Chapter 05

CHAPTER 5

The Time Value of Money

CHAPTER ORIENTATION

In this chapter, the concept of the time value of money is introduced; that is, a dollar today is worth more than a dollar received a year from now. If we are to compare projects and financial strategies logically, 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 so that during the second period the interest is earned on the original principal plus the interest earned during the first period.

A. Mathematically, the future value of an investment if compounded annually at a rate of r for n years will be

FVn = PV (l + r)n

where n = the number of years during which the compounding occurs

r = 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 either by increasing the number of years we let it compound or by compounding it at a higher rate.

87 ©2011 Pearson Education, Inc. Publishing as Prentice Hall

88 ♦ Keown/Martin/Petty Instructor’s Manual with Solutions

2. If the compounded period is less than one year, the future value of an investment can be determined as follows:

mn

mr 1PV nFV ⎟⎠⎞

⎜⎝⎛ +=

where m = the number of times compounding occurs during the year

II. Determining the present value, that is, the value in today’s dollars of a sum of money to be received in the future, involves nothing other than inverse compounding. The differences in these techniques are simply different points of view.

A. Mathematically, the present value of a sum of money to be received in the future can be determined by the following equation:

PV = FVn ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ nr 11

where n = the number of years until payment will be received,

r = the opportunity rate or discount rate

PV = the present value of the future sum of money

FVn = 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 both the number of years until the payment will be received and the opportunity rate.

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 money at 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 and compounding each one of the individual deposits to the future or by using the following compound annuity equation:

FVn = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∑−

=+

1 n

0 t tr 1 = PMT ⎥

⎦

⎤⎢⎣

⎡ −+rr n 1)1(

where PMT = the annuity value deposited at the end of each year

r = the annual interest (or discount) rate

n = the number of years for which the annuity will last

©2011 Pearson Education, Inc. Publishing as Prentice Hall

Foundations of Finance, Seventh Edition ♦ 89

FVn = the future value of the annuity at the end of the nth year

B. Pension funds, insurance obligations, and interest received from bonds all involve 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 discounting each one of the individual cash flows back to the present or by using the following present value of an annuity equation:

PV = PMT ( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∑= +

n

1 t tr 11 = PMT

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+

−

ir)(1

11 n

where PMT = the annuity withdrawn at the end of each year

r = the annual interest or discount rate

PV = the present value of the future annuity

n = the number of years for which the annuity will last

C. This procedure of solving for PMT, the annuity value when r, n, and PV are known, is also the procedure used to determine what payments are associated with 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, the annual annuity.

IV. A perpetuity is an annuity that continues forever; that is, every year from now on, this investment 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 = r

pp

where PV = the present value of the perpetuity

pp = the constant dollar amount provided by the perpetuity

r = the annual interest or discount rate

V. To aid in the calculations of present and future values, tables are provided at the end of the text.

A. To aid in determining the value of FVn in the compounding formula



FVn = PV (l + r)n

The value of (l + r)n is referred to as FUTURE VALUE FACTORr,n calculated as (1 + r)n.

B. To aid in the computation of present values

PV = FVn ( )nr 1

1

+

With ( )nr 1

1

+, equal to the Present Value Factor

C. Because of the time-consuming nature of compounding an annuity,

FVn = PMT where (∑−

=

+1 n

0 t

tr 1 )

= ( )∑−

=

+1 n

0 t

tr 1 ⎥⎦

⎤⎢⎣

⎡ −+rr n 1)1(

and is referred to as the ANNUITY PRESENT VALUE FACTORr,n

D. To simplify the process of determining the present value of an annuity

PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1 where

( )∑= +

n

1 t tr 1

1 is referred to as the ANNUITY PRESENT

VALUE FACTORr,n=⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+−

ii)(1

1 n⎤⎡ 1



ANSWERS TO END-OF-CHAPTER QUESTIONS

5-1. The concept of time value of money is a recognition that a dollar received today is

worth more than a dollar received a year from now, or at any future date. It exists because there are investment opportunities on money; that is, we can place our dollar received today in a savings account and one year from now have more than a dollar.

5-2. Compounding and discounting are inverse processes of each other. In compounding, money is moved forward in time, while in discounting, money is moved back in time. This can be shown mathematically in the compounding equation:

FVn = PV (1 + r)n

We can derive the discounting equation by multiplying each side of this equation

by ( )n

11 r+

and we get:

PV = FVn ( )nr 11+

5-3. We know that FVn = PV(1 + r)n

Thus, an increase in r will increase FVn, and a decrease in n will decrease FVn. 5-4. Bank C, which compounds continuously, pays the highest interest. This occurs

because, while all banks pay the same interest, 5%, bank C compounds the 5% continuously. Continuous compounding allows interest to be earned more frequently than any other compounding period.

5-5. An annuity is a series of equal dollar payments for a specified number of years. Examples of annuities include mortgage payments, interest payments on bonds, fixed lease payments, and any fixed contractual payment. A perpetuity is an annuity that continues forever; that is, every year from now on this investment pays the same dollar amount. The difference between an annuity and a perpetuity is that a perpetuity has no termination date, whereas an annuity does.

5-6. This problem involves a comparison of three websites, all of which are very good.



SOLUTIONS TO END-OF-CHAPTER PROBLEMS

5-1. a. FVn = PV (1 + r)n

FV10 = $5,000(1 + 0.10)10

FV10 = $5,000 (2.594) FV10 = $12,970

Or: N = 10 I/Y = 10 PV = –5,000 PMT = 0 CPT FV = $12,969

b. FVn = PV (1 + r)n

FV7 = $8,000 (1 + 0.08)7 FV7 = $8,000 (1.714) FV7 = $13,712




c. FV12 = PV (1 + r)n

FV12 = $775 (1 + 0.12)12

FV12 = $775 (3.896) FV12 = $3,019.40

Or: N = 12 I/Y = 12 PV = –775 PMT = 0 CPT FV = $3,019

d. N = 5

I/Y = 5 PV = –21,000 PMT = 0 CPT FV = $26,802.

5-2. I/Y = 5

PV = –500,000 PMT = 0 FV = 1,039.50 CPT N = 15

b. I/Y = 9

PV = –35.0 PMT = 0 FV = 53.87 CPT N = 5

c. I/Y = 20

PV = –100.0 PMT = 0 FV = 298.60 CPT N = 6



d. I/Y = 2 PV = –53.0 PMT = 0 FV = 78.76 CPT N = 20

5-3. a. N = 12

CPT I/Y = 12 PV = –500 PMT = 0 FV = 1,948

b. N = 7

CPT I/Y = 4.999 PV = –300 PMT = 0 FV = 422.10

c. N = 20

CPT I/Y = 9 PV = –50 PMT = 0 FV = 280.20

d. N = 5

CPT I/Y = 20 PV = –200 PMT = 0 FV = 497.60

5-4. a. N = 10

I/Y = 10 CPT PV = –308.43 PMT = 0 FV = 800



b. N = 5 I/Y = 5 CPT PV = –235.06 PMT = 0 FV = 300

c. N = 8

I/Y = 3 CPT PV = –789 PMT = 0 FV = 1,000

d. N = 8

I/Y = 20 CPT PV = –233 PMT = 0 FV = 1,000

5-5. a. N = 10

I/Y = 5 PV = 0 PMT = –500 CPT FV = 6,289

b. N = 5

I/Y = 10 PV = 0 PMT = –100 CPT FV = 610.51

c. N = 7

I/Y = 7 PV = 0 PMT = –35 CPT FV = 302.89



d. N = 3 I/Y = 2 PV = 0 PMT = –25 CPT FV = 76.51

5-6. a. N = 10 I/Y = 7 CPT PV = 17,559 PMT = –2,500 FV = 0

b. N = 3

I/Y = 3 CPT PV = –198 PMT = 70 FV = 0

c. N = 7

I/Y = 6 CPT PV = –1,563.06 PMT = 280 FV = 0

d. N = 10

I/Y = 10 CPT PV = –3,072.28 PMT = 500 FV = 0

5-7. a. FVn = PV (1 + r)n compounded for 1 year

FV1 = $10,000 (1 + 0.06)1 FV1 = $10,000 (1.06) FV1 = $10,600

Or:



N = 1 I/Y = 6 PV = –10,000 PMT = 0 CPT FV = $10,600

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 + r)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)15 FV15 = $10,000 (3.172) FV15 = $31,720


compounded for 1 year at 10%

FV1 = $10,000 (1 + 0.1)1 FV1 = $10,000 (1 + 1.100) FV1 = $11,000





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

FV5 = $10,000 (1.611) FV5 = $16,110



FV15 = $10,000 (1 + 0.1)15 FV15 = $10,000 (4.177) FV15 = $41,770


c. There is a positive relationship between both the interest rate used to compound

a present sum and the number of years for which the compounding continues and the future value of that sum.

5-8. a. N = 35 I/Y = 4 PMT = 0 FV = 2,000,000 CPT PV = –$506,831



b. N = 35 I/Y = 14 PMT = 0 FV = 2,000,000 CPT PV = –$20,387

5-9. N = 5 I/Y = 6 PV = –5,000 PMT = 0 CPT FV = $6,691

b. FVn = PV mn

mr 1 ⎟⎠⎞

⎜⎝⎛ +

FV5 = $5,000 (1 + 0.06

2 )2×5

FV5 = $5,000 (1 + 0.03)10

FV5 = $5,000 (1.344) FV5 = $6,720

Or: Using a financial calculator where you can change the number of times compounding occurs per year, you can solve this problem in one of two ways.

One way to solve this problem if you’re using a Texas Instruments BAII-Plus calculator is to first make P/Y = 2

N = 5 × 2 = 10

I/Y = 6

PV = –5,000

PMT = 0

CPT FV = $6,720

OR if you don’t want to use P/Y button (that is, set P/Y=1)

N = 5 × 2 = 10

I/Y = 6/2

PV = –24

PMT = 0

CPT FV = $6,720



Now solving for bimonthly compounding (six times per year):

FVn = PV mn

mr 1 ⎟⎠⎞

⎜⎝⎛ +

FV5 = $5,000 (1 + 0.06

6 )6(5)

FV5 = $5,000 (1 + 0.01)30 FV5 = $5,000 (1.348) FV5 = $6,740

Using a financial calculator where you can change the number of times compounding occurs per year, you can solve this problem in one of two ways.


N = 5 × 6 = 30

I/Y = 6

PV = –5,000

PMT = 0

CPT FV = $6,739 OR if you don’t want to use P/Y button (that is, set P/Y=1)

N = 5 × 6 = 30

I/Y = 6/6

PV = –5,000

PMT = 0

CPT FV = $6,739

c. FVn = PV (1 + r)n

FV5 = $5,000 (1 + 0.12)5

FV5 = $5,000 (1.762)

FV5 = $8,810




FV5 = PV mn

mr 1 ⎟⎠⎞

⎜⎝⎛ +

FV5 = $5,000 ⎟⎠⎞

⎜⎝⎛ +

212.0 1 2×5

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

FV5 = $5,000 (1.791)

FV5 = $8,955

Or: Using a financial calculator where you can change the number of times compounding occurs per year, you can solve this problem in one of two ways.


N = 5 × 2 = 10 I/Y = 12 PV = –5,000 PMT = 0 CPT FV = $8,954

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 5 × 2 = 10 I/Y = 12/2 PV = –24 PMT = 0 CPT FV = $8,954

Now solving for bimonthly compounding (six times per year):

FV5 = PV mn

mr 1 ⎟⎠⎞

⎜⎝⎛ +

FV5 = $5,000 6(5)0.121

6⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $5,000 (1 + 0.02)30

FV5 = $5,000 (1.811)

FV5 = $9,055





N = 5 × 6 = 30 I/Y = 12 PV = –5,000 PMT = 0 CPT FV = $9,057

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 5 × 6 = 30 I/Y = 12/6 PV = –5,000 PMT = 0 CPT FV = $9,057

d. FVn = PV (1 + r)n

FV12 = $5,000 (1 + 0.06)12

FV12 = $5,000 (2.012)

FV12 = $10,060


e. An increase in the stated interest rate will increase the future value of a given sum. Likewise, an increase in the length of the holding period will increase the future value of a given sum.



5-10. Annuity A:

N = 12

I/Y = 11

CPT PV = –$55,185

PMT = –8,500

FV = 0

Since the cost of this annuity is $50,000 and its present value is $55,182, given an 11% opportunity cost, this annuity has value and should be accepted.

Annuity B:

N = 25

I/Y = 11

CPT PV = –$58,952

PMT = –7,000

FV = 0

Since the cost of this annuity is $60,000 and its present value is only $58,954, given an 11% opportunity cost, this annuity should not be accepted.

Annuity C:

N = 20

I/Y = 11

CPT PV = –$63,707 PMT = –8,000

FV = 0

Since the cost of this annuity is $70,000 and its present value is only $63,704, given an 11% opportunity cost, this annuity should not be accepted.

5-11. Year 1: FVn = PV (1 + r)n

FV1 = 15,000 (1 + 0.2)1

FV1 = 15,000 (1.200)

FV1 = 18,000 books



OR: N = 1 I/Y = 20 PV = –15,000 PMT = 0 CPT FV = 18,000 books

Year 2: FVn = PV (1 + r)n

FV 2 = 15,000 (1 + 0.2)2

FV 2 = 15,000 (1.440)

FV2 = 21,600 books



FV3 = 15,000 (1.20)3

FV3 = 15,000 (1.728)

FV3 = 25,920 books




Book sales

25,000

20,000

15,000

1 2 3

years

The sales trend graph is not linear, because this is a compound growth trend. Just as compound interest occurs when interest paid on the investment during the first period is added to the principal of the second period, interest is earned on the new sum. Book sales growth was compounded; thus, the first year the growth was 20% of 15,000 books, the second year 20% of 18,000 books, and the third year 20% of 21,600 books.

5-12. FVn = PV (1 + r)n

FV1 = 47(1 + 0.12)1

FV1 = 47(1.12)

FV1 = 52.6 Home Runs in 2010

OR: N = 1 I/Y = 12 PV = –47 PMT = 0 CPT FV = 52.6 home runs

FV2 = 47(1 + 0.12)2

FV2 = 47(1.21)


OR:



N = 2 I/Y = 12 PV = –47 PMT = 0 CPT FV = 58.96 home runs

FV3 = 47(1 + 0.12)3

FV3 = 47(1.331) FV3 = 66.03 Home Runs in 2012.


FV3 = 47(1 + 0.12)4 FV4 = 47(1.464) FV4 = 73.96 Home Runs in 2013.


FV5 = 47(1 + 0.12)5 FV5 = 47(1.611) FV5 = 82.83 Home Runs in 2014.




5-13. N = 25

I/Y = 9

PV = 60,000

CPT PMT = –$6,108.38

FV = 0

5-14. N = 15

I/Y = 6

PV = 0

CPT PMT = –$644.44

FV = 15,000

5-15. N = 10

CPT I/Y = 8.0% PV = –500

PMT = 0

FV = 1,079.50

5-16. N = 10 I/Y = 5 PV = –100,000 PMT = 0 CPT FV = $162,889

How much must be invested annually to accumulate $162,889?

N = 10 I/Y = 10

PV = 0

CPT PMT = –$10,221 FV = $162,889



5-17. N = 10 I/Y = 9 PV = 0 CPT PMT = –$658,200.90 FV = 10,000,000

5-18. One dollar at 12.0% compounded monthly for one year

FVn = PV (1 + r)n

FV12 = $1(1 + .01)12

= $1(1.127)

= $1.127



N = 1 × 12 = 12 I/Y = 12 PV = –1 PMT = 0 CPT FV = $1.1268


N = 1 × 12 = 12

I/Y = 12/12

PV = –1

PMT = 0

CPT FV = $1.1268

One dollar at 13.0% compounded annually for one year

FVn = PV (1 + r)n

FV1 = $1(1 + .13)1

= $1(1.13)

= $1.13

OR



N = 1

I/Y = 13

PV = –1

PMT = 0

CPT FV = $1.13 The loan at 12% compounded monthly is more attractive.

5-19. Investment A

N = 5 I/Y = 20 CPT PV = –29,906 PMT = 10,000 FV = 0

Investment B

Step 1: 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.

Step 1:

N = 6 I/Y = 20 CPT PV = –33,255 PMT = 10,000 FV = 0

Step 2:

N = 4 I/Y = 20 CPT PV = 16,037 PMT = 0 FV = –33,255



Investment C

PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

= $10,000 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 1)20. 1(

1+ $50,000

⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 6)20. 1(

1

+ $10,000 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 10)20. 1(

1

= $10,000(.833) + $50,000(.335) + $10,000(.162)

= $8,330 + $16,750 + $1,620

= $26,700

OR: Simply calculate the present value of all three single cash flows and then add them together: N = 1 I/Y = 20 CPT PV = 8,333 PMT = 0 FV = 10,000 N = 6 I/Y = 20

CPT PV = –16,745 PMT = 0 FV = 50,000 N = 10 I/Y = 20

CPT PV = –1,615 PMT = 0 FV = 10.000

Then add all the present values together: $8,333 + $16,745 + $1,615 = $26,693



5-20. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $1,000 ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 7)10. 1(

1

PV = $1,000(.513)

PV = $513 OR:

N = 7 I/Y = 10 CPT PV = –513 PMT = 0 FV = 1,000

5-21. a. PV = r

PP

PV = 08.0

300$

PV = $3,750

b. PV = r

PP

PV = 12.0000,1$

PV = $8,333.33

c. PV = r

PP

PV = 09.0

100$

PV = $1,111.11

d. PV = r

PP

PV = 05.095$

PV = $1,900



5-22. Using a financial calculator where you can change the number of times compounding occurs per year, you can solve this problem in one of two ways.


CPT N = 18 and since P/Y = 2, there are two periods per year, so N = 9 years I/Y = 16 PV = –1 PMT = 0 FV = 4


CPT N = 18 and since the interest rate is expressed in semi-annual terms, there are two periods per year, so N=9 years

I/Y = 16/2

PV = –1

PMT = 0

FV = 4

5-23. Step 1 (First, discount the annuity back to the beginning of year 11, which is the end of year 10.):

N = 5 I/Y = 6 CPT PV = –42,124 PMT = 10,000 FV = 0

Step 2 (Then, discount this equivalent sum to present.): N = 10 I/Y = 6 CPT PV = 23,473 PMT = 0 FV = –42,124



Step 3 (Then, determine the present value of the $20,000 withdrawal at the end of year 15):

N = 15 I/Y = 6 CPT PV = 8,345 PMT = 0 FV = –20,000

Step 4: (Add the present values together):

Thus, you would have to deposit $23,473 + $8,345 or $31,818 today. 5-24. N = 10

I/Y = 10 PV = –40,000 CPT PMT = $6,510 FV = 0

5-25. N = 5

CPT I/Y = 19.86% PV = –30,000

PMT = 10,000

FV = 0

5-26. N = 5

CPT I/Y = 22.0% PV = –10,000

PMT = 0

FV = 27,027

5-27. N = 5 I/Y = 12 PV = –25,000 CPT PMT = $6,935 FV = 0



5-28. The present value of $10,000 in 12 years at 11 percent is:

N = 12 I/Y = 11 CPT PV = 2,858 PMT = 0 FV = –10,000

The present value of $25,000 in 25 years at 11% is:

N = 25 I/Y = 11 CPT PV = 8,345 PMT = 0 FV = –1,840

5-29. N = 5 I/Y = 12 PV = 0 CPT PMT = $3,148 FV = –20,000

5-30. a. N = 15

I/Y = 7

PV = 0

CPT PMT = $1,990

FV = –50,000

b. N = 15 I/Y = 7 CPT PV = 18,122 PMT = 0 FV = –50,000

c. The contribution of the $10,000 deposit toward the $50,000 goal is

N = 10 I/Y = 7 PV = –10,000 PMT = 0 CPT FV = 19,672



Thus, only $30,328 need be accumulated by annual deposit. N = 15 I/Y = 7 PV = 0 CPT PMT = $1,207 FV = –30,328

5-31. This problem can be subdivided into (1) the compound value of the $100,000 in the savings account, (2) the compound value of the $300,000 in stocks, (3) the additional savings due to depositing $10,000 per year in the savings account for 10 years, and (4) the additional savings due to depositing $10,000 per year in the savings account at the end of years 6-10. (Note the $20,000 deposited in years 6-10 is covered in parts 3 and 4.)

(1) N = 10 I/Y = 7 PV = –100,000 PMT = 0 CPT FV = 196,715

(2) N = 10 I/Y = 12 PV = –300,000 PMT = 0 CPT FV = 931,754

(3) Compound annuity of $10,000, 10 years N = 10 I/Y = 7 PV = 0 PMT = –10,000 CPT FV = 138,164

(4) Compound annuity of $10,000 (years 6-10) N = 5 I/Y = 7 PV = 0 PMT = –10,000 CPT FV = 57,507



At the end of ten years, you will have $196,715 + $931,754 + $138,164 + $57,507 = $1,324,140.

N = 20 I/Y = 10 PV = –1,324,140 CPT PMT = 155,533 FV = 0

5-32. N = 20 I/Y = 15 PV = –100,000 CPT PMT = 15,976 FV = 0

5-33. N = 30 I/Y = 10 PV = –150,000 CPT PMT = 15,912 FV = 0

5-34. This is an annuity due problem:

At 10%: N = 20

I/Y = 10 CPT PV = 425,678 x 1.1 = 468,246 PMT = –50,000 FV = 0

At 20%: N = 20

I/Y = 20 CPT PV = 243,479 x 1.2 = 292,175 PMT = –50,000 FV = 0

5-35. N = 46

CPT I/Y = 30.14% PV = –0.12 PMT = 0 FV = 22,000



5-36. a. (note: 1/36 is the same as 12/35 – that is the first second of the first day of the year can be thought of as the last day of the previous year).

$50,000 per year $250,000

$50,000

$100,000

1/11 1/16 1/21 1/26 1/31 1/36

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(ANNUITY PRESENT VALUE FACTOR 10%, 19 yr.

– ANNUITY PRESENT VALUE FACTOR10%, 4 yr.)

+ $250,000(PRESENT VALUE FACTOR10%, 20 yr.)

+ $50,000(PRESENT VALUE FACTOR10%, 23 yr. + PRESENT VALUE FACTOR10%, 2

+ $100,000 (PRESENT VALUE FACTOR10%, 25 yr.)

= $50,000 (8.365 - 3.170) + $250,000 (.149)

+ $50,000 (0.112 + .102) + $100,000 (.092)

= $259,750 + $37,250 + $10,700 + $9,200

= $316,900

b. If you live longer than expected, you could end up with no money later on in life. 5-37.

rate (r) = 8% number of periods (n) = 7 payment (PMT) = $0 present value (PV) = $900 type (0 = at end of period) = 0

Future value = $1,542.44 Excel formula: = FV(rate,number of periods,payment,present value,type) Notice that present value ($500) took on a negative value.



5-38. 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 =

r = 11.18%

Excel formula: = RATE(number of periods,payment,present value,future value,type,guess)

Notice that present value ($30,000) took on a negative value. 5-39. Two things to keep in mind when you're working this problem: first, you'll have to

convert the annual rate of 8 percent into a monthly rate by dividing it by 12, and second, you'll have to convert the number of periods into months by multiplying 25 times 12 for a total of 300 months.

Excel formula: = PMT(rate, number of periods, present value, future value, type)

rate (r) = 8%/12 number of periods (n) = 300 present value (PV) = $300,000 future value (FV) = $0 type (0 = at end of period) = 0

monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them. You can also use Excel to calculate the interest and principal portion of any loan amortization payment. You can do this using the following Excel functions:

Calculation: Formula: Interest portion of payment = IPMT(rate, period, number of periods,

present value, future value, type)

Principal portion of payment = PPMT(rate, period, number of periods, present value, future value, type)

where period refers to the number of an individual periodic payment.

Thus, if you would like to determine how much of the 48th monthly payment went toward interest and principal you would solve as follows:

Interest portion of payment 48: ($1,884.37)

The principal portion of payment 48: ($431.08)



5-40. a. N = 384

I/Y = 6

PV = –24

PMT = 0

CPT FV = 125.217 billion dollars b. One way to solve this problem if you’re using a Texas Instruments BAII-Plus

calculator is to first make P/Y = 12

N = 384 x 12 = 4,608

I/Y = 6

PV = –24

PMT = 0

CPT FV = 229.893 billion dollars OR if you don’t want to use P/Y button (that is, set P/Y=1)

N = 384 x 12 = 4,608

I/Y = 6/12 = .5

PV = –24

PMT = 0

CPT FV = 229.893 billion dollars c. N = 10

I/Y = 10

CPT PV = –23.13 billion dollars PMT = 0

FV = 60. billion

d. N = 10

CPT I/Y = 14.87% PV = –15 billion PMT = 0

FV = 60. billion

e. N = 40

I/Y = 7

PV = –28. billion

CPT PMT = 2.10 billion dollars FV = 0



5-41. 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 6 years?

N = 6

I/Y = 7.5

CPT PV = –11,605.50 dollars PMT = 0

FV = 17,910.78

5-42. N = 45

I/Y = 8.75

PV = 0

CPT PMT = –2,054.81 dollars

FV = 1,000,000

5-43. First, we must calculate what Mr. Burns will need in 20 years. Once we know how much he will need, 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 present value, you must multiply your answer, in this case –$4.192 billion times (1 + r) 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 dollars

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



5-44. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money 5 years ago and we want to know what it will be worth in 20 years)?

N = 25

I/Y = 7.5

PV = –100,000

PMT = 0

CPT FV = 609,833.96 dollars Now we determine what the additional $1,500 per year will grow to (note that since Homer will be making these investments at the beginning of each year for 20 years we have an annuity due, thus, once you calculate the present value, you must multiply your answer, in this case $64,957.02 times (1 + r)):

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

5-45. Since this problem involves monthly payments we must first divide the annual interest rate by 12 to convert it to monthly terms. Then, N becomes the number of months or compounding periods,

N = 60

I/Y = 6.2/12

PV = –25,000

CPT PMT = 485.65 dollars FV = 0

OR: If you’re using a Texas Instruments BAII-Plus calculator you can solve this problem by using the P/Y function and make P/Y = 12:

N = 5 × 12 = 60

I/Y = 6.2

PV = –25,000

CPT PMT = 485.65 FV = 0



5-46. Since this problem involves monthly payments there are two ways to solve it:

One way to solve this problem, if you’re using a Texas Instruments BAII-Plus calculator, is to first make P/Y = 12

N = 36

CPT I/Y = 11.62 PV = –999

PMT = 33

FV = 0


N = 36

CPT I/Y = 0.9683 x 12 = 11.62% (remember, we just calculated the monthly interest rate because n was expressed in months, so to calculate the annual rate we multiply it times 12)

PV = –999

PMT = 33

FV = 0

5-47. First, what will be the monthly payments if Suzie goes for the 4.9 percent financing? Since this problem involves monthly payments there are two ways to solve it:


N = 60

I/Y = 4.9

PV = –25,000

CPT PMT = $470.64 FV = 0

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 60

I/Y = 4.9/12

PV = –25,000

CPT PMT = $470.64 FV = 0



Now, calculate how much the monthly payments would be if Suzie took the $1,000 cash back and reduced the amount owed from $25,000 to $24,000. Again, since this problem involves monthly payments there are two ways to solve it:


N = 60 I/Y = 6.9 PV = -24,000 CPT PMT = $474.10 FV = 0

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 60 I/Y = 6.9/12 PV = –24,000 CPT PMT = $470.10 FV = 0

5-48. Since this problem involves quarterly payments there are two ways to solve it:


N = 16 I/Y = 6.4% PV = 0 PMT = –1,000 CPT FV = $18,071.11

OR if you don’t want to use P/Y button (that is, set P/Y=1) N = 16 I/Y = 6.4/4 PV = 0 PMT = –1,000 CPT FV = $18,071.11

5-49. Since this problem involves monthly payments there are two ways to solve it:


CPT N = 41.49 (rounded up to 42 months) I/Y = 12.9 PV = –5000 PMT = 150 FV = 0



OR if you don’t want to use P/Y button (that is, set P/Y = 1) CPT N = 41.49 (rounded up to 42 months) I/Y = 12.9/12 PV = –5000 PMT = 150 FV = 0

5-50. 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 = –140,000 PMT = 0 FV = 420,000

c. Since this problem involves monthly payments there are two ways to solve it. One way to solve this problem, if you’re using a Texas Instruments BAII-Plus calculator, is to use the P/Y = function and 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 FV = 420,000


N = 120

I/Y = 6 /12

PV = –140,000

CPT PMT = $1,008.57

FV = 420,000



d. Since this problem involves monthly payments there are two ways to solve it. One way to solve this problem, if you’re using a Texas Instruments BAII-Plus calculator, is to use the P/Y = function and 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


N = 120

CPT I/Y = 0.7069% x 12 = 8.48% PV = –140,000

PMT = –500

FV = 420,000 5-51. a. Future value of $300 deposited every three months at 9% compounded

quarterly N = 35 x 4 = 140 I/Y = 9.0/4 = 2.25 PV = 0 PMT = –300 CPT FV = $287,138

b. Future value of $15,000 invested for 15 years at 9% compounded quarterly N = 15 x 4 = 60 I/Y = 9.0/4 = 2.25 PV = –15,000 PMT = 0 CPT FV = $57,002

Adding the answers in parts a and b together we get: $344,140



5-52. N = 20 CPT I/Y = 13.00 PV = –21,074.25 PMT = 3000 FV = 0

5-53. At retirement, Milhouse wants: $300,000 for a boat and an annuity of $80,000 per year for 15 years at 6%. First, calculate the PV of the annuity (which is how much he will need to withdraw $80,000 each year), then add $300,000:

N = 50 I/Y = 6% CPT PV = –$1,260,948 PMT = $80,000 FV = 0

Now add the $300,000 he wants to buy a boat to the $1,260,948, and at the end of 43 years he will need to have $1,560,948 Next, determine how much he needs to deposit at the end of each year for 43 years at 9% in order to accumulate $1,560,948:

N = 43 I/Y = 9 PV = 0 CPT PMT = –3,540.80 FV = 1,560,948

5-54. N = 200

I/Y = 3.98 PV = –12,345 PMT = 0 CPT FV = $30,300,773

5-55. N = 7

CPT I/Y = 15% PV = –4,510 PMT = 0 FV = 12,000



5-56. CPT N = 45 I/Y = 4.5% PV = –45,530 PMT = 0 FV = 330,000

5-57. N = 28 I/Y = 7% CPT PV = –60,000. PMT = 0 FV = 398,930

SOLUTION TO MINI CASE

a. Discounting is the inverse of compounding. We really have only one formula to move a single cash flow through time. In some instances, we are interested in bringing that cash flow back to the present (finding its present value) when we already know the future value. In other cases, we are merely solving for the future value where we know the present value.

b. The present value of an annuity factor is actually derived from the present value factor. This can be seen by examining the value of each. The present value factor gives values of

nr) (11+

for various values of i and n, while the present value of an annuity factor gives values of

( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

for various values of r and n. Thus, the value in the present value of annuity for an n-year annuity for any discount rate r is merely the sum of the first n values of the present value factor.

c. (1) N = 10 I/Y = 8 PV = –5,000 PMT = 0 CPT FV = $10,795



(2) CPT N = 15 I/Y = 10 PV = –400 PMT = 0 FV = $1,671

(3) N = 10 CPT I/Y = 15 PV = –1,000 PMT = 0 FV = $4,046

d. One way to solve this problem if you’re using a Texas Instruments BAII-Plus calculator is to use the P/Y = function and make P/Y = 2. Then, N becomes the number of months or compounding periods:

N = 10

I/Y = 10

PV = –1,000

PMT = 0

CPT FV = 1,629 OR if you don’t want to use P/Y button (that is, set P/Y=1)

N = 10

I/Y = 10/2

PV = –1,000

PMT = 0

CPT FV = 1,629 e. An annuity due is an annuity in which the payments occur at the beginning of each

period as opposed to occurring at the end of each period, which is when the payment occurs in an ordinary annuity.

f. N = 7

I/Y = 10

CPT PV = –4,868

PMT = 1,000

FV = 0



PV(annuity due)

N = 7

I/Y = 10

CPT PV = –4,868 x 1.1 = 5,355

PMT = 1,000

FV = 0

g. N = 7

I/Y = 10

PV = 0

PMT = –1,000

CPT FV = 9,487

FVn(annuity due)

N = 7

I/Y = 10

PV = 0

PMT = –1,000

CPT FV = 9,487 x 1.1 = 10,436 h. N = 25

I/Y = 10

PV = –100,000

CPT PMT = 11,017 FV = 0

i. PV = r

PP

= 08.000,1$

= $12,500



j. Step 1 (First, discount the annuity back to the beginning of year 10, which is the end of year 9.):

N = 10 I/Y = 10 CPT PV = –6,144.57 PMT = 1,000 FV = 0

Step 2 (Then, discount this equivalent sum to present.): N = 9 I/Y = 10 CPT PV = 2,605 PMT = 0 FV = –6,144.57

k. Step 1 (First, discount the perpetuity back to the beginning of year 10, which is the end of year 9.):

= 10.000,1$

= $10,000

Step 2 (Then, discount this equivalent sum to present.):

N = 9

I/Y = 10

CPT PV = 4,241

PMT = 0

FV = –10,000



ALTERNATIVE PROBLEMS AND SOLUTIONS ALTERNATIVE PROBLEMS

5-1A. (Compound Interest) To what amount will the following investments accumulate?

a. $4,000 invested for 11 years at 9% compounded annually

b. $8,000 invested for 10 years at 8% compounded annually

c. $800 invested for 12 years at 12% compounded annually

d. $21,000 invested for 6 years at 5% compounded annually

5-2A. (Compound Value Solving for n) How many years will the following take?

a. $550 to grow to $1,043.90 if invested at 6% compounded annually

b. $40 to grow to $88.44 if invested at 12% compounded annually

c. $110 to grow to $614.79 if invested at 24% compounded annually

d. $60 to grow to $73.80 if invested at 3% compounded annually

5-3A. (Compound Value Solving for r) At what annual rate would the following have to be invested?

a. $550 to grow to $1,898.60 in 13 years

b. $275 to grow to $406.18 in 8 years

c. $60 to grow to $279.66 in 20 years

d. $180 to grow to $486.00 in 6 years

5-4A. (Present Value) What is the present value of the following future amounts?

a. $800 to be received 10 years from now discounted back to present at 10%

b. $400 to be received 6 years from now discounted back to present at 6%

c. $1,000 to be received 8 years from now discounted back to present at 5%

d. $900 to be received 9 years from now discounted back to present at 20%

5-5A. (Compound Annuity) What is the accumulated sum of each of the following streams of payments?

a. $500 a year for 10 years compounded annually at 6%

b. $150 a year for 5 years compounded annually at 11%

c. $35 a year for 8 years compounded annually at 7%

d. $25 a year for 3 years compounded annually at 2%



5-6A. (Present Value of an Annuity) What is the present value of the following annuities?

a. $3,000 a year for 10 years discounted back to the present at 8%

b. $50 a year for 3 years discounted back to the present at 3%

c. $280 a year for 8 years discounted back to the present at 7%

d. $600 a year for 10 years discounted back to the present at 10%

5-7A. (Compound Value) Trish Nealon, who recently sold her Porsche, placed $20,000 in a savings account paying annual compound interest of 7%.

a. Calculate the amount of money that will have accrued if she leaves the money in the bank for 1, 5, and 15 years.

b. If she moves her money into an account that pays 9% or one that pays 11%, rework part a using these new interest rates.

c. What conclusions can you draw about the relationships between interest rates, time, and future sums from the calculations you have done above?

5-8A. (Compound Interest with Nonannual Periods) Calculate the amount of money that will be in each of the following accounts at the end of the given deposit period:

Compounding Period Annual (Compounded Deposit Amount Interest Every Period Account Deposited Rate Month) (Years)

Korey Stringer $2,000 12% 2 2 Eric Moss 50,000 12 1 1 Ty Howard 7,000 18 2 2 Rob Kelly 130,000 12 3 2 Matt Christopher 20,000 14 6 4 Juan Porter 15,000 15 4 3

5-9A. (Compound Interest with Nonannual Periods)

a. Calculate the future sum of $6,000, given that it will be held in the bank 5 years at an annual interest rate of 6%.

b. Recalculate part a using a compounding period that is (1) semiannual and (2) bimonthly.

c. Recalculate parts a and b for a 12% annual interest rate.

d. Recalculate part a using a time horizon of 12 years (annual interest rate is still 6%).



e. With respect to the effect of changes in the stated interest rate, and holding periods on future sums in parts c and d, what conclusions do you draw when you compare these figures with the answers found in parts a and b?

5-10A. (Solving for r in Annuities) Ellen Denis, a sophomore mechanical engineering student, receives a call from an insurance agent, who believes that Ellen is an older woman ready to retire from teaching. He talks to her about several annuities that she could buy that would guarantee her an annual fixed income. The annuities are as follows:

Initial Payment into Duration Annuity Amount of Money of Annuity Annuity (at t = 0) Received per year (Years) A $50,000 $8500 12 B $60,000 $7000 25 C $70,000 $8000 20

If Ellen could earn 12% on her money by placing it in a savings account, should she place it instead in any of the annuities? Which ones, if any? Why?

5-11A. (Future Value) Sales of a new marketing book were 10,000 copies this year and were expected to increase by 15% per year. What are expected sales during each of the next three years? Graph this sales trend and explain.

5-12A. (Future Value) Reggie Jackson, formerly of the New York Yankees, hit 41 home runs in 1980. If his home-run output grew at a rate of 12% per year, what would it have been over the following 5 years?

5-13A. (Loan Amortization) Stefani Moore purchased a new house for $150,000. She paid $30,000 down and agreed to pay the rest over the next 25 years in 25 equal annual payments that included principal payments plus 10% compound interest on the unpaid balance. What will these equal payments be?

5-14A. (Solving for PMT in an Annuity) To pay for your child’s education, you wish to have accumulated $25,000 at the end of 15 years. To do this, you plan on depositing an equal amount in the bank at the end of each year. If the bank is willing to pay 7% compounded annually, how much must you deposit each year to obtain your goal?

5-15A. (Solving for r in Compound Interest) If you were offered $2,376.50 ten years from now in return for an investment of $700 currently, what annual rate of interest would you earn if you took the offer?

5-16A. (Present Value and Future Value of an Annuity) In 10 years, you plan to retire and buy a house in Marco Island, Florida. The house you are looking at currently costs $125,000 and is expected to increase in value each year at a rate of 5%. Assuming you can earn 10% annually on your investment, how much must you invest at the end of each of the next 10 years to be able to buy your dream home when you retire?



5-17A. (Compound Value) The Knutson Corporation needs to save $15 million to retire a $15 million mortgage that matures on December 31, 2002. To retire this mortgage, the company plans to put a fixed amount into an account at the end of each year for 10 years, with the first payment occurring on December 31, 1993. The Knutson Corporation expects to earn 10% annually on the money in this account. What annual contribution must it make to this account to accumulate the $15 million by December 31, 2002?

5-18A. (Compound Interest with Nonannual Periods) After examining the various personal loan rates available to you, you find that you can borrow funds from a finance company at 24% compounded monthly or 26% compounded annually. Which alternative is the most attractive?

5-19A. (Present Value of an Uneven Stream of Payments) You are given three investment alternatives to analyze. The cash flows from these three investments are as follows:

Investment End of Year A B C

1 $15,000 $20,000 2 15,000 3 15,000 4 15,000 5 15,000 $15,000 6 15,000 60,000 7 15,000 8 15,000 9 15,000 10 15,000 20,000

Assuming a 20% discount rate, find the present value of each investment.

5-20A. (Present Value) The Shin Corporation is planning to issue bonds that pay no interest but can be converted into $1,000 at maturity, 8 years from their purchase. To price these bonds competitively with other bonds of equal risk, it is determined that they should yield 9%, compounded annually. At what price should the Shin Corporation sell these bonds?

5-21A. (Perpetuities) What is the present value of the following?

a. A $400 perpetuity discounted back to the present at 9%

b. A $1,500 perpetuity discounted back to the present at 13%

c. A $150 perpetuity discounted back to the present at 10%

d. A $100 perpetuity discounted back to the present at 6%



5-22A. (Solving for n with Nonannual Periods) About how many years would it take for your investment to grow sevenfold if it were invested at 10% compounded annually?

5-23A. (Complex Present Values) How much do you have to deposit today so that beginning 11 years from now you can withdraw $10,000 a year for the next 5 years (periods 11 through 15) plus an additional amount of $15,000 in that last year (period 15)? Assume an interest rate of 7%.

5-24A. (Loan Amortization) On December 31, Loren Billingsley bought a yacht for $60,000, paying $15,000 down and agreeing to pay the balance in 10 equal annual installments that include both principal and 9% interest on the declining balance. How big would the annual payments be?

5-25A. (Solving for r in an Annuity) You lend a friend $45,000, which your friend will repay in 5 equal annual payments of $9,000 with the first payment to be received one year from now. What rate of return does your loan receive?

5-26A. (Solving for r in Compound Interest) You lend a friend $15,000, for which your friend will repay you $37,313 at the end of 5 years. What interest rate are you charging your “friend”?

5-27A. (Loan Amortization) To purchase some new machinery, a firm borrows $30,000 from the bank at 13% compounded annually. This loan is to be repaid in equal annual installments at the end of each year over the next 4 years. How much will each annual payment be?

5-28A. (Present Value Comparison) You are offered $1,000 today, $10,000 in 12 years, or $25,000 in 25 years. Assuming that you can earn 11% on your money, which should you choose?

5-29A. (Compound Annuity) You plan to buy some property in Florida five years from today. To do this, you estimate that you will need $30,000 at that time for the purchase. You would like to accumulate these funds by making equal annual deposits in your savings account, which pays 10% annually. If you make your first deposit at the end of this year and you would like your account to reach $30,000 when the final deposit is made, what will be the amount of your deposit?

5-30A. (Complex Present Value) You would like to have $75,000 in 15 years. To accumulate this amount, you plan to deposit an equal sum in the bank each year, which will earn 8% interest compounded annually. Your first payment will be made at the end of the year.

a. How much must you deposit annually to accumulate this amount?

b. If you decide to make a lump-sum deposit today instead of the annual deposits, how large should this lump-sum deposit be? (Assume you can earn 8% on this deposit.)

c. At the end of 5 years you will receive $20,000 and deposit this in the bank toward your goal of saving $75,000 at the end of 15 years. In addition to this



deposit, how much must you deposit in equal annual deposits to reach your goal? (Again, assume that you can earn 8% on this deposit.)

5-31A. (Comprehensive Present Value) You are trying to plan for retirement in 10 years, and currently you have $150,000 in a savings account and $250,000 in stocks. In addition, you plan to add to your savings by depositing $8,000 per year in your savings account at the end of each of the next 5 years and then $10,000 per year at the end of each year for the final 5 years until retirement.

a. Assuming your savings account returns 8% compounded annually while your investments in stocks will return 12% compounded annually, how much will you have at the end of 10 years? (Ignore taxes.)

b. If you expect to live for 20 years after you retire, and at retirement you deposit all of your savings in a bank account paying 11 percent, how much can you withdraw each year after retirement (20 equal withdrawals beginning one year after you retire) to end up with a zero balance at death?

5-32A. (Loan Amortization) On December 31, Eugene Chung borrowed $200,000, agreeing to repay this sum in 20 equal annual installments that included both the principal and 10% interest on the declining balance. How large will the annual payments be?

5-33A. (Loan Amortization) To buy a new house, you must borrow $250,000. To do this, you take out a $250,000, 30-year, 9% mortgage. Your mortgage payments, which are made at the end of each year (one payment each year), include both principal and 9% interest on the declining balance. How large will your annual payments be?

5-34A. (Present Value) The state lottery’s million-dollar payout provides for one million dollars to be paid over 24 years in $40,000 amounts. The first $40,000 payment is made immediately with the remaining 24 payments occurring at the end of each of the next 24 years. If 10% is the appropriate discount rate, what is the present value of this stream of cash flows? If 20% is the appropriate discount rate, what is the present value of the cash flows?

5-35A. (Solving for i in Compound Interest—Financial Calculator Needed) In March 1963, issue number 39 of Tales of Suspense was issued. The original price for that issue was 12 cents. By March of 1997, 34 years later, the value of this comic book had risen to $2,000. What annual rate of interest would you have earned if you had bought the comic in 1963 and sold it in 1997?

5-36A. (Comprehensive Present Value) You have just inherited a large sum of money and you are trying to determine how much you should save for retirement and how much you can spend now. For retirement, you will deposit today (January 1, 1997) a lump sum in a bank account paying 10% compounded annually. You do not plan to touch this deposit until you retire in 5 years (January 1, 2002), and you plan to live for 20 additional years and then to drop dead on December 31, 2021. During your retirement, you would like to receive income of $60,000 per year to be received on the first day of each year, with the first payment on January 1, 2002, and the last payment on January 1, 2021. Complicating this objective is your desire to have one final 3-year fling during



which time you’d like to track down all the original members of the “Mr. Ed Show” and “The Monkeys” and get their autographs. To finance this you want to receive $300,000 on January 1, 2017 and nothing on January 1, 2018, and January 1, 2019, as you will be on the road. In addition, after you pass on (January 1, 2022), you would like to have a total of $100,000 to leave to your children.

a. How much must you deposit in the bank at 10% on January 1, 1997 in order to achieve your goal? (Use a time line in order to answer this question.)

b. What kinds of problems are associated with this analysis and its assumptions? SOLUTIONS TO ALTERNATIVE PROBLEMS

5-1A. a. FVn = PV (1 + r)n

FV11 = $4,000(1 + 0.09)11

FV11 = $4,000 (2.580)

FV11 = $10,320


FV10 = $8,000 (1 + 0.08)10

FV10 = $8,000 (2.159)

FV10 = $17,272


FV12 = $800 (1 + 0.12)12

FV12 = $800 (3.896)

FV12 = $3,117


FV6 = $21,000 (1 + 0.05)6

FV6 = $21,000 (1.340)

FV6 = $28,140



5-2A. a. FVn = PV (1 + r)n

$1,043.90 = $550 (1 + 0.06)n

1.898 = FUTURE VALUE FACTOR6%, n yr. Thus, n = 11 years (because the value of 1.898 occurs in the 11-year

row of the 6% column of Appendix B).


$88.44 = $40 (1 + .12)n

2.211 = FUTURE VALUE FACTOR12%, n yr. Thus, n = 7 years


$614.79 = $110 (1 + 0.24)n



$78.30 = $60 (1 + 0.03)n


5-3A a. FVn = PV (1 + r)n

$1,898.60 = $550 (1 + r)13

3.452 = FUTURE VALUE FACTORi%, 13 yr. Thus, i = 10% (because the Appendix B value of 3.452 occurs in the

12-year row in the 10% column)


$406.18 = $275 (1 + r)8

1.477 = FUTURE VALUE FACTORi%, 8 yr. Thus, i = 5%


$279.66 = $60 (1 + r)20

4.661 = FUTURE VALUE FACTORi%, 20 yr.

Thus, i = 8%




$486.00 = $180 (1 + r)6

2.700 = FUTURE VALUE FACTORi%, 6 yr. Thus, i = 18%

5-4A. a. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $800 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 10)1.0 1(

1

PV = $800 (0.386)

PV = $308.80

b. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $400 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 6)06.0 1(

1

PV = $400 (0.705)

PV = $282.00

c. PV = FVn ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ n)1 1(

1

PV = $1,000 ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 8)05.0 1(

1

PV = $1,000 (0.677)

PV = $677

d. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $900 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 9)2.0 1(

1

PV = $900 (0.194)

PV = $174.60



5-5A. a. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

FV = $500 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 10

0 t

t0.06 1

FV10 = $500 (13.181)

FV10 = $6,590.50

b. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

FV5 = $150 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 5

0 t

t11.0 1

FV5 = $150 (6.228)

FV5 = $934.20

c. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

FV7 = $35 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 8

0 t

t0.07 1

FV7 = $35 (10.260)

FV7 = $359.10

d. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

FV3 = $25 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 3

0 t

t02.01

FV3 = $25 (3.060)

FV3 = $76.50



5-6A. a. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $3,000 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

10

1 t t08.0 1

1

PV = $3,000 (6.710)

PV = $20,130

b. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $50 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛∑= +

3

1 t t0.03 11

PV = $50 (2.829)

PV = $141.45

c. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $280 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

8

1 t t07.0 1

1

PV = $280 (5.971)

PV = $1,671.88

d. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $600 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

10

1 t t0.1 1

1

PV = $600 (6.145)

PV = $3,687.00



5-7A. a. FVn = PV (1 + r)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


FV15 = $20,000 (1 + 0.09)15

FV15 = $20,000 (3.642)

FV15 = $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


FV15 = $20,000 (1 + 0.11)15

FV15 = $20,000 (4.785)

FV15 = $95,700

c. There is a positive relationship between both the interest rate used to compound a present sum and the number of years for which the compounding continues, and the future value of that sum.

5-8A. FVn = PV mnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

Account PV r m n mnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

PVmnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

Korey Stringer 2,000 12% 6 2 1.268 $2,536 Eric Moss 50,000 12 12 1 1.127 56,350 Ty Howard 7,000 18 6 2 1.426 9,982 Rob Kelly 130,000 12 4 2 1.267 164,710 Matt Christopher 20,000 14 2 4 1.718 34,360 Juan Porter 15,000 15 3 3 1.551 23,265

5-9A. a. FVn = PV (1 + r)n

FV5 = $6,000 (1 + 0.06)5

FV5 = $6,000 (1.338)

FV5 = $8,028

b. FVn = PV mnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 2(5)0.061

2⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 (1 + 0.03)10

FV5 = $6,000 (1.344)

FV5 = $8,064



FVn = PV mnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 6(5)0.061

6⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 (1 + 0.01)30

FV5 = $6,000 (1.348)

FV5 = $8,088


FV5 = $6,000 (1 + 0.12)5

FV5 = $6,000 (1.762)

FV5 = $10,572

FV5 = PV ⎟⎠⎞

⎜⎝⎛ +

mr1

mn

FV5 = $6,000 2(5)0.121

2⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 (1 + 0.06)10

FV5 = $6,000 (1.791)

FV5 = $10,746

FV5 = PV mnr1

m⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 6(5)0.121

2⎛ ⎞+⎜ ⎟⎝ ⎠

FV5 = $6,000 (1 + 0.02)30

FV5 = $6,000 (1.811)

FV5 = $10,866


FV12 = $6,000 (1 + 0.06)12

FV12 = $6,000 (2.012)

FV12 = $12,072

e. An increase in the stated interest rate will increase the future value of a given sum. Likewise, an increase in the length of the holding period will increase the future value of a given sum.



5-10A. Annuity A: PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $8,500 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

12

1 t t0.12 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% opportunity cost, this annuity has value and should be accepted.

Annuity B: PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $7,000 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

25

1 t t12.0 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 a 12% opportunity cost, this annuity should not be accepted.

Annuity C: PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

PV = $8,000 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

20

1 t t12.0 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% opportunity cost, this annuity should not be accepted.

5-11A. Year 1: FVn = PV (1 + r)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


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 is earned on the new sum. Book sales growth was compounded; thus, the first year the growth was 15% of 10,000 books, the second year 15% of 11,500 books, and the third year 15% of 13,220 books.

5-12A. FVn = PV (1 + r)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)


FV3 = 41(1 + 0.12)3

FV3 = 41(1.405)



FV3 = 57.605 Home Runs in 1983.

FV3 = 41(1 + 0.12)4

FV4 = 41(1.574)

FV4 = 64.534 Home Runs in 1984 (for what at that time would have been a new major league record).

FV5 = 41(1 + 0.12)5

FV5 = 41(1.762)

FV5 = 72.242 Home Runs in 1985 (again for a new major league record, but not up to Barry Bonds’ 73 homers in 2001).

Actually, Reggie never hit more than 41 home runs in a year. In 1982, he hit 15 only; in 1983 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.

5-13A. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

$120,000 = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

+∑=

25

1 t 1.0 11

t

$120,000 = PMT(9.077)

Thus, PMT = $13.220.23 per year for 25 years

5-14A. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

$25,000 = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 15

0 t

t07.0 1

$25,000 = PMT(25.129)

Thus, PMT = $994.87



5-15A. FVn = PV (1 + r)n

$2,376.50 = $700 (FUTURE VALUE FACTORr%, 10 yr.)

3.395 = FUTURE VALUE FACTORi%, 10 yr Thus, i = 13%

5-16A. The value of the home in 10 years

FV10 = PV (1 + .05)10

= $125,000(1.629)

= $203,625

How much must be invested annually to accumulate $203.625?

$203,625 = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 10

0 t

t10. 1

$203,625 = PMT(15.937)

PMT = $12,776.87

5-17A. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

$15,000,000 = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 10

0 t

t.10 1

$15,000,000 = PMT(15.937)

Thus, PMT = $941,206

5-18A. One dollar at 24.0% compounded monthly for one year

FVn = PV (1 + r)n

FV12 = $1(1 + .02)12 = $1(1.268) = $1.268 One dollar at 26.0% compounded annually for one year

FVn = PV (1 + r)n

FV1 = $1(1 + .26)1 = $1(1.26) = $1.26 The loan at 26% compounded monthly is more attractive.



5-19A. Investment A

PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

i

= $15,000 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

5

1 t t20 . 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.

PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

= $15,000 ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

6

1 t t20. 1

1

= $15,000(3.326)

= $49,890—then discount the equivalent sum back to present.

PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

= $49,890 ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 4)20.1(

1

= $49,890(.482)

= $24,046.98



Investment C

PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

= $20,000 ⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 1)20.1(

1+ $60,000

⎟⎟⎠

⎞⎜⎜⎝

⎛

+ 6)20.1(

1

+ $20,000 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 10)20.1(

1

= $20,000(.833) + $60,000(.335) + $20,000(.162)

= $16,660 + $20,100 + $3,240

= $40,000

5-20A. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $1,000 ⎟⎟⎠

⎞

⎜⎜⎝

⎛

+ 8)09.1(

1

= $1,000(.502)

= $502

5-21A. a. PV = r

PP

PV = $4000.09

PV = $4,444

b. PV = r

PP

PV = $1,5000.13

PV = $11,538

c. PV = r

PP

PV = $1500.10

PV = $1,500



d. PV = r

PP

PV = $1000.06

PV = $1,667

5-22A. FVn = PV mn

mr 1 ⎟⎠⎞

⎜⎝⎛ +

7 = 12n0.101

2⎛ ⎞+⎜ ⎟⎝ ⎠

7 = (1 + 0.05)2n

7 = FUTURE VALUE FACTOR5%, 2n yr.

A value of 7.040 occurs in the 5% column and 40-year row of the table in Appendix B. Therefore, 2n = 40 years, and n = approximately 20 years.

5-23A. The Present value of the $10,000 annuity over years 11-15.

PV = PMT ( ) ( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

+∑∑==

10

1 t t

15

1 t t .07 1

1 - 07. 1

1

= $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)07.1(

1

= $15,000(.362)

= $5,430

Thus, you would have to deposit $20,840 + $5,430 or $26,270 today.

5-24A. PV = PMT ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

10

1 t t09. 1

1

$45,000 = PMT(6.418)

PMT = $7,012



5-25A. PV = PMT ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

5

1 t ti 1

1

$45,000 = $9,000 (ANNUITY PRESENT VALUE FACTORr%, 5 yr.)

5.0 = ANNUITY PRESENT VALUE FACTORi%, 5 yr. i = 0%

5-26A. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr)(11

$15,000 = $37,313 (PRESENT VALUE FACTORr%, 5 yr.)

.402 = PRESENT VALUE FACTOR20%, 5 yr. Thus, i = 20%

5-27A. PV = PMT ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+∑=

n

1 t tr 1

1

$30,000 = PMT ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

4

1 t t13. 1

1

$30,000 = PMT(2.974)

PMT = $10,087

5-28A. The present value of $10,000 in 12 years at 11% is:

PV = FVn t

1(1 r)

⎛ ⎞⎜ ⎟+⎝ ⎠

PV = $10,000 12

1(1 .11)

⎛ ⎞⎜ ⎟+⎝ ⎠

PV = $10,000 (.286)

PV = $2,860



The present value of $25,000 in 25 years at 11% is:

PV = $25,000 25

1(1 .11)

⎛ ⎞⎜ ⎟+⎝ ⎠

= $25,000 (.074)

= $1,850

Thus, take the $10,000 in 12 years.

5-29A. FVn = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

$30,000 = PMT ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 5

0 t

t10. 1

$30,000 = PMT(6.105)

PMT = $4,914

5-30A. a. FV = ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

$75,000 = ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 15

0 t

t08. 1

$75,000 = PMT (ANNUITY FUTURE VALUE FACTOR8%, 15 yr.)

$75,000 = PMT(27.152)

PMT = $2,762.23 per year

b. PV = FVn ⎟⎟⎠

⎞⎜⎜⎝

⎛+ nr) (11

PV = $75,000 (PRESENT VALUE FACTOR8%, 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 + r)n

FVn = $20,000 (FUTURE VALUE FACTOR8%. 10 yr.)

FV10 = $20,000(2.159) = $43,180

Thus, only $31,820 need be accumulated by annual deposit.

FV = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

$31,820 = PMT (ANNUITY FUTURE VALUE FACTOR8%, 15 yr.)

$31,820 = PMT [27.152]

PMT = $1,171.92 per year

5-31A. This problem can be subdivided into (1) the compound value of the $150,000 in the savings account, (2) the compound value of the $250,000 in stocks, (3) the additional savings due to depositing $8,000 per year in the savings account for 10 years, and (4) the additional 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 c and d.)

a. Future value of $150,000 FV10 = $150,000 (1 + .08)10 FV10 = $150,000 (2.159) FV10 = $323,850 b. Future value of $250,000 FV10 = $250,000 (1 + .12)10 FV10 = $250,000 (3.106) FV10 = $776,500 c. Compound annuity of $8,000, 10 years

FV10 = PMT ( ) ⎟⎠

⎞⎜⎝

⎛+∑

−

=

1 n

0 t

tr 1

= $8,000 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 10

0 t

t08. 1

= $8,000 (14.487)

= $115,896



d. Compound annuity of $2,000 (years 6-10)

FV5 = $2,000 ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

−

=

1 5

0 t

t08. 1

= $2,000 (5.867)

= $11,734

At the end of 10 years, you will have $323,850 + $776,500 + $115,896 + $11,734 = $1,227,980.

PV = PMT ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+∑=

20

1 t t11. 1

1

$1,227,980 = PMT (7.963)

PMT = $154,210.72

5-32A. PV = PMT (ANNUITY PRESENT VALUE FACTORr%, n yr.)

$200,000 = PMT (ANNUITY PRESENT VALUE FACTOR10%, 20 yr.)

$200,000 = PMT(8.514)

PMT = $23,491

5-33A. PV = PMT (ANNUITY PRESENT VALUE FACTORr%, n yr.)

$250,000 = PMT (ANNUITY PRESENT VALUE FACTOR9%, 30 yr.)

$250,000 = PMT(10.274)

PMT = $24,333

5-34A. At 10%:

PV = $40,000 + $40,000 (ANNUITY PRESENT VALUE FACTOR10%, 24 yr.)

PV = $40,000 + $40,000 (8.985)

PV = $40,000 + $359,400

PV = $399,400



At 20%:

PV = $40,000 + $40,000 (ANNUITY PRESENT VALUE FACTOR20%, 24 yr.)

PV = $40,000 + $40,000 (4.938)

PV = $40,000 + $197,520

PV = $237,520

5-35A. FV = PMT (FUTURE VALUE FACTORr%, n yr.)

$2,000 = .12(FUTURE VALUE FACTORi%, 35 yr.)

Solving using a financial calculator:

r = 32.70%

5-36A. a.

$60,000 per year $300,000

$60,000

$100,000

1/06 1/11 1/16 1/21 1/26 1/31

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 (ANNUITY PRESENT VALUE FACTOR10%, 19 yr.

– ANNUITY PRESENT VALUE FACTOR10%, 4 yr.)


+ $60,000 (PRESENT VALUE FACTOR10%, 23 yr. + PRESENT VALUE F


= $60,000 (8.365-3.170) + $300,000 (.149)

+ $60,000 (0.112 + .102) + $100,000 (.092)

= $311,700 + $44,700 + $12,840 + $9,200

= $378,440

b. If you live longer than expected, you could end up with no money later on in life.



5-37A. rate (r) = 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. 5-38A. 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 = r = 11.18%

Excel formula: =RATE(number of periods, payment, present value, future value, type, guess)

Notice that present value ($30,000) took on a negative value.

5-39A. To buy a new house you take out a 25 year mortgage for $300,000. What will your monthly interest rate payments be if the interest rate on your mortgage is 8 percent?

Two things to keep in mind when you're working this problem: first, you'll have to convert the annual rate of 8 percent into a monthly rate by dividing it by 12, and second, you'll have to convert the number of periods into months by multiplying 25 times 12 for a total of 300 months. Excel formula: =PMT(rate, number of periods, present value, future value, type) rate (r) = 8%/12 number of periods (n) = 300 present value (PV) = $300,000 future value (FV) = $0 type (0 = at end of period) = 0 monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them.



You can also use Excel to calculate the interest and principal portion of any loan amortization payment. You can do this using the following Excel functions:

Calculation: Formula: Interest portion of payment =IPMT(rate, period, number of periods, present value,

future value, type)

Principal portion of payment =PPMT(rate, period, number of periods, present value, future value, type)

Where period refers to the number of an individual periodic payment.

Thus, if you would like to determine how much of the 48th monthly payment went toward interest and principal you would solve as follows:

Interest portion of payment 48: ($1,884.37)

The principal portion of payment 48: ($431.08)

5-40A. a. N = 379

I/Y = 6

PV = –24

PMT = 0

CPT FV = 93.57 billion dollars

b. For this problem, first make P/Y = 12

N = 379 x 12 = 4,548

I/Y = 6

PV = –24

PMT = 0

CPT FV = 170.40 billion dollars

c. N = 10

I/Y = 10

CPT PV = –77.108 billion dollars PMT = 0

FV = 200. billion



d. N = 10

CPT I/Y = 14.87% PV = –15 billion

PMT = 0

FV = 60. billion

e. N = 40

I/Y = 7

PV = –30. billion

CPT PMT = 2.25 billion dollars

FV = 0

5-41A. 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 6 years?

N = 6

I/Y = 7.5

CPT PV = –11,605.50 dollars PMT = 0

FV = 17,910.78

5-42A. N = 45

I/Y = 8.75

PV = 0


FV = 1,000,000



5-43A. First, we must calculate what Mr. Burns will need in 20 years. Once we know how much he will need, we can calculate how much he needs to deposit each year in order to come up with that amount (note: once you calculate the present value, 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

5-44A. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money 5 years ago and we want to know what it will be worth in 20 years)?

N = 25

I/Y = 7.5

PV = –100,000

PMT = 0


Now we determine what the additional $1,500 per year will grow to (note that since Homer will be making these investments at the beginning of each year for 20 years we have an annuity due, thus, once you calculate the present value, you must multiply your answer, in this case $64,957.02 times (1 + r)):

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



5-45A. 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 = 6.2

PV = –25,000

CPT PMT = 485.65 dollars

FV = 0


N = 36

CPT I/Y = 11.62%

PV = –999

PMT = 33

FV = 0

5-47A. 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 = 4.9

PV = –25,000


Now, calculate how much the monthly payments would be if Suzie took the $1,000 cash 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




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 N = 41.49 (rounded up to 42 months)

I/Y = 12.9

PV = -5000

PMT = 150

FV = 0

5-50A. 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 = –140,000

PMT = 0

FV = 420,000




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


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

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Chapter 05