2007

Page 1F. MICHAUX

CORPORATE FINANCE

2007

Page 2F. MICHAUX

GENERAL AGENDA

Valuation and Discounted Cash Flow Method

Valuing Bonds

Valuing Stocks

2007

Page 3F. MICHAUX

VALUATION AND DISCOUNTED CASH FLOW

METHOD

2007

Page 4F. MICHAUX

TIME VALUE OF MONEY

BASIC PROBLEM FACED BY FINANCIAL MANAGER IS

HOW TO VALUE FUTURE CASH FLOWS?

I HAVE TO SPEND MONEY TODAY TO BUILD A PLANT WHICH WILL GENERATE CASH

FLOWS IN THE FUTURE

2007

Page 5F. MICHAUX

IF I HAD THE DOLLAR TODAY

I COULD INVEST IT

EARN INTEREST DURING THE YEAR

SO THAT I’D HAVE MORE THAN A DOLLAR IN A YEAR’S TIME

A DOLLAR TODAY

IS WORTH MORE

THAN A DOLLAR IN THE FUTURE

2007

Page 6F. MICHAUX

INVESTING FOR ONE PERIOD

I INVEST $100 TODAY AT r = .1 PER YEAR

AT END OF YEAR, I RECEIVE $110 (FV)

FV=100(1+r)

2007

Page 7F. MICHAUX

PRESENT VALUE

Present Value = PV

PV = Discount Factor X C1

2007

Page 8F. MICHAUX

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of any cash flow.

DFr t

1

1( )

PRESENT VALUE

2007

Page 9F. MICHAUX

Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time.

t

tt r

CCDFPV

1

PRESENT VALUE

2007

Page 10F. MICHAUX

PRESENT VALUESEXAMPLE: SAVING FOR A NEW

COMPUTERYou will need $3,000 in a year’s time to buy a

computerYou can earn interest at 8% per yearHow much do you need to set aside now?

PV OF $3,000 = 3,000/1.08 = 3,000 x .926 = $2,777.77.926 is the 1-YEAR DISCOUNT FACTOR

at the end of 1 year$2,777.77 grows to 2,777.77 x 1.08 = $3,000

2007

Page 11F. MICHAUX

OFTEN CALLED DISCOUNT FACTOR

CALCULATING DISCOUNTED CASH FLOWS

r DISCOUNT RATEHURDLE RATE

OPPORTUNITY COST OF CAPITAL

1

( )1 r

2007

Page 12F. MICHAUX

WE HAVE ASSUMED THAT FUTURE CASH FLOWS ARE KNOWN WITH

CERTAINTY

IF FUTURE CASH FLOWS ARE NOT CERTAINUSE EXPECTED FUTURE CASH FLOWS

USE HIGHER DISCOUNT RATEEXPECTED RATE OF RETURN ON OTHER

INVESTMENTS OF COMPARABLE RISK WHICH IS NOT AVAILABLE TO US BECAUSE WE INVESTED IN THE PROJECT

SAFE DOLLAR IS WORTH MORE THAN A RISKY DOLLAR

2007

Page 13F. MICHAUX

INTEREST RATE DOES NOT HAVE TO BE FOR A YEAR

INTEREST RATE per period WHERE THE PERIOD IS ALWAYS SPECIFIED

THE EQUATION FV=PV(1+r)

GIVES THE FV at the end of the period, WHEN I INVEST P AT AN INTEREST RATE OF r PER PERIOD

2007

Page 14F. MICHAUX

EXAMPLE

r =.02 PER QUARTER

P=$100

HOW MUCH DO I HAVE AT THE END OF THE QUARTER?

FV=PV(1+r)=100x1.02=102

2007

Page 15F. MICHAUX

TWO RULES FOR ACCEPTING OR REJECTING

PROJECTS

1. INVEST IN PROJECTS WITH POSITIVE NPV

2. INVEST IN PROJECTS OFFERING RETURN

GREATER THAN

OPPORTUNITY COST OF CAPITAL

2007

Page 16F. MICHAUX

RATE OF RETURN RULE

RETURN = PROFIT = 400 - 350 = 14.3%

INVESTMENT 350

ACCEPT PROJECT BECAUSE RATE OF RETURN IS GREATER THAN THE OPPORTUNITY COST OF CAPITAL, 7%

2007

Page 17F. MICHAUX

VALUING AN OFFICE BUILDING

STEP 1: FORECAST CASH FLOWSCost of building, C0 = 350Sale price in Year 1, C1 = 400

STEP 2: ESTIMATE OPPORTUNITY COST OF CAPITALIf equally risky investments in the capital marketoffer a return of 7%, then cost of capital, r = 7%

STEP 3: Discount future cash flows

C1 400PV = = = 374 1 + r 1.07STEP 4: Accept project if PV of payoff exceeds investmentNPV = -350 + 374 = +24

2007

Page 18F. MICHAUX

ONE-PERIOD PROJECT: RETURN UNCERTAIN

INVEST $1,000 NOW. RECEIVE EXPECTED UNCERTAIN CASH FLOW AFTER 1 YEAR, WHOSE EXPECTED VALUE IS $1,300

INVESTORS CAN BUY EQUALLY RISKY SECURITIES WITH 35% EXPECTED RETURN.

DECISION:1. DON'T INVEST BECAUSE 30% PROJECT RETURN IS LESS THAN

35% OPPORTUNITY COST.2. DON'T INVEST BECAUSE NET PRESENT VALUE IS

NEGATIVE.

1,300 NET PRESENT VALUE = 1.35 - 1,000 = 963 - 1,000 = -37

VALUE OF FIRMWILL FALL BY $37

IF WE ACCEPT THE PROJECT

2007

Page 19F. MICHAUX

INVESTING FOR MORE THAN ONE PERIOD

I INVEST P=$100 FOR 2 YEARS AT r =.1 PER YEAR.

AT END OF YEAR 1, I HAVE FV1=100X1.1=110 IN MY ACCOUNT, WHICH IS MY BEGINNING PRINCIPAL FOR YEAR 2.

AT THE END OF YEAR 2, I WILL HAVE

FV2 =FV1(1+r) =P(1+r)(1+r) =P(1+r)2

=121I WILL EARN $10 INTEREST IN YEAR 1,

$11 INTEREST IN YEAR 2, ALTHOUGH r = .1 IN BOTH YEARS.

WHY?

2007

Page 20F. MICHAUX

FUTURE VALUE OF $121 HAS FOUR PARTS

FV2=P(1+r)2=P+2rP+Pr2

P=100 RETURN OF PRINCIPAL

2rP=20 SIMPLE INTEREST ON PRINCIPAL FOR 2 YEARS AT 10% PER YEAR

Pr 2=1 INTEREST EARNED IN YEAR 2 ON $10 INTEREST PAID IN YEAR 1

AMOUNT OF SIMPLE INTEREST CONSTANT EACH YEAR

AMOUNT OF COMPOUND INTEREST INCREASES EACH YEAR

2007

Page 21F. MICHAUX

FV OF PRINCIPAL, P,

AT END OF n YEARS IS

FVn=PV(1+R)n

2007

Page 22F. MICHAUX

COMPOUND INTEREST

INTEREST EARNED ON PRINCIPAL AND

REINVESTED INTEREST OF PRIOR

PERIODS

2007

Page 23F. MICHAUX

SIMPLE INTEREST

INTEREST EARNED ON THE

ORIGINAL PRINCIPAL ONLY

2007

Page 24F. MICHAUX

Compound Interest

i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183

2007

Page 25F. MICHAUX

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

Year

r = 5%

r = 10%

r = 15%

FUTURE VALUE

FUTURE VALUEYear

125

1020

5%1.0501.1031.2761.6292.653

10%1.1001.2101.3312.5946.727

15%1.1501.3232.0114.04616.37

0 2 4 6 8 10 12 14 16 18 20

20

15

10

5

0

FUTURE VALUE OF $1

YEARS

2007

Page 26F. MICHAUX

PRESENT VALUE

PRESENT VALUE OF $1

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10 12 14 16 18 20

r = 5%

r = 10%

r = 15%

PRESENT VALUE

Year 5% 10% 15% 1 .952 .909 .870 2 .907 .826 .756 5 .784 .621 .497 10 .614 .386 .247 20 .377 .149 .061

YEARS

2007

Page 27F. MICHAUX

FUTURE VALUE• COMPOUND

PRINCIPAL AMOUNT

FORWARD INTO THE

FUTURE

PRESENT VALUE• DISCOUNT

A FUTURE VALUE BACK TO THE

PRESENT

2007

Page 28F. MICHAUX

BASIC RELATIONSHIP BETWEEN PV AND FV

PVFV

(1 )0

t

t r

2007

Page 29F. MICHAUX

Present ValuesExample

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.

000,300000,100000,150

2Year 1Year 0Year

2007

Page 30F. MICHAUX

Present Values

Example - continued

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.

400,18$

900,261000,300873.2

500,93000,100935.1

000,150000,1500.10Value

Present

Flow

Cash

Factor

DiscountPeriod

207.11

07.11

TotalNPV

2007

Page 31F. MICHAUX

PV0 == CC

r

C

r

C

r0

1

1

2

2

2

t

t

t(1 ) (1 ).......

(1 )

DISCOUNTED CASH FLOW (DCF) EQUATION

2007

Page 32F. MICHAUX

NPV =

Crt

t

t(1 )

NET PRESENT VALUE OF A PROJECTWHERE THE SUMMATION IS OVER ALL THE CASH

FLOWS GENERATED BY THE PROJECT,

INCLUDING INITIAL NEGATIVE CASH FLOWS AT THE START OF THE PROJECT, C0 ETC.

2007

Page 33F. MICHAUX

EXAMPLE

C0 = -500, C1 = +400, C2 = +400

r1 = r2 = .12

NPV = -500 + +

= -500 + 400 (.893) + 400 (.794)

= -500 + 357.20 + 318.80 = +176

400 400 1.12 (1.12)2

2007

Page 34F. MICHAUX

PV = C

rC

rCr(1 ) (1 )

.......(1 )2 n

Cr

r

r(1 )

11

(1 )

11

(1 )

n

Cr

r

11

(1 )n

2007

Page 35F. MICHAUX

PERPETUITIES

Cr

r

11

(1 )n

CASH FLOWS LAST FOREVER

PV

Cr

== AS n GETS VERY LARGE

2007

Page 36F. MICHAUX

ALTERNATIVE WAY TO VALUE A PERPETUITY

IF I LEAVE AN AMOUNT OF MONEY, P, IN THE BANK,

I CAN EARN ANNUAL INTEREST OF C = rP FOREVER

P Cr

2007

Page 37F. MICHAUX

C r EXAMPLE:

SUPPOSE YOU WANT TO ENDOW A CHAIR AT YOUR OLD UNIVERSITY, WHICH WILL PROVIDE $100,000

EACH YEAR FOREVER. THE INTEREST RATE IS 10%

$100,000 PV = = $1,000,000 .10

A DONATION OF $1,000,000 WILL PROVIDE AN ANNUAL INCOME OF .10 X $1,000,000 = $100,000 FOREVER.

PV =

VALUING PERPETUITIES

2007

Page 38F. MICHAUX

Cr

Cr

Cr

Cr

Cr

C

r

C

rC

rr

Cr

Cr

1 2

2

3

3

4

4

1 1

2

1

2

3

1 1

1

1 (1 ) (1 ) (1 ).........

(1 )

(1 g)

(1 )

(1 g)

(1 )....

(1 )1

1(1 g)

(1 )(1 ) (1 g)

g

PV

GROWING PERPETUITIES

2007

Page 39F. MICHAUX

GROWING PERPETUITIES

SUPPOSE YOU WISH TO ENDOW A CHAIR AT

YOUR OLD UNIVERSITY WHICH WILL PROVIDE

$100,000 PER YEAR GROWING AT 4% PER YEAR

TO TAKE INTO ACCOUNT INFLATION. THE

INTEREST RATE IS 10% PER YEAR.

PVC

r g100,000.10 .04

$1,666,6671

2007

Page 40F. MICHAUX

PV = C

rCr

Cr(1 ) (1 )

.......(1 )2 n

Cr

r

11

(1 )n

FOUR VARIABLES, PV, r, n, C

IF WE KNOW ANY THREE, SOLVE FOR THE FOURTH

2007

Page 41F. MICHAUX

Asset Year of payment Present Value 1 2 . . t+1 . . Perpetuity (first payment year 1)

Perpetuity (first payment year t + 1)

Annuity from year 1 to year t

(1+r)1

t)Cr(-

Cr

(1+r))rC 1

( t

Cr

PRICE AN ANNUITY AS EQUAL TO THE DIFFERENCE BETWEEN TWO

PERPETUITIES

2007

Page 42F. MICHAUX

CALCULATING PV WHEN I KNOW C, r, N OR HOW MUCH AM I PAYING FOR MY CAR?EXAMPLE: I BUY A CAR WITH THREE END-OF-YEAR PAYMENTS OF $4,000 THE INTEREST RATE IS 10% A YEAR

1 1PV = $4,000 x - = $4,000 x 2.487 = $9,947.41 .10 .10(1.10)3

ANNUITY TABLE

NUMBER INTEREST RATE OF YEARS 5% 8% 10% 1 .952 .926 .909 2 1.859 1.783 1.736 3 2.723 2.577 2.487 5 4.329 3.993 3.791 10 7.722 6.710 6.145

2007

Page 43F. MICHAUX

LOANSEXAMPLE: AMORTIZATION SCHEDULE FOR 5-YEAR , $5,000

LOAN, 9% INTEREST RATE, ANNUAL PAYMENTS IN ARREARS.

SOLVE FOR PMT AS ORDINARY ANNUITY PMT=$1,285.46

WE KNOW THE TOTAL PAYMENT, WE CALCULATE THE

INTEREST DUE IN EACH PERIOD AND BACK CALCULATE THE

AMORTIZATION OF PRINCIPAL

1 1PMT = $5,000 / - $5,000 / 3.889 = $1,285.46 .09 .09(1.09)5

2007

Page 44F. MICHAUX

AMORTIZATION SCHEDULEYEAR BEGINNING TOTAL INTEREST PRINCIPAL ENDING ...............BALANCE PAYMENT PAID PAID BALANCE

1 5,000 1,285.46 450.00 835.46 4,164.542 4,165 1,285.46 374.81 910.65 3,253.883 3,254 1,285.46 292.85 992.61 2,261.274 2,261 1,285.46 203.51 1,081.95 1,179.325 1,179 1,285.46 106.14 1,179.32 0

INTEREST DECLINES EACH PERIODAMORTIZATION OF PRINCIPAL INCREASES OVER

TIME

2007

Page 45F. MICHAUX

0

2000

4000

6000

8000

10000

12000

14000

1 6 11 16 21 26

Year

$

AMORTIZING LOAN

Year

$

AMORTIZATION

INTEREST

30

2007

Page 46F. MICHAUX

GENERAL RESULT

EAR ==

r IS THE QUOTED ANNUAL RATE,

COMPOUNDED m TIMES PER YEAR

(1m

) 1m r

EAREQUIVALENT ANNUALLY COMPOUNDED RATE

2007

Page 47F. MICHAUX

ANNUAL PERCENTAGE RATE (APR)

EXAMPLE: CAR LOAN CHARGES INTEREST

AT 1% PER MONTH

APR OF 12% PER YEAR BUT

EAR=(1+.01)12-1=12.6825% PER YEAR

THIS IS THE RATE YOU ACTUALLY PAY

2007

Page 48F. MICHAUX

6% INTEREST RATE COMPOUNDING EAR APR FREQUENCY

YEAR 1 6.000% 6.000%

QUARTER 4 6.136% 6.000%

MONTH 12 6.168% 6.000%

DAY 365 6.183% 6.000%

MINUTE 525,600 6.184% 6.000%

CONTINUOUSLY - 6.184% 6.000%

2007

Page 49F. MICHAUX

GENERAL RESULT

EAR =(1m

) 1m r

= = eerr – 1 – 1 AS m INCREASES WITHOUT LIMIT

$1 INVESTED CONTINUOUSLY

AT AN INTEREST RATE r FOR t YEARS BECOMES ert -1

2007

Page 50F. MICHAUX

10% PER YEAR CONTINUOUSLY COMPOUNDED

EAR = e.1 - 1 = 10.51709%

2007

Page 51F. MICHAUX

NOMINAL AND REAL RATES OF INTEREST

NOMINAL CASH FLOW FROM BANK IS $1,100

IF INFLATION IS 6% OVER THE YEAR, REAL CASH

FLOW IS

REAL CASH FLOW =

$1,1001.06

$1,037.74

NOMINAL CASH FLOW(1 AVERAGE INFLATION RATE )t

2007

Page 52F. MICHAUX

NOMINAL AND REAL RATES OF INTEREST

20-YEAR

$1,000 INVESTMENT

10% PER YEAR INTEREST RATE

EXPECTED AVERAGE FUTURE INFLATION 6% / YEAR

FUTURE NOMINAL CASH FLOW = $1,000x1.120

= $6,727.50

FUTURE REAL CASH FLOW

$6,727.50

1.06$2,097.67

20

2007

Page 53F. MICHAUX

NOMINAL RATE OF RETURN 10%

REAL RATE OF RETURN1.11.06

1 3.774%

FISHER EQUATION

(1+ rnominal) = (1+ rreal)(1+EXPECTED INFLATION RATE)

= 1 + rreal + EXPECTED INFLATION RATE + rreal (EXPECTED INFLATION RATE)

APPROXIMATELY, rnominal = rreal +EXPECTED INFLATION RATE

2007

Page 54F. MICHAUX

Internal Rate of Return

Example

You can purchase a turbo powered machine tool gadget for $4,000. The investment will generate $2,000 and $4,000 in cash flows for two years, respectively. What is the IRR on this investment?

0)1(

000,4

)1(

000,2000,4

21

IRRIRRNPV

%08.28IRR

2007

Page 55F. MICHAUX

VALUING BONDS

2007

Page 56F. MICHAUX

BONDSINTEREST ONLY LOANS, PRINCIPAL OR FACE VALUE

OR PAR VALUE REPAID AT END OF LOAN

STATED INTEREST RATE CALLED COUPON

DENOMINATIONS (OR PAR VALUES) OF CORPORATE BONDS TYPICALLY $1,000. GOVERNMENT BONDS USUALLY HAVE GREATER PAR VALUES. BOND SELLING AT PAR IS SELLING “FLAT.”

MATURITY SOMETIMES CASUALLY USED FOR REMAINING LIFE OF BOND OR CURRENT MATURITY

MOST US BONDS PAY INTEREST SEMIANNUALLY

PRICE OFTEN STATED AS PERCENTAGE OF PAR VALUE

2007

Page 57F. MICHAUX

6%, 5 year bonds

CASH FLOWS AT END OF EACH YEAR (IGNORING

SEMIANNUAL PAY)

1995 1996 1997 1998 1999

60 60 60 60 1,060

SIMILAR BONDS RETURN 6.9%

PV60

(1.069)60

(1.069)

60

(1.069)

60

(1.069)

1,060

(1.069)9632 3 4 5

BOND IS SELLING AT 96.3 (PERCENT OF PAR VALUE)

2007

Page 58F. MICHAUX

AFTER BOND IS ISSUED, INTEREST RATES ON SIMILAR BONDS CHANGE

BUT CASH FLOWS FROM BOND STAY SAME

• PRICE OF BOND WILL VARY BECAUSE THE PRICE IS THE PV OF THE REMAINING CASH FLOWS

• DISCOUNT RATES CHANGE WITH CHANGES IN YIELD TO MATURITY (YTM) OR YIELD ON SIMILAR BONDS!

2007

Page 59F. MICHAUX

PV(BOND) = PV (COUPON PAYMENTS) + PV (FINAL PAYMENT)

PV(COUPON PAYMENTS) IS THE PV OF AN ANNUITY

601

.0691

.069(1.069)

1,000

(1.069)( )5 5

= 246.67 + 716.33= $963

PV(BOND)

2007

Page 60F. MICHAUX

YTM

• TURN THE QUESTION AROUND • ASK WHAT RETURN, r, DO INVESTORS EXPECT WHEN A

5-YEAR, 6% COUPON BOND IS PRICED AT 96.3?• WE NEED TO FIND THE VALUE OF r THAT SATISFIES

THE EQUATION

96360

(1+R)60

(1+R)

60

(1+R)

60

(1+R)

1,060

(1+R)2 3 4 5

r IS THE YIELD TO MATURITY (YTM) OR YIELDWE ASSUME A FLAT TERM STRUCTURE OF INTEREST RATES

2007

Page 61F. MICHAUX

INTEREST RATE RISK

WHEN MARKET INTEREST RATES RISE, BOND PRICES FALL.

WHEN MARKET INTEREST RATES FALL, BOND PRICES RISE.

BOND PRICE SENSITIVITY TO CHANGES IN INTEREST RATESGREATER

1. LONGER CURRENT MATURITY

2. LOWER THE COUPON RATE.

2007

Page 62F. MICHAUX

WHY ARE LONGER MATURITY BONDS MORE SENSITIVE TO CHANGESIN MARKET INTEREST RATES?

• MORE OF THE PRICE OF THE BOND IS DERIVED

FROM CASH FLOWS (INTEREST AND PRINCIPAL)

THAT OCCUR LATER IN TIME

• AND THEREFORE HAVE TO BE DISCOUNTED MORE

• MORE SENSITIVE TO CHANGES IN INTEREST

RATES

2007

Page 63F. MICHAUX

EXAMPLE:

1

(12)60

rIS MORE SENSITIVE TO CHANGES IN r THAN

1

(12)2

r

2007

Page 64F. MICHAUX

BONDS MAKE SEMI-ANNUAL COUPON PAYMENTS

• ANNUAL COUPON RATE IS QUOTED AS TWICE THE SEMIANNUAL COUPON RATE– 6% COUPON BOND PAYS $30 TWICE A YEAR

• BOND YIELD IS QUOTED AS TWICE THE SEMIANNUAL BOND YIELD

PV30

(1.0345)30

(1.0345)

1,030

(1.0345)2 10 . . . . . . . . . . . . . . . . . . . . . . . . . .

2007

Page 65F. MICHAUX

VALUE OF A BOND

ANNUAL COUPON C, ANNUAL YIELD TO MATURITY r

PV 2

(12)

2

(12)

............ 2

(12)2 2n

C

r

C

r

C

r

04/19/23 2007

Page 66F. MICHAUX

TERM STRUCTURE

Spot Rate - The actual interest rate today (t=0)

Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time.

Future Rate - The spot rate that is expected in the future.

Yield To Maturity (YTM) - The IRR on an interest bearing instrument.

YTM (r)

Year

1981

1987 & present

1976

1 5 10 20 30

2007

Page 67F. MICHAUX

TERM STRUCTURE

WE DISCOUNT CASH FLOW AT TIME 1 BY r1

RATE APPROPRIATE FOR 1-PERIOD LOANRATE FIXED TODAY, 1-PERIOD SPOT RATE

WE DISCOUNT CASH FLOW AT TIME 2 BY r2

RATE APPROPRIATE FOR 2-PERIOD LOANRATE FIXED TODAY, 2-PERIOD SPOT RATE

TERM STRUCTURE OF INTEREST RATES DESCRIBED BY SERIES OF INTEREST RATES r1 r2 ETC

t)t(11

2)2

(11

)1

(11PV

rrr

....

2007

Page 68F. MICHAUX

YIELD TO MATURITY• INSTEAD OF DISCOUNTING EACH PAYMENT AT

DIFFERENT RATE OF INTEREST– FIND SINGLE RATE OF INTEREST, r WHICH

GIVES SAME PV– YTM– REALLY IRR

PV1

(1 )1

(1 ).......

1(1 )2 t

r r r

BOND TABLES SHOW BOND PRICESFOR DIFFERENT COUPONS AND YTM

BOND PRICES QUOTED AS PERCENT OF FACE VALUE

04/19/23 2007

Page 69F. MICHAUX

DURATION

Year CF PV@YTM % of Total PV % x Year

1 105 96.77 .090 0.090

2 105 89.19 .083 0.164

3 105 82.21 .076 0.227

4 105 75.77 .070 0.279

5 1105 734.88 .681 3.406

1078.82 1.00 4.166 Duration

Example (Bond 1)

Calculate the duration of our 10.5% bond @ 8.5% YTM

04/19/23 2007

Page 70F. MICHAUX

Year CF PV@YTM % of Total PV % x Year

1 90 82.95 .081 0.081

2 90 76.45 .075 0.150

3 90 70.46 .069 0.207

4 90 64.94 .064 0.256

5 1090 724.90 .711 3.555

1019.70 1.00 4.249 Duration

Example (Bond 2)Given a 5 year, 9.0%, $1000 bond, with a 8.5% YTM, what is

this bond’s duration?

DURATION

2007

Page 71F. MICHAUX

DURATION AND VOLATILITY

• DURATION MEASURES AVERAGE TIMING OF CASH FLOWS– DURATION =1 x [PV(C1)/V] + 2 x [PV(C2)/V]

+ . . .

• BONDS WITH LONGER DURATION ALSO HAVE GREATER VOLATILITY

VOLATILITY(%) = DURATION/(1 + YIELD)

VOLATILITY OF BOND(1) (%) = 4.166/1.085 = 3.84

VOLATILITY OF BOND(2) (%) = 4.249/1.085 = 3.92

2007

Page 72F. MICHAUX

VALUING STOCKS

2007

Page 73F. MICHAUX

WHY IS IT IMPORTANT TO HAVE A THEORY OF THE

VALUATION OF COMMON STOCKS?

• MANAGERS SHOULD BE MAKING DECISIONS WHICH INCREASE SHARE PRICE– NEED TO UNDERSTAND HOW SHARE PRICE IS

DETERMINED• CASES WHERE WE CANNOT DIRECTLY OBSERVE

STOCK PRICE– WE ARE TRYING TO VALUE

• A DIVISION OF A COMPANY • PRIVATELY HELD FIRM FOR POSSIBLE SALE

04/19/23 2007

Page 74F. MICHAUX

STOCKS & STOCK MARKET

Common Stock - Ownership shares in a publicly held corporation.

Secondary Market - market in which already issued securities are traded by investors.

Dividend - Periodic cash distribution from the firm to the shareholders.

P/E Ratio - Price per share divided by earnings per share.

04/19/23 2007

Page 75F. MICHAUX

STOCKS & STOCK MARKET

Book Value - Net worth of the firm according to the balance sheet.

Liquidation Value - Net proceeds that would be realized by selling the firm’s assets and paying off its creditors.

Market Value Balance Sheet - Financial statement that uses market value of assets and liabilities.

2007

Page 76F. MICHAUX

IF I AM GOING TO HOLD A STOCK FOREVER

• PRICE OF THE STOCK

=PV(EXPECTED FUTURE DIVIDENDS)

04/19/23 2007

Page 77F. MICHAUX

VALUING COMMON STOCKS

Dividend Discount Model - Computation of today’s stock price which states that share value equals the present value of all expected future dividends.

H - Time horizon for your investment.

PDiv

r

Div

r

Div P

rH H

H01

12

21 1 1

( ) ( )

...( )

04/19/23 2007

Page 78F. MICHAUX

Example

Current forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return?

VALUING COMMON STOCKS

04/19/23 2007

Page 79F. MICHAUX

Example

Current forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return?

PV

PV

300

1 12

324

1 12

350 94 48

1 12

00

1 2 3

.

( . )

.

( . )

. .

( . )

$75.

VALUING COMMON STOCKS

2007

Page 80F. MICHAUX

LET’S SEE HOW MUCH SOMEONE WILL PAY FOR THE STOCK TODAY

• HOW MUCH SHOULD THE PERSON WHO BUYS IT FROM ME PAY FOR THE STOCK NOW (P0)

• IF SHE IS GOING TO RECEIVE A DIVIDEND AT THE END OF THE PERIOD (DIV1)

• AND THEN SHE IS GOING TO SELL IT (AT A PRICE P1)?

2007

Page 81F. MICHAUX

DIV1

DIVDIV

11

DIV1

DIV

1

1 1

12 2

1 22

22( ) ( )

Pr

Pr

r

r r

P

r1

P0

2007

Page 82F. MICHAUX

WE HAVE NOW SUCCEEDED IN RELATING TODAY’S PRICE TO:

• EXPECTED DIVIDENDS IN YEARS 1 AND 2, DIV1 AND DIV2

• EXPECTED PRICE AT END OF YEAR 2, P2

• WE CAN REPEAT THE PROCESS

2007

Page 83F. MICHAUX

LET’S SEE HOW MUCH SOMEONE WILL PAY FOR THE STOCK IN TWO

YEAR’S TIME

• HOW MUCH SHOULD THE PERSON PAY FOR THE STOCK IN TWO YEAR’S TIME (P2)

• IF SHE IS GOING TO RECEIVE A DIVIDEND AFTER ONE YEAR (DIV3)

• AND THEN SHE IS GOING TO SELL IT • (AT A PRICE P3)?

2007

Page 84F. MICHAUX

DIV DIV( ) ( )

DIV( )

DIVDIV

( )( )

DIV DIV( )

DIV( ) ( )

1 22

22

12

3 3

2

1 22

33

33

1 1 1

11

1 1 1 1

r rP

r

r

Pr

1 r

r r rP

r

P0

2007

Page 85F. MICHAUX

Pr r r

P

r01 2

233

H HH

DIV DIV

( )

DIV

( )......

DIV

( )

1 1 1 1

DIVHH(1 r)=

P

rH

H( )1

NOW THE PRICE OF THE STOCK IS OBVIOUSLY INDEPENDENT OF THE TIME HORIZON, H.AS WE GO OUT FURTHER IN TIME, MORE OF THE PRICE IS ACCOUNTED FOR BY THE DIVIDEND TERMS, SO THAT THE PRESENT VALUE OF THE TERMINAL PRICE BECOMES LESS IMPORTANT.

2007

Page 86F. MICHAUX

0%

25%

50%

75%

100%

0 1 2 3 4 10 20 50 100

Price Dividends

AS WE GO OUT FURTHER IN TIME, PRESENT VALUE OF THE DIVIDEND TERMS INCREASES

AND THE PRESENT VALUE OF THE TERMINAL PRICE DECLINES

Horizon period

DIVIDENDS INCREASE BY 10% A YEARCAPITALIZATION RATE IS 15%

Present value of

2007

Page 87F. MICHAUX

1. BY CONSIDERING HOW MUCH A BUYER WILL PAY FOR THE STOCK

WHEN IT IS REPEATEDLY SOLD, WE FIND THAT THE STOCK PRICE IS THE PV OF ALL FUTURE DIVIDENDS.

2. WE OBTAIN THE SAME RESULT INDEPENDENTLY OF THE ASSUMPTIONS WE MAKE

ABOUT THE LENGTH OF SUCCESSIVE HOLDING PERIODS.

04/19/23 2007

Page 88F. MICHAUX

VALUING COMMON STOCK

If we forecast no growth, and plan to hold out stock indefinitely, we will then value the stock as a PERPETUITY.

r

EPSor

r

DivPPerpetuity 11

0

Assumes all earnings are paid to shareholders.

04/19/23 2007

Page 89F. MICHAUX

Constant Growth - A version of the dividend growth model in which dividends grow at a constant rate (Gordon Growth Model).

VALUING COMMON STOCK

04/19/23 2007

Page 90F. MICHAUX

Example- continued

If the same stock is selling for $100 in the stock market, what might the market be assuming about the growth in dividends?

$100$3.

.

.

00

12

09

g

g

Answer

The market is assuming the dividend will grow at 9% per year, indefinitely.

VALUING COMMON STOCK

04/19/23 2007

Page 91F. MICHAUX

• If a firm elects to pay a lower dividend, and reinvest the funds, the stock price may increase because future dividends may be higher.

Payout Ratio - Fraction of earnings paid out as dividends

Plowback Ratio - Fraction of earnings retained by the firm.

VALUING COMMON STOCK

04/19/23 2007

Page 92F. MICHAUX

Growth can be derived from applying the return on equity to the percentage of earnings plowed back into operations.

g = return on equity X plowback ratio

“g” can also be estimated from historical growth rates in:

• dividends• eps (earnings per share)

VALUING COMMON STOCK

2007

Page 93F. MICHAUX

ESTIMATING THE CAPITALIZATION RATE

OR REQUIRED RATE OF RETURNIf dividends are expected to grow at a constant rate, g

DIV1

P0 = r - g

DIV1

so that r = + g P0MARKET CAPITALIZATION RATE

=DIVIDEND YIELD, (D1 /P0)

+ EXPECTED RATE OF GROWTH IN

DIVIDENDS, g

04/19/23 2007

Page 94F. MICHAUX

Example

Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?

VALUING COMMON STOCK

04/19/23 2007

Page 95F. MICHAUX

Example

Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to blow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?

P0

5

1267

.$41.

No Growth With Growth

VALUING COMMON STOCK

04/19/23 2007

Page 96F. MICHAUX

Example

Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to blow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?

P0

5

1267

.$41.

No Growth With Growth

g

P

. . .

. .$75.

20 40 08

3

12 08000

VALUING COMMON STOCK

04/19/23 2007

Page 97F. MICHAUX

Example - continuedIf the company did not plowback some earnings, the stock price would remain at $41.67. With the plowback, the price rose to $75.00.

The difference between these two numbers (75.00-41.67=33.33) is called the Present Value of Growth Opportunities (PVGO).

VALUING COMMON STOCK

04/19/23 2007

Page 98F. MICHAUX

Present Value of Growth Opportunities (PVGO) - Net present value of a firm’s future investments.

Sustainable Growth Rate - Steady rate at which a firm can grow: plowback ratio X return on equity.

VALUING COMMON STOCK

2007

Page 99F. MICHAUX

SUPERNORMAL GROWTH

FIRM MAY HAVE A CURRENT HIGH RATE OF GROWTH WHICH CANNOT BE SUSTAINED– SUPERNORMAL GROWTH

DO NOT USE THE SUPERNORMAL GROWTH RATE IN CALCULATING – COST OF EQUITY– FAIR MARKET PRICE

2007

Page 100F. MICHAUX

• DIVIDEND DIV0 AT t=0 GROWING AT A SUPERNORMAL GROWTH RATE gS TO DIVt AT t, AND THEN GROWING AT A NORMAL GROWTH RATE gn

• WHAT IS THE PRICE OF THE STOCK TODAY?

PRICE TODAY, P0 =

PV OF DIVIDENDS IN SUPERNORMAL GROWTH PERIOD

+ PV OF CONSTANT GROWTH DIVIDENDS

SUPERNORMAL GROWTH

2007

Page 101F. MICHAUX

Pr r

P

r01 2

23 3

3

DIV1

DIV

(1 )

DIV

(1 )

DIV =DIV (1+g )

DIV =DIV (1+g )

DIV =DIV (1+g )

DIVg

1 0 s

2 0 s2

2 0 s3

34

n

Pr

SUPERNORMAL GROWTH

2007

Page 102F. MICHAUX

INCOME V.S. GROWTH STOCKS

• Investors in utility stocks expect dividend income. Hence, a high payout ratio of about 40%-50% is normal.

• Technology stocks can have zero payout ratio.

2007

Page 103F. MICHAUX

New Economy v.s. Old Economy Stocks

• New economy stocks have high P/E• Old economy stocks have high Div/P

(Autumn 1999)

P/E Div/PAdmiral 77.8 0.2

Lynx Group 44.7 0.8

Cable & Wireless 75.4 0.9

B.T. 36.8 1.6

Power Gen 8.7 8.6

UU 7.3 7.1

Hyder Water 3.7 19.8

2007

Page 104F. MICHAUX

Problems with DGM

• Theoretical– Relationship between current and future

dividends (M&M) and share price– Determinants of dividend growth

• Practical– Accounting information– Accounting earnings v.s. economic earnings– Economic Value Added, Cash flow Return on

Equity

2007

Page 105F. MICHAUX

DIVIDENDS IRRELEVANT?

• In 1950s 9/10 US companies paid dividends

• Today only 1/5 US company pays dividend• Higher capital gains tax?• Share buybacks?• Fashion in bull market?• Stock market still punishes companies

trimming or suspending dividends by 6% and 25% drop in share price respectively.

2007

Page 106F. MICHAUX

P/E ratio and DGMDivide each side of 2nd equation by EPS1

)(

)1(

)(

)1(

1

0

10

10

OEplowbackxRr

plowback

E

P

OEplowbackxRr

EPSplowbackP

gr

DIVP

2007

Page 107F. MICHAUX

FCF and PV

Valuing a Business

The value of a business is usually computed as the discounted value of FCF out to a valuation horizon (H).

• The valuation horizon is sometimes called the terminal value and is calculated like PVGO.

HH

HH

r

PV

r

FCF

r

FCF

r

FCFPV

)1()1(...

)1()1( 22

11

2007

Page 108F. MICHAUX

FCF and PV

• Free Cash Flows (FCF) should be the theoretical basis for all PV calculations.

• FCF is a more accurate measurement of PV than either Div or EPS.

• The market price does not always reflect the PV of FCF.

• When valuing a business for purchase, always use FCF.

2007

Page 109F. MICHAUX

FCF and PVValuing a Business

HH

HH

r

PV

r

FCF

r

FCF

r

FCFPV

)1()1(...

)1()1( 22

11

PV (free cash flows) PV (horizon value)

04/19/23 2007

Page 110F. MICHAUX

FCF and PV

Example

Given the cash flows for Concatenator Manufacturing Division, calculate the PV of near term cash flows, PV (horizon value), and the total value of the firm. r=10% and g= 6%

66613132020202020(%) growth .EPS

1.891.791.681.59.23-.20-1.39-1.15-.96-.80- FlowCash Free

1.891.781.681.593.042.693.462.882.402.00Investment

3.783.573.363.182.812.492.071.731.441.20Earnings

51.3173.2905.2847.2643.2374.2028.1740.1400.1200.10ValueAsset

10987654321

Year

04/19/23 2007

Page 111F. MICHAUX

FCF and PVExample - continued

Given the cash flows for Concatenator Manufacturing Division, calculate the PV of near term cash flows, PV (horizon value), and the total value of the firm. r=10% and g= 6%

.

4.2206.10.

59.1

1.1

1 value)PV(horizon 6

6.3

1.1

23.

1.1

20.

1.1

39.1

1.1

15.1

1.1

96.

1.1

.80-PV(FCF) 65432

04/19/23 2007

Page 112F. MICHAUX

FCF and PV

Example - continued

Given the cash flows for Concatenator Manufacturing Division, calculate the PV of near term cash flows, PV (horizon value), and the total value of the firm. r=10% and g= 6%

.

$18.8

22.4-3.6

value)PV(horizonPV(FCF)s)PV(busines

2007

Page 113F. MICHAUX

Company Value

• Enterprise Value

• Equity Value

• Equity Value = Enterprise Value – Debt