Modigliani-Miller theorem
Encyclopedia
The Modigliani–Miller theorem (of Franco Modigliani
, Merton Miller
) forms the basis for modern thinking on capital structure
. The basic theorem states that, under a certain market price process (the classical random walk
), in the absence of tax
es, bankruptcy
costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is unaffected by how that firm is financed. It does not matter if the firm's capital is raised by issuing stock
or selling debt. It does not matter what the firm's dividend policy
is. Therefore, the Modigliani–Miller theorem is also often called the capital structure irrelevance principle.
Modigliani was awarded the 1985 Nobel Prize in Economics for this and other contributions.
Miller was a professor at the University of Chicago
when he was awarded the 1990 Nobel Prize in Economics, along with Harry Markowitz
and William Sharpe
, for their "work in the theory of financial economics," with Miller specifically cited for "fundamental contributions to the theory of corporate finance."
of Carnegie Mellon University
. The story goes that Miller and Modigliani were set to teach corporate finance for business students despite the fact that they had no prior experience in corporate finance. When they read the material that existed they found it inconsistent so they sat down together to try to figure it out. The result of this was the article in the American Economic Review and what has later been known as the M&M theorem.
was originally proven under the assumption of no taxes. It is made up of two propositions which can also be extended to a situation with taxes.
Consider two firms which are identical except for their financial structures. The first (Firm U) is unlevered: that is, it is financed by equity only. The other (Firm L) is levered: it is financed partly by equity, and partly by debt. The Modigliani–Miller theorem states that the value of the two firms is the same.
where
is the value of an unlevered firm = price of buying a firm composed only of equity, and 
is the value of a levered firm = price of buying a firm that is composed of some mix of debt and equity. Another word for levered is geared, which has the same meaning.
To see why this should be true, suppose an investor is considering buying one of the two firms U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does. The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.
This discussion also clarifies the role of some of the theorem's assumptions. We have implicitly assumed that the investor
's cost of borrowing money is the same as that of the firm, which need not be true in the presence of asymmetric information, in the absence of efficient markets, or if the investor has a different risk profile to the firm.
Proposition II:.
A higher debt-to-equity ratio leads to a higher required return on equity, because of the higher risk involved for equity-holders in a company with debt. The formula is derived from the theory of weighted average cost of capital
(WACC).
These propositions are true assuming the following assumptions:
These results might seem irrelevant (after all, none of the conditions are met in the real world), but the theorem is still taught and studied because it tells something very important. That is, capital structure
matters precisely because one or more of these assumptions is violated. It tells where to look for determinants of optimal capital structure and how those factors might affect optimal capital structure.
where
This means that there are advantages for firms to be levered, since corporations can deduct interest payments. Therefore leverage lowers tax
payments. Dividend
payments are non-deductible.
Proposition II:
where
The same relationship as earlier described stating that the cost of equity rises with leverage, because the risk to equity rises, still holds. The formula however has implications for the difference with the WACC
. Their second attempt on capital structure included taxes has identified that as the level of gearing increases by replacing equity with cheap debt the level of the WACC drops and an optimal capital structure does indeed exist at a point where debt is 100%
The following assumptions are made in the propositions with taxes:
Miller and Modigliani published a number of follow-up papers discussing some of these issues.
The theorem was first proposed by F. Modigliani and M. Miller in 1958.
When misinterpreted in practice, the theorem can be used to justify near limitless financial leverage
while not properly accounting for the increased risk, especially bankruptcy risk, that excessive leverage ratios bring. Since the value of the theorem primarily lies in understanding the violation of the assumptions in practice, rather than the result itself, its application should be focused on understanding the implications that the relaxation of those assumptions bring.
The formula's use of EBIT / Cost of Capital to calculate a company's value is extremely limiting. It also uses the weighted average cost of capital formula, which calculates the value based on E + D, where E = the value of equity and D = the value of debt. Modigliani and Miller are equating two different formulas to arrive at a number which maximizes a firm's value. It is inappropriate to say that a firm's value is maximized when these two different formulas cross each other because of their striking differences. The formula essentially says a firm's value is maximized when a company has earnings * the discount rate multiple = book value. Modigliani and Miller equate E + D = EBIT / Cost of Capital. This seems to over-simplify the firm's valuation.
Franco Modigliani
Franco Modigliani was an Italian economist at the MIT Sloan School of Management and MIT Department of Economics, and winner of the Nobel Memorial Prize in Economics in 1985.-Life and career:...
, Merton Miller
Merton Miller
Merton Howard Miller was the co-author of the Modigliani-Miller theorem which proposed the irrelevance of debt-equity structure. He shared the Nobel Memorial Prize in Economic Sciences in 1990, along with Harry Markowitz and William Sharpe...
) forms the basis for modern thinking on capital structure
Capital structure
In finance, capital structure refers to the way a corporation finances its assets through some combination of equity, debt, or hybrid securities. A firm's capital structure is then the composition or 'structure' of its liabilities. For example, a firm that sells $20 billion in equity and $80...
. The basic theorem states that, under a certain market price process (the classical random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
), in the absence of tax
Tax
To tax is to impose a financial charge or other levy upon a taxpayer by a state or the functional equivalent of a state such that failure to pay is punishable by law. Taxes are also imposed by many subnational entities...
es, bankruptcy
Bankruptcy
Bankruptcy is a legal status of an insolvent person or an organisation, that is, one that cannot repay the debts owed to creditors. In most jurisdictions bankruptcy is imposed by a court order, often initiated by the debtor....
costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is unaffected by how that firm is financed. It does not matter if the firm's capital is raised by issuing stock
Stock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
or selling debt. It does not matter what the firm's dividend policy
Dividend policy
Dividend policy is concerned with taking a decision regarding paying cash dividend in the present or paying an increased dividend at a later stage. The firm could also pay in the form of stock dividends which unlike cash dividends do not provide liquidity to the investors, however, it ensures...
is. Therefore, the Modigliani–Miller theorem is also often called the capital structure irrelevance principle.
Modigliani was awarded the 1985 Nobel Prize in Economics for this and other contributions.
Miller was a professor at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
when he was awarded the 1990 Nobel Prize in Economics, along with Harry Markowitz
Harry Markowitz
Harry Max Markowitz is an American economist and a recipient of the John von Neumann Theory Prize and the Nobel Memorial Prize in Economic Sciences....
and William Sharpe
William Forsyth Sharpe
William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
, for their "work in the theory of financial economics," with Miller specifically cited for "fundamental contributions to the theory of corporate finance."
Historical background
Miller and Modigliani derived the theorem and wrote their groundbreaking article when they were both professors at the Graduate School of Industrial Administration (GSIA)Tepper School of Business
The Tepper School of Business is a private business school located on Carnegie Mellon University’s campus in Pittsburgh, Pennsylvania, USA.The school consistently ranks highly among the top business schools in the U.S., as well as in a wide range of specializations, such as finance,...
of Carnegie Mellon University
Carnegie Mellon University
Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania, United States....
. The story goes that Miller and Modigliani were set to teach corporate finance for business students despite the fact that they had no prior experience in corporate finance. When they read the material that existed they found it inconsistent so they sat down together to try to figure it out. The result of this was the article in the American Economic Review and what has later been known as the M&M theorem.
Propositions
The theoremTheorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
was originally proven under the assumption of no taxes. It is made up of two propositions which can also be extended to a situation with taxes.
Consider two firms which are identical except for their financial structures. The first (Firm U) is unlevered: that is, it is financed by equity only. The other (Firm L) is levered: it is financed partly by equity, and partly by debt. The Modigliani–Miller theorem states that the value of the two firms is the same.
Without taxes
Proposition I:where
is the value of an unlevered firm = price of buying a firm composed only of equity, and is the value of a levered firm = price of buying a firm that is composed of some mix of debt and equity. Another word for levered is geared, which has the same meaning.To see why this should be true, suppose an investor is considering buying one of the two firms U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does. The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.
This discussion also clarifies the role of some of the theorem's assumptions. We have implicitly assumed that the investor
Investor
An investor is a party that makes an investment into one or more categories of assets --- equity, debt securities, real estate, currency, commodity, derivatives such as put and call options, etc...
's cost of borrowing money is the same as that of the firm, which need not be true in the presence of asymmetric information, in the absence of efficient markets, or if the investor has a different risk profile to the firm.
Proposition II:.
-
is the required rate of return on equity, or cost of equityCost of equityIn finance, the cost of equity is the return a firm theoretically pays to its equity investors, i.e., shareholders, to compensate for the risk they undertake by investing their capital. Firms need to acquire capital from others to operate and grow...
. -
is the company unlevered cost of capital (ie assume no leverage). -
is the required rate of return on borrowings, or cost of debt. -
is the debt-to-equity ratio.
A higher debt-to-equity ratio leads to a higher required return on equity, because of the higher risk involved for equity-holders in a company with debt. The formula is derived from the theory of weighted average cost of capital
Weighted average cost of capital
The weighted average cost of capital is the rate that a company is expected to pay on average to all its security holders to finance its assets....
(WACC).
These propositions are true assuming the following assumptions:
- no taxes exist,
- no transaction costs exist, and
- individuals and corporations borrow at the same rates.
These results might seem irrelevant (after all, none of the conditions are met in the real world), but the theorem is still taught and studied because it tells something very important. That is, capital structure
Capital structure
In finance, capital structure refers to the way a corporation finances its assets through some combination of equity, debt, or hybrid securities. A firm's capital structure is then the composition or 'structure' of its liabilities. For example, a firm that sells $20 billion in equity and $80...
matters precisely because one or more of these assumptions is violated. It tells where to look for determinants of optimal capital structure and how those factors might affect optimal capital structure.
With taxes
Proposition I:where
-
is the value of a levered firm. -
is the value of an unlevered firm. -
is the tax rate (
) x the value of debt (D) - the term
assumes debt is perpetual
This means that there are advantages for firms to be levered, since corporations can deduct interest payments. Therefore leverage lowers tax
Tax
To tax is to impose a financial charge or other levy upon a taxpayer by a state or the functional equivalent of a state such that failure to pay is punishable by law. Taxes are also imposed by many subnational entities...
payments. Dividend
Dividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
payments are non-deductible.
Proposition II:
where
-
is the required rate of return on equity, or cost of levered equity = unlevered equity + financing premium. -
is the company cost of equity capital with no leverage (unlevered cost of equity, or return on assets with D/E = 0). -
is the required rate of return on borrowings, or cost of debt. -
is the debt-to-equity ratio. -
is the tax rate.
The same relationship as earlier described stating that the cost of equity rises with leverage, because the risk to equity rises, still holds. The formula however has implications for the difference with the WACC
Weighted average cost of capital
The weighted average cost of capital is the rate that a company is expected to pay on average to all its security holders to finance its assets....
. Their second attempt on capital structure included taxes has identified that as the level of gearing increases by replacing equity with cheap debt the level of the WACC drops and an optimal capital structure does indeed exist at a point where debt is 100%
The following assumptions are made in the propositions with taxes:
- corporations are taxed at the rate
on earnings after interest, - no transaction costs exist, and
- individuals and corporations borrow at the same rate
Miller and Modigliani published a number of follow-up papers discussing some of these issues.
The theorem was first proposed by F. Modigliani and M. Miller in 1958.
Economic consequences
While it is difficult to determine the exact extent to which the Modigliani–Miller theorem has impacted the capital markets, the argument can be made that it has been used to promote and expand the use of leverage.When misinterpreted in practice, the theorem can be used to justify near limitless financial leverage
Leverage (finance)
In finance, leverage is a general term for any technique to multiply gains and losses. Common ways to attain leverage are borrowing money, buying fixed assets and using derivatives. Important examples are:* A public corporation may leverage its equity by borrowing money...
while not properly accounting for the increased risk, especially bankruptcy risk, that excessive leverage ratios bring. Since the value of the theorem primarily lies in understanding the violation of the assumptions in practice, rather than the result itself, its application should be focused on understanding the implications that the relaxation of those assumptions bring.
Criticisms
The main problem with the Modigliani and Miller (1958) is that they assume shareholders are the owners of the public corporations. This assumption has been refuted by legal scholars since Berle and Means (1932). Shareholders are neither the owners, residual claimants (i.e. owners of the profit), or the investors as 99.9% are in the secondary market.The formula's use of EBIT / Cost of Capital to calculate a company's value is extremely limiting. It also uses the weighted average cost of capital formula, which calculates the value based on E + D, where E = the value of equity and D = the value of debt. Modigliani and Miller are equating two different formulas to arrive at a number which maximizes a firm's value. It is inappropriate to say that a firm's value is maximized when these two different formulas cross each other because of their striking differences. The formula essentially says a firm's value is maximized when a company has earnings * the discount rate multiple = book value. Modigliani and Miller equate E + D = EBIT / Cost of Capital. This seems to over-simplify the firm's valuation.