Ch5 - Solutions

©2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 5 Time Value of Money: The Basics

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

FV10 = $5000(1 + 0.10)10

FV10 = $5000 (2.594)

FV10 = $12,970

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

FV7 = $8000 (1 + 0.08)7 FV7 = $8000 (1.714)

FV7 = $13,712

(c) FV12 = PV (1 + i)n

FV12 = $775 (1 + 0.12)12

FV12 = $775 (3.896)

FV12 = $3019.40

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

FV5 = $21,000 (1 + 0.05)5

FV5 = $21,000 (1.276)

FV5 = $26,796.00

5-2. (a) FVn = PV (1 + i)n compounded forward for 1 year at 6%

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

FV1 = $10,000 (1.06)

FV1 = $10,600

compounded forward for 5 years at 6%

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

FV5 = $10,000 (1.338)

FV5 = $13,380

compounded forward for 15 years at 6% FV15 = $10,000 (1 + 0.06)15

FV15 = $10,000 (2.397)

FV15 = $23,970

Solutions to End of Chapter Problems—Chapter 5 131


(b) FVn = PV (1 + i)n compounded forward for 1 year at 8%

FV1 = $10,000 (1 + 0.08)1 FV1 = $10,000 (1.080) FV1 = $10,800


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 forward 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

compounded forward for 15 years at 10% FV15 = $10,000 (1 + 0.1)15

FV15 = $10,000 (4.177)

FV15 = $41,770 (c) There is a positive relationship between both the interest rate used to compound a present sum

and the number of years for which the compounding continues and the eventual future sum that results.

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

N = ln (FVn/PV) / ln(1 + i)

N = ln(30,000/20,000) / ln (1.07)

N = 5.9918 years (b) FVn = PV (1 + i)n

FV10.25 = 20,000 (1.07)10.25

FV10.25 = 40,014.16

132 Titman/Keown/Martin • Financial Management, Eleventh Edition


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

N = ln (FVn/PV) / ln(1 + i)

N = ln (30,000/20,000)/ln (1.11)

N = 3.89 years

N = ln (FVn/PV)/ln(1 + i)

N = ln (30,000/20,000)/ln (1.03)

N = 13.72 years (d) There is an inverse relationship between the interest rate and the time required to achieve a

certain future sum as a result of compounded interest.

5-4. FVn = PV (1 + i)n

FV200 = 12,345(1.0398)200

FV200 = 30,300,773.41

5-5. FVn = PV (1 + im )mn

Account PV i m n (1 + im )mn PV (1 + i

m )mn

Theodore Logan III $1,000 10% 1 10 2.594 $2,594

Vernell Coles 95,000 12% 12 1 1.127 107,065

Thomas Elliott 8,000 12% 6 2 1.268 10,144

Wayne Robinson 120,000 8% 4 2 1.172 140,640

Eugene Chung 30,000 10% 2 4 1.477 44,310

Kelly Cravens 15,000 12% 3 3 1.423 21,345

5-6. (a) FVn = PV (1 + i)n

FV5 = $5000 (1 + 0.06)5

FV5 = $5000 (1.338)

FV5 = $6690

(b) FVn = PV (1 + im )mn

FV5 = $5,000 2 5

0.061

2

× +

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

FV5 = $5,000 (1.344)

FV5 = $6,720

FVn = PV (1 + im )mn

FV5 = 5,000 6 5

0.061

2

× +

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

FV5 = $6,740



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

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

FV5 = $5,000 (1.762)

FV5 = $8,810

FV5 = PV (1 + im )mn

FV5 = $5000 2 5

0.121

2

× +

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

FV5 = $5,000 (1.791)

FV5 = $8,955


FV5 = $5,000 6 5

0.121

2

× +

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

FV5 = $5,000 (1.811)

FV5 = $9,055

(d) FVn = PV (1 + i)n FV12 = $5,000 (1 + 0.06)12

FV12 = 5,000 (2.012)

FV12 = $10,060 (e) An increase in the stated interest rate will increase the future value of a given sum. Likewise, an

increase in the length of the holding period will increase the future value of a given sum.

5-7. (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 + im )mn

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 (1 + im )mn

FV5 = $6,000 6 5

0.061

6

× +

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 = $6000 2 5

0.121

2

× +

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

FV5 = $6,000 (1.791)

FV5 = $10,746


FV5 = $6,000 6 5

0.121

6

× +

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

FV5 = $6,000 (1.811)

FV5 = $10,866

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

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

FV12 = $6,000 (2.012)

FV12 = $12,072 (e) An increase in the stated interest rate will increase the future value of a given sum. Likewise,

an increase in the length of the holding period will increase the future value of a given sum. Furthermore, at any stated annual interest rate, the more compounding periods per year, the higher the future value of a given sum.

5-8. Year 1: FVn = PV (1 + i)n

FV1 = 15,000(1 + 0.20)1

FV1 = 15,000(1.20)

FV1 = 18,000 books

Year 2: FVn = PV (1 + i)n FV2 = 15,000(1 + 0.20)2

FV2 = 15,000(1.44)

FV2 = 21,600 books

Year 3: FVn = PV (1 + i)n

FV3 = 15,000(1 + 0.20)3

FV3 = 15,000(1.728)

FV3 = 25,920 books

The sales trend graph is not linear because this is a compound growth trend. With compound interest, interest paid on the investment during the first period is added to the principal of the second period, and interest in the second period is earned on the new total. 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.



5-9. Year 1: FVn = PV (1 + i)n

FV1 = 10,000(1 + 0.15)1

FV1 = 10,000(1.15)

FV1 = 11,500 headphones


FV2 = 10,000(1 + 0.15)2

FV2 = 10,000(1.322)



FV3 = 10,000(1 + 0.15)3

FV3 = 10,000(1.521)


The sales trend graph is not linear because this is a compound growth trend. With compound interest, interest paid on the investment during the first period is added to the principal of the second period, and interest in the second period is earned on the new total. Headphone sales growth was compounded; thus the first year the growth was 15% of 10,000 headphones, the second year 15% of 11,500 headphones, and the third year 15% of 13,220 headphones.



5-10. FVn = PV (1 + i)n

FV35 = 3,500(1.11)35

FV35 = 3,500 (1.575)

FV35 = 135,012

FV40 = 3500 (1.11)40

FV40 = 3500 (65.001)

FV40 = 227,503

5-11. YEAR Beginning Value Compound Interest End Value 1 10,000 1,100 11,100 2 11,100 1,221 12,321 3 12,321 1,355.31 13,676.31

Simple interest is the same 1,100 per year based on the original principal. The compound interest in year 3 is 1,355.31, which is 255.31 more than simple interest.

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

PV = 2,000,000/(1.04)35 PV = 506,830.94

N@14% = ln (FVn/PV)/ln (1 + i)

N = ln (2,000,000 / 506,830.94)/ln (1.14)

N = 1.373/0.131

N = 10.48 years

5-13. (a) N = ln (FVn/PV)/ln (1+ i)

N = ln (1,039.5/500)/ln (1.05)

N = 0.732/ 0.049

N = 15 years

(b) N = ln (FVn/PV)/ln (1 + i)

N = ln (53.87/35)/ln (1.09) N = 0.432/0.086

N = 5 years

(c) N = ln (FVn/PV)/ln (1 + i)

N = ln (298.60/100)/ln (1.20)

N = 1.094/0.182

N = 6 years

(d) N = ln (FVn/PV)/ln (1 + i)

N = ln (78.76/53)/ln (1.02)

N = 0.396/0.02

N = 19.8 years



5-14. (a) FVn = PV (1 + i)n

$1,948 = $500 (1 + i)12

3.896 = (1 + i)12

(3.896)1/12 = 1 + i

1.12 = 1+i

i = 0.12 or 12%

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

$422.10 = $300 (1 + i)7

1.407 = (1 + i)7

(1.407)1/7 = 1 + i

1.05 = 1 + i

i = 0.05 or 5%

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

$280.20 = $50 (1 + i)20

5.604 = (1 + i)20

(5.604)1/20 = 1 + i

i = 0.09 or 9%

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

$497.60 = $200 (1 + i)5

2.488 = (1 + i)5 (2.488)1/5 = 1 + i

i = 0.20 or 20%

5-15. (a) PV = 1

(1 )n nFV

i

+

PV = $800 10

1

(1 0.1)

+

PV = $800 (0.386)

PV = $308.80

(b) PV = FVn 1

(1 )ni

+

PV = $300 5

1

(1 0.05)

+

PV = $300 (0.784)

PV = $235.20

(c) PV = FVn 1

(1 )ni

+

PV = $1,000 8

1

(1 0.03)

+

PV = $1,000 (0.789)



PV = $789

(d) PV = FVn 1

(1 )ni

+

PV = $1,000 8

1

(1 0.02)

+

PV = $1,000 (0.233)

PV = $233

5-16. FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (12,000/4,510)1/7 − 1

I = 0.15 or 15%

5-17. (a) PV = FVn 1

(1 )ni

+

PV = 1,000/(1.10)30

PV = 57.31

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

I = (FVn/PV)1/n − 1

I = (1000/365)1/30

I = 3.42%

5-18. FVn = PV (1 + i)n

N = ln (FVn/PV)/ln(1 + i)

N = ln (330,000/45,530)/ln (1.045) N = 45 years

5-19. PV = FVn 1

(1 )ni

+

PV = 398,930/(1.07)28

PV = 59,999.95

5-20. FVn = PV (1 + i)n

I = (FVn/PV)1/n −1

I = (5,200/7,600)1/7 − 1

I = −5.28%

5-21. (a) FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (20,000/900)1/30 − 1 I = 10.89%



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

I = (FVn/PV)1/n − 1

I = (3500/900)1/10 − 1

I = 14.55%

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

I = (FVn/PV)1/n − 1

I = (20,000/3,500)1/20 − 1

I = 9.11%

5-22. FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (1079.5/500)1/10 − 1

Thus, i = 8%

5-23. FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (2376.5/700)1/10 − 1

Thus, i = 13%

5-24. FVn = PV (1 + i)n

N = ln (FVn/PV)/ln (1 + i)

Annual rate = 16% with semiannual compounding

Periodic rate = (16%/2) = 8%

N = ln(4/1)/ln(1.08)

N = 18.01 semiannual periods = 9.00 years

5-25. FVn = PV (1 + i)n

N = ln (FVn/PV)/ln (1 + i) Annual Rate = 10% with semiannual compounding

Periodic Rate = (10%/2) = 5%

N = ln (7/1)/ln(1.05)

N = 39.88 semiannual periods = 19.94 years

5-26. I = (FVn/PV)1/n − 1

I = (27,027/10,000)1/5 − 1

Thus, i = 22%

5-27. I = (FVn/PV)1/n − 1

I = (37,313/15,000)1/5 − 1

Thus, i = 20%



5-28. PV = FVn 1

(1 )ni

+

PV = 300,000/(1.11)13

PV = 77,254.27

The better choice is the 100,000 today.

5-29. PV = FVn 1

(1 )ni

+

For 10,000 12 years from now:

PV = 10,000/(1.11)12

PV = 2,858.40

For 25,000 25 years from now:

PV = 25,000/(1.11)25

PV = 1,840.20

1,000 today vs. 10,000 12 years from now (PV = 2,858.40) vs. 25,000 25 years from now (PV = 1,840.20), the best choice is 10,000 in 12 years.

5-30. FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (9500/0.12)1/43 − 1

I = 29.99%

or solve with financial calculator:

N = 43 CPT I/Y = 29.99% PV = − 0.12

PMT = 0

FV = 9,500

5-31. FVn = PV (1 + i)n

I = (FVn/PV)1/n − 1

I = (9,000/0.12)1/47 − 1

I = 26.98%

or solve with financial calculator:

N = 47

CPT I/Y = 26.98% PV = − 0.12

PMT = 0

FV = 9,000



5-32. 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

5-33. 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)

= FV(0.08, 7, 0, −900, 0)

= 1,542.44

Notice that present value ($900) was entered as a negative value in the Excel formula.

5-34. 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

i = 11.18%

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

= RATE(20,0, −30000,250000,0)

= 0.1118 = 11.18%

Notice that present value ($30,000) was entered as a negative value in the excel formula.

5-35. EAR = (1 + i/m)mn − 1

EAR = (1 + 0.12/12) 12 − 1

EAR = 0.127 or 12.7%

The loan at 12% compounded monthly is more attractive than the 13% loan compounded annually.

5-36. EAR = (1 + i/m)mn − 1

EAR = (1 + 0.24/12)12 − 1

EAR = 0.268 or 26.8%

The loan at 26% compounded annually is more attractive.



5-37. Since the first part of this problem involves daily 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%

Now let’s look at monthly compounding; 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 EAR for both CDs.

EAR = (1 + i/m)mn − 1

EAR = (1 + (0.0495/365))365 − 1

EAR = 5.0742%

EAR = (1 + i/m)mn − 1

EAR = (1 + (0.05/12))12 − 1

EAR = 5.1162%

5-38. EAR = (1 + i/m)mn − 1

EAR = (1 + (0.078/12))12 − 1

EAR = 8.08%

So, the better deposit rate is at Burns Bank (8.08% vs. 8.00%)

Date post:	23-Oct-2014
Category:	Documents
Upload:	chama2020
View:	652 times
Download:	3 times

Ch5 - Solutions

Documents