2007
Page 1F. MICHAUX
CORPORATE FINANCE
2007
Page 2F. MICHAUX
GENERAL AGENDA
Valuation and Discounted Cash Flow Method
Valuing Bonds
Valuing Stocks
2007
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VALUATION AND DISCOUNTED CASH FLOW
METHOD
2007
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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
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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)
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PRESENT VALUE
Present Value = PV
PV = Discount Factor X C1
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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
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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
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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
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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
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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
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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%
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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
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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?
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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
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FV OF PRINCIPAL, P,
AT END OF n YEARS IS
FVn=PV(1+R)n
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Page 22F. MICHAUX
COMPOUND INTEREST
INTEREST EARNED ON PRINCIPAL AND
REINVESTED INTEREST OF PRIOR
PERIODS
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SIMPLE INTEREST
INTEREST EARNED ON THE
ORIGINAL PRINCIPAL ONLY
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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
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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
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PRESENT VALUE
PRESENT VALUE OF $1
0
0,2
0,4
0,6
0,8
1
1,2
0 2 4 6 8 10 12 14 16 18 20
r = 5%
r = 10%
r = 15%
PRESENT VALUE
Year 5% 10% 15% 1 .952 .909 .870 2 .907 .826 .756 5 .784 .621 .497 10 .614 .386 .247 20 .377 .149 .061
YEARS
2007
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FUTURE VALUE• COMPOUND
PRINCIPAL AMOUNT
FORWARD INTO THE
FUTURE
PRESENT VALUE• DISCOUNT
A FUTURE VALUE BACK TO THE
PRESENT
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Page 28F. MICHAUX
BASIC RELATIONSHIP BETWEEN PV AND FV
PVFV
(1 )0
t
t r
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Page 29F. MICHAUX
Present ValuesExample
Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.
000,300000,100000,150
2Year 1Year 0Year
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Page 30F. MICHAUX
Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.
400,18$
900,261000,300873.2
500,93000,100935.1
000,150000,1500.10Value
Present
Flow
Cash
Factor
DiscountPeriod
207.11
07.11
TotalNPV
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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
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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.
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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
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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
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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
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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
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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
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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
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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%
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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
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Page 50F. MICHAUX
10% PER YEAR CONTINUOUSLY COMPOUNDED
EAR = e.1 - 1 = 10.51709%
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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
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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.
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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
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Page 63F. MICHAUX
EXAMPLE:
1
(12)60
rIS MORE SENSITIVE TO CHANGES IN r THAN
1
(12)2
r
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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 . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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