After studying this chapter, Corporate Finance of Money...Corporate Finance Chapter 2 The Time Value of Money Dr. Chainarin Srinutchasart 1 After studying this chapter, • You will

FIN 3701 Chapter 2 :The Time Value of Money

1

Assumption University of Thailand

FIN3701CorporateFinance

Chapter 2The Time Value

of Money

Dr. Chainarin Srinutchasart

1

After studying this chapter,• You will understand the concept of future

value, with both annual and intra-yearcompounding.

• Your will be able to distinguish betweenfuture value and present value concepts.

• Your will be able to calculate the futurevalue and present value of a singlepayment and an annuity.

• You will see how to utilize future valueand present value tables.

2

Principles Applied in This Chapter

• Principle 1: Money Has a Time Value

• Principle 3: Cash Flows Are the Sourceof Value.

3

Corporate Finance addresses thefollowing 3 questions:

1. What long-term investments should thefirm engage in?

2. How can the firm raise money for therequired investments? (Alternatives:Bonds, Stocks, Preferred Stocks=whatis the appropriate price?)

3. How much short-term cash flow does acompany need to pay its bills? and howto raise it

4

We know that receiving $1 today is worth morethan $1 in the future. This is due toopportunity costs.The opportunity cost of receiving $1 in thefuture is the interest (based upon inflation,economy and other risks) we could haveearned if we had received the $1 sooner.

Today Future

So, interest rate = Rf + Inflation + Risk Premium

5

Intuition Behind Present ValueThere are three reasons why a dollar tomorrow isworth less than a dollar today.• Individuals prefer present consumption to

future consumption. To induce people to give uppresent consumption you have to offer them morein the future.

• When there is monetary inflation. the value ofcurrency decreases over time. The greater theinflation the greater the difference in valuebetween a dollar today and a dollar tomorrow.

• If there is any uncertainty (risk) associated withthe cash flow in the future, the less that cash flowwill be valued.

Interest rate = Rf + Inflation rate + Risk premium6


2

Using Timelines to Visualize Cashflows

• A timeline identifies the timing andamount of a stream of cash flows alongwith the interest rate.

• A timeline is typically expressed in years,but it could also be expressed as months,days or any other unit of time.

7

Time Line Example

• The 4-year timeline illustrates the following:• The interest rate is 10%.• A cash outflow of $100 occurs at the beginning

of the first year (at time 0), followed by cashinflows of $30 and $20 in years 1 and 2, a cashoutflow of $10 in year 3 and cash inflow of $50in year 4.

The end ofperiod

0 1 2 3 4

YearsCash flow -$100 $30 $20 -$10 $50

i=10%

8

The Time Value of MoneyCompounding

andDiscounting Single Sums

9

If we can measure thisopportunity cost, we can:• Translate $1 today into its equivalent in the future

(compounding).

• Translate $1 in the future into its equivalent today(discounting).

Today Future

?

Today Future

?

10

Simple Interest and CompoundInterest• What is the difference between simple

interest and compound interest?• Simple interest: Interest is earned only

on the principal amount.• Compound interest: Interest is earned

on both the principal and accumulatedinterest of prior periods.

11

Simple Interest and CompoundInterest (cont.)• Example: Suppose that you deposited

$500 in your savings account that earns5% annual interest. How much will youhave in your account after two yearsusing (a) simple interest and (b)compound interest?

12


3

Simple Interest and CompoundInterest (cont.)Simple Interest• Interest earned

• = 5% of $500 = .05×500 = $25 per year• Total interest earned = $25×2 = $50• Balance in your savings account:

• = Principal + accumulated interest• = $500 + $50 = $550

13

Simple Interest and CompoundInterest (cont.)Compound interest• Interest earned in Year 1

• = 5% of $500 = $25• Interest earned in Year 2

• = 5% of ($500 + accumulated interest)• = 5% of ($500 + 25) = .05×525 =

$26.25• Balance in your savings account:

• = Principal + interest earned• = $500 + $25 + $26.25 = $551.25

14

Compound Interest andFuture Value

15

Future Value - single sums• If you deposit $100 in an account earning

6%, how much would you have in theaccount after 1 year?

16

Future Value - single sums• If you deposit $100 in an account earning 6%,

how much would you have in the accountafter 1 year?

PV = FV =

0 1Calculator Solution:P/Y = 1 I = 6N = 1 PV = -100FV = $106

-100 106

17

Future Value - single sums• If you deposit $100 in an account earning 6%,

how much would you have in the accountafter 1 year?

PV = FV =

0 1Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 1 ) (use FVIF table, or)FV = PV (1 + i)n

FV = 100 (1.06)1 = $106

-100 106

18


4

19


6%, how much would you have in theaccount after 5 years?

20

Future Value - single sums• If you deposit $100 in an account earning 6%, how

much would you have in the account after 5years?

PV = FV =0 5

Calculator Solution:P/Y = 1 I = 6N = 5 PV = -100FV = $133.82

-100 133.82

21

Future Value - single sums• If you deposit $100 in an account earning 6%, how

much would you have in the account after 5years?

PV = FV =0 5

-100 133.82

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 5 ) (use FVIF table, or)FV = PV (1 + i)nFV = 100 (1.06)5 = $133.82

22


6% with quarterly compounding, howmuch would you have in the account after5 years?

23

Future Value - single sums• If you deposit $100 in an account earning 6% with

quarterly compounding, how much would youhave in the account after 5 years?

PV = FV =

0 20Calculator Solution:P/Y = 4 I = 6N = 20 PV = -100FV = $134.68

-100 134.68

24


5


quarterly compounding, how much would youhave in the account after 5 years?

PV = FV =

0 20

-100 134.68

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.015)20 = $134.6825


6% with monthly compounding, howmuch would you have in the account after5 years?

26


monthly compounding, how much would youhave in the account after 5 years?

PV = FV =0 20

Calculator Solution:P/Y = 12 I = 6N = 60 PV = -100FV = $134.89

-100 134.89

27


monthly compounding, how much would youhave in the account after 5 years?

PV = FV =0 20

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .005, 60) (use FVIF table, or)FV = PV (1 + i)nFV = 100 (1.005)60 = $134.89

-100 134.89

28

Future Value - continuouscompounding

• What is the FV of $1,000 earning 8% withcontinuous compounding, after 100years?

29

Future Value - continuouscompounding• What is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

PV = FV =

0 100Mathematical Solution:FV = PV (e in)FV = 1000 (e .08x100) = 1000 (e 8)FV = $2,980,957.99

-1000 $2.98m

30


6

Present Value and Annuities

31

Present Value - single sums• If you receive $100 one year from now, what is the

PV of that $100 if your opportunity cost is 6%?

32



PV = FV =0 1

Calculator Solution:P/Y = 1 I = 6N = 1 FV = 100PV = -94.34

100-94.34

33



PV = FV =0 1

Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 1 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34

100-94.34

34

35

Present Value - single sums• If you receive $100 five years from now, what is

the PV of that $100 if your opportunity cost is 6%?

PV = FV =

0 5Calculator Solution:P/Y = 1 I = 6N = 5 FV = 100PV = -74.73

100-74.73

36


7

Present Value - single sums• If you receive $100 five years from now, what is

the PV of that $100 if your opportunity cost is 6%?

PV = FV =

0 5Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 5 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73

100-74.73

37

Present Value - single sums• What is the PV of $1,000 to be received 15 years

from now if your opportunity cost is 7%?

PV = FV =

0 15Calculator Solution:P/Y = 1 I = 7N = 15 FV = 1,000PV = -362.45

1000-362.45

38

Present Value - single sums• What is the PV of $1,000 to be received 15 years

from now if your opportunity cost is 7%?

PV = FV =0

Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .07, 15 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.07)15 = $362.45

-362.

15

100045

39

Present Value - single sums• If you sold land for $11,933 that you bought 5

years ago for $5,000, what is your annual rate ofreturn?

Calculator Solution:P/Y = 1 N = 5PV = -5,000 FV = 11,933I = 19%

PV = FV =0 5

-5,000 11,933

40

Present Value - single sums• If you sold land for $11,933 that you bought 5

years ago for $5,000, what is your annual rate ofreturn?

Mathematical Solution:PV = FV (PVIF i, n )5,000 = 11,933 (PVIF ?, 5 )PV = FV / (1 + i)n

5,000 = 11,933 / (1+ i)5

.419 = ((1/ (1+i)5)2.3866 = (1+i)5

(2.3866)1/5 = (1+i) i = .19

41

Present Value - single sums• Suppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How longwill it take for your account to grow to $500?

Calculator Solution:• P/Y = 12 FV = 500• I = 9.6 PV = -100• N = 202 months

PV = FV =

0 5

-100 500

42


8

Present Value - single sums• Suppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How longwill it take for your account to grow to $500?

Mathematical Solution:PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)1.60944 = .007968 N N = 202 months

43

The Time Value of Money

Compounding and DiscountingCash Flow Streams

0 1 2 3 4

44

Annuities

45

Annuities• Annuity: a sequence of equal cash

flows, occurring at the end of each period.

0 1 2 3 4

46

Examples of Annuities:• If you buy a bond, you will receive equal

semi-annual coupon interest paymentsover the life of the bond.

• If you borrow money to buy a house or acar, you will pay a stream of equalpayments.

47

Future Value - annuity• If you invest $1,000 each year at 8%, how

much would you have after 3 years?

0 1 2 3

Calculator Solution:P/Y = 1 I = 8 N = 3PMT = -1,000FV = $3,246.40

1000 1000 1000

48


9

Future Value - annuity• If you invest $1,000 each year at 8%, how

much would you have after 3 years?Mathematical Solution:FV = PMT (FVIFA i, n )FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)

FV = PMT (1 + i)n - 1i

FV = 1,000 (1.08)3 - 1 = $3246.40.08

4950

Present Value - annuity• What is the PV of $1,000 at the end of

each of the next 3 years, if the opportunitycost is 8%?

0 1 2 3

Calculator Solution:P/Y = 1 I = 8 N = 3PMT = -1,000PV = $2,577.10

1000 1000 1000

51

Present Value - annuity• What is the PV of $1,000 at the end of each of the

next 3 years, if the opportunity cost is 8%?Mathematical Solution:PV = PMT (PVIFA i, n )PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)

1PV = PMT 1 - (1 + i)n

i

1PV = 1000 1 - (1.08 )3 = $2,577.10

.0852

53

Growing Annuities,Perpetuities,Growing Perpetuities

54


10

Growing AnnuityA growing stream of cash flows with a fixedmaturity

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

55

Growing Annuity: ExampleA defined-benefit retirement plan offers to pay$20,000 per year for 40 years and increase theannual payment by 3% each year. What is thepresent value at retirement if the discount rateis 10%?

0 1

$20,000

57.121,265$10.1

03.11

03.10.

000,20$40

PV

2

$20,000×(1.03)

40

$20,000×(1.03)39

56

Growing Annuity: Example

• You are evaluating an income generatingproperty. Net rent is received at the end ofeach year.

• The first year's rent is expected to be$8,500, and rent is expected to increase7% each year.

• What is the present value of theestimated income stream over the first 5years if the discount rate is 12%?

57

Growing Annuity: Example (Cont.)

0 1 2 3 4 5

500,8$ )07.1(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

58

Perpetuities

59

Perpetuities• Suppose you will receive a fixed payment

every period (month, year, etc.) forever.This is an example of a perpetuity.

• You can think of a perpetuity as anannuity that goes on forever.

60


11

Present Value of a Perpetuity• When we find the PV of an annuity, we

think of the following relationship:

PV = PMT (PVIFA i, n )

61

Mathematically,

(PVIFA i, n ) =

We said that a perpetuity is an annuitywhere n = infinity. What happens tothis formula when n gets very, verylarge?

1 – (1 + i)

i

1n

62

Mathematically,

1 – (1 + i)

i

1n

this becomes zero.

So we’re left with PVIFA =1i

63

Present Value of a Perpetuity• So, the PV of a perpetuity is very simple

to find:

PMTiPV =

64

What should you be willing to pay in orderto receive $10,000 annually forever, if yourequire 8% per year on the investment?

PMT $10,000i .08

PV = =

= $125,000

65

Growing Perpetuities

66


12

Growing Perpetuity• A growing stream of cash flows that lasts

forever

0…

1

C

2

C×(1+g)

3

C ×(1+g)2

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

67

Growing Perpetuity: Example• The 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 valueof this promised dividend stream?

0…

1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

00.26$05.10.

30.1$

PV

68

Annuities Due and UnevenCash Flows

69

Ordinary Annuity vs.Annuity Due

$1000 $1000 $1000

4 5 6 7 8

70

Annuities Due• Annuity due is an annuity in which all the

cash flows occur at the beginning of theperiod. For example, rent payments onapartments are typically annuity due asrent is paid at the beginning of the month.

71

Earlier, we examined this“ordinary” annuity:

• Using an interest rate of 8%, we find that:• The Future Value (at 3) is $3,246.40.• The Present Value (at 0) is $2,577.10.

0 1 2 3

1000 1000 1000

72


13

Earlier, we examined this“ordinary” annuity:

• Same 3-year time line,• Same 3 $1000 cash flows, but• The cash flows occur at the beginning of each

year, rather than at the end of each year.• This is an “annuity due.”

0 1 2 3

1000 1000 1000

73

Future Value - annuity due

0 1 2 3

• If you invest $1,000 at the beginning of eachof the next 3 years at 8%, how much wouldyou have at the end of year 3?

-1000 -1000 -1000

Calculator Solution:Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000FV = $3,506.11

74

Future Value - annuity due• If you invest $1,000 at the beginning of each

of the next 3 years at 8%, how much wouldyou have at the end of year 3?Mathematical Solution: Simply compound theFV of the ordinary annuity one more period:FV = PMT (FVIFA i, n ) (1 + i)FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)

FV = PMT (1 + i)n - 1i

FV = 1,000 (1.08)3 - 1 = $3,506.11.08

(1 + i)

(1.08)

75

Calculator Solution:Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000PV = $2,783.26

Present Value - annuity due

0 1 2 3

• What is the PV of $1,000 at the beginning ofeach of the next 3 years, if your opportunitycost is 8%?

1000 1000 1000

76

Present Value - annuity dueMathematical Solution: Simply compound the FVof the ordinary annuity one more period:PV = PMT (PVIFA i, n ) (1 + i)PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)

1PV = PMT 1 - (1 + i)n

i

1PV = 1000 1 - (1.08 )3 = $2,783.26

.08(1.08)

(1 + i)

77

Uneven Cash Flows

78


14

• Is this an annuity?• How do we find the PV of a cash flow stream

when all of the cash flows are different? (Usea 10% discount rate.)

-10,000 2,000 4,000 6,000 7,000

Uneven Cash Flows

79

• Sorry! There’s no quickie for this one. Wehave to discount each cash flow backseparately.

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000

Uneven Cash Flows

80

Period CF PV (CF)0 -10,000 -10,000.001 2,000 1,818.182 4,000 3,305.793 6,000 4,507.894 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000

NPV

81

Examples

82

Example• Cash flows from an investment are

expected to be $40,000 per year at theend of years 4, 5, 6, 7, and 8. If yourequire a 20% rate of return, what is thePV of these cash flows?

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

83

• This type of cash flow sequence is oftencalled a “deferred annuity.”

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

84


15

How to solve:• 1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

Or,85

• 2) Find the PV of the annuity:• PV : End mode; P/YR = 1; I = 20; PMT =

40,000; N = 5PV = $119,624

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

3

86

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

$119,624

Then discount this single sum back to time 0.PV: End mode; P/YR = 1; I = 20;

N = 3; FV = 119,624;Solve: PV = $69,226

87

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

$119,624

• The PV of the cash flow stream is $69,226.

69,226

88

Retirement Example• After graduation, you plan to invest $400

per month in the stock market. If youearn 12% per year on your stocks, howmuch will you have accumulated whenyou retire in 30 years?

0 1 2 3 . . . 360

400 400 400 400

89

0 1 2 3 . . . 360

400 400 400 400

Using your calculator,

P/YR = 12N = 360

PMT = -400I%YR = 12

FV = $1,397,985.65

90


16

Retirement ExampleIf you invest $400 at the end of each month for thenext 30 years at 12%, how much would you have atthe end of year 30?

Mathematical Solution:

FV = PMT (FVIFA i, n )FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table)

FV = PMT (1 + i)n - 1i

FV = 400 (1.01)360 - 1 = $1,397,985.65.01

91

House Payment Example• If you borrow $100,000 at 7% fixed

interest for 30 years in order to buy ahouse, what will be your monthly housepayment?

92

0 1 2 3 . . . 360

? ? ? ?


P/YR = 12N = 360

I%YR = 7PV = $100,000

PMT = -$665.30

93

House Payment ExampleMathematical Solution:PV = PMT (PVIFA i, n )100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table)

1PV = PMT 1 - (1 + i)n

i

1100,000 = PMT 1 - (1.005833 )360 PMT=$665.30

.00583394

Example• Upon retirement, your goal is to spend 5

years traveling around the world. To travel instyle will require $250,000 per year at thebeginning of each year.

• If you plan to retire in 30 years, what are theequal monthly payments necessary toachieve this goal? The funds in yourretirement account will compound at 10%annually.

95

• How much do we need to have by the endof year 30 to finance the trip?

• PV30 = PMT (PVIFA .10, 5) (1.10) == 250,000 (3.7908) (1.10) == $1,042,470

27 28 29 30 31 32 33 34 35

250 250 250 250 250

96


17

27 28 29 30 31 32 33 34 35

250 250 250 250 250


Mode = BEGINPMT = -$250,000

N = 5I%YR = 10P/YR = 1

PV = $1,042,466

97

27 28 29 30 31 32 33 34 35

250 250 250 250 250

• Now, assuming 10% annualcompounding, what monthly paymentswill be required for you to have$1,042,466 at the end of year 30?

1,042,466

98

27 28 29 30 31 32 33 34 35

250 250 250 250 250


Mode = ENDN = 360

I%YR = 10P/YR = 12

FV = $1,042,466PMT = -$461.17

1,042,466

99

• So, you would have to place $461.17 inyour retirement account, which earns10% annually, at the end of each of thenext 360 months to finance the 5-yearworld tour.

100

Amortized Loans,Making Interest RatesComparable,Some Complications

101

Amortized Loans• An amortized loan is a loan paid off in

equal payments – consequently, the loanpayments are an annuity.

• Examples: Home mortgage loans, Autoloans

102


18

Amortized Loans (cont.)• Example Suppose you plan to get a

$9,000 loan from a furniture dealer at18% annual interest with annualpayments that you will pay off in over fiveyears.

• What will your annual payments be onthis loan?

103

Amortized Loans (cont.)• Using a Financial Calculator• Enter

• N = 5• i/y = 18.0• PV = 9000• FV = 0• PMT = -$2,878.00

104

The Loan Amortization Schedule

Year AmountOwed onPrincipal atthe Beginningof the Year (1)

AnnuityPayment(2)

InterestPortionof theAnnuity(3) = (1) ×18%

Repayment of thePrincipalPortion oftheAnnuity(4) =(2) –(3)

Outstanding LoanBalance atYear end,After theAnnuityPayment (5)=(1) – (4)

1 $9,000 $2,878 $1,620.00 $1,258.00 $7,742.00

2 $7,742 $2,878 $1,393.56 $1,484.44 $6,257.56

3 $6257.56 $2,878 $1,126.36 $1,751.64 $4,505.92

4 $4,505.92 $2,878 $811.07 $2,066.93 $2,438.98

5 $2,438.98 $2,878 $439.02 $2,438.98 $0.00

105

The Loan Amortization Schedule(cont.)

• We can observe the following from thetable:• Size of each payment remains the

same.• However, Interest payment declines

each year as the amount owed declinesand more of the principal is repaid.

106

Making Interest RatesComparable

107

Which is the better loan:• 8% compounded annually, or• 7.8% compounded annually?

Which is the better loan:• 8% compounded annually, or• 7.8% compounded monthly?

108


19

Annual Percentage Rate (APR)• The annual percentage rate (APR)

indicates the amount of interest paid orearned in one year without compounding.APR is also known as the nominal orstated interest rate. This is the raterequired by law.

109

Comparing Loans using EAR(or APY> Annual Percentage Yield)

• We cannot compare two loans based on APR ifthey do not have the same compounding period.

• To make them comparable, we calculate theirequivalent rate using an annual compoundingperiod. We do this by calculating the effectiveannual rate (EAR)

110

Comparing Loans using EAR(cont.)• Example Calculate the EAR for a loan that

has a 5.45% quoted annual interest ratecompounded monthly.

• EAR = [1+.0545/12]12 - 1= 1.0558 – 1= .05588 or 5.59%

111

Which is the better loan:• 8% compounded annually, or• 7.85% compounded quarterly?• We can’t compare these nominal (quoted)

interest rates, because they don’t include thesame number of compounding periods peryear!

We need to calculate the APY or EAR.

112

• Find the APY for the quarterly loan:

• The quarterly loan is more expensive than the8% loan with annual compounding!

APY = ( 1 + ) m - 1quoted ratem

APY = ( 1 + ) 4 - 1.07854

Annual Percentage Yield (APY)

APY = .0808, or 8.08%

113

Complications:Inflation,Compounding Frequency,Currencies

114


20

Complication 1Inflation• How does inflation affect DCF analysis?

NPV = CF +0

CF1

(1+r)

CF2

(1+r)2

CF3

(1+r)3

CF4

(1+r)4

CF5

(1+r)5+ + + + + …

Discounting Rule• Treat inflation consistently: Discount real

cashflows at the real interest rate andnominal cashflows at the nominal interestrate.

115

Complication 1

CashflowsNorminal = actual cashflowsReal = cashflows expressed in today's purchasing power

real CF = normal CF / (1+ inflation rate)

Discount ratesNorminal = actual interest ratesReal = interest rates adjusted for inflation1 + real int. rate = (1+ nominal int. rate)/(1+ inflation rate)Approximation: real int. rate normal int.rate - inflationrate

Terminology

t tt

116

ExampleThis year you earned $100,000. You expect your earningsto grow 2% annually, in real terms, for the remaining 20years of your career. Interest rates are currently 5% andinflation is 2%. What is the present value of your income?Real interest rate = 1.05/ 1.02 - 1 = 2.94%Real cashflows

Cashflow 102,000 104,040 ….. 148,595

1.0294 1.02942 ….. 1.029420

PV 99,086 98,180 ….. 83,219

Year 1 2 ….. 20

_●●

Present value= $1,818,674117

Complication 2Compounding frequencyOn many investments or loans, interest is credited orcharged more often than once a year.ExamplesBank accounts - dailyMortgages and leases - monthlyBonds – semiannuallyImplicationEffective annual rate (EAR) can be much differentthan the stated annual percentage rate (APR)

118

ExampleCar Loan'Finance charge on the unpaid balance, computeddaily, at the rate of 6.75% per year.‘If you borrow $10,000 to be repaid in one year, howmuch would you owe in a year?Daily interest rate = 6.75 / 365 = 0.0185%Day 1: balance=10,000.00 x 1.000185=10,001.85Day 2: balance=10,000.85 x 1.000185=10,003.70Day 365: balance=10,000.00 x (1.000185)365=10.698.50

EAR=6.985%119

Effective annual rateEAR = (1+APR/k)k – 1APR = quoted annual percentage ratek = number of compounding intervals each yearWhat happens as k gets big?In the limit as k → ∞, interest is 'continuouslycompounded‘EAR = eAPR – 1'e' is the base of the natural logarithm2.7182818

120


21

Complication 2, cont.Discounting ruleIn applications, interest is normallycompounded at the same frequency aspayments.If so, just divide the APR by number ofcompounding intervals.BondsMake semiannual payments, interestcompounded semiannually Discountsemiannual cashflows by APR/2MortgagesMake monthly payments, interest compoundedmonthly Discount monthly cashflows by APR/12

121

Complication 3CurrenciesHow do we discount cashflows in foreign currencies?

Discounting ruleDiscount each currency at its own interest rate: discount $'sat the U.S. interest rate, €'s at the U.K. interest rate, ....This gives PV of each cashflow stream in its own currency.Convert to domestic currency at the current exchange rate.

PV = CF +0

CF1

(1+r)

CF2

(1+r)2

CF3

(1+r)3

CF4

(1+r)4

CF5

(1+r)5+ + + + + …

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Currencies, cont.LogicYou have $1 now. How many pounds can youconvert this to in one year? The current exchangerate is 1.6$/€ and the U.K.interest rate is 5%.Today: $1 = €0.625One year: €0.625 x 1.05=€0.6563Implication: $1 today is worth 0.6263 pounds in oneyear.The discounting rule simply reverses this procedure.It starts with pounds in one year, then converts to $today.

123

ExampleYour firm just signed a contract to deliver 2,000batteries in each of the next 2 years to a customer inJapan, at a per unit price of ¥800. It also signed acontract to deliver 1,500 in each of the next 2 yearsto a customer in Britain, at a per unit price of £6.2.Payment is certain and occurs at the end of the year.

The British interest rate is r£ = 5% and the Japaneseinterest rate is r¥=3.5%. The exchange rates are s¥/$= 118 and s$/£ = 1.6.

What is the value of each contract?

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ExampleJapanCFt = 2,000 x 800 = ¥ 1,600,000

PV contract = 3,039,511 x (1/118 ¥/$ ) = $ 25,759

BritainCFt = 1,500 x 6.2 = £ 9,300

PV contact = 17,293 x 1.6$/£ = $ 27,668

PV contract = = ¥ 3,039,5111,600,000 1,600,000

1.035 1.0352

+

PV contract = = ¥ 17,2939,3001.05 1.052

+ 9,300

Source: NYU125

Reference• Sheridan Titman, Arthur J. Keown, John D. Martin,

Financial Management: Principles andApplications(12thed). New Jersey: Pearson &Prentice Hall Inc, 2014.

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After studying this chapter, Corporate Finance of Money...Corporate Finance Chapter 2 The Time Value of Money Dr. Chainarin Srinutchasart 1 After studying this chapter, • You will

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