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The **Modigliani–Miller theorem** (of Franco Modigliani, Merton Miller) is an influential element of economic theory; it forms the basis for modern thinking on capital structure.^{ [1] } The basic theorem states that in the absence of taxes, 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.^{ [2] } Since the value of the firm depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt, the Modigliani–Miller theorem is often called the **capital structure irrelevance principle**.

**Franco Modigliani** was an Italian-American economist and the recipient of the 1985 Nobel Memorial Prize in Economics. He was a professor at University of Illinois at Urbana–Champaign, Carnegie Mellon University, and MIT Sloan School of Management.

**Merton Howard Miller** was an American economist, and the co-author of the Modigliani–Miller theorem (1958), 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 F. Sharpe. Miller spent most of his academic career at the University of Chicago's Booth School of Business.

**Capital structure** in corporate finance is the way a corporation finances its assets through some combination of equity, debt, or hybrid securities.

- Historical background
- The theorem
- Without taxes
- Proposition I
- Proposition II
- With taxes
- Proposition I 2
- Proposition II 2
- Notes
- Further reading
- External links

The key Modigliani-Miller theorem was developed in a world without taxes. However, if we move to a world where there are taxes, when the interest on debt is tax-deductible, and ignoring other frictions, the value of the company increases in proportion to the amount of debt used.^{ [3] } And the source of additional value is due to the amount of taxes saved by issuing debt instead of equity.

**Tax deduction** is a reduction of income that is able to be taxed and is commonly a result of expenses, particularly those incurred to produce additional income. Tax deductions are a form of tax incentives, along with exemptions and credits. The difference between deductions, exemptions and credits is that deductions and exemptions both reduce taxable income, while credits reduce tax.

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 F. Sharpe, for their "work in the theory of financial economics", with Miller specifically cited for "fundamental contributions to the theory of corporate finance".

**The University of Chicago** is a private research university in Chicago, Illinois. Founded in 1890, the school is located on a 217-acre campus in Chicago's Hyde Park neighborhood, near Lake Michigan. The University of Chicago holds top-ten positions in various national and international rankings.

**Harry Max Markowitz** is an American economist, and a recipient of the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences.

**William Forsyth Sharpe** is an American economist. He 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.

Miller and Modigliani derived and published their theorem when they were both professors at the Graduate School of Industrial Administration (GSIA) of Carnegie Mellon University. Despite limited prior experience in corporate finance, Miller and Modigliani were assigned to teach the subject to current business students. Finding the published material on the topic lacking, the professors created the theorem based on their own research^{[ citation needed ]}. The result of this was the article in the *American Economic Review* and what has later been known as the M&M theorem.

The **Tepper School of Business** is the business school of Carnegie Mellon University. It is located in the university’s 140-acre (0.57 km^{2}) campus in Pittsburgh, Pennsylvania, US.

**Carnegie Mellon University** (**CMU**) is a private research university based in Pittsburgh, Pennsylvania. Founded in 1900 by Andrew Carnegie as the **Carnegie Technical Schools**, the university became the **Carnegie Institute of Technology** in 1912 and began granting four-year degrees. In 1967, the Carnegie Institute of Technology merged with the Mellon Institute of Industrial Research to form Carnegie Mellon University. With its main campus located 3 miles (5 km) from Downtown Pittsburgh, Carnegie Mellon has grown into an international university with over a dozen degree-granting locations in six continents, including campuses in Qatar and Silicon Valley, and more than 20 research partnerships.

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.

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.^{ [4] }

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 than the firm.

here

*is the expected rate of return on equity, or cost of equity.**is the expected 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 (WACC).

These propositions are true under the following assumptions:

- 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 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

*is the value of a levered firm.**is the value of an unlevered firm.**is the tax rate () x the value of debt (D)"*

`Derivation of - Amount of Annual Interest= Debt x Interest Rate Annual Tax Shield= Debt x Interest Rate x Tax Rate Capitalisation Value (Perpetual Firm) = (Debt × Interest Rate x Tax Rate) ÷ Interest Rate`

- 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 payments. Dividend payments are non-deductible.

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. 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.

- ↑ Titman, Sheridan (2002). "The Modigliani and Miller Theorem and the Integration of Financial Markets".
*Financial Management*.**31**(1): 101–115. doi:10.2307/3666323. JSTOR 3666323. - ↑ MIT Sloan Lecture Notes, Finance Theory II, Dirk Jenter, 2003
^{[ unreliable source? ]} - ↑ Fernandes, Nuno. Finance for Executives: A Practical Guide for Managers. NPV Publishing, 2014, p. 82.
- ↑ Arnold G. (2007)

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations .(March 2009) (Learn how and when to remove this template message) |

- Brealey, Richard A.; Myers, Stewart C. (2008) [1981].
*Principles of Corporate Finance*(9th ed.). Boston: McGraw-Hill/Irwin. ISBN 978-0-07-340510-0. - Stewart, G. Bennett (1991).
*The Quest for Value: The EVA management guide*. New York: HarperBusiness. ISBN 978-0-88730-418-7. - Modigliani, F.; Miller, M. (1958). "The Cost of Capital, Corporation Finance and the Theory of Investment".
*American Economic Review*.**48**(3): 261–297. JSTOR 1809766. - Modigliani, F.; Miller, M. (1963). "Corporate income taxes and the cost of capital: a correction".
*American Economic Review*.**53**(3): 433–443. JSTOR 1809167. - Miles, J.; Ezzell, J. (1980). "The weighted average cost of capital, perfect capital markets and project life: a clarification".
*Journal of Financial and Quantitative Analysis*.**15**(3): 719–730. CiteSeerX 10.1.1.455.6733 . doi:10.2307/2330405. JSTOR 2330405. - Sargent, Thomas J. (1987).
*Macroeconomic Theory*(Second ed.). London: Academic Press. pp. 157–162. ISBN 978-0-12-619751-8. - Suresh P. Sethi, Extension of the Miller and Modigliani theory to allow for share repurchases, Mathematical Finance Letters, Vol 2017 (2017), Article ID 3 http://scik.org/index.php/mfl/article/view/3138
- Sethi, S.P.; Derzko, N.A.; Lehoczky, J.P. (1991). "A Stochastic Extension of the Miller-Modigliani Framework".
*Mathematical Finance*.**1**(4): 57–76. doi:10.1111/j.1467-9965.1991.tb00019.x. - Sethi, S.P. (1996). "When Does the Share Price Equal the Present Value of Future Dividends?".
*Economic Theory*.**8**: 307–319.

In finance, **discounted cash flow** (**DCF**) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. All future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value of the cash flows in question.

In finance, the **net present value** (**NPV**) or **net present worth** (**NPW**) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

This aims to be a complete article **list of economics topics**:

The **weighted average cost of capital** (**WACC**) is the rate that a company is expected to pay on average to all its security holders to finance its assets. The WACC is commonly referred to as the firm's cost of capital. Importantly, it is dictated by the external market and not by management. The WACC represents the minimum return that a company must earn on an existing asset base to satisfy its creditors, owners, and other providers of capital, or they will invest elsewhere.

The APV was introduced in 1974 bij Stewart Myers. According to Myers, the value of the levered firm is equal to the value of the firm with no debt plus the present value of the tax savings due to the tax deductability of interest payments, the so called value of the tax shield (VTS). Myers proposes calculating the VTS by discounting the tax savings at the cost of debt (Kd). The argument is that the risk of the tax saving arising from the use of debt is the same as the risk of the debt. The method is to calculate the NPV of the project as if it is all-equity financed. Then the base-case NPV is adjusted for the benefits of financing. Usually, the main benefit is a tax shield resulted from tax deductibility of interest payments. Another benefit can be a subsidized borrowing at sub-market rates. The APV method is especially effective when a leveraged buyout case is considered since the company is loaded with an extreme amount of debt, so the tax shield is substantial.

In Economics and Accounting, the **cost of capital** is the cost of a company's funds, or, from an investor's point of view "the required rate of return on a portfolio company's existing securities". It is used to evaluate new projects of a company. It is the minimum return that investors expect for providing capital to the company, thus setting a benchmark that a new project has to meet.

In corporate finance, **free cash flow** (**FCF**) or **free cash flow to firm** (**FCFF**) is a way of looking at a business's cash flow to see what is available for distribution among all the securities holders of a corporate entity. This may be useful to parties such as equity holders, debt holders, preferred stock holders, and convertible security holders when they want to see how much cash can be extracted from a company without causing issues to its operations.

**Return on capital** (**ROC**), or **return on invested capital** (**ROIC**), is a ratio used in finance, valuation and accounting, as a measure of the profitability and value-creating potential of companies after taking into account the amount of initial capital invested. The ratio is calculated by dividing the after-tax operating income (NOPAT) by the book value of both debt and equity capital less cash/equivalents.

In finance, **leverage** is any technique involving the use of debt rather than fresh equity in the purchase of an asset, with the expectation that the after-tax profit to equity holders from the transaction will exceed the borrowing cost, frequently by several multiples — hence the provenance of the word from the effect of a lever in physics, a simple machine which amplifies the application of a comparatively small input force into a correspondingly greater output force. Normally, the lender will set a limit on how much risk it is prepared to take and will set a limit on how much leverage it will permit, and would require the acquired asset to be provided as collateral security for the loan. For example, for a residential property the finance provider may lend up to, say, 80% of the property's market value, for a commercial property it may be 70%, while on shares it may lend up to, say, 60% or none at all on certain volatile shares.

The following outline is provided as an overview of and topical guide to finance:

A **tax shield** is the reduction in income taxes that results from taking an allowable deduction from taxable income. For example, because interest on debt is a tax-deductible expense, taking on debt creates a tax shield. Since a tax shield is a way to save cash flows, it increases the value of the business, and it is an important aspect of business valuation.

In corporate finance, **Hamada’s equation**, named after Robert Hamada, is used to separate the financial risk of a levered firm from its business risk. The equation combines the Modigliani-Miller theorem with the capital asset pricing model. It is used to help determine the levered **beta** and, through this, the optimal capital structure of firms.

The **dividend discount model** (**DDM**) is a method of valuing a company's stock price based on the theory that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. In other words, it is used to value stocks based on the net present value of the future dividends. The equation most widely used is called the **Gordon growth model** (**GGM**). It is named after Myron J. Gordon of the University of Toronto, who originally published it along with Eli Shapiro in 1956 and made reference to it in 1959. Their work borrowed heavily from the theoretical and mathematical ideas found in John Burr Williams 1938 book "The Theory of Investment Value."

In finance, the **capital structure substitution theory ** (**CSS**) describes the relationship between earnings, stock price and capital structure of public companies. The CSS theory hypothesizes that managements of public companies manipulate capital structure such that earnings per share (EPS) are maximized. Managements have an incentive to do so because shareholders and analysts value EPS growth. The theory is used to explain trends in capital structure, stock market valuation, dividend policy, the monetary transmission mechanism, and stock volatility, and provides an alternative to the Modigliani–Miller theorem that has limited descriptive validity in real markets. The CSS theory is only applicable in markets where share repurchases are allowed. Investors can use the CSS theory to identify undervalued stocks.

**Dividend policy** is concerned with financial policies regarding paying cash dividend in the present or paying an increased dividend at a later stage. Whether to issue dividends, and what amount, is determined mainly on the basis of the company's unappropriated profit and influenced by the company's long-term earning power. When cash surplus exists and is not needed by the firm, then management is expected to pay out some or all of those surplus earnings in the form of cash dividends or to repurchase the company's stock through a share buyback program.

**Corporate finance** is an area of finance that deals with sources of funding, the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to allocate financial resources. The primary goal of corporate finance is to maximize or increase shareholder value. Although it is in principle different from managerial finance which studies the financial management of all firms, rather than corporations alone, the main concepts in the study of corporate finance are applicable to the financial problems of all kinds of firms.

The term **shareholder yield** captures the three ways in which the management of a public company can distribute cash to shareholders: cash dividends, stock repurchases and debt reduction.

In corporate finance, **free cash flow to equity** (FCFE) is a metric of how much cash can be distributed to the equity shareholders of the company as dividends or stock buybacks—after all expenses, reinvestments, and debt repayments are taken care of. Whereas dividends are the cash flows actually paid to shareholders, the FCFE is the cash flow simply available to shareholders. The FCFE is usually calculated as a part of DCF or LBO modelling and valuation. The FCFE is also called the levered free cash flow.

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