

Model: The irrelevance of capital structure in perfect capital markets 1 Introduction This appendix presents proofs of a
modern version of the Modigliani and Miler propositions on the irrelevance of
capital structure in perfect capital markets. Note: Footnotes and highlighted
text can be clicked for more info. The propositions on capital structure made by Modigliani and Miller (M&M) are among the most important contributions in the theory of corporate finance. The theorems were first stated in the seminal papers [1958, 1963] on the cost of capital, corporate valuation and capital structure. The first versions were very simple. They assumed that debt could be only of the riskfree type, and that all firms belonged to the same risk class. The latter assumption meant that the expected future cash flow in all firms should be perfectly correlated, and only allowed to vary by a scale factor. E.g. CF_{i,t} = aCF_{j,t}, where CF is cash flow in firm i and j, respectively, a is the scale factor, and t count the future periods. These assumptions are certainly unrealistic, but fortunately, later developments in the theory of corporate finance made them redundant. Independent of the capital structure theory Sharpe [1963, 1964] and Treynor [1961] built the first model that could price risky assets on individual terms. This model has now come to be known as the capital asset pricing model (CAPM). Rubinstien [1973] merged the CAPM model with the original M&M framework on capital structure. This was possible because the M&M framework and the CAPM model are built on almost the exact same assumptions on perfect capital markets. The gain was that the more realistic assumption of risky debt replaced the one of riskfree debt. Equally, the assumption that firms belonging to the same risk class were replaced by the assumption that firms have the same beta b risk. E.g. b_{i} = COV[k_{i},k_{m}]/VAR[k_{m}], where COV[k_{i},k_{m}] is the covariance between the returns of the asset under valuation (k_{i}) and the market portfolio (k_{m}), and VAR[k_{m}] is the variance of the market portfolio. The equal beta assumption is much less restrictive, because there are indefinitely many and not perfectly correlated future cash flows that produce the same beta. Rubinsteins model represented an important progress, but it could not generalize all of the M&M propositions on capital structure (To be exact, Rubinstein was the first to prove the basic M&M irrelevance theorem using a meanvariance approach, but Stiglitz [1969] had already introduced risky debt by using a statepreference approach. But this was only of theoretical relevance, since the nature of the state preference theory prevent it from being tested empirically). Hsia [1981] generalized the remaining propositions and this progress was also contingent on another and independent discovery in the field of option pricing. This discovery was provided by Black and Scholes [1973] who developed the first option pricing model (OPM) in continuous time. This model is also built on the assumptions of perfect capital markets, so Hsia was able to merge the CAPM, and the OPM with the M&M framework on capital structure and thereby generalize the remaining propositions. Rubinstein’s and Hsia’s versions are perhaps the most general that have been produced so far. The price to be paid for this generalization is that the proofs behind the M&M propositions have become considerably more complicated from a mathematical point of view. This text reproduces the full Rubinstein argument. The proofs of Hsia arguments are by large omitted but the results will be stated. However, both the Rubinstein and the Hsia argument is ‘proved’ by a simple and easily comprehendible numeric example that illustrates all of the generalized M&M propositions. 2 Theory What
Rubinstein generalized, was the most basic of the M&M propositions: The
proposition on the irrelevance of capital structure. Proposition I: “The market
value of any firm is independent of its capital structure and is given by
capitalizing its expected return at the rate r
appropriate to its class”, Modigliani and Miller [1958, page 268]. In modern
terms, capital structure is irrelevant, and firm value is equal to the
present value of the free cash flow discounted at the relevant cost of
capital. Rubinstein
used the following assumptions explicitly or implicitly: 1)
Capital markets are perfect ·
Markets are frictionless. I.e., there is no
transaction cost (Implication: the borrowing rate equals the lending rate),
no taxes, and all assets are perfectly divisible and marketable (Implication:
e.g. human capital is nonexisting (it is not divisible and it can’t be owned
as an asset), and there are no constraining regulations. ·
Perfect competition in product and securities
markets. Among other things, this imply that all producers and consumers are
price takers. ·
Information efficiency. That is, information is
costless, and is received simultaneously by all individuals, both corporate
insiders and outsiders (this implies no signaling). ·
Agents are perfectly rational and use it to maximize
there utility 2)
There is no cost to bankruptcy. 3)
Firms issue risky debt and equity, and have the same
beta b
risk. 4)
All cash flow streams are perpetuities, and no
growth is allowed. 5)
Allowing for proportional corporate tax eases the
notax assumption above. 6)
Managers always maximize the shareholders’ wealth
(imply no agency costs). 7)
Homemade leverage is a perfect substitute for
corporate leverage. That is, there is no difference between corporate and
personal borrowing (necessary for arbitrage arguments). The assumptions above are primarily associated with
the original M&M framework. First and foremost, in order to apply the
CAPM model, the following assumptions are needed.[1] 8) Investors are risk averse individuals. 9) Investors have homogenous expectations (beliefs) about asset returns (Implication: all investors perceive identical opportunity sets). 10) Asset returns are distributed by the normal distribution. 11) A risk free asset exists, and investors may borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate (k_{f}). 12) There are a definite number of assets and their quantities are fixed over time. These
were the assumptions. The following proofs draw heavily on Copeland and
Weston [1988, pages 462471]. According to the CAPM
model, the equilibrium cost of capital for debt (b for bonds) and for
equity (s for stocks) in any firm is: _{} (1) _{} (2) where, · E[k_{b}] is the expected cost of
capital on bonds. · E[k_{s}] is the expected cost of
capital on stocks. · k_{f} is the expected return on a
riskfree portfolio; that is, a portfolio that yields constant return period
after period with certainty. · E[k_{m}] is the expected return on the
market portfolio. · b_{b}, or b_{s} is the systematic risk. That is, the risk that can’t be
diversified away. b measures the size of risk contrary to (E[k_{m}]
 k_{f}) that measures the price of risk. · COV(k_{b},k_{m}), or COV(k_{s},k_{m})
is the expected covariance in returns between the capital being priced and
the overall market portfolio. · VAR(k_{m}) is the expected variance of
the market portfolio. Assuming
that there is no growth and that investment equals depreciation (I=Dep), the
return on equity, k_{s}, of any firm can also be expressed by its
accounting definition. _{} (3) where, ·
NOI is net operating income. That is, revenues less
variable and fixed cash expenses from operations and less deprecations. ·
B is the market value of debt. ·
t_{c} is
the corporate tax rate. ·
S^{L} is the market value of equity in a
levered firm. Taking the expectation to (3)
yields _{} (4) Substituting (3) into the COV[k_{s},k_{m}]
from (2) yields _{} _{} (5) Substituting
(5) into (2) yields _{} (6) Substituting
(6) into (4) yields _{} _{} (7) <=> E[NOI](1t_{c}) 
E[k_{b}]B(1t_{c}) = _{} (8) Making
an analog argument for an nonlevered firm, where B=0 and S^{L}=V^{U}
(V^{U} is the market value of an nonlevered firm) yields: _{} (9) Substituting
E[NOI](1t_{c})
from (8) into (9) yields _{} _{} =>[2] _{} _{} =>[3] _{} _{} <=> k_{f}
(V^{L}B) = k_{f} V^{U }k_{f} B(1t_{c})
<=> V^{L}B = V^{U } B(1t_{c})
<=> V^{L} = V^{U }+ Bt_{c} (10) Q.E.D. If t_{c} =
0, then firm value is independent of leverage, and proposition 1 is proved to
hold. Hsia
[1981] generalized the rest of the M&M propositions. They will not be
proved, only presented. Essentially, what Hsia does, is extending the
Rubinstein framework by the BlackScholes OPM formula. Then, he uses the fact
that the value of equity in a levered firm is identical to the value of a
call option written on the firm, and that the value of risky debt in a
levered firm is the same as the value of riskfree debt plus a “short” put
option on the value of the firm. Both options are of the European type, and
has an exercise price equal to the future value of the riskfree debt, and a
maturity equal to the maturity of the riskfree debt. The rest of the Hsia
argument is purely mathematical[4]. Proposition II: The required
rate of return on equity, or cost of equity (k_{s}), increases with
financial leverage (B/[B+S]). This
proposition can be proved by the following formula on the cost of equity: _{} (11) where, ·
N(.) is the cumulative normal probability of the
unit normal variate, d_{1}. ·
k_{V} is the return on the firm’s asset. ·
T is the maturity time of the firm’s bonds. ·
s is
the standard variate of returns on the firm’s assets. ·
D is the future value of the zero coupon bond, or
face value. Proposition III: The required
rate of return on debt or cost of debt (k_{b}) increases with
financial leverage (B/[B+S]). This
proposition can be proved by the following formula on the cost of debt: _{} (12) Proposition IV: In a world
without taxes, the weighted average cost of capital (WACC) is invariant to
changes in the capital structure of the firm. Proposition
4 can easily be proved by inserting (11) and (12) in the definition for the
average cost of capital: WACC = k_{b}*B/V + k_{s}*S/V, and V
= B+S = {k_{f} + N(d_{1})[k_{V}  k_{f}]V/B}*B/V + { k_{f}
+ N(d_{1})[k_{V}  k_{f}]V/S }*S/V = k_{f}{(B+S)/V} + [k_{V} 
k_{f}]*{ N(d_{1}) + N(d_{1})} =
k_{f} + [k_{V}  k_{f}]*{ 1 N(d_{1}) + N(d_{1})} = k_{V} (k_{V} is the same as r in proposition
1)_{ }(13) Q.E.D. The
WACC formula may be generalized to include any number of financial
instruments and different tax assumptions. 3 Illustrating
the M&M propositions The generalized Modigliani and Miller theorems may seem complicated when looking at the mathematics. But fortunately, the propositions are very simple to illustrate in a figure. This is done in figure I below. 



Note that all four theorems are fulfilled in figure G. Also very interestingly, the figure version of the M&M framework does not require returns to follow the normal distribution. Actually, any underlying risk distribution may do! What matters, is that firm cash flow is unrelated to capital structure, and this is always so in the perfect M&M world. One might think that the M&M theorems are trivial. But they are not. Before the invention of the M&M, framework many corporate treasurers in fact believed that leveraging did increase corporate risk. By verbal ‘logic’, they argued something like this: “If you agree that levering up the corporate capital structure makes the remaining equity riskier, and you also agree that any junior debt added to the capital structure must, necessarily, be riskier debt carrying a lower rating, and a higher interest than on any debt outstanding before the higher leveraging, then you must also agree that leveraging does increase corporate risk, thereby decreasing the market value of the firm”. Propositions 2 and 3 do indeed agree that both the risk of debt and equity increase when leverage increases. However, proposition 4 proofs that this does not increase the corporate risk, and when this is true, proposition 1 about corporate value must also be valid. The oldfashioned treasurer simply ‘forgets’ to take into account the changing weights. Evidence Despite all the propositions of the M&M world, when the real world is considered, all firms and branches behave ‘as if’ an optimal capital structure existed. Empirically, branches and firms have capital structures that do not change much over time. Another stylized fact is that some firms are entirely financed by equity, but no firms are entirely debt financed. Neither of these observations are consistent with the M&M propositions. There are several extensions of the M&M capital structure theory that may explain the existence of an optimal capital structure. Click here to go to the hypothesis exhibition in order to find some of these explanations. 

 Copyright 19972018, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice. 
[1] For proof of the CAPM model click
here.
[2] Use the fact that the value of a
levered firm, V^{L}, is equal to the value of the firms stocks, S^{L},
plus the value of its bonds, B: V^{L}= S^{L} + B <=> S^{L}
= V^{L } B.
[3] Substituting (E1) into E[k_{b}].
[4] Proof of the BlackScholes OPM
model may be found in Copeland and Weston [1988, pages 296299]. In the same
book [pages 464469] some of the Hsia proofs may be found.