Fundamental of Corporate Finance,chpt5,6

8/3/2019 Fundamental of Corporate Finance,chpt5,6

1/14

Chapter 5,6

Time Value of Money

Prepared and Taught by Lecturer : YIN SOKHNG

E-mail: [email protected]

Tel:(855) 16889872 / 17989972

Chapter Outline

5.1 The One-Period Case

5.2 The Multiperiod Case

5.3 Compounding Periods & Continuous

Compounding

5.4 Valuing Level Cash Flows: Annuities and

Perpetuities

5.5 Loan Amortization

2Instructed by YIN SOKHENG, Master in Finance

3

5.1 The One-Period Case:

Future Value

If you were to invest $10,000 at 5-percent interest forone 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 iscall the Future Value (FV).

Instructed by YIN SOKHENG, Master in Finance


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4

In the one-period case, the formula for FV

can be written as:

FV= C0(1 + r)T

Where C0 is cash flow today (time zero) and

ris the appropriate interest rate.


5

Present Value

If you were to be promised $10,000 due in one

year when interest rates are at 5-percent, your

investment be worth $9,523.81 in todays

dollars.

05.1

000,10$81.523,9$

The amount that a borrower would need to set aside

today to to 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).


6

In the one-period case, the formula for PVcan be

written as:

r

CPV

1

1

Where C1 is cash flow at date 1 and

ris the appropriate interest rate.



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7

5.2 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,

ris the appropriate interest rate, and

Tis the number of periods over which the cash is

invested.


8

Suppose that Jay Ritter invested in the initialpublic offering of the Modigliani company.Modigliani pays a current dividend of $1.10,which is expected to grow at 40-percent peryear 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


9

Future Value and Compounding

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

10

0 1 2 3 4 5

10.1$

3)40.1(10.1$

02.3$

)40.1(10.1$

54.1$

2)40.1(10.1$

16.2$

5)40.1(10.1$

92.5$

4)40.1(10.1$

23.4$


11

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$53.943,9$


12

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?

TrCFV )1(0

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

2000,5$

000,10$)10.1( T

2ln)10.1ln( T

years27.70953.0

6931.0

)10.1ln(

2lnT



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13

What Rate Is Enough?

Assume the total cost of a college educationwill be $50,000 when your child enterscollege in 12 years. You have $5,000 to investtoday. What rate of interest must you earn onyour investment to cover the cost of yourchilds education?

TrCFV )1(0

12)1(000,5$000,50$ r

10000,5$

000,50$)1(12 r 12110)1( r

2115.12115.1110121 r


14

5.3 Compounding Periods & Continuous

CompoundingCompounding an investment m times a year for T

years provides for future value of wealth:Tm

m

rCFV

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$2

12.150$ 6

32

FV


15

Effective Annual Interest Rates

A reasonable question to ask in the aboveexample 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:

93.70$)06.1(50$)2

12.1(50$ 632 FV

93.70$)1(50$ 3 EARInstructed by YIN SOKHENG, Master in Finance


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Effective Annual Interest Rates (continued)

So, investing at 12.36% compounded annually

is the same as investing at 12% compounded

semiannually.

93.70$)1(50$ 3 EARFV

50$93.70$)1(

3 EAR

1236.150$

93.70$31

EAR


17

Effective Annual Interest Rates (continued)

Find the Effective Annual Rate (EAR) of an 18%

APR loan that is compounded monthly.

What we have is a loan with a monthly interest

rate of 1 percent.

This is equivalent to a loan with an annual interest

rate of 19.56 percent

19561817.1)015.1(12

18.11 12

12

mn

m

r


18

Continuous Compounding

The general formula for the future value of an

investment compounded continuously over many

periods can be written as:FV= C0e

rT

Where

C0 is cash flow at date 0,

ris the stated annual interest rate,

Tis 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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5.4 Valuing Level Cash Flows: Annuities

and Perpetuities

Annuityfinite series of equal payments that

occur at regular intervals If the first payment occurs at the end of the period,

it is called anordinary annuity

If the first payment occurs at the beginning of the

period, it is called anannuity due

Perpetuityinfinite series of equal payments


20

Ordinary Annuity vs. Annuity Due

Ordinary Annuity

Annuity starts at

end of period

Annuity Due

Annuity starts

NOW


21

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.

Perpetuity

A constant stream of cash flows that lasts forever.

Growing perpetuity

A stream of cash flows that grows at a constant

rate forever.



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Ordinary Annuities

r

rCFV

r

r

CPV

T

T

1)1(

)1(

11

Note: C = PMT


23

Annuities Due

rr

rCFV

rr

rCPV

T

T

11)1(

1)1(

11

Note: C = PMT


24

Ordinary Annuity

A constant stream of cash flows with a fixed maturity.

The formula for the present value of an ordinary

annuity is:

Tr

C

r

C

r

C

r

CPV

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

Trr

CPV

)1(

11

0 1

C

2

C

3

C

T

C



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Annuity Intuition

An annuity is valued as the difference between twoperpetuities:

one perpetuity that starts at time 1

less a perpetuity that starts at time T+ 1

0 1

C

2

C

3

C

T

C

Tr

r

C

r

CPV

)1(


26

Annuity: Example

If you can afford a $400 monthly car payment, howmuch car can you afford if interest rates are 7% on 36-month loans?

59.954,12$)1207.1(

1112/07.

400$36

PV

0 1

$400

2

$400

3

$400

36

$400


27

What is the present value of a four-year annuity of

$100 per year that makes its first payment two years

from today if the discount rate is 9%?

22.297$09.1

97.327$

0PV

0 1 2 3 4 5

$100 $100 $100 $100 $323.97$297.22

97.327$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$ 4321

4

1

1 t tPV



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28

Growing Annuity

A growing stream of cash flows with a fixed maturity.

The formula for the present value of a growing

annuity:

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

0 1

C

2

C(1+g)

3

C(1+g)2

T

C(1+g)T-1


29

Growing Annuity

A defined-benefit retirement plan offers to pay $20,000per year for 40 years and increase the annual paymentby three-percent each year. What is the present value atretirement if the discount rate is 10 percent?

57.121,265$10.103.11

03.10.000,20$

40

PV

0 1

$20,000

2

$20,000(1.03)

40

$20,000(1.03)39


30

PV of Growing AnnuityYou are evaluating an income property that is providing

increasing rents. Net rent is received at the end of each year. The

first year's rent is expected to be $8,500 and rent is expected to

increase 7% each year. Each payment occur at the end of the year.What is the present value of the estimated income stream over the

first 5 years if the discount rate is 12%?

0 1 2 3 4 5

500,8$

2)07.1(500,8$

095,9$ 65.731,9$

3)07.1(500,8$

87.412,10$

4)07.1(500,8$

77.141,11$

$34,706.26

)07.1(500,8$



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PV of a delayed growing annuityYour firm is about to make its initial public offering ofstock and your job is to estimate the correct offering price.Forecast dividends are as follows.

Year: 1 2 3 4

Dividends pershare

$1.50 $1.65 $1.82 5% growththereafter

If investors demand a 10% return on investments of this risk

level, what price will they be willing to pay?

Year 0 1 2 3

Cash flow $1.50 $1.65 $1.82

4

$1.821.05


32

PV of a delayed growing annuity

Year 0 1 2 3

Cash flow $1.50 $1.65 $1.82 dividend + P3

PV of cash flow $32.81

22.38$05.10.

05.182.13

P

81.32$)10.1(

22.38$82.1$

)10.1(

65.1$

)10.1(

50.1$320

P

= $1.82 + $38.22


33

Perpetuity

A 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( r

C

r

C

r

CPV

r

CPV



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34

Perpetuity: Example

What is the value of a British consol that promises topay 15 each year?

The interest rate is 10-percent.

0

1

15

2

15

3

15

15010.

15PV


35

Growing Perpetuity

A 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( r

gC

r

gC

r

CPV

gr

CPV


36

Growing Perpetuity: Example

The expected dividend next year is $1.30 and dividendsare expected to grow at 5% forever.

If the discount rate is 10%, what is the value of thispromised dividend stream?

0

1

$1.30

2

$1.30(1.05)

3

$1.30(1.05)2

00.26$05.10.

30.1$

PV



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5.5 Loan Amortization

1. Interest only loan, the principal is repaid all at once.

e.g. corporate bond, T-note, T-bond

2. Principal and Interest Loans: Interest declines as youroutstanding principal declines (principal fixed).e.g.

small, smallest , and medium loan

3. Fixed payment loan:loan is repaid with equal (monthly) paymentseach payment is combination of principal and interest

e.g. car loan, mortgage loan


38

Amortization Table:

Principal and Interest Loan

. Interest declines as your outstanding

principal declines (principal fixed).

. Suppose we borrow $5,000 to be repaid in 5

annual payments with a 9% annual interest

rate.


39

Amortization Schedule

Year BeginningBalance

TotalPayment

InterestPaid

PrincipalPaid

EndingBalance

1 $5,000 $ 1,450 $ 450 $ 1,000 $ 4,000

2 $ 4000 $ 1,360 $ 360 $ 1,000 $ 3,000

3 $ 3000 $ 1,270 $ 270 $ 1,000 $ 2,000

4 $ 2000 $ 1,180 $ 180 $ 1,000 $ 1,000

5 $ 1000 $ 1,090 $ 90 $ 1,000 $--0--

Total $ 6,350 $ 1,350 $ 5,000



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Amortization Table:

Fixed Payment Loan

Show breakdown of the payment into interest and

principal

Step 1: Determine the amount of each loan payment.

Solve for present value of annuity

Step 2: Determine how much of each payment is

interest versus principal.

Suppose our 5 years, 9 percent, $5,000 loan (car

loan) was amortized. How would the amortization

schedule look?


41

Fixed Payment Loan

)1()1(

11

)1(

11

rr

r

PVPMT

r

r

PVPMT

T

T

(Ordinary Annuities)

(Annuities due)


42

Amortization ScheduleYear Beginning

BalanceTotal

Payment(PMT)Interest

PaidPrincipal

PaidEndingBalance

1 $5,000.00 $ 1,285.46 $ 450.00 $ 835.46 $ 4,164.54

2 $ 4,164.54 $ 1,285.46 $ 374.81 $ 910.60 $ 3,253.88

3 $ 3,253.88 $ 1,285.46 $ 292.85 $ 992.61 $ 2,261.27

4 $ 2,261.27 $ 1,285.46 $ 203.51 $ 1,081.95 $ 1,179.32

5 $ 1,179.32 $ 1,285.46 $ 106.14 $ 1,179.32 $--0--

Total $6,427.30 $1,427.31 $5,000.00


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Fundamental of Corporate Finance,chpt5,6