STP #5.5. You just bought a corporate bond at $863.75...

STP #5.5. You just bought a corporate bond at $863.75 today. In five years the

bond will mature and you will receive $1,000. What is the rate of return

on this bond?

Price of bond today 863.75$

Number of years till bond matures 5

Future value of bond $1,000

PV n FVIF i% FV

Prove the future value$1,000

Trial & Error Method to solve for i% that results in a FV of $1,000.

STP #5.4. Twenty-five years ago, Amanda Carter investec $10,000 in an account paying an

annual interest rate of 5.75 percent. What is the value of the investment today?

What is the interest on interest earned on this investment?

Amount invested 25 years ago $10,000

Number of years invested 25

Annual interest rate 5.75%

PV

n

i% FV Value of investment today

interest per year

Sinple interest over 25 years

Interest on Interest

STP #5.3. What is the future value of an investment of $3,000 after three years

with compounding at the following rates and frequencies:

a. 8.75 percent compounded monthly

b. 8.625 percent compounded daily

c. 8.5 percent compounded continuously

Investment $3,000

Number of years 3

PV 3000

i%

n

m 12FV

PV

i%

n

m 365 FV

PV

i%

n

m 100000

FV

STP #5.2. Michael Carter is expecting an inheritance of $1.25 million in four years.

If he had the money today, he could earn interest at an annual rate of 7.35

percent. What is the present value of this inheritance?

Future value of inheritance expected $1,250,000

Number of years until inheritance 4

Annual interest rate 7.35%

FV

n

i%

PV = FV(FVIF)

STP #5.1. Santiago Hernandez is planning to invest $25,000 in a money market account for

two years. The account pays interest of 5.75 percent compounded on a monthly basis.

How much money will Santiago Hernandez have at the end of two years?

Amount of investment 25,000$

Term (years) 2

Interest rate per year 5.75%

Frequency of compounding per year 12

Monthly Compounding

PV

n

i%

m

FV = PV(FVIF)

FV

5.37 You will be graduating in two years, and are thinking about your future.

You know that you will want to buy a house five years after you graduate and that you

will want to put down $60,000. As of right now, you have $8,000 in your savings account.

You are also fairly certain that once you graduate, you can go work in the family

business and earn $32,000 a year, with a 5 percent raise every year. You plan to live with

your parents for the first two years after graduation, which will enable you to minimize

your expenses and put away $10,000 each year. The next three years, you will have to live

out on your own as your younger sister will be graduating from college and has

already announced her plan to move back into the family house. Thus, you will be able

to save only 13 percent of your annual salary. Assume that you will be able to invest

savings from your salary at 7.2 percent. At what interest rate will you need

to invest the current savings account balance in order to achieve your goal?

Hint: Draw a time line that shows all the cash flows for years 0 through 7.

Remember, you want to buy a house seven years from now and your first salary will be

in year 3.

Number of years until graduation 2

Number of years until you buy a house 7

Down payment required on house $60,000

Current savings $8,000

Annual income in family business $32,000

Annual raise in salary 5%

Annual savings after graduation while living at home $10,000

Number of years after graduation that you will live at home 2

Annual saving (as a percent of salary)after moving out 13%

Rate of return on invested savings from salary 7.20%

Cash flow 0 1 2 3 4 5 6 7 Future value

Down payment required on house $60,000.00

Current saving $8,000

Salary $32,000

Annual Saving $10,000 $10,000

PVIF @ 7.20% 0.932835821 0.870182669 0.811737564 0.757217877 0.70635996

$9328.36 $8701.83

NPV

FV = PV(FVIF)

Amount that current savings account balance must amount to to reach down payment goal

PV $8,000

FV

n 7

FV = PV (1+ i%)^n

18984.93 = 8,000(1+i%)^7

FV/PV

2.3731 = (1+i%)^7

Raise 2.3731 to 1/7th power Rate of return that current savings will have to earn to reach goal i%

Year

5.35 Sam Bradford, a number 1 draft pick of the St. Louis Rams,

and his agent are evaluating three contract options. Each option offers a signing bonus

and a series of payments over the life of the contract. Bradford uses a 10.25 percent rate of return

to evaluate the contracts. Given the cash flows for each option, which one

should he choose?

10.25% 10.25% 10.25%

Yea

r

Cash

Flow

Type

Option A Option B Option C

A PVIF PV B PVIF PV C PVIF PV

0 Signing

Bonus

$3,100,000 $4,000,000 $4,250,000 0

1 Annual

Salary

$650,000 $825,000 $550,000 1

2 Annual

Salary

$715,000 $850,000 $625,000

2

3 Annual

Salary

$822,250 $925,000 $800,000

3

4 Annual

Salary

$975,000 $1,250,000 $900,000

4

5 Annual

Salary

$1,100,000 $1,000,000

5

6 Annual

Salary

$1,250,000

6

Interest rate 10.25%

5.36 Surmec, Inc., has reported sales of $2.1 million last year.

The company’s primary business line is manufacturing of nuts and bolts.

Since this is a mature industry, the analysts are certain that the sales will grow

at a steady rate of 7 percent a year. The company reports

net income that represents 23 percent of sales. The management would like to buy

a new fleet of trucks but can only do so once the profit reaches $620,000 a year.

At the end of what year will they be able to buy the new fleet of trucks?

What will the sales and profit be that year?

Last year's sales $2,100,000

Annual growth rate of sales expected 7%

Net income as a percent of sales 23%

Target net income needed before fleet of trucks can be purchased 620,000$

Last year’s net income = Last Year’s sales * Net Income as % of sales PV

FV

i%

FV = PV(1+i%)^n

620,000 = 483,000(1.07)^n

FV/PV

1.2836 = (1.07)^n

Trial & Error Method

(1.07) ^ 3.69 = 1.28359Number of years after which the fleet of truck can be purchased = n =

Sales of the year after which trucks can be purchased = FV = PV (FVIF)

PV

i%

4 n = round up to 4.0

Profit of the year after which trucks can be purchased = #VALUE!

5.34 When you were born your parents set up a bank account in your name

with an initial investment of $5,000. You are turning 21 in a few days

and will have access to all your funds. The account was earning 7.3 percent

for the first seven years then the rates went down to 5.5 percent for six years.

The economy was doing well at the end of the 1990s and your account was

earning 8.2 percent for three years in a row. Unfortunately, the next two

years you earned only 4.6 percent. Finally, as the economy recovered,

your return jumped to 7.6 percent for the last three years.

a. How much money was in your account before the rates went

down drastically (end of year 16)?

b. How much money is in your account now (end of year 21)?

c. What would be the balance now if your parents made another

deposit of $1,200 at the end of year 7?

Initial investment = $5,000

Age at time of investment = 0 years

Current age = 21 years

Interest rate (years 1-7)= 7.30%





n i% FV

FV at the end of year 7 7 7.30% $1,200.00

FV at the end of year 8 8 5.50% $1,266.00

FV at the end of year 9 9 5.50%






FV at the end of year 15 15 8.20% FV at the end of year 16 16 8.20% [a]




FV at the end of year 20 20 7.60% FV at the end of year 21 21 7.60% [b]

FV at the end of year 21

FV of additional deposits from year 7 thru year 21 FV at the end of year 21 [c]

5.33 You have $12,000 in cash. You can deposit it today in a

mutual fund earning 8.2 percent semiannually, or you can wait,

enjoy some of it, and invest $11,000 in your brother’s business in two

years. Your brother is promising you a return of at least 10 percent

on your investment. Whichever alternative you choose, you will need to

cash in at the end of ten years. Assume your brother is trustworthy

and both investments carry the same risk. Which one will you choose?

Alternative A

Cash available for investment = $12,000

Rate of return on mutual fund= 8.20%

Compounding frequency (per year)= 2

Term of investment = 10 years

Alternative B

Cash available for investment after 2 years = $11,000

Rate of return promised by brother = 10%

Compounding frequency (per year)= 1

Term of investment= 8 years

Annual Compounding Semi-annual compounding

PV

i%

m

n FV

PV

i%

m

n

FV

Choose alternative A

5.32 Patrick Seeley has $2,400 that he is looking to invest.

His brother approached him with an investment opportunity that could double

his money in four years. What interest rate would the investment have to

yield in order for Patrick’s brother to deliver on his promise?

Patrick's investment = $2,400

Period of investment = 4 years

Value at end of term= $4,800

PV

n

FV

FV = PV(FVIF)

FV/PV = (1+ i%)^n

FV/PV

2.000 = (1+i%)^4

Raise 2.0 to 1/4th power

1.1892 = (1 + i%)

i% =

5.31 You have $2,500 you want to invest in your classmate’s

start-up business. You believe the business idea to be great and hope to

get $3,700 back at the end of three years. If all goes according to plan,

what will your return on investment be?

Investment needed= 2,500$

Period of investment= 3 years

Expected terminal value= $3,700

PV

n

FV

FV = PV(FVIF)

3,700= 2,500(1+I%)^3

FV/PV

1.48 = (1+I%)^3

RAISE 1.48 to 1/3rd power

i% =

5.22 Multiple compounding periods: Samantha is looking to invest some money,

so she can collect $5,500 at the end of three years. Which investment

should she make given the following choices:

a. 4.2 percent compounded daily

b. 4.9 percent compounded monthly

c. 5.2 percent compounded quarterly

d. 5.4 percent compounded annually

Amount of money she would like to have after 3 years= $5,500

Term of investment = 3 yearsm FV

a. Interest rate= 4.20% Compounding frequency (per year)= 365 i%

n

Amount of investment required today = m PV

b. Interest rate= 4.90% Compounding frequency (per year)= 12 FV

i%

Amount of investment required today = n

m

PV

FV

c. Interest rate= 5.20% Compounding frequency (per year)= 4 i%

n


FV

d. Interest rate= 5.40% Compounding frequency (per year)= 1 i%

n


She should invest at 5.4% compounded annually i.e. option D.

5.21 Multiple compounding periods: Find the present value of $3,500

under each of the following rates and periods:

a. 8.9 percent compounded monthly for five years.

b. 6.6 percent compounded quarterly for eight years.

c. 4.3 percent compounded daily for four years.

d. 5.7 percent compounded continuously for three years.

Future value = $3,500 PV = FV(PVIF)

a. Interest rate = 8.90% i%

Term = 5 years n

Compounding frequency (per year)= 12 mPresent value = PV

b. Interest rate = 6.60% i%

Term = 8 years n


c. Interest rate = 4.30% i%

Term = 4 years n


d. Interest rate = 5.70% i%

Term = 3 years n


5.29. Present value: Caroline Weslin needs to decide whether to accept

a bonus of $1,820 today or wait two years and receive $2,100 then.

She can invest at 6 percent. What should she do?

Bonus offered today = PV = 1,820$

Bonus after 2 years = FV = 2,100$

Rate of return = I = 6%

Period = n= 2 years

PV

FV

i%

n

FV=PV(FVIF)FV =

Since 2,044.95 < 2,100.00

Do not accept the bonus today

PV = FV (PVIF)PV=

Accept the alternative since it has a higher PV

5.25. Time to grow: Neon Lights Company is a private company with sales

of $1.3 million a year. Management wants to go public but has to wait until the

sales reach $2 million. If sales are expected to grow at a steady 12 percent

annually, when is the earliest that Neon Lights can go public?

Current annual sales = $1,300,000

Sales level at which they can go public= $2,000,000

Expected annual growth rate of sales = 12%

How long will it take for them to reach their goal?

Time = 3.8 years

Alternative method:

Since FV=PV*(1+r)n

==> n = Ln(FV/PV)/Ln(1+r)

n= 3.8 years

PV

FV

g

FV=PV(1+g)^n

FV/PV

1.5384 = (1.12)^n


1.12 raised to the 3.8 power =

n =

5.24. Time to grow: You are able to deposit $850 in a bank CD today,

and you will withdraw the money only once the balance is $1,000.

If the bank pays 5 percent interest, how long will it take you to attain your goal?

Present value of deposit today = $850

Amount of withdrawal required = $1,000

Interest rate paid by bank per year= 5%

How long will it take to reach your goal?

Time = 3.331 years

Alternative method:

Since FV=PV*(1+r)n

==> n = Ln(FV/PV)/Ln(1+r)

n= $3.33 years

PV

FV

i%

FV=PV(1+i%)^n

FV/PV

1.1764 = (1.05) ^ n


1.05 raised to 3.331 = 1.176414492

n years

5.23. Time to grow: Zephyr Sales Company has currently reported sales

of $1.125 million. If the company expects its sales to grow at 6.5

percent annually, how long will it be before the company can double

its sales? Use a financial calculator to solve this problem.

Year 0 Sales 1,125,000

Growth rate of sales 6.50% per yearPV

g

FV

FV=PV(1+i%)^n

FV/PV

2.0 =(1.065)^n


n = years

5.28. Growth rate: Infosys Technologies, Inc., an Indian technology company,

reported a net income of $419 million this year. Analysts expect the company’s

earnings to be $1.468 billion in five years. What is the expected growth rate in the company’s earnings?

Year 0 Net Income = 419,000,000 PV

Year 5 Net Income = 1,468,000,000 FV

Number of years = 5 N

FV = PV(1+i%)^n

FV/PV = (1+i%)^n

FV/PV =

RAISE TO THE 1/5th Power

(FV/PV)^.2 =

EXPECTED g =

Albegraic method:

Since: Year 5 Sales = Year 0 Sales*(1+g)^5

==>g =[ (Year 5 Sales)/(Year 0 Sales)]^(1/5)-1

g = 28.50%

5.27. Growth rates: Xenix Corp had sales of $353,866 in 2011. If management expects its

sales to be at $476,450 in three years, what is the rate at which the company’s

sales are expected to grow?

2008 Sales = 353,866$ PV

2011 Sales= $476,450 FV

Number of years= 3 n

FV=PV(FVIF)

FV/PV=

1.3464 = (1+i%)^3

cube root of 1.3464 = Growth rate =

5.20. Multiple compounding periods: Find the future value of an investment

of $2,500 made today for the following rates and periods:

a. 6.25 percent compounded semiannually for 12 years

b. 7.63 percent compounded quarterly for 6 years

c. 8.9 percent compounded monthly for 10 years

d. 10 percent compounded daily for 3 years

e. 8 percent compounded continuously for 2 years

Adjusted Data

Present value of investment = $2,500 PV

a. interest rate = 6.25% i%

frequency of compounding= 2 times per year m

term = 12 years n

FV=PV(FVIF)Future value of investment = FV

b. interest rate = 7.63% i%


term = 6 years n


c. interest rate = 8.90% i%


term = 10 years n


d. interest rate = 10.00% i%


term = 3 years n


e. interest rate = 8.00% i%


term = 2 years n


5.19. Growth rate: You decide to take advantage of the current online dating

craze and start your own Web site. You know that you have 450 people

who will sign up immediately and, through a careful marketing research and

analysis, determine that membership can grow by 27 percent in the first two years,

22 percent in year 3, and 18 percent in year 4. How many members do

you expect to have at the end of four years?

# of members in Year 0= 450

Growth rate in Years 1-2 27%

Growth rate in Year 3 22%

Growth rate in Year 4 18%

PV

n 0 1 2 3 4

g 0.00% 27.00% 27.00% 22.00% 18.00%FV members

YEAR 0 1 2 3 4

5.18. Growth rate: CelebNav, Inc., had sales last year of $700,000,

and the analysts are predicting a good year for the start-up, with sales growing

20 percent a year for the next three years. After that, the sales should grow

11 percent per year for two years, at which time the owners are planning

to sell the company. What are the projected sales for the last year

of the company’s operation?

Sales in Year 0 $700,000

Growth rate in years 1-3 20%

Growth rate in years 4 and 5 11%

PV

n 0 1 2 3 4 5

g 0.00% 20.00% 20.00% 20.00% 11.00% 11.00%FV

5.17. Growth rate: Your finance textbook sold 53,250 copies in its first year.

The publishing company expects the sales to grow at a rate of 20 percent for

the next three years, and by 10 percent in the fourth year. Calculate the total

number of copies that the publisher expects to sell in years 3 and 4.

Draw a time line to show the sales level for each of the next four years.

Sales in Year 0 53,250

Growth rate in year 0 0%

Growth rate in years 1-3 20%

Growth rate in year 4 10%

PV

n 0 1 2 3 4

g 0% 20.00% 20.00% 20.00% 10.00%FV

5.16. Number of periods: You invest $150 in a mutual fund today that pays

9 percent interest annually. How long will it take to double your money?

Present value of investment = 150$

Rate of interest = 9%

Future value of investment = 300$

PV

i%

FV

n

FV=PV(1.09)^n

300 = 150(1.09)^n

2.00 = (1.09)^n

1.09^8 = 1.99 or 2.0

n = 8

5.15. Interest rate: You are in a desperate need of cash and turn to your uncle,

who has offered to lend you some money. You decide to borrow $1,300

and agree to pay back $1,500 in two years. Alternatively, you could borrow

from your bank that is charging 6.5 percent interest annually. Should you go with

your uncle or the bank?

Amount borrowed from uncle= 1,300$

Period of loan = 2

Amount paid back to uncle = 1,500$

Rate of interest of bank loan = 6.50%

PV

n

FV

i%

FV=PV(1+i%)^n

1,500 = 1,300(1+i%)^2

Divide FV by PV

Revised FV model (FV/PV) = (1+i%)^2

Square root of FV/PV

Square root of FV/PV - 1.0

Decision to Borrow from Bank

5.14. Present value: Elizabeth Sweeney wants to accumulate $12,000 by the

end of 12 years. If the annual interest rate is 7 percent, how much will

she have to invest today to achieve her goal?

Amount of money to be accumulated= 12,000$

Term of investment = 12

Rate of interest = 7.00%

FV

n

i%

PV=FV(PVIF)

PV

5.13. Present value: You want to buy some bonds that have a value

of $1,000 at the end of seven years. The bonds are said to pay 4.5 percent

interest annually. How much should you pay for them today?

Future value of bonds = 1,000$

Maturity of bonds = 7


FV

n

i%

PV=FV(PVIF)

PV

5.12. Present value: Tracy Chapman is saving to buy a house in five years.

She plans to put 20 percent down at that time, and she believes that she will

need $35,000for the downpayment. If Tracy can invest in a fund that pays 9.25 percent annually,

how much will she need to invest today?

Amount needed for down payment = $35,000.00

Number of years until home purchase= 5

Rate of return on fund investment = 9.25%

FV

n

i%

PV=FV(PVIF)

PV

5.11. Present value: You brother has asked you for a loan and has promised

to pay back $7,750 at the end of three years. If you normally invest

to earn 6 percent per year, how much will you be willing to lend to your brother?

Expected rate of return = 6.00%

Term of loan = 3 years

Future payment to be received = 7,750$

i%

n

FV

PV=FV(PVIF)

PV =

5.10. Present value: Maria Addai has been offered a future payment of

$750 two years from now. If she can earn 6.5 percent

compounded annually on her investment, what should she pay for this

investment today?

Opportunity cost 6.50%

Term = 2 years

Future payment to be received 750$

i%

n

FV

PV=FV(PVIF)

PV =

5.9. Present value: Roy Gross is considering an investment that pays 7.6 percent.

How much will he have to invest today such that the investment will be worth

$25,000 in six years?

Rate of return= 7.60%

Term = 6 years

Terminal value = 25,000$

i%

n

FV

PV=FV(PVIF)

PV =

5.8. Growth rates: Joe Maur, a catcher for the Minnesota Twins is

expected to hit 15 home runs in 2012. If his home-run-hitting ability is

expected to grow by12 percent every year for the following five years,

how many home runs is he expected to hit in 2017?

Number of home runs expected in 2007= 15

Growth rate in hitting home runs = 12%

Number of years of growth = 5

PV

g

n

FV=PV(FVIF)

FV =

Home runs

5.7. Multiple compounding periods: Find the future value of an investment of $100,000

made today for five years and paying 8.75 percent for the following compounding periods:

a. Quarterly

b. Monthly

c. Daily

d. Continuous

Investment = 100,000$

Interest rate= 8.75%

Term = 5

Compound Period m

Quarterly Compounding Monthly Compounding Daily Copounding Continuous Compounding

a b c d

PV

i%

n

m

FV=PV(FVIF) FV=PV(FVIF) FV=PV(FVIF) FV=PV(FVIF)

5.6. Future value: Your birthday is coming up and instead of any presents,

your parents promised to give you $1,000 in cash. Since you have a part-time job

and, thus, don’t need the cash immediately, you decide to invest the money

in a bank CD that pays 5.2 percent quarterly for the next two years.

How much money can you expect to gain in this period of time?

Amount received as a gift= 1,000$

Interest rate (per year)= 5.20%

Frequency of compounding per year = 4

Term of investment (years)= 2

Annually Quarterly

Compounded Compounded

PV

i%

n

m

FV=PV(FVIF)

FV =

5.5. Future value: Your bank pays 5 percent interest semiannually on your

savings account. You don’t expect the current balance of $2,700 to change over

the next four years. How much money can you expect to have at the end of this period?


Current balance= 2,700$

Term (years)= 4

Frequency of compounding (per year)= 2

Annually Semi-annually


i%

PV

n

m

FV=PV(FVIF)

FV =

5.4. Future value: Kate Eden received a graduation present of $2,000 that she

is planning on investing in a mutual fund that earns 8.5 percent each year.

How much money can she collect in three years?

Graduation gift = 2,000$

Rate of return per year = 8.50%

Period of investment = 3 years

PV

i%

n

FV=PV(FVIF)

FV =

5.2. Future value: Ted Rogers is investing $7,500 in a bank CD that pays a 6 percent

annual interest. How much will the CD be worth at the end of five years?

Present value of investment $7,500

Rate of return per year 6.00%

Term (years) 5

Compounding per year 1

PV

i%

n

m

FV=PV(FVIF)

FV =

5.3. Future value: Your aunt is planning to invest in a bank deposit that will pay

7.5 percent interest semiannually. If she has $5,000 to invest, how much will she

have at the end of four years?


Interest rate per year 7.50%

Term (years) 4

Compounding frequency 2

Annually Semiannually


PV

i%

n

m

FV=PV(FVIF)

FV =

#5.26 Time to Grow: You have just inherited $550,000. You plan to save this money and continue to

live off the money that you are earning in your current job. If the $550,000 is everything that you

have other than an old car and some beat-up furniture, and you can invest the money in a bond

that pays 4.6 percent interest annually, how long will it be before you are a millionaire?

PV given

i% given

FV given

n = ? Trial&Error

PV FVIF FV FV

Prove this FV to find n

Time to Grow: You have just inherited $550,000. You plan to save this money and continue to

live off the money that you are earning in your current job. If the $550,000 is everything that you

have other than an old car and some beat-up furniture, and you can invest the money in a bond

that pays 4.6 percent interest annually, how long will it be before you are a millionaire?

Prove this FV to find n

#5.30 Refer to problem 5.29. Congress and the president have decided to increase the Federal tax rate in an

effort to reduce the budget deficit. Suppose that Caroline Weslin will pay 35 percent of her bonus

to the Federal government for taxes if she accepts the bonus today and 40 percent if she receives

her bonus in two years. Will the increase in tax rates affect her decision?

Refer to problem #5.29

Her bonus today is $1,820 today

She will pay 35.00% in taxes if she accepts the bonus today

She will pay $637.00 in taxes.

Net cash flow today is $1,183.00 PV of left over amount.

Assume she receives her bonus of $2,100.00 two years from

Year Bonus CF PVIF PV

0

1

2 $2,100

40.00% She will pay

She will pay this in taxes

PV of left over amount

Yes, it make a difference if the tax rate increases. Note the differences in the two present values.

Accepting the bonus today and paying the current tax rate is the most advantangeous decision.

Refer to problem 5.29. Congress and the president have decided to increase the Federal tax rate in an

effort to reduce the budget deficit. Suppose that Caroline Weslin will pay 35 percent of her bonus

to the Federal government for taxes if she accepts the bonus today and 40 percent if she receives

her bonus in two years. Will the increase in tax rates affect her decision?

in taxes if she accepts the bonus today

today and a discount rate of 6.00%

in taxes if she accepts bonus 2 years from now.

She will pay this in taxes

PV of left over amount

Yes, it make a difference if the tax rate increases. Note the differences in the two present values.

Accepting the bonus today and paying the current tax rate is the most advantangeous decision.

5.1. Future value: Chuck Tomkovick is planning to invest $25,000 today in a mutual

fund that will provide a return of 8 percent each year. What will be the

value of the investment in ten years?


Rate of return per year 8.00%

Term (years) 10

Compounding per year 1

PV

i%

n

m

FV=PV(FVIF)

FV =

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