Chapter 4

NET PRESENT VALUE

FORMULA

Future value: FV= C0(1+r )t

Present value: PV= C t

(1+r )t

Effective Annual Rate: EAR= (1+ rm

)m

−1

Future value with EAR : FV= C0(1+EAR)T

Future value with EAR : PV= CT/(1+EAR)T

FV= C0erT

PV= C/r

Annuity

FV= C*[ (1+r )T−1

r¿

PV= C*[ 1−(1+r )−T

r¿

Growing Annuity

PV= C*[ 1−( 1+g1+r )T

r−g¿

PROBLEMS

1. Compute the future value of $1,000 compounded annually for

a. 10 years at 5 percent.

b. 10 years at 7 percent.

c. 20 years at 5 percent.

d. Why is the interest earned in part c not twice the amount earned in part a?

Solution:

FV= C0(1+r )t

a. FV= 1000 x (1+0.05)10 = 1628.89

b. FV= 1000 x (1+0.07)10= 1967.15

c. FV= 1000 x (1+0.05)20= 2653.3

d. Interest has compounded over the interest earned during first 10 years. The interest earned in first 10 years is 628.89, which was reinvested, which has compounded at 1.05 times each year. Hence the interest earned in part c is not twice the amount earned in part a, rather its more than double the amount (2.63 times).

2. Calculate the present value of the following cash flows discounted at 10 percent.

a. $1,000 received seven years from today.

b. $2,000 received one year from today.

c. $500 received eight years from today.

Solution:

PV= C t

(1+r )t

a. PV= 1000/ (1+0.10)7 = 513.16

b. PV= 2000/ (1+0.10)1= 1818.18

c. PV= 500/ (1+0.10)8= 233.25

3. Would you rather receive $1,000 today or $2,000 in 10 years if the discount rate is 8 percent?

Solution:

In order to compare these two, we need to calculate the present value of $2000.

PV= C t

(1+r )t

=2000

(1+.08)10

= 926.4

This value is less than $1000. Hence, $1000 should be received today instead of the 2nd offer.

4. The government has issued a bond that will pay $1,000 in 25 years. The bond will pay no interim coupon payments. What is the present value of the bond if the discount rate is 10 percent?

Solution:

PV of bond = Face value

(1+r )t

= 1000

(1+.1)25

= 92.3

5. A firm has an estimated pension liability of $1.5 million due 27 years from today. If the firm can invest in a risk-free security that has a stated annual interest rate of 8 percent, how much must the firm invest today to be able to make the $1.5 million payment?

Solution:

FV= C0(1+r )t

1.5 M= C0(1+0.08)27

C0= $187,780,23

The firm must invest $187,780,23 today.

6. You have won the Florida state lottery. Lottery officials offer you the choice of the following alternative payouts:

Alternative 1: $10,000 one year from now.

Alternative 2: $20,000 five years from now.

Which alternative should you choose if the discount rate is:

a. 0 percent?

b. 10 percent?

c. 20 percent?

d. What discount rate makes the two alternatives equally attractive to you?

Solution:

PV= C t

(1+r )t

Alternative 1(PV of $10,000 one

year from now)

Alternative 2(PV of $20,000 five

years from now)

Decision

a. 0 percent 10,000 20,000 Alternative 2b. 10 percent 9,090.91 12,418.43 Alternative 2c. 20 percent 8,333.33 8,037.55 Alternative 1

Let, at discount rate ‘r’, present value of the two alternatives will be equal.

1000

(1+r )1= 2000

(1+r )5

(1+r )5

(1+r )1= 2

(1+r) = 1.189

R= 0.189 or 18.9%

8. Suppose you bought a bond that will pay $1,000 in 20 years. No intermediate coupon payments will be made. If the appropriate discount rate for the bond is 8 percent,

a. what is the current price of the bond?

b. what will the price be 10 years from today?

c. what will the price be 15 years from today?

Solution:

a. Current price is equal to the present value.

PV= 1000

(1+.08)20

= 214.55b. Price of the bond 10 years from today:

FV10= 214.55*(1+.08)10 = 463.2c. Price of the bond 15 years from today:

FV10= 214.55*(1+.08)15 = 680.599. Ann Woodhouse is considering the purchase of a house. She expects that she will own the

house for 10 years and then sell it for $5 million. What is the most she would be willing to pay for the house if the appropriate discount rate is 12 percent?

Solution:

The maximum payment for the house today is equal to the present value of $5 million to make the investment worthwhile.

PV= 5,000,000

(1+.12)10= 1,609,866.18

10. You have the opportunity to make an investment that costs $900,000. If you make this investment now, you will receive $120,000 one year from today, $250,000 and $800,000 two and three years from today, respectively. The appropriate discount rate for this investment is 12 percent.

a. Should you make the investment?

b. What is the net present value (NPV) of this opportunity?

c. If the discount rate is 11 percent, should you invest? Compute the NPV to support your

answer.

Solution:

b. NPV = -900k + 120k

(1+.12)1 +

250k

(1+.12)2 +

800k

(1+.12)3

=-24.13 k

a. The NPV of the investment is negative and should not be made.

c. NPV = -900k + 120k

(1+.11)1 +

250k

(1+.11)2 +

800k

(1+.11)3

= -4.033 k

Even at 11% discount rate, the NPV of the investment is negative, so no investment should be made.

11. You have the opportunity to invest in a machine that will cost $340,000. The machine will generate cash flows of $100,000 at the end of each year and require maintenance costs of $10,000 at the beginning of each year. If the economic life of the machine is five years and the relevant discount rate is 10 percent, should you buy the machine? What if the relevant discount rate is 9 percent?

Solution:

At 10%,

NPV of cash inflows = 100k

(1+.1)1 + 100k

(1+.1)2 + 100k

(1+.1)3 + 100k

(1+.1)4 + 100k

(1+.1)5 = 379.08 k

NPV of cash outflows= 340k

(1+.1)0 +

10k

(1+.1)0+ 10k

(1+.1)1 +

10k

(1+.1)2 +

10k

(1+.1)3 +

10k

(1+.1)4 = 381. 7 k

So, NPV of the investment = -381. 7 + 379.08 = -2.62 k

At 10% discount rate the investment provides a negative NPV. The investment should not be made.

At 9%,

NPV of cash inflows = 100k

(1+.09)1 +

100k

(1+.09)2 +

100k

(1+.09)3 +

100k

(1+.09)4 +

100k

(1+.09)5 = 388.96 k

NPV of cash outflows= 340k

(1+.09)0 +

10k

(1+.09)0+

10k

(1+.09)1 +

10k

(1+.09)2 +

10k

(1+.09)3 +

10k

(1+.09)4 =

382. 4 k

So, NPV of the investment = -382.4 + 388.96 = 6.56 k

At 9% discount rate the investment provides a positive NPV. The investment can be made.

12. Today a firm signed a contract to sell a capital asset for $90,000. The firm will receive payment five years from today. The asset costs $60,000 to produce.

a. If the appropriate discount rate is 10 percent, is the firm making a profit on this item?

b. At what appropriate discount rate will the firm break even?

Solution:

a. NPV= -60,000 + 90,000

(1.1)5 =-41117.08

The firm is not making profit on this item.

b. In order to break-even, NPV should be equal to 0.

NPV= 0 = -60,000 + 90,000

(1+r )5

r= 0.0845 or, 8.45%15. Suppose you deposit $1,000 in an account at the end of each of the next four years. If the

account earns 12 percent, how much will be in the account at the end of seven years?

Solution:

Explanation FV= C0(1+r )t

The deposit at the end of year 1 will be compounded over 6 years.

1000* (1.12)6 1973.82

The deposit at the end of year 2 will be compounded over 5 years.

1000* (1.12)5 1762.34

The deposit at the end of year 3 will be compounded over 4 years.

1000* (1.12)4 1573.52

The deposit at the end of year 4 will be compounded over 3 years.

1000* (1.12)3 1404.93

Total Sum of all the future values

6714.61

16. What is the future value three years hence of $1,000 invested in an account with a stated annual interest rate of 8 percent,

a. compounded annually?

b. compounded semiannually?

c. compounded monthly?

d. compounded continuously?

e. Why does the future value increase as the compounding period shortens?

Solution:

EAR= (1+ rm

)m

−1

Future value formula EAR= (1+ r

m)m

−1Calculation FV

a. compounded annually

FV= C0(1+r )t 8% 1000*(1.08)3 1259.7

b. compounded semiannually

FV= C0(1+EAR)T 0.0816 1000*(1.0816)3 1265.32

c. compounded monthly

FV= C0(1+EAR)T 0.0824 1000*(1.0824)3 1268.19

d. compounded continuously

FV= C0erT N/A 1000* e0.08*3 1271.25

e. The shorter the compounding period, the more frequently interest is earned and for more periods, the investment is compounded.

18. Calculate the present value of $5,000 in 12 years at a stated annual interest rate of 10 percent, compounded quarterly.

Solution:

PV= CT/(1+EAR)T

EAR = (1+ rm )m

−1

= (1+ .1/4)4-1

=0.1038

PV= CT/(1+EAR)T

= 5000/ (1.1038)12

=1528.57

19. Bank America offers a stated annual interest rate of 4.1 percent, compounded quarterly, while Bank USA offers a stated annual interest rate of 4.05 percent, compounded monthly.In which bank should you deposit your money?

Solution:

Bank America Bank USAStated Annual Interest Rate (SAIR)

4.1% 4.05%

Compounded Quarterly Monthly

EAR=(1+ rm

)m

−14.16% 4.13%

We should deposit our money in Bank America which offers higher effective annual rate.

20. What is the price of a British CONSOL that pays $120 annually if the next payment occurs one year from today? The market interest rate is 15%.

Solution:

PV= C/r

=120/0.15 = 800

21. Assuming an interest rate of 10 percent, calculate the present value of the following streams of yearly payments:

a. $1,000 per year forever, with the first payment one year from today.

b. $500 per year forever, with the first payment two years from today.

c. $2,420 per year forever, with the first payment three years from today.

Solution:

PV= C/r

Explanation Calculation Present valuea. $1000: first

payment one year from today

1000/0 .1 10,000

b. $500: first payment two years from today

By applying the PV formula for perpetuity, we get the present value at the end of year 1, to get the value at year 0, we need to discount in by (1+r)

(500 /0.1)(1+.1)

4545.45

c. $2420: first payment three years from today

By applying the PV formula for perpetuity, we get the present value at the end of year 2, to get the value at year 0, we need to discount in by (1+r)2

(2420 /0.1)¿¿

20000

22. Given an interest rate of 10 percent per year, what is the value at date t= 5 (i.e., the end of year 5) of a perpetual stream of $120 annual payments starting at date t= 9?

Solution:

PV= C/r

By applying this formula we get value of the stream at t=8.

Value at Year 8= 120/ 0.1 =1200

To get value at t=5, we need to discount it back to 3 years: 1200/ (1.1)3= 901.6

23. Harris, Inc., paid a $3 dividend yesterday. If the firm raises its dividend at 5 percent every year and the appropriate discount rate is 12 percent, what is the price of Harris stock?

Solution:

Next divided= 3*(1.05)= $3.15

PV= C/(r-g)

Price of the stock= $3.15/ (0.12-0.05) = $45

26. IDEC Pharmaceuticals is considering a drug project that costs $100,000 today and is expected to generate end-of-year annual cash flow of $50,000 forever. At what discount rate would IDEC be indifferent between accepting or rejecting the project?

Solution:

IDEC will be indifferent when,

NPV =0= -100,000 + 50,000/r

r= 50,000/100,000

r= 0.5 or 50%

29. Should you buy an asset that will generate income of $1,200 per year for eight years? The price of the asset is $6,200 and the annual interest rate is 10 percent.

Solution:

PV= C*[ 1−(1+r )−T

r¿

= 1200*[ 1−(1+0.1 )−8

0.1¿

= 6401.9, which is higher than 6200. So the project will generate a positive NPV (201.9).

Decision: Buy the asset.

30. What is the present value of end-of-year cash flows of $2,000 per year, with the first cash flow received three years from today and the last one 22 years from today? Use a discount rate of 8 percent.

Solution:

Cash flows are received for 19 years.

PV= C*[ 1−(1+r )−T

r¿

Value at the end of year 2 =2000 *[ 1−(1+0.08 )−19

0.08¿ =19,207.2

We need to discount it for 2 years to get the present value, PV = 119,207.2/ (1.08)2 = 16,467.1

31. What is the value of a 15-year annuity that pays $500 a year? The annuity’s first payment is at the end of year 6 and the annual interest rate is 12 percent for years 1 through 5 and 15 percent thereafter.

Solution:

Cash flows are received for 15 years.

PV= C*[ 1−(1+r )−T

r¿

Value at the end of year 2 =500 *[ 1−(1+0.15 )−15

0.15¿ =2923.7

We need to discount it for 5 years at the discount rate of 12% to get the present value, PV = 2923.7/ (1.12)5 = 165932. You are offered the opportunity to buy a note for $12,800. The note is certain to pay $2,000 at

the end of each of the next 10 years. If you buy the note, what rate of interest will you receive?

Solution:

NPV= 0 = -12,800+ 2000* [ 1−(1+r )−10

r¿

12,800/2,000 = 6.4= 1−(1+r )−10

r¿

From, annuity factor table, r= 0.09 or 9%33. You need $25,000 five years from now. You budget to make equal payments at the end of every

year into an account that pays an annual interest rate of 7 percent.

a. What are your annual payments?

b. Your rich uncle died and left you $20,000. How much of it must you put into the same account as a lump sum today to meet your goal?

Solution:

a. FV= C*[ (1+r )T−1

r¿

15%12%

0 5 6 15

25000= C* [ (1+.07)5−1

0.07¿

C= 4347.27

b. The lump sum amount must be equal to the present value of 25000, which is 25000/ (1.07)5= 17,824.7.

34. Nancy Ferris bought a building for $120,000. She paid 15 percent down and agreed to pay the balance in 20 equal annual installments. What are the equal installments if the annual interest rate is 10 percent?

Solution:

Down payment= 0.15* $120,000= 18,000

Balance= 102,000

102,000= C*[ 1−(1+0.1 )−20

0.1¿

C= $11,980.9

35. You have recently won the super jackpot in the Illinois state lottery. On reading the fine print, you discover that you have the following two options:

a. You receive $160,000 at the beginning of each year for 31 years. The income would be taxed at a rate of 28 percent. Taxes are withheld when the checks are issued.

b. You receive $1,750,000 now, but you do not have access to the full amount immediately. The $1,750,000 would be taxed at 28 percent. You are able to take $446,000 of the after-tax amount now. The remaining $814,000 will be placed in a 30- year annuity account that pays $101,055 on a before-tax basis at the end of each year.

Using a discount rate of 10 percent, which option should you select?

Solution:

PV= C*[ 1−(1+r )−T

r¿

a. After tax payment at the beginning of each year= 160,000*(1-.28)= 115,200

PV= 115,200+ 115,200* [ 1−(1+0.1 )−30

0.1¿ = 1,201,180.55

b. After tax payment (immediate)= 446,000 After tax payment each year= 101,055*(1-.28)= 72,759.6

PV= 72,759.6* [ 1−(1+0.1 )−30

0.1¿

=685,898.5

Present value of total payment= 446,000 + 685,898.5 = 1,131,897.5

The present value of option a is higher, so that one should be chosen. 36.You are saving for the college education of your two children. They are two years apart in age; one will begin college in 15 years, the other will begin in 17 years. You estimate your children’s college expenses to be $21,000 per year per child. The annual interest rate is 15 percent. How much money must you deposit in an account each year to fund your children’s education? You will begin payments one year from today. You will make your last deposit when your oldest child enters college.Solution:

PV= C*[ 1−(1+r )−T

r¿

PV = 21000 *[ 1−(1+0.15)−4

0.15¿

= 59,954.55

This is the beginning value of the children’s college expenses. For the 1st child this is the PV 14 years from now, and for the 2nd one 16 years from now.

First child,

PV of 59,954.55= 59,954.55/(1.15)14 = 8473.3

2nd child,

PV of 59,954.55= 59,954.55/(1.15)16 = 6407.03

Total present value = 8473.3 + 6407.03= 14,880.33

Total number of deposits= 15

To find the deposit amount each year,

15%

0 15 17

PV = 14,880.33= C*[ 1−(1+0.15)−15

0.15¿

C= 2544.8

37.A well-known insurance company offers a policy known as the “Estate Creator Six Pay.” Typically the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company.First birthday $750; Fourth birthday $800Second birthday $750 ; Fifth birthday $800Third birthday $750 ; Sixth birthday $800No more payments are made after the child’s sixth birthday. When the child reaches age 65, he or she receives $250,000. If the relevant interest rate is 6 percent for the first six years and 7 percent for all subsequent years, is the policy worth buying?

Solution:

NPV of the investment = (-750)*[ 1−(1+0.06)−3

0.06¿+ {(-800)*[

1−(1+0.06)−3

0.06¿}/(1.06)3 +

25000/(1.07)64= -750* 2.673 – (800* 2.673)/(1.06)3 + 25000/(1.07)59 + [250,000/(1.07)59]/ (1.06)6

= -2004.75 -1795.44 + 3254.33= -2545.86The negative NPV suggests that the policy is not worth buying.

38.Your company is considering leasing a $120,000 piece of equipment for the next 10 years. Your company can buy the equipment outright or lease it. The annual lease payments of $15,000 are due at the beginning of each year. The lease includes an option for your company to buy the equipment for $25,000 at the end of the leasing period (i.e., 10 years). Should your company accept the lease offer if the appropriate discount rate is 8 percent a year?Solution:

birth1st bd

2nd bd

3rd bd

4th bd

5th bd 6th bd

65th bd

-750 -750 -750 -800 -800 -800 25,000

6%

7%

The first payment will be at t= 0.

PV of annual lease payment= 15000+ 15000*[ 1−(1+0.08)−9

0.08¿

= 15000 + 15000*6.2469

= 108,703.5

PV of equipment price = 25000/ (1.08)10 =11,579.84

Total present value of lease payments = 108,703.5 + 11,579.84 =120,283.34

This value is higher than the present buying price of the equipment. Therefore, buying the equipment now would be a better option.

39. You are saving for your retirement. You have decided that one year from today you will deposit 2 percent of your annual salary in an account which will earn 8 percent per year. Your salary last year was $50,000, and it will increase at 4 percent per year throughout your career. How much money will you have for your retirement, which will begin in 40 years?

Solution:

Current salary = $50,000

2% of which is $50,000*0.02= 1000

This amount will grow by 4% per year for next 40 years. The amount =1000*1.04= 1040

The formula for growing annuity will be applied here.

PV= C*[ 1−( 1+g1+r )T

r−g¿

= 1040*[ 1−( 1+.041+.08 )40

0.08−0.04¿

=$20,254.18Future value of this amount at t=40, is the amount that I’ll have for my retirement.$20,254.18* (1.08)40 = $440,012.3740. You must decide whether or not to purchase new capital equipment. The cost of the machine is

$5,000. It will produce the following cash flows. The appropriate discount rate is 10 percent.Year Cash Flow1 $ 7002 9003 1,0004 1,0005 1,0006 1,000

7 1,2508 1,375

Solution:Year Cash Flow

($)Discount factor

Present value

0 -5000 1 -50001 700 1/(1+0.1)1 636.362 900 1/(1+0.1)2 743.83 1,000 1/(1+0.1)3 751.314 1,000 1/(1+0.1)4 683.015 1,000 1/(1+0.1)5 620.926 1,000 1/(1+0.1)6 564.477 1,250 1/(1+0.1)7 641.458 1,375 1/(1+0.1)8 641.45

NPV 282.77

Decision: Positive NPV suggests to purchase the equipment.

41. Your younger brother has come to you for advice. He is about to enter college and has two options open to him. His first option is to study engineering. If he does this, his undergraduate degree would cost him $12,000 a year for four years. Having obtained this, he would need to gain two years of practical experience: in the first year he would earn $20,000, in the second year he would earn $25,000. He would then need to obtain his master’s degree, which will cost $15,000 a year for two years. After that he will be fully qualified and can earn $40,000 per year for 25 years.His other alternative is to study accounting. If he does this, he would pay $13,000 a year for four years and then he would earn $31,000 per year for 30 years.The effort involved in the two careers is the same, so he is only interested in the earnings the jobs provide. All earnings and costs are paid at the end of the year. What advice would you give him if the market interest rate is 5 percent? A day later he comes back and says he took your advice, but in fact, the market interest rate was 6 percent. Has your brother made the right choice?

Solution:

Formula Engineering AccountingPresent value of undergraduate cost

PV= C*[ 1−(1+r )−T

r¿

-12000* [ 1−(1+.05 )−4

0.05¿= -42,552

-13000* [ 1−(1+.05 )−4

0.05¿= -46,098

Present value of two year’s

PV=C/(1+r)t 20,000/(1.05)5+ 25,000/(1.05)6

earnings =34,325.91Present value of Master’s cost

PV= {C*[ 1−(1+r )−T

r¿}/(1+r )

T2{-15000* [ 1−(1+.05 )−2

0.05¿}/ (1.05)6= -20,812.694

Present value of earning of 25 years

PV= {C*[ 1−(1+r )−T

r¿}/(1+r )

T2{40000* [ 1−(1+.05 )−25

0.05¿}/ (1.05)8= 381,572.25

{31000* [ 1−(1+.05 )−30

0.05¿}/ (1.05)4= 392,056.8076

352,533.466 345,958.8076

Option 1 provides higher NPV so studying Engineering would be the suggestion.

The same procedure for 6% and then compare the NPVs.

42. Ms. Adams has received a job offer from a large investment bank as an assistant to the vice president. Her base salary will be $35,000. She will receive her first annual salary payment one year from the day she begins to work. In addition, she will get an immediate $10,000 bonus for joining the company. Her salary will grow at 4 percent each year. Each year she will receive a bonus equal to 10 percent of her salary. Ms. Adams is expected to work for 25 years. What is the present value of the offer if the discount rate is 12 percent?

Solution:

PV= C*[ 1−( 1+g1+r )T

r−g¿

As salary will increase by 4% each year, so will the bonus.Salary at year 1= 35000Bonus = 35000*0.1 = 3500Total payment = (35000+ 3500)= 38500

PV of the offer = 10,000 + 38500* [ 1−( 1+.041+.12 )25

0.12−0.04¿ = 415,783.6

43. Southern California Publishing Company is trying to decide whether or not to revise its popular textbook, Financial Psychoanalysis Made Simple. They have estimated that the revision will cost $40,000. Cash flows from increased sales will be $10,000 the first year.

r=10%

g=4%3500

251

35000

10000

0

These cash flows will increase by 7 percent per year. The book will go out of print five years from now. Assume the initial cost is paid now and all revenues are received at the end of each year. If the company requires a 10 percent return for such an investment, should it undertake the revision?

Solution:

NPV= -40,000 + 10,000* [ 1−( 1+.071+.1 )5

0.1−0.07¿ = 3,041.91

44. Ernie Els wants to save money to meet two objectives. First, he would like to be able to retire 30 years from now with a retirement income of $300,000 per year for 20 years beginning at the end of the 31 years from now. Second, he would like to purchase a cabin in the mountains 10 years from now at an estimated cost of $350,000. He can afford to save only $40,000 per year for the first 10 years. He expects to earn 7 percent per year from investments. Assuming he saves the same amount each year, what must Ernie save annually from years 11 to 30 to meet his objectives?

Solution:

Present value of retirement income of $300,000 for 20 years:

PV20= C*[ 1−(1+r )−T

r¿

=300,000* [ 1−(1+.07 )−20

0.07¿

=300,000* 10.594 = 3,178,200This needs to be discounted again to get the present value today,PV= 3,178,200/ (1.07)30 =417,510.97Present value of price of cabin:PV= 350,000/(1.07)10 = 177,922.25Present value of saving of first 10 years:

PV= 40,000*[ 1−(1+.07 )−10

0.07¿ = 280,943.26

Let, he must save amount ‘C’ from year 11 to year 30 and he replenishes all his savings after the 50 years.PV = {C* [ 1−(1+.07 )−20

0.07¿}/ (1.07)10 = 5.38545C

The PV of savings and PV of withdrawals (retirement and cabin) should be equal, in other words NPV must be equal to zero.(280,943.26+ 5.385C) =417,510.97+177,922.25

5.38545C = 314,489.96

C= 58,396.23

Yearly Saving amount= $58,396.2345.Jack Ferguson has signed a three-year contract to work for a computer software company. He expects to receive a base salary of $5,000 a month and a bonus of $10,000 at year-end. All payments are made at the end of periods. What is the present value of the contract if the stated annual interest rate, compounded monthly, is 12 percent?46.Peter Green bought a $15,000 Honda Civic with 20 percent down and financed the rest with a four-year loan at 8 percent stated annual interest rate, compounded monthly. What is his monthly payment if he starts the payment one month after the purchase?47.On September 1, 1998, Susan Chao bought a motorcycle for $10,000. She paid $1,000 down and financed the balance with a five-year loan at a stated annual interest rate of 9.6 percent, compounded monthly. She started the monthly payment exactly one month after the purchase, i.e., October, 1998. In the middle of October, 2000, she got a new job and decided to pay off the loan. If the bank charges her 1 percent prepayment penalty based on the loan balance, how much should she pay the bank on November 1, 2000?48.When Marilyn Monroe died, ex-husband Joe DiMaggio vowed to place fresh flowers on her grave every Sunday as long as he lived. A bunch of fresh flowers that the former baseball player thought appropriate for the star cost about $5 when she died in 1962.Based on actuarial tables, “Joltin’ Joe” could expect to live for 30 years after the actress died. Assume that the stated annual interest rate, compounded weekly, is 10.4 percent.Also, assume that the rate of inflation is 3.9 percent per year, when expressed as a stated annual inflation rate, compounded weekly. Assuming that each year has exactly 52 weeks, what is the present value of this commitment?Solution:

Total weeks in 30 years= 30*52= 1560

EAR= (1+ 0.10452

)52

−1 = 0.1095

Effective annual inflation rate= (1+ 0.03952

)52

−1 = 0.04

PV= C*[ 1−( 1+g1+r )T

r−g¿

=5*[ 1−( 1+.041+.1095 )30

.1095−.04¿

=56.6149.In January 1984, Richard “Goose” Gossage signed a contract to play for the San Diego Padres that guaranteed him a minimum of $9,955,000. The guaranteed payments were $875,000 for 1984, $650,000 for 1985, $800,000 in 1986, $1 million in 1987, $1 million in 1988, and $300,000 in 1989. In addition, the contract called for $5,330,000 in deferred money payable at the rate of $240,000 per year from 1990 through 2006 and then $125,000 a year from 2007 through 2016. If the effective annual rate of interest is 9 percent and all payments are made on July 1 of each year, what would the present value of these guaranteed payments be on January 1, 1984? Assume an interest rate of 4.4 percent per six months. If he were to receive an equal annual salary at the end of each of the five years from 1984 through 1988, what would his equivalent annual salary be? Ignore taxes throughout this problem.Solution:

Year Cash Flow ($) Discount factor Present value1 Jul 1984 875,000 1/(1+0.044)0 875,0001985 650,000 1/(1+0. 09)1 596,330.281986 800,000 1/(1+0. 09)2 673,3441987 1,000,000 1/(1+0. 09)3 772,183.481988 1,000,000 1/(1+0. 09)4 708,425.21989 300,000 1/(1+0. 09)5 194,979.421990- 2006 240,000

1−(1+.09 )−17

0.09¿ /(1.09)5

1,332,672

2007-2016 125,000 1−(1+.09 )−10

0.09¿ /(1.09)22

120,477.65

Value on 1 Jul 1984 5,273,412We need to discount in back to 1 January 1984 using the 4.4% semi-annual rate:PV= 5,273,412/(1.044) = 5,051,160.9550.Mike Bayles has just arranged to purchase a $400,000 vacation home in the Bahamas with a 20% down payment. The mortgage has an 8% annual percentage rate (APR) and calls for equal monthly payments over the next 30

years. His first payment will be due one month from now. However, the mortgage has an 8-year balloon payment, meaning that the loan must be paid off then. There were no other transaction costs or finance charges. How big will Justin’s balloon payment be in 8 years?Solution:

Down payment= 400,000* 20%= $80,000

EAR= (1+ rm )m

−1

=(1+ .0812

)12

−1

=0.083 or 8.3%

PV of equal monthly payments= 400,000-80,000= 320,000

PV= C*[ 1−(1+EAR )−T

EAR¿

320,000= C*[ 1−(1+.083 )−30∗12

0.083¿

C= 26,560The balloon payment will be equal to the value of payments of last (30-8)=22 years at the end of year 8.PV= 26560*[ 1−(1+.083 )−22∗12

0.083¿

=320,000, this is amount of the balloon payment. 51.You want to lease a set of golf clubs from Pings Ltd. for $4,000. The lease contract is in the form of 24 months of equal payments at a 12% annual percentage rate (APR). Suppose payments are due in the beginning of the month and your first payment is due immediately. What will your monthly lease payment be?Solution:Monthly annual rate= 12%/12= 1%PV= 4000 = C+ C*[ 1−(1+.01 )−23

0.01¿

C= 185.43 (that’s the monthly lease payment)52.A 10-year annuity pays $900 per year, with payments made at the end of each year. The first $900 will be paid 5 years from now. If the APR is 8% and interest is compounded quarterly, what is the present value of this annuity?

Solution:

EAR= (1+ rm

)m

−1

= (1+.08/4)4-1 =0.0824PV4= 900*[ 1−(1+.0824 )−10

0.0824¿ =5,9575.71

PV= 5,9575.71/(1.0824)4 = 4,353.553.Paul Adams owns a health club in downtown LA. He charges his customers an annual fee of $400 and has an existing customer base of 500. Paul plans to raise the annual fee by 10 percent every year and expects the club membership to grow at a constant rate of 3 percent for the next five years. The overall expenses of running the health club are $80,000 a year and are expected to grow at the inflation rate of 2 percent annually. After five years, Paul plans to buy a luxury boat for $500,000, close the health club, and travel the world in his boat for the rest of his life. What is the annual amount that Paul can spend while on his world tour if he will have no money left in the bank when he dies? Assume Paul has a remaining life of 15 years and earns 6 percent on his saving.Solution: Current Year 1Earnings $400*500= $200,000 (400* 1.1)*( 500*1.03)=226,600Annual expense $80,000 80,000*1.02= 81,600Net income $120,000 145000Growth rate of net income = (145000-12000)/12000 =.2083 or 20.83%Compounding factor* Future value at year 5Year 1 145000 (1.08)^4 197,270.9Year 2 145000*1.2083= 175,203.5 (1.08)^3 220705.95Year 3 175,203.5*1.2083= 211,698.4 (1.08)^2 246925Year 4 211,698.4*1.2083=255,795.16 (1.08)^1 276258.77Year 5 255,795.16*1.2083=309,077.3 (1.08)^0 309,077.3FV of net income from club at end of year 5 2,223,321.4Income form year 1 will earn interest for 4 years, income from year 2 for 3 years and so on.

Value of boat at year 5= 500,000Remaining money= 2,223,321.4-500000= 473,083.48Let, the annual amount that Paul can spend while on his world tour= C, PV5 of which is = C*[ 1−(1+.08 )−15

0.08¿= 473,083.48

C= 55,270.13Current earnings= 400*500= 200,000Earnings next year= (400*1.1)*(500*1.03)= 226,600Accumulated growth rate of earnings= (226,600-200,000)/ 200,000 =0.133Current expense= 80,000Expense next year = 80,000*1.02 = 81,600Earnings (Previous year’s earnings*1.133)

Expenses (Previous year’s expenses*1.02)Net Income available for saving

Compounding factor* Future value after 5 yearsYear 1 226,600 81,600 145000 (1.08)^4 197,270.9Year 2 256,737.8 83,232 173505. (1.08)^3 218,566.3Year 3 290,883.93 84,896.64 205987.29 (1.08)^2 240,263.6Year 4 329,571.49 86,594.573 242976.92 (1.08)^1 262,415.1Year 5 373,404.498 88,326.46 285087.038 (1.08)^0 285,087.038FV of net income from club at end of year 5 1,203,603

*Income form year 1 will earn interest for 4 years, income from year 2 for 3 years and so on.Value of boat at year 5= 500,000Remaining money= 1,203,603-500,000= 703,603Let, the annual amount that Paul can spend while on his world tour= C, PV5 of which is = C*[ 1−(1+.08 )−15

0.08¿= 703,603

C= 82,201.62