CHAPTER 2
Valuation and Financing
Decisions in an Ideal
Capital Market
Guidance...…………………………………………………………… 2-3 to 2-4
PowerPoint slides……………………………………………………. 2-5 to 2-17
Guidance
Chapter 2 is the most mathematically intensive chapter in the text. Two weeks of lectures are
required to cover it fully. Ignoring the model reconciliation discussion in Section 2.8 can reduce
lecture time. Also, coverage of the Black-Scholes model in Section 2.7 can be reduced by
introducing the model but ignoring the expected return and risk formulas in the latter portion of
the section (though we return to these formulas in later chapters). In addition, the more technical
discussion of the limitations of the M&M structure to actually prove Proposition II can be skirted
if necessary.
However, we advise against ignoring the Binomial Pricing Model, because it serves well
to illustrate: (a) the concept of riskless arbitrage; (b) the rational partitioning of a firm’s value
among debt and equity claims; and (c) the effects of financial decisions on the values of a firm’s
debt and equity securities (which we illustrate using the Binomial Pricing Model in Chapter 3).
It is important to discuss the 5 assumptions of the Ideal Capital Market (Section 2.2)
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at the outset of lecturing because: (a) they establish the student’s perspective on the nature of
capital markets; (b) they are required for all of the pricing models discussed in this chapter; and
(c) the real-world factors discussed in later chapters represent direct violations of these
assumptions, as we discuss in Chapter 3.
It is also important to discuss Modern Portfolio Theory (and particularly the risk-reducing
benefits of diversification) in relation to corporate financial management because, as we discuss
later: (a) the corporate form allows for separation of ownership and control, and by extension for
shareholders to become diversified, which in turn reduces all firms’ costs of capital (which is
best illustrated via the CAPM); (b) a manager may be required to be undiversified, and thus
exposed to greater risk than the firm’s shareholders; and (c) the topic relates to the concept of
ownership structure, which we introduce in Chapter 3 and discuss throughout the text.
It is important to introduce the CAPM in part because the SML is used in the valuation of
stocks in Chapter 9.
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CHAPTER
2
Valuation and Financing
Decisions in an Ideal Capital
Market
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Finally, Appendix A can be ignored or used as an alternative (original) means of
illustrating M&M Proposition I. Appendix B provides values of the cumulative normal
distribution function, which is needed (absent a calculator or computer that provides such values)
for the BSOPM.
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Assumptions of the Ideal Capital Market
• Assumption 1: Capital Markets are Frictionless
(i.e., no taxes or transaction costs).
• Assumption 2: All Market Participants Share
Homogeneous Expectations.
• Assumption
3: Alland
market
participants
Modigliani
Miller’s
(M&M) are
atomistic.
Proposition I
• Assumption 4: The Firm’s Investment Program is
• Fixed
M&Mand
Proposition
I: The market value of a
Known.
firm (V) is invariant
to theFinancing
amountisofFixed.
leverage
• Assumption
5: The Firm’s
[i.e., debt (D) relative to equity (E)] used to
finance its assets.
Proof is based on an arbitrage argument. If VU 
2-3
EU=VLD+EL does not hold for a given firm, an
arbitrageur can purchase the firm (or a portion of
it), alter its capital structure, and then sell it for an
immediate profit.
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2-4
Modigliani and Miller’s (M&M)
Proposition II…
• M&M Proposition II: The expected return on
a firm’s equity increases with the firm’s
leverage.
…implies both equations below (for a given firm)
This is a corollary to Proposition I. The value of a
firm can remain invariant to changes in capital
structure only if the firm’s Weighted Average Cost
of Capital (WACC), which is the discount rate
used to determine the value of the firm’s assets, is
 EL 
 D 
constant
asrD
leverage varies…
WACC
 rA 
 rLE 
  constan t
D  E L


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 D  EL 
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 D 
rLE  rA  
 (rA  rD )
EL 
2-5
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A Numerical Example
• Firm Z is currently an all-equity firm with VU EU=$100,
and rA  rUE=10%.
• Firm Z recapitalizes by issuing debt with a value of
D=$35, using the proceeds to pay a dividend to stockholders.
• The value of firm Z remains $100: VL D+EL=$35+$65.
• Stockholders are indifferent to the change because they
maintain $100 in value: $35 in dividends and $65 in stock.
• If the cost of debt capital is rD=7%, then the cost of
levered equity is:
 35 
rLE  10 %    (10 %  7%)  11 .62 %
 65 
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Dividends are also Irrelevant in an Ideal
Capital Market
• If a firm’s investment and debt policy are
fixed, then as the firm increases its dividend, it must
eventually finance such payments by issuing additional
shares. However, a dividend is just a partial liquidation of
the original shareholders’ interest in the firm. Hence, a
dividend and the simultaneous issuance of new shares (at a
fair market price) is just a forced sale of a portion of the
original shareholders’ claim on the firm to new
shareholders. But this sale occurs at a fair market price, so
it is a matter of indifference to all parties, including the
original shareholders who can, if they wish, use their
dividend cash to restore their proportional claim on the
firm by purchasing some of the issued shares.
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2-8
Modern Portfolio Theory
" MPT involves two basic constructs:
 the statistical effects of diversification on the expected
return and risk of a portfolio; and
 the attitudes of investors toward risk; specifically, it is
assumed that investors are averse to risk, but are
sufficiently tolerant of risk to bear it if sufficient
compensation (i.e., higher expected return), is provided.
" MPT assumes that investors are concerned only
with the expected return and standard deviation of
their overall portfolio. MPT addresses the task of
identifying the portfolio that maximizes an
investor s expected utility given the investor s
willingness to tradeoff risk and expected return.
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2-7
Statistics for a Portfolio of Two Securities
" Expected return:
 rp = wArA + wB rB
" Example: rA=10%; rB =15%; wA =.7; wB =.3
 rp = 0.7(10%)+0.3(15%)=11.5%
" Standard deviation:
1/ 2
p  w2A2A  w2B2B  2wAwBABAB



" Example: A=30%; B=50%; =0.33; wA =.7
 p=[0.72302+0.32502+2(0.7)(0.3)(30)(50)(0.33)]1/2 =29.6%
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Statistics for an N-Security Portfolio
• Expected return (general)
N

rp   w i ri
i1
• Expected return (equally weighted)

rp 
1 N
 ri
N i 1
• Standard deviation (general)
N N

 p  [   w i w j ij ]1 / 2
i 1 j1
• Standard deviation (equally weighted)

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p  [
1
1
* (  i2 )  (1  ) * (  ij )]1 / 2 ,
N
N
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2-8
FIGURE 2-1 Annual Return Standard Deviation,p, for an EquallyWeighted Portfolio as a Function of the Number of Risky Securities in
the Portfolio, N, for Alternative Values of the Average Pairwise
Correlation () Between the Securities' Returns
45
40
35
 = 0.50
p (%)
30
25
= 0.25
20
= 0.10
15
10
5
0
1
2
3
4
5
7
10 15 20 30 40 55 70 100 150 225 300 450
Number of Securities (N)
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2-9
The Capital Asset Pricing Model (CAPM)…
• In the CAPM equilibrium, all investors hold
the market portfolio, which includes all risky
assets (each held in proportion to its relative
outstanding total market value), in
combination with the risk-free asset.
The details of an individual investor’s complete
portfolio (C) can be described using the Capital
Market Line (CML).
The expected return on an individual security is a
function of its systematic risk, measured by 
(beta). This relationship is the Security Market
Line (SML).
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2-10
…The Capital Asset Pricing Model (CAPM)
• The Capital Market Line (CML):

 
rc  rf   c  (rM  rf )
M 
• The Security Market Line (SML):

ri  rf   i (rM  rf )
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The Binomial Pricing Model…
• Assumed distribution of future stock price:
 With u>1:
VTu  uV
 With d=1/u:
VTd  dV
 A useful formula for u:
u  e
T
 With: V=firm value; and p=prob. of an up jump:
 VTu  V 
 VTd  V 
rA  p 
  (1  p) 

V
V




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…The Binomial Pricing Model…
• Payoff on default-risky debt of a levered firm:
 If up jump:
D Tu  X
 If down jump: D T  VT  X
d
d
• Payoff on default-risky levered equity:
 If up jump:
E uLT  VTu  X
 If down jump:
E dLT  0
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…The Binomial Pricing Model…
• We can value the firm’s equity and debt by
creating a riskfree hedge portfolio with a long
position in the levered firm’s assets and a short
position in  units of the firm’s levered equity.
The value of  must be chosen so that the
portfolio has the same payoff in both the up
and down states:
[VTu  E uLT ]  [VTd  E dLT ]
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…The Binomial Pricing Model
• The value of the equity of a levered firm:
VTu  VTd
 Given hedge ratio:   u
E LT  E dLT
the value of the equity is:
1  u 1 
u 
E L  V[ ]  VT    E LT  /(1  rf ) T
  

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The Put-Call Parity Theorem
• By arbitrage, the following relationship must
hold among the value of an underlying asset
(V), the values of put (P) and call (C) options
on the underlying asset that share the same
exercise price (X) and expiration date (T), and
the present value of a risk-free pure-discount
bond that pays the amount X at date T:
V+P=C+X/(1+rf)T
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The Black-Scholes Option Pricing Model…
• Assumption: The underlying asset’s
instantaneous returns are normally
distributed with constant per-annum
mean  and standard deviation of 
• The equity of a levered firm is a call
option on the firm’s assets, where X, the
promised payment on pure-discount
debt, is the exercise price, and the time to
expiration of the call option is T, the time
to maturity of the debt.
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…The Black-Scholes Option Pricing Model
• The formula is:
C  V[ N (d )]  e  rf T X[ N (d   T )]
• where N(d) is the cumulative normal
distribution function and:
V
ln ( )  [rf  ( 2 / 2)]T
X
d
 T
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FIGURE
Distribution
of IBM's
Monthly
Stock
Returns,
FIGURE2-4
2-6 The
Combinations
of Leverage
(D/V)
and Asset
Standard
Deviation () Yielding Standard Deviations of 1%, 5%, 10%, and 20%
1980-2000
Mean (1.21%)
16
100%
Std. Deviation (7.56%)
14
80%
12
6%
5%
Normal Distribution
Probability Density (%)
for the Firm's 5-Year Pure Discount Debt
18
Frequency
4%
10
60%
D/V 8
3%
Normal
Distribution
40%
6
D=20%
4
20%
2
D=1%
2%
1%
D=5%
D=10%
0
0%
0%
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-22
-20
-18
-16
-14
-12
-10-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0%
20%
40%
60%
80%
100%
Return (%, Integer Categories)

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FIGURE 2-5 Annual Return Standard Deviations for the Equity (EL)
FIGURE
An Illustration of a Firm-Specific Security Market Line
and
Debt (2-7
D) of a Levered Firm, and Market Debt Ratio, D/V, all as
Functions of the Promised Payment, X, on 5-Year Pure-Discount Debt.
(Standard deviation of assets is =20%.)
140%
30%
Annual Expected Return
Equity (X=90)
120%
25%
100%
20%
80%
D/V
LE
15%
60%
Equity (X=50)

10%
40%
20%5%
0%
0%
0
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Assets
Debt (X=90)
D
Debt (X=50)
0%
50
20%
100
40%
150
200
60%
80%
X
Annual Return Standard Deviation
2-25
2-27
250
100%
300
120%
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