
=
t15
0t
.12)(1
$20,000 = PMT(6.353)
PMT = $3,148.12
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5-32A. (a) FVn = PMT
+∑
−
=
t1n
0t
i)(1
$50,000 = PMT
+
∑
−
=
t115
0t
.07)(1
$50,000 = PMT (FVIFA7%, 15 yr.)
$50,000 = PMT(25.129)
PMT = $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.
FVn = PMT
+∑−
=
t1n
0t
i)(1
$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 $100,000in 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 savingsaccount for 10 years, and (4) the additional savings due to depositing $10,000 per year in the savings account at the end of years 6-10. (Note the $20,000deposited 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
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(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
+∑
−
=
t1n
0t
i)(1
= $10,000
+∑
−
=
t110
0t
.07)(1
= $10,000 (13.816)
= $138,160
(4) Compound annuity of $10,000 (years 6 - 10)
FV5 = $10,000
+∑
−
=
t15
0t
.07)(1
= $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
+∑= t
20
1t .10)(11
$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
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5-36A. At 10%:
PV = $50,000 + $50,000 (PVIFA10%, 19 yr.)
PV = $50,000 + $50,000 (8.365)
PV = $50,000 + $418,250PV = $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. PV (annuity due) = PMT(PVIFAi,n)(l+i)= $1,000(PVIFA10%,10 years)(1+.10)
= $1,000(6.145)(1.10)
= $6,759.50
PV (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
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5-40A. FVn = PV (FVIFi%, n yr.)
$6,500 = .12(FVIFi%, 37 yr.)
solving using a financial calculator:
i = 34.2575%
5-41A. (a)
$50,000 per year $250,000 $50,000
$100,000
1/04 1/09 1/14 1/19 1/24 1/29
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.
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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 ($900) 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,futurevalue,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 toconvert the annual rate of 8 percent into a monthly rate by dividing it by 12, andsecond, you'll have to convert the number of periods into months by multiplying 25times 12 for a total of 300 months.
Excel formula: =PMT(rate,number of periods,present value,future value,type)
rate (i) = 8%/12number of periods (n) = 300 present value (PV) = $300,000future value (FV) = $0type (0 = at end of period) = 0
monthly mortgage payment = ($2,315.45)
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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 loanamortization payment. You can do this using the following Excel functions:
Calculation: Formula:
Interest portion of payment =IPMT(rate,period,number of periods,presentvalue,future value,type)
Principal portion of payment =PPMT(rate,period,number of periods,presentvalue,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 wenttoward interest and principal you would solve as follows:
Interest portion of payment 48: ($1,884.37)
The principal portion of payment 48: ($431.08)
5-45A.a. N = 378
I/Y = 6
PV = -24
PMT = 0
CPT FV = 88.27 billion dollars
b. N = 10
I/Y = 10
CPT PV = -77.108 billion dollarsPMT = 0
FV = 200. billion
c. N = 10
CPT I/Y = 14.87%
PV = -50 billion
PMT = 0
FV = 200. billion
d. N = 40I/Y = 7
PV = -100. billion
CPT PMT = 7.5 billion dollars
FV = 0
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5-46A. What will the car cost in the future?
N = 6
I/Y = 3
PV = -15,000
PMT = 0
CPT FV = 17,910.78 dollars
How much must Bart put in an account today in order to have $17,910.78 in 6years?
N = 6
I/Y = 7.5
CPT PV = -11,605.50 dollars
PMT = 0
FV = 17,910.78
5-47A. N = 45
I/Y = 8.75
PV = 0
CPT PMT = -2,054.81 dollars
FV = 1,000,000
5-48A.First, we must calculate what Mr. Burns will need in 20 years, then we will know whathe needs in 20 years and we can then calculate how much he needs to deposit each
year in order to come up with that amount (note: once you calculate the presentvalue, you must multiply your answer, in this case -$4.192 billion times (1 + i) because this is an annuity due):
N = 10
I/Y = 20
CPT PV = -4.1925 billion × 1.20 = -5.031 billion dollars
PMT = 1 billion
FV = 0
Next, we will determine how much Mr. Burns needs to deposit each year for 20 years
to reach this goal of accumulating $5.031 billion at the end of the 20 years:
N = 20
I/Y = 20
PV = 0
CPT PMT = -26.9 million dollars
FV = 5.031 billion
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5-49A. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money5 years ago and we want to know what it will be worth in 20 years)?
N = 25
I/Y = 7.5
PV = -100,000PMT = 0
CPT FV = 609,833.96 dollars
Now we determine what the additional $1,500 per year will grow to (note that sinceHomer will be making these investments at the beginning of each year for 20 yearswe have an annuity due, thus, once you calculate the present value, you must multiplyyour answer, in this case $64,957.02 times (1 + i)):
N = 20
I/Y = 7.5
PV = 0
PMT = -1,500
CPT FV = 64,957.02 × 1.075 = 69,828.80 dollars
Finally, we must add the two values together:
$609,833.96 + $69,828.80 = $679,662.76
5-50A. Since this problem involves monthly payments wemust first, make P/Y = 12. Then, N becomes the number of monthsor compounding periods,
N = 60I/Y = 6.2
PV = -25,000
CPT PMT = 485.65 dollars
FV = 0
5-51A. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 36
CPT I/Y = 11.62%
PV = -999
PMT = 33
FV = 0
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5-52A. First, what will be the monthly payments if Suzie goes for the 4.9 percent financing?Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 60
I/Y = 4.9PV = -25,000
CPT PMT = 470.64 dollars
FV = 0
Now, calculate how much the monthly payments would be if Suzie took the $1,000cash back and reduced the amount owed from $25,000 to $24,000. Again, since this problem involves monthly payments we must first, make P/Y = 12.
N = 60
I/Y = 6.9
PV = -24,000
CPT PMT = 474.10 dollars
FV = 0
5-53A. Since this problem involves quarterly compounding we must first, make P/Y = 4.Then, N becomes the number of quarters or compounding periods,
N = 16
I/Y = 6.4%
PV = 0
PMT = -1000
CPT FV = 18,071.11 dollars
5-54A. There are several ways you could solve this problem. One way would be to calculatethe future value of an amount, say $100, deposited in each of these CDs would growto at the end of a year. Let’s try this first. Since the first part of this problem involvesdaily compounding we must first, make P/Y = 365. Then, N becomes the number of days in a year,
N = 365
I/Y = 4.95
PV = -100
PMT = 0
CPT FV = 105.0742 or 5.0742%
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Now, let’s look at monthly compounding we must first, and again, we’ll see what$100 will grow to at the end of a year. First, we make P/Y = 12.
N = 12
I/Y = 5.0
PV = -100
PMT = 0
CPT FV = 105.1162 or 5.1162%
An alternative approach would be to use the ICONV button on a Texas InstrumentsBA II-Plus calculator. That button calculates the APY, or annual percentage yield,also called the effective rate.
5-55A. Since this problem involves monthly payments we must first,make P/Y = 12. Then, N becomes the number of months orcompounding periods,
CPT N = 41.49 (rounded up to 42 months)
I/Y = 12.9
PV = -5000
PMT = 150
FV = 0
5-56A. a. Since this problem begins using annual payments, make sure your calculator is set to P/Y=1.
N = 12
CPT I/Y = 8.37%
PV = -160,000
PMT = 0
FV = 420,000
b. Again, since this problem begins using annual payments, make sure your calculator is set to P/Y=1
N = 10
CPT I/Y = 11.6123%
PV = -140,000PMT = 0
FV = 420,000
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c. Since this problem now involves monthly payments we must first, make P/Y= 12. Then, N becomes the number of months or compounding periods,
N = 120
I/Y = 6
PV = -140,000
CPT PMT = -1,008.57 dollars
FV = 420,000
d. Since this problem now involves monthly payments we must first, make P/Y= 12. Then, N becomes the number of months or compounding periods.Also, since Professor ME will be depositing both the $140,000 (immediately)and $500 (monthly), they must have the same sign,
N = 120
CPT I/Y = 8.48%
PV = -140,000
PMT = -500
FV = 420,000
SOLUTION TO INTEGRATIVE PROBLEM
1. Discounting is the inverse of compounding. We really only have one formula tomove 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 wealready know the future value. In other cases we are merely solving for the futurevalue when we know the present value.
2. The values in the present value of an annuity table (Table 5-8) are actually derivedfrom the values in the present value table (Table 5-4). This can be seen, byexamining the value, represented in each table. The present value table gives valuesof
for various values of i and n, while the present value of an annuity table gives values
of
ti)(1
1n
1t +∑=
for various values of i and n. Thus the value in the present value of annuity table for an n-year annuity for any discount rate i is merely the sum of the first n values in the present value table.
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3. (a) FVn = PV (1 + i)n
FV10 = $5,000(1 + 0.08)10
FV10 = $5,000 (2.159)
FV10 = $10,795
(b) 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).
(c) 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 10year 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 paymentoccurs 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
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7. FVn = 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
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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
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$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 inthe 13 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
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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
+∑
−
=
t1n
0t
i)(1
FV10 = $500
+∑−
=
t110
0t
0.06)(1
FV10 = $500 (13.181)
FV10 = $6,590.50
(b) FVn = PMT
+∑
−
=
t1n
0t
i)(1
FV5 = $150
+∑
−
=
t15
0t
0.11)(1
FV5 = $150 (6.228)
FV5 = $934.20
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(c) FVn = PMT
+∑
−
=
t1n
0t
i)(1
FV8 = $35
+∑−
=
t18
0t
0.07)(1
FV8 = $35 (10.260)
FV8 = $359.10
(d) FVn = PMT
+∑
−
=
t1n
0t
i)(1
FV3 = $25
+∑
−
=
t13
0t
0.02)(1
FV3 = $25 (3.060)
FV3 = $76.50
5-6B. (a) PV = PMT
+∑= t
n
1t i)(1
1
PV = $3,000
+∑= t
10
1t 0.08)(1
1
PV = $3,000 (6.710)
PV = $20,130
(b) PV = PMT
+
∑= t
n
1t i)(1
1
PV = $50
+∑= t
3
1t 0.03)(1
1
PV = $50 (2.829)
PV = $141.45
(c) PV = PMT
+∑= t
n
1t i)(1
1
PV = $280
+∑= t
8
1t 0.07)(1
1
PV = $280 (5.971)
PV = $1,671.88
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(d) PV = PMT
+∑= t
n
1t i)(1
1
PV = $600
+
∑=
t
10
1t 0.1)(1
1
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%
FV5 = $20,000 (1 + 0.09)15
FV5 = $20,000 (3.642)
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FV5 = $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%
FV5 = $20,000 (1 + 0.11)15
FV5 = $20,000 (4.785)
FV5 = $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 compoundingcontinues and the future value of that sum.
5-8B. FVn = PV (1 + )mn
Account PV i m n (1 + )mn PV(1 + )mn
Korey Stringer 2,000 12% 6 2 1.268 $2,536Erica Moss 50,000 12% 12 1 1.127 56,350
Ty Howard 7,000 18% 6 2 1.426 9,982Rob Kelly 130,000 12% 4 2 1.267 164,710Mary Christopher 20,000 14% 2 4 1.718 34,360Juan Diaz 15,000 15% 3 3 1.551 23,265
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
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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 = PVmn
FV5 = $6,0002X5
FV5 = $6,000 (1 + 0.06)10
FV5 = 6,000 (1.791)
FV5 = $10,746
FV5 = PV mn
FV5 = $6,0006X5
FV5 = $6,000 (1 + 0.02)30FV5 = $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 givensum. Likewise, an increase in the length of the holding period will increasethe future value of a given sum.
5-10B. Annuity A: PV = PMT
+∑= t
n
1t i)(1
1
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PV = $8,500
+∑= t
12
1t 0.12)(1
1
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
+∑= t
n
1t i)(1
1
PV = $7,000
+∑= t
25
1t 0.12)(1
1
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 a12 percent opportunity cost, this annuity should not be accepted.
Annuity C: PV = PMT
+∑= t
n
1t i)(1
1
PV = $8,000
+∑= t
20
1t 0.12)(1
1
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
FV2 = 10,000(1 + 0.15)2
FV2 = 10,000(1.322)
FV2 = 13,220 books
Year 3: FVn = PV (1 + i)n
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FV3 = 10,000(1 + 0.15)3
FV3 = 10,000(1.521)
FV3 = 15,210 books
Book sales
20,000
15,000
10,000
1 2 3years
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 isearned on the new sum. Book sales growth was compounded; thus, the firstyear 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)
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FV3 = 57.605 Home Runs in 1983.
FV4 = 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 leaguerecord).
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
+∑= t
n
1t i)(1
1
$120,000 = PMT
+∑= t
25
1t 0.1)(1
1
$120,000 = PMT(9.077)
Thus, PMT = $13,220.23 per year for 25 years
5-14B. FVn = PMT
+∑
−
=
t1n
0t
i)(1
$25,000 = PMT
+∑
−
=
t115
0t
0.07)(1
$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
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How much must be invested annually to accumulate $203,625?
$203,625 = PMT
+∑
−
=
t110
0t
.10)(1
$203,625 = PMT(15.937)
PMT = $12,776.87
5-17B. FVn = PMT
+∑
−
=
t1n
0t
i)(1
$15,000,000 = PMT
+∑
−
=
t110
0t
.10)(1
$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
FV1 = $1(1 + .02)1
= $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 annually is more attractive.
5-19B. Investment A
PV = PMT
+∑= t
n
1t i)(1
i
= $15,000
+∑= t
5
1t .20)(1
1
= $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.
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PV = PMT
+∑= t
n
1t i)(1
1
= $15,000
+
∑=
t
6
1t .20)(1
1
= $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 =
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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 inAppendix 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.
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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
+−
+ ∑∑==
t
10
1tt
15
1t .07)(11
.07)(11
= $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.
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5-26B. PV = PMT
+∑= t
10
1t .09)(1
1
$45,000 = PMT(6.418)
PMT = $7,012
5-27B. PV = PMT
+∑= t
5
1t i)(1
1
$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
+∑= t
n
1t i)(1
1
$30,000 = PMT
+∑= t
4
1t .13)(1
1
$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
+ ni)(1
1
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
+∑
−
=
t1n
0t
i)(1
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$30,000 = PMT
+∑
−
=
t15
0t
.10)(1
$30,000 = PMT(6.105)
PMT =$4,914
5-32B. (a) FVn = PMT
+∑
−
=
t1n
0t
i)(1
$75,000 = PMT
+∑
−
=
t115
0t
.08)(1
$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.
FVn = PMT
+∑
−
=
t1n
0t
i)(1
$31,820 = PMT (FVIFA8%, 15 yr.)
$31,820 = PMT [27.152]
PMT = $1,171.92 per year
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5-33B.(a) This problem can be subdivided into (1) the compound value of the $150,000in 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 savingsaccount for 10 years, and (4) the additional savings 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
+∑
−
=
t1n
0t
i)(1
= $8,000
+∑
−
=
t110
0t
.08)(1
= $8,000 (14.487)
= $115,896
(4) Compound annuity of $2,000 (years 6-10)
FV5 = $2,000
+∑
−
=
t15
0t
.08)(1
= $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
+∑= t
20
1t .11)(1
1
$1,227,980 = PMT (7.963)
PMT = $154,210.72
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5-34B. PV = PMT (PVIFAi%, n yr.)
$200,000 = PMT (PVIFA10%, 20 yr.)
$200,000 = PMT(8.514)
PMT = $23,4915-35B. 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 (4.937)
PV = $40,000 + $197,480
PV = $237,480
5-37B FVn(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. PV(annuity due) = PMT(PVIFAi,n)(l + i)
= $1000 (PVIFA12%, 15 years)(1 + .12)
= $1000(6.811)(1.12)
= $7,628.32
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PV(annuity due) = PMT(PVIFAi,n)(l + i)
= $1000(PVIFA15%, 15 years)(l + .15)
= $1000(5.847)(1.15)
= $6,724.055-39B. PV = PMT(PVIFAi,n)(PVIFi,n)
= $1000(PVIFA15%,10 years)(PVIF15%, 7 years)
= $1000(5.019)(.376)
= $1,887.14
5-40B. FVn = PV (FVIFi%, n yr.)
$3,500 = .12(FVIFi%, 38 yr.)
solving using a financial calculator:
i = 31.0681%
5-41B. (a)
$60,000 per year $300,000 $60,000
$100,000
1/05 1/10 1/15 1/20 1/25 1/30
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 (PVIF
10%, 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