Time Value of Money_ST

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Denzil Watson and Antony Head, Corporate Finance: Principles and Practice, 5th Edition, © Pearson Education Limited 2010

Chapter 1

The finance function

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Key concepts

• Relationship between risk and return.• Time value of money.

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Risk and return

• Risk refers to the possibility that actual outcome may differ from expected outcome.

• Risk can be measured by standard deviation.• Investors require increasing compensation

(return) for taking on increasing risk.

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Risk and return (Cont.)

• Return refers to the financial rewards gained as a result of making an investment.

• Return of an investment in forms of dividend payments and capital gains (share price increases), or interest payment.

• Return on an investment can be measured over a standard period such as 1 year.

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Risk and return (Cont.)

• Shareholder return is annual dividend (D1) plus share price increase (P1 – P0).

• Relative return in percentage terms is100 × [(P1 – P0) + D1]/P0.

• This is called total shareholder return.

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Time value of money

• Refers to the fact that the value of money changes over time, and relevant to anyone expecting to pay or receive money over a period of time, due to 3 main factors:– Time– Inflation– Risk

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Time value of money

1. The One-Period Case2. The Multi-period Case3. Compounding Periods4. Simplifications5. What Is a Firm Worth?6. Summary and Conclusions

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1. The One-Period Case: Future Value• If you were to invest $10,000 at 5-percent interest

for one year, your investment would grow to $10,500

$500 would be interest ($10,000 × .05)$10,000 is the principal repayment ($10,000 × 1)$10,500 is the total due. It can be calculated as:

$10,500 = $10,000×(1.05).

The total amount due at the end of the investment is called the Future Value (FV).

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• In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r)

Where C0 is cash flow at date 0 and r is the

appropriate interest rate.

FV = $10,500

Year 0 1

C0 = $10,000$10,000 x 1.05

C0×(1 + r)

1. The One-Period Case: Future Value

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• If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment would be worth $9,523.81 in today’s dollars.

05.1000,10$81.523,9$

The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is call the Present Value (PV) of $10,000.

Note that $10,000 = $9,523.81×(1.05).


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• In the one-period case, the formula for PV can be written as:

rCPV

1

1

Where C1 is cash flow at date 1 and r is the appropriate interest rate.

C1 = $10,000

Year 0 1

PV = $9,523.81$10,000/1.05

C1/(1 + r)


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1. The One-Period Case: Net Present Value

• The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.

• Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?

81.23$81.523,9$500,9$

05.1000,10$500,9$

NPVNPV

NPV

Yes!

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In the one-period case, the formula for NPV can be written as:

P VC o s tN P V

If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5-percent, our FV would be less than the $10,000 the investment promised and we would be unambiguously worse off in FV terms as well:

$9,500×(1.05) = $9,975 < $10,000.

1. The One-Period Case: Net Present Value (Cont.)

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2. The Multiperiod Case: Future Value

• The general formula for the future value of an investment over many periods can be written as:

FV = C0×(1 + r)T

Where C0 is cash flow at date 0,

r is the appropriate interest rate, and

T is the number of periods over which the cash is invested.

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• Suppose that Jay Ritter invested in the initial public offering of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year for the next five years.

• What will the dividend be in five years?

FV = C0×(1 + r)T

$5.92 = $1.10×(1.40)5

2. The Multiperiod Case: Future Value

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Future Value and Compounding• Notice that the dividend in year five, $5.92,

is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30

This is due to compounding.

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0 1 2 3 4 5

1 0.1$

3)4 0.1(1 0.1$

0 2.3$

)4 0.1(1 0.1$

5 4.1$

2)4 0.1(1 0.1$

1 6.2$

5)4 0.1(1 0.1$

9 2.5$

4)4 0.1(1 0.1$

2 3.4$

Future Value and Compounding

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Present Value and Compounding• How much would an investor have to set

aside today in order to have $20,000 five years from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

5)15.1(000,20$53.943,9$

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How Long is the Wait?

If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

TrCF V )1(0 T)1 0.1(0 0 0,5$0 0 0,1 0$

2000,5$000,10$)10.1( T

2ln)1 0.1ln ( T

years 27.70953.06931.0

)10.1ln(2ln

T

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Assume the total cost of a university education will be $50,000 when your child enters university in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education? About 21.15%.

What Rate Is Enough?

TrCF V )1(0 1 2)1(0 0 0,5$0 0 0,5 0$ r

10000,5$000,50$)1( 1 2 r 1 211 0)1( r

2 1 1 5.12 1 1 5.111 0 1 21 r

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3. Compounding PeriodsCompounding an investment m times a year

for T years provides for future value of wealth: Tm

mrCFV

10

For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to

93.70$)06.1(50$212.150$ 6

32

FV

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Effective Annual Interest RatesA reasonable question to ask in the above example is what is the effective annual rate of interest on that investment?

The Effective Annual Interest Rate (EAR) is the annual rate that would give us the same end-of-investment wealth after 3 years:

9 3.7 0$)0 6.1(5 0$)21 2.1(5 0$ 632 F V

9 3.7 0$)1(5 0$ 3 E A R

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Effective Annual Interest Rates (Cont.)

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually.

9 3.7 0$)1(5 0$ 3 E A RF V

50$93.70$)1( 3 EAR

1236.150$

93.70$ 31

EAR

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Continuous Compounding (Advanced)• The general formula for the future value of an

investment compounded continuously over many periods can be written as:

FV = C0×erT

Where C0 is cash flow at date 0,

r is the stated annual interest rate,

T is the number of periods over which the cash is invested, and

e is a transcendental number approximately equal to 2.718. ex is a key on your calculator.

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4. Simplifications• Perpetuity

– A constant stream of cash flows that lasts forever.

• Growing perpetuity– A stream of cash flows that grows at a constant rate

forever.

• Annuity– A stream of constant cash flows that lasts for a fixed

number of periods.

• Growing annuity– A stream of cash flows that grows at a constant rate for

a fixed number of periods.

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PerpetuityA constant stream of cash flows that lasts forever.

0

…1

C

2

C

3

C

The formula for the present value of a perpetuity is:

32 )1()1()1( rC

rC

rCPV

rCPV

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Perpetuity: ExampleWhat is the value of a British CONSOL that promises

to pay £15 each year, every year until the sun turns into a red giant and burns the planet to a crisp?

The interest rate is 10-percent.

0

…1

£15

2

£15

3

£15

£15010.

£15PV

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Growing PerpetuityA growing stream of cash flows that lasts forever.

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

The formula for the present value of a growing perpetuity is:

3

2

2 )1()1(

)1()1(

)1( rgC

rgC

rCPV

grCPV

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Growing Perpetuity: ExampleThe expected dividend next year is $1.30 and

dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this

promised dividend stream?

0

…1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

0 0.2 6$0 5.1 0.

3 0.1$

PV

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AnnuityA constant stream of cash flows with a fixed maturity.

0 1

C

2

C

3

C

The formula for the present value of an annuity is:

TrC

rC

rC

rCPV

)1()1()1()1( 32

TrrCPV

)1(11

T

C

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Annuity: ExampleIf you can afford a $400 monthly car payment, how much

car can you afford if interest rates are 7% on 36-month loans?

0 1

$400

2

$400

3

$400

59.954,12$)1207.1(

1112/07.

400$3 6

PV

36

$400

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Growing AnnuityA growing stream of cash flows with a fixed maturity.

0 1

C

The formula for the present value of a growing annuity:

T

T

rgC

rgC

rCPV

)1()1(

)1()1(

)1(

1

2

T

rg

grCPV

)1(11

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

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Growing AnnuityA retirement plan offers to pay $20,000 per year for

40 years and increase the annual payment by 3-percent each year. What is the present value at retirement if the discount rate is 10-percent?

0 1

$20,000

57.121,265$10.103.11

03.10.000,20$ 4 0

PV

2

$20,000×(1.03)

40

$20,000×(1.03)39

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5. What Is a Firm Worth?

• Conceptually, a firm should be worth the present value of the firm’s cash flows.

• The tricky part is determining the size, timing, and risk of those cash flows.

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6. Summary and Conclusions• Two basic concepts, future value and present

value.• Interest rates are commonly expressed on an

annual basis, but semi-annual, quarterly, monthly and even continuously compounded interest rate arrangements exist.

• The formula for the net present value of an investment that pays $C for N periods is:

N

ttN r

CCrC

rC

rCCN PV

1020 )1()1()1()1(

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6. Summary and Conclusions (Cont.)• We presented four simplifying formulae:

rCPV :P erpetuity

grCPV

:P erpetuity G row ing

TrrCPV

)1(11:Annuity

T

rg

grCPV

)1(11 :Annuity Growing

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1. You are planning to move next year. You recently told your best friend that you have decided to sell all of your current furniture just prior to moving as you do not want to pay to transport it across the country. Your friend offered to pay you $2,000 next year for this furniture. What is your friend's offer worth today if you can earn 6 percent on your money?

A) $1,779.99 B) $1,818.22 C) $1,886.79 D) $2,000.00 E) $2,120.00

Multiple choices

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1. C

Multiple choices

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2. All else constant, the present value _____ when the discount rate increases.

A) increases B) decreases C) remains constant D) can either increase or decrease E) can either remain constant or decrease

Multiple choices

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2. B

Multiple choices

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3.Terry is considering purchasing a used tractor and restoring

it. He estimates that his total restoration cost including his labor, in today's dollars, will be $21,400. He also estimates that he can resell the tractor 1 year from now at a price of $24,000. Should Terry undertake this project if he requires a 9.5 percent rate of return? Why or why not?

A) Yes; because the price of $24,000 provides a return in excess of 9.5 percent

B) Yes; because the selling price is greater than the cost C) No; because the present value of the sales price is less

than $21,400 D) No; because the net present value is -$102.16 E) No; because the net present value is $517.81

Multiple choices

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3. A.

Multiple choices

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4. Paul is considering a business investment which is guaranteed to pay $4,000 one year from today. What is the maximum amount Paul should pay today for this investment if he wants to earn a 9.75 percent rate of return?

A) $3,411.68 B) $3,644.65 C) $3,661.67 D) $3,689.02 E) $3,701.08

Multiple choices

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4. B

Multiple choices

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5. Ann is considering buying an office building at a cost of $199,900. She estimates that she can resell the building after one year at a price of $229,500. What discount rate approximately equates those two prices?

A) 83 percent B) 60 percent C) 14.81 percent D) 10 percent E) 33 percent

Multiple choices

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5. C

Multiple choices

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6. The Corner Art Store owns a painting which they display in their showroom. The painting is currently valued at $1,350 but is currently not for sale. The value of the painting is increasing by 7.4 percent annually. Which one of the following prices best represents the value of the painting next year should the store decide to sell it at that time?

A) $1,350 B) $1,400 C) $1,450 D) $1,500 E) $1,550

Multiple choices

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6. C

Multiple choices

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7. On your 5th birthday, your grandparents opened an investment account in your name and made an initial deposit of $2,500. The account pays 4.5 percent interest. How much will you have in the account on your 21st birthday if you don't add or withdraw any money before then?

A) $4,711.68 B) $5,002.10 C) $5,055.93 D) $5,207.19 E) $5,211.14

Multiple choices

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7. C

Multiple choices

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8. Today, you are investing $27,500 in a new project. The

project will return $8,000, $9,500, and $11,000 at the end of each of the next three years, respectively. What is the net present value of this project at a 9 percent discount rate?

A) -$3,670.57 B) -$3,209.11 C) $3,711.09 D) $3,787.88 E) $3,901.11

Multiple choices

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8. A

Multiple choices

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9.Your goal is to have $15,000 in your savings account 3 years from now so that you can purchase a home. Which one of the following statements is correct given this situation? Assume that you only make one initial deposit into the savings account.

A) The higher the interest rate on your savings, the larger the amount that you need to deposit today to fund this account.

B) If you deposit $7,500 today and earn 7 percent interest, you will reach your goal in 3 years.

C) If you have $10,000 to deposit today, you will need to earn at least 15 percent interest to reach your goal.

D) The less money you have to deposit today into the account, the greater the interest rate must be if you are to reach your goal of $15,000.

E) You will have to deposit $12,460 today if the interest you can earn is 4.7 percent.

Multiple choices

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9. D

Multiple choices

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10. What is the future value of $12,000 received today if it is invested at 10.5 percent compounded annually for 25 years?

A) $131,484.77 B) $145,625.76 C) $147,475.83 D) $152,521.75 E) $153,374.89

Multiple choices

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10. B

Multiple choices

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11. Consider an eight-year investment costing $55,000. It is expected to pay $3,000 per year in each of the next five years and $15,000 per year in the last three years. If the required discount rate is 12 percent, what is the net present value of this investment?

A. $86,257.28 B. ($23,742.72)C. $5,000 D. $60,000.00

Multiple choices

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11. B

Multiple choices

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Jim makes a deposit of $12,000 in a bank account. The deposit is to earn interest annually at the rate of 9 percent for seven years.

a)How much will Jim have on deposit at the end of seven years?

b)Assuming the deposit earned a 9 percent rate of interest compounded quarterly, how much would he have at the end of seven years?

c)In comparing parts (a) and (b), what are the respective effective annual yields? Which alternative is better?

Exercises 1

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John is considering the purchase of a lot. He can buy the lot today and expects the price to rise to $15,000 at the end of 10 years. He believes that he should earn an investment yield of 10 percent annually on this investment. The asking price for the lot is $7,000. Should he buy it?

Exercises 2

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An investor can make an investment in a real estate development and receive an expected cash return of $45,000 after six years. Based on a careful study of other investment alternatives, she believes that an 18 percent annual return compounded quarterly is a reasonable return to earn on this investment. How much should she pay for it today?

Exercises 3

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Your uncle has given you a bond that will pay $500 at the end of each year forever into the future. If the market yield on this bond is 8.25 percent, how much is it worth today?

Exercises 4

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Suppose you have an investment that is expected to generate a $20,000 cash flow next year and that this is expected to increase by 5 percent per year forever into the future.

a)If your required rate of return on this investment is 18 percent, how much is it worth to you today?

b)Suppose now that you do not know how fast the cash flows will grow in the future, but that you expect them to grow at a constant rate. Suppose also that this investment is currently priced at $200,000. If the required rate of return is still 18 percent, how fast does the market expect the annual cash flows to grow?

Exercises 5

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