Modigliani–Miller Theorem

The Modigliani–Miller theorem is a set of two propositions on corporate capital structure. It was first proposed by Franco Modigliani and Merton Miller in 1958.

Two Propositions (No Taxes)

Modigliani and Miller made some very serious assumptions. The most important two are that there are no taxes and no costs of financial distress. Additional assumptions will be discussed in the next section. The two propositions are still true when the no taxes assumption is relaxed.

Proposition I : the market value of any firm is independent of its capital structure.

This means a firm cannot change its total value just by splitting its cash flows into different streams: The firm’s value is determined by its real assets, not by how it is financed. Thus capital structure is irrelevant as long as the firm’s investment decisions are taken as given.

Firms can not create value simply by changing the company’s capital structure.

Proposition II: the cost of equity is a linear function of the company’s debt/equity ratio.

More specifically, expected return on equity = expected return on assets + (expected return on assets – expected return on debt) * debt-equity ratio.

The mathematical representation (which can be derived from the WACC formula) is:

$latex r_E = r_0 + (r_0 – r_D)(D/E) $

According to this proposition, as the company increases its use of debt financing, the cost of equity rises. We know from MM Proposition I that the value of the company is unchanged and the weighted average cost of capital remains constant if the company changes its capital structure. What Proposition II then means is that the cost of equity increases in such a manner as to exactly offset the increased use of cheaper debt in order to maintain a constant WACC.

The risk of the equity depends on two factors: the risk of the company’s operations (business risk) and the degree of financial leverage (financial risk). Business risk determines the cost of capital, whereas the capital structure determines financial risk.

The expected rate of return on the common stock of a levered firm increases in proportion to the debt–equity ratio (D/E), expressed in market values; the rate of increase depends on the spread between $latex r_A$, the expected rate of return on a portfolio of all the firm’s securities, and $latex r_D$, the expected return on the debt.

Note that $latex r_E = r_A$ if the firm has no debt.


  • Expectations are homogeneous. This means investors agree on the expected cash flows from a given investment. This means that all investors have the same expectations with respect to the cash flows from an investment in bonds or stocks.

  • Bonds and shares of stock are traded in perfect capital markets. This means that there are no transactions costs, no taxes, no bankruptcy costs, and everyone has the same information. In a perfect capital market, any two investments with identical cash flow streams and risk must trade for the same price.

  • Investors can borrow and lend at the risk-free rate.

  • There are no agency costs. This means that managers always act to maximize shareholder wealth.

  • The financing decision and the investment decision are independent of each other. This means that operating income is unaffected by changes in the capital structure.

  • No costs of asymmetric information

  • debtholders have prior claim to assets and income relative to equityholders, the cost of debt is less than the cost of equity

Two Propositions (With Taxes)

Weighted Average Cost of Capital

The weighted average cost of capital (WACC) is a weighted average of the after-tax required rates of return on a company’s common stock, preferred stock, and long-term debt, where the weights are the fraction of each source of financing in the company’s target capital structure.


$latex r_{WACC} = \displaystyle\frac{D}{D+E} \times r_D \times (1-t) + \displaystyle\frac{E}{D+E} \times r_E$


  • $latex r_{WACC}$ is the WACC
  • $latex r_D$ is the before-tax marginal cost of debt
  • $latex r_E$ is the cost of equity
  • D denotes the market value of the shareholders’ outstanding debt
  • E denotes the market value of the shareholders’ outstanding equity
  • t is the marginal tax rate

Capital Structure

A company’s capital structure is the mix of debt and equity the company uses to finance its business. The goal of a company’s capital structure decision—the choice between how much debt and how much equity a company uses in financing its investments—is to determine the financial leverage or capital structure that maximizes the value of the company by minimizing the weighted average cost of capital. The weighted average cost of capital (WACC) is given by the weighted average of the marginal costs of financing for each type of financing used.



A portfolio is a specific weighting of assets.


Portfolio weight is often represented by greek letter $latex \omega$ (omega).

Why Do We Need Portfolios

Diversification is protection against ignorance. It makes little sense if you know what you are doing. – Warren Buffett

The primary purpose of a portfolio is to reduce risk through diversification.

If you know with certainty that stock A is going to provide the highest return in the future, you can simply buy A. (This is technically also a portfolio with 100% weight on A and 0% weight on everything else.) But if you know two stock A and B are going to provide the highest and second highest return, but not which is which, then the best way to invest is to split your money between A and B. This is diversification.

In real life, if you can pick stocks like Warren Buffett, you can have a highly concentrated portfolio and no need to diversify. But it is extremely difficult to constantly pick good stocks.

What Makes A “Good” Portfolio

This is where different theories come in. We need to know what investors want and what they want to avoid.

One of the most influential theories is the modern portfolio theory (MPT).

Risk–Return Trade-Off

Investors invest for anticipated future returns, but those returns rarely can be predicted precisely. There will almost always be risk associated with investments. Actual or realized returns will almost always deviate from the expected return anticipated at the start of the investment period. For example, in 1931 (the worst calendar year for the market since 1926), the S&P 500 index fell by 46%. In 1933 (the best year), the index gained 55%. You can be sure that investors did not anticipate such extreme performance at the start of either of these years.

Naturally, if all else could be held equal, investors would prefer investments with the highest expected return. However, the no-free-lunch rule tells us that all else cannot be held equal. If you want higher expected returns, you will have to pay a price in terms of accepting higher investment risk. If higher expected return can be achieved without bearing extra risk, there will be a rush to buy the high-return assets, with the result that their prices will be driven up. Individuals considering investing in the asset at the now-higher price will find the investment less attractive. Its price will continue to rise until expected return is no more than commensurate with risk. At this point, investors can anticipate a “fair” return relative to the asset’s risk, but no more. Similarly, if returns were independent of risk, there would be a rush to sell high-risk assets and their prices would fall. The assets would get cheaper (improving their expected future rates of return) until they eventually were attractive enough to be included again in investor portfolios. We conclude that there should be a risk–return trade-off in the securities markets, with higher-risk assets priced to offer higher expected returns than lower-risk assets.

Working Capital

Working capital is the difference between current assets and current liabilities. It illustrates the amount of liquid assets that a company has to build its business.

Working capital are funds used for operational liquidity. A firm often has day to day liquidity needs that require short term liquidity. Large quantities of working capital indicate potential to expand quickly. Firms needing to make quick equipment purchases through working capital frequently arise. Firms that do not have sufficient working capital (or negative amounts) may find themselves having to borrow the money from other means, slowing down growth.

While having a surplus of working capital is good for growth, excess amounts of working capital may mean the company could find better uses of its current assets.


Working Capital = Current Assets – Current Liabilities

Basic Principles of Capital Budgeting

Capital budgeting has a rich history and sometimes employs some pretty sophisticated procedures. Fortunately, capital budgeting relies on just a few basic principles. Capital budgeting usually uses the following assumptions:

Decisions are based on cash flows. The decisions are not based on accounting concepts, such as net income. Furthermore, intangible costs and benefits are often ignored because, if they are real, they should result in cash flows at some other time.

Timing of cash flows is crucial. Analysts make an extraordinary effort to detail precisely when cash flows occur.

Cash flows are based on opportunity costs. What are the incremental cash flows that occur with an investment compared to what they would have been without the investment?

Cash flows are analyzed on an after-tax basis. Taxes must be fully reflected in all capital budgeting decisions.

Financing costs are ignored. This may seem unrealistic, but it is not. Most of the time, analysts want to know the after-tax operating cash flows that result from a capital investment. Then, these after-tax cash flows and the investment outlays are discounted at the “required rate of return” to find the net present value (NPV). Financing costs are reflected in the required rate of return. If we included financing costs in the cash flows and in the discount rate, we would be double-counting the financing costs. So even though a project may be financed with some combination of debt and equity, we ignore these costs, focusing on the operating cash flows and capturing the costs of debt (and other capital) in the discount rate.

Capital budgeting cash flows are not accounting net income. Accounting net income is reduced by noncash charges such as accounting depreciation. Furthermore, to reflect the cost of debt financing, interest expenses are also subtracted from accounting net income. (No subtraction is made for the cost of equity financing in arriving at accounting net income.) Accounting net income also differs from economic income, which is the cash inflow plus the change in the market value of the company. Economic income does not subtract the cost of debt financing, and it is based on the changes in the market value of the company, not changes in its book value (accounting depreciation). We will further consider cash flows, accounting income, economic income, and other income measures at the end of this reading.

In assumption 5 above, we referred to the rate used in discounting the cash flows as the “required rate of return.” The required rate of return is the discount rate that investors should require given the riskiness of the project. This discount rate is frequently called the “opportunity cost of funds” or the “cost of capital.” If the company can invest elsewhere and earn a return of r, or if the company can repay its sources of capital and save a cost of r, then r is the company’s opportunity cost of funds. If the company cannot earn more than its opportunity cost of funds on an investment, it should not undertake that investment. Unless an investment earns more than the cost of funds from its suppliers of capital, the investment should not be undertaken. The cost-of-capital concept is discussed more extensively elsewhere. Regardless of what it is called, an economically sound discount rate is essential for making capital budgeting decisions.

Although the principles of capital budgeting are simple, they are easily confused in practice, leading to unfortunate decisions. Some important capital budgeting concepts that managers find very useful are given below.

A sunk cost is one that has already been incurred. You cannot change a sunk cost. Today’s decisions, on the other hand, should be based on current and future cash flows and should not be affected by prior, or sunk, costs.

An opportunity cost is what a resource is worth in its next-best use. For example, if a company uses some idle property, what should it record as the investment outlay: the purchase price several years ago, the current market value, or nothing? If you replace an old machine with a new one, what is the opportunity cost? If you invest $10 million, what is the opportunity cost? The answers to these three questions are, respectively: the current market value, the cash flows the old machine would generate, and $10 million (which you could invest elsewhere).

An incremental cash flow is the cash flow that is realized because of a decision: the cash flow with a decision minus the cash flow without that decision. If opportunity costs are correctly assessed, the incremental cash flows provide a sound basis for capital budgeting.

An externality is the effect of an investment on other things besides the investment itself. Frequently, an investment affects the cash flows of other parts of the company, and these externalities can be positive or negative. If possible, these should be part of the investment decision. Sometimes externalities occur outside of the company. An investment might benefit (or harm) other companies or society at large, and yet the company is not compensated for these benefits (or charged for the costs). Cannibalization is one externality. Cannibalization occurs when an investment takes customers and sales away from another part of the company.

Conventional versus nonconventional cash flows—A conventional cash flow pattern is one with an initial outflow followed by a series of inflows. In a nonconventional cash flow pattern, the initial outflow is not followed by inflows only, but the cash flows can flip from positive to negative again (or even change signs several times). An investment that involved outlays (negative cash flows) for the first couple of years that were then followed by positive cash flows would be considered to have a conventional pattern. If cash flows change signs once, the pattern is conventional. If cash flows change signs two or more times, the pattern is nonconventional.

Several types of project interactions make the incremental cash flow analysis challenging. The following are some of these interactions:

Independent versus mutually exclusive projects. Independent projects are projects whose cash flows are independent of each other. Mutually exclusive projects compete directly with each other. For example, if Projects A and B are mutually exclusive, you can choose A or B, but you cannot choose both. Sometimes there are several mutually exclusive projects, and you can choose only one from the group.

Project sequencing. Many projects are sequenced through time, so that investing in a project creates the option to invest in future projects. For example, you might invest in a project today and then in one year invest in a second project if the financial results of the first project or new economic conditions are favorable. If the results of the first project or new economic conditions are not favorable, you do not invest in the second project.

Unlimited funds versus capital rationing. An unlimited funds environment assumes that the company can raise the funds it wants for all profitable projects simply by paying the required rate of return. Capital rationing exists when the company has a fixed amount of funds to invest. If the company has more profitable projects than it has funds for, it must allocate the funds to achieve the maximum shareholder value subject to the funding constraints.

Source: Institute, CFA. 2018 CFA Program Level II Volume 3 Corporate Finance. CFA Institute, 07/2017. VitalBook file.