Financial Management Chapter 05 IM 10th Ed

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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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

(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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Prof. Rushen Chahal

the 12 year row in the 12 percent column)

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Prof. Rushen Chahal

(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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Prof. Rushen Chahal

(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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Prof. Rushen Chahal

compounded for 5 years

FV5 = $10,000 (1 + 0.06)5

FV5 = $10,000 (1.338)

FV5 = $13,380


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


FV15 = $10,000 (1 + 0.08)15FV15 = $10,000 (3.172)

FV15 = $31,720


FV1 = $10,000 (1 + 0.1)1

FV1 = $10,000 (1 + 1.100)

FV1 = $11,000


FV5 = $10,000 (1 + 0.1)5

FV5 = $10,000 (1.611)

FV5 = $16,110


FV15 = $10,000 (1 + 0.1)15

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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

$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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Prof. Rushen Chahal

= $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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Prof. Rushen Chahal

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)


$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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Prof. Rushen Chahal

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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Prof. Rushen Chahal

$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

112



Prof. Rushen Chahal

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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Prof. Rushen Chahal


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

114



Prof. Rushen Chahal

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

115



Prof. Rushen Chahal

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.

116



Prof. Rushen Chahal

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)

117



Prof. Rushen Chahal

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

118



Prof. Rushen Chahal

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

119



Prof. Rushen Chahal

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


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

120



Prof. Rushen Chahal

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


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


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


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%

121



Prof. Rushen Chahal

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

122



Prof. Rushen Chahal

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.

123



Prof. Rushen Chahal

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

124



Prof. Rushen Chahal

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

125



Prof. Rushen Chahal

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

126



Prof. Rushen Chahal

$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

127



Prof. Rushen Chahal

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

128



Prof. Rushen Chahal

(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

129



Prof. Rushen Chahal

(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


FV5 = $20,000 (1 + 0.07)5

FV5 = $20,000 (1.403)

FV5 = $28,060


FV15 = $20,000 (1 + 0.07)15

FV15 = $20,000 (2.759)

FV15 = $55,180

(b) FVn = PV (1 + i)n


FV1 = $20,000 (1 + 0.09)1

FV1 = $20,000 (1.090)

FV1 = $21,800


FV5 = $20,000 (1 + 0.09)5

FV5 = $20,000 (1.539)

FV5 = $30,780


FV5 = $20,000 (1 + 0.09)15

FV5 = $20,000 (3.642)

130



Prof. Rushen Chahal

FV5 = $72,840


FV1 = $20,000 (1 + 0.11)1

FV1 = $20,000 (1.11)

FV1 = $22,200


FV5 = $20,000 (1 + 0.11)5

FV5 = $20,000 (1.685)

FV5 = $33,700


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

131



Prof. Rushen Chahal

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

132



Prof. Rushen Chahal

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


FV2 = 10,000(1 + 0.15)2

FV2 = 10,000(1.322)

FV2 = 13,220 books


133



Prof. Rushen Chahal

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)

134



Prof. Rushen Chahal

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

135



Prof. Rushen Chahal

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.

136



Prof. Rushen Chahal

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 =

137



Prof. Rushen Chahal

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)


$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.

138



Prof. Rushen Chahal

Investment B:

PV = FVn (PVIFi,n)


$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.

139



Prof. Rushen Chahal

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

140



Prof. Rushen Chahal

$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)


(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]


141



Prof. Rushen Chahal

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


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

142



Prof. Rushen Chahal

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

143



Prof. Rushen Chahal

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

Documents

Financial Management Chapter 05 IM 10th Ed