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Review
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Principle of separation
Present discounted value the real investment. (equivalence).
Decide whether to undertake it (optimization).
Select the appropriate financial investment or disinvestment (optimization).
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Representation
Nature’s move,plus the contestant’sguess.
pr =
2/3
guess w
rong
guess right
pr = 1/3
switch and winor
stay and lose
switch and loseorstay and win
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No arbitrage condition:
Price of bond = price of zero-coupon bond + price of stripped coupon.
Otherwise, a money machine, one way or the other.
Riskless increase in wealth
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Pie theory
The bond is the whole pie. The strip is one piece, the zero is the
other. Together, you get the whole pie. No arbitrage pricing requires that the
values of the pieces add up to the value of the whole pie.
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No arbitrage principle
Market prices must admit no profitable, risk-free arbitrage.
No money pumps. Otherwise, acquisitive investors would
exploit the arbitrage indefinitely.
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Definition of a call option
A call option is the right but not the obligation to buy 100 shares of the stock at a stated exercise price on or before a stated expiration date.
The price of the option is not the exercise price.
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Example
A share of IBM sells for 74. The call has an exercise price of 76. The value of the call seems to be zero. In fact, it is positive and in one example
equal to 2.
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t = 0 t = 1
S = 74
S = 80, call = 4
S = 70, call = 0Pr. = .5
Pr. = .5
Value of call = .5 x 4 = 2
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Definition of a put option
A call option is the right but not the obligation to sell 100 shares of the stock at a stated exercise price on or before a stated expiration date.
The price of the option is not the exercise price.
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Put call parity at expiration
Equivalence at expiration s + p = X + c Values at time t in caps:
S + P = Xe-r(T-t) + C Write S - Xe-r(T-t) = C - P
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Puts and calls before expiration
S, P, and C are the market values at time t before expiration T.
Xe-r(T-t) is the market value at time t of the exercise money to be paid at T
Traders tend to ignore r(T-t) because it is small relative to the bid-ask spreads.
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No arbitrage pricing impliesput call parity
If the relation does not hold, a risk-free arbitrage is available.
If S - Xe-r(T-t) > C – P, then Sell the stock short, and also sell the put.
Use the proceeds to buy the call and a bond paying X at expiration. The position is riskless. It nets the arbitrageur a positive sum. That violates no arbitrage pricing.
Similarly if inequality is in the other direction.
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“Real” options
The option to abandon is a put option. Deciding to delay or not, the firm
exercises or not its call on the project.
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Review item
What is the interest rate?
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Do write:
The interest rate is the premium for current delivery of money.
P0 is the price of current money in current money, namely 1.
P1 is the price of time-one money in terms of current money, something <1.
P 11
0 P
Pr
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Review item
When a firm creates value through a financial transaction, who gets the increase?
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Answer
Old equity means the shareholders at the time the decision is made.
Old equity gets the gains. Why? Old equity has no competitors.
Everyone else is competitive and must accept a market return.
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Example
Coupons sell for 450 Principal sells for 500 The bond MUST sell for 950. Otherwise, an arbitrage opportunity exists. For instance, if the bond sells for 920… Buy the bond, sell the stripped components.
Profit 30 per bond, indefinitely. Similarly, if the bond sells for 980 …
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Confirmation in an excel spread sheet.
Time contribution balance0 3.25473 3.254731 3.25473 6.7047442 3.25473 10.36176… …15 3.25473 83.557116 3.25473 91.8253
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Incremental cash flows
Cash flows that occur because of undertaking the project
Not sunk barges … oops, I mean costs. Opportunity cost Side effects Working capital
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Net working capital
= cash + inventories + receivables - payables
a cost at the start of the project (in dollars of time 0,1,2 …)
a revenue at the end in dollars of time T-2, T-1, T.
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Internal rate of return
Definition: IRR is the discount rate that makes NPV = 0
CFCF
r
CF
r
CF
rT
T01 2
21 1 10
( ). . .
( )
That is, IRR is the r such that
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IRR’s at r=1 and r=2.
NPV
r
100% 200%
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Scale problems in IRR
Time 0 1 IRR NPV(r=.1)
Littledam
-100 200 1 81.8181...
Bigdam
-1000 1500 .5 363.63...
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Decision Tree for Stewart Pharmaceutical
Do not test
Test
Failure
Success
Do not invest
Invest
Invest
The firm has two decisions to make:To test or not to test.To invest or not to invest.
0$NPV
NPV = $3.4 B
NPV = $0
NPV = –$91.46 m
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The Option to Abandon: Example
The firm has two decisions to make: drill or not, abandon or stay.
Do not drill
Drill
0$NPV
Failure
Success: PV = $575
Sell the rig; salvage value
= $250
Sit on rig; stare at empty hole:
PV = $0.
Traditional NPV analysis overlooks the option to abandon.
- $300
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Beta measures risk
How much risk is added depends on the relation of AM and
Define beta
2M
AMA
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Covariance
It measures the tendency of two assets to move together.
Variance is a special case -- the two assets are the same.
Variance = expectation of the square of the deviation of one asset.
Covariance = expectation of the product of the deviations of two assets.
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Rf
E[RM] Security
marke
t line
1
Rate of returnexpected by the market
E[Rj]
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Example of beta and NPV
Wingmar Inc. has a beta of 2. The Market risk premium is 8.5% The risk-free rate is 4%. Wingmar has a project with cash flows -
100, 60, 80. The project is typical of Wingmar’s core
business. Should the project be undertaken?
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The Long-Term Financial Deficit (2002)
Sources of Cash Flow (100%)
Internal cash flow (retained earnings plus depreciation)
97%
Long-term debt and
equity 3%
Uses of Cash Flow (100%)
Capital spending
98%
Net working capital
plus other uses 2%
Internal cash flow
External cash flow
Financial deficit
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Event Studies: Dividend Omissions
Cumulative Abnormal Returns for Companies Announcing Dividend Omissions
0.146 0.108
-0.72
0.032-0.244
-0.483
-3.619
-5.015-5.411-5.183
-4.898-4.563-4.747-4.685-4.49
-6
-5
-4
-3
-2
-1
0
1
-8 -6 -4 -2 0 2 4 6 8
Days relative to announcement of dividend omission
Cum
ulat
ive
abno
rmal
ret
urns
(%
)
Efficient market response to “bad news”
S.H. Szewczyk, G.P. Tsetsekos, and Z. Santout “Do Dividend Omissions Signal Future Earnings or Past Earnings?” Journal of Investing (Spring 1997)
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Relationship among Three Different Information Sets
All informationrelevant to a stock
publicly availableinformation
past prices
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EPS and ROE under Proposed Capital Structure
Shares Outstanding = 240
Bust Normal Boom
EBIT $1,000 $2,000 $3,000
Interest 640 640 640
Net income $360 $1,360 $2,360
EPS $1.50 $5.67 $9.83
ROA 5% 10% 15%
ROE 3% 11% 20%
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MM Proposition II no tax
Debt-to-equityratio (B/S)
Cost of capital: r
(%)
.r0
rS
rWACC
rB
LBS S
Brrrr )( 00
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MM II and WACC
Debt-to-equityratio (B/S)
Cost of capital: r
(%)
.r0
rS
rB.. rWACC
.
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Channels$ of operatingcash flows
TC
Corporatetaxes
1-TB (1-TC)(1-TS)
TB TS
Personaltaxes
Debtchannel
Equitychannel
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Value asequity
Value asdebt
Operating C.F.’s ofthe whole economy
...
tax cut
increasedequity
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Summary: APV, FTE, and WACC
APV WACC FTE
Initial Investment All All Equity Portion
Cash Flows UCF UCF LCF
Discount Rates r0 rWACC rS
PV of financing Yes No No
Which is best?Use WACC and FTE when the debt ratio is constantUse APV when the level of debt is known.
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Review item
Two assets have the same expected return.
Each has a standard deviation of 2%. The correlation coefficient is .5. What is the standard deviation of an
equally weighted portfolio?
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Review item
A firm has a project with positive NPV. The project costs 100M to start. The firm has only 50M. What should it do?
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Answer
Raise the money in the capital market. It can because NPV is market
valuation.
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Capital asset pricing model
jfMfj RRERRE ][)(
T-bill rate is known.Market premium is known, approximately 8.5%.Estimate beta as in the project
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Security Market LineExpected returnon security (%)
Beta ofsecurity
Rm
Rf
1
M.
0.8
S.
T. Security market line (SML)
S is overvalued.Its price falls
T is undervalued.Its price rises
.. .
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EPS and ROE under Proposed Capital Structure
Shares Outstanding = 240
Recession Expected Expansion
EBIT $1,000 $2,000 $3,000
Interest 640 640 640
Net income $360 $1,360 $2,360
EPS $1.50 $5.67 $9.83
ROA 5% 10% 15%
ROE 3% 11% 20%
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Proposition II of M-M
rB is the interest rate
rs is the return on (levered) equity r0 is the return on unlevered equity
B is value of debt SL is value of levered equity
rs = r0 + (B / SL) (r0 - rB)
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MM Proposition II no tax
Debt-to-equityratio (B/S)
Cost of capital: r(%)
.r0
rS
rWACC
rB
LBS S
Brrrr )( 00
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MM II (with taxes)
Corporate taxes, not personal rB = interest rate rS = return on equity r0 = return on unlevered equity B = value of debt SL = value of levered equity Previously, without taxes
rS = r0 + (B/SL)(r0 - rB)
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Effect of tax shield
Increase of equity risk is partly offset by the tax shield
rS = r0 + (1-TC)(r0 - rB)(B/SL) Leverage raises the required return less
because of the tax shield.
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MM II and WACC
Debt-to-equityratio (B/S)
Cost of capital: r(%)
.r0
rS
rB.0.200=
0.100
. rWACC
.0.2351
200370
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Optimal Debt and Value
Debt (B)
Value of firm (V)
0
Present value of taxshield on debt
Present value offinancial distress costs
Value of firm underMM with corporatetaxes and debt
VL=VU+TCB=
V=Actual value of firm
VU=Value of firm with no debt
B*
Maximumfirm value
Optimal amount of debt
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ChannelsOperating CashFlows = $1
Debtchannel
Equitychannel
TC
(1-TC)(1-TS)
TB
1 - TB
TS
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Value asequity
Value asdebt
Operating C.F.’s ofthe whole economy
D of Institutions D of rich investors
V* = 1/RB V* = 1/RS
asdebt
as equity
Miller: Tax-class clienteles
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Value asequity
Value asdebt
Operating C.F.’s ofthe whole economy
tax reform
increaseddebt
...
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Separation theorem interpreted for dividends (Figure 18.4)
C1
C0
s lo p e = - (1 + r)
L o w -d iv id e n d firm
H ig h -d iv id e n dfirm
w
F u tu rere tu rno r
d iv id e n d n o
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Dividend equilibrium
$ of operatingcash flows
HiDivvalueper $1
LoDivvalueper $1
mq ili riuo iv
EL
mEquilibriuHiD iv
u bD
V*=1/Rh V*=1/RL
...
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Review item
What is the weighted average cost of capital?
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Answer
Give the definitions and the formula. rB = bond rate
rS = expected return on shares
B = market value of bonds S = market value of shares TC = corporate tax rate
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Pay-off pitch
rWACC =(S/(S+B))rS + (B/(S+B))(1-TC)rB
Now say that it applies when (1) the physical project has the same
risk as the firm (2) it is financed like the firm.
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Review item
Does a good project have IRR greater than the hurdle rate, or less?
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Answer
IRR is the discount rate that makes NPV(IRR) = 0.
The hurdle rate is the market rate for the risk-class.
Investing means cash flows are first negative, then positive.
Financing (in this context) means cash flows are first positive, then negative.
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More answer
Other sign patterns, IRR is not useful. Investing, a good project has IRR >
hurdle rate. Financing, a good project has hurdle
rate > IRR.
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