The capital structure of a firm: Modigliani and Miller

A firm is using financial leverage (or gearing) when it uses debt to increase its own value. In general, whenever a firm changes his value because of a shift in capital structure, it is influencing its stockholders. When a firm maximizes its market value it is also maximizing the interests of its stockholders.

MODIGLIANI AND MILLER THEORIES

According to Modigliani and Miller, when you consider perfect markets, any combination of securities that keeps its value constant should be equally likeable by the firm. Therefore the firm’s value is independent of its choices of capital structure. To prove this, consider two firms: one levered (uses debt), and one unlevered (without debt). Let’s denote the levered firm as firm A, its value Va is the same as the total value of its equity Ea. Firm B uses debt and, because of that, the value of its stocks is computed as its value less the total value of its debt Eb=Vb-Db. If you want to risk as little as possible you will invest in firm A. Suppose that you buy shares making up the 5% of the company’s value. If this is the case, you will collect at the end of the year 5% of the profits of the company. Now suppose that you want to buy 5% of company B and that you buy equal amounts of the debt and equity. You will receive thanks to the debt 5% of the total interest paid by the firm to all debtholders and, thanks to the shares 5% of the profits. But from the profits in this case you have to subtract the debt the company has to pay back. So you will in the end receive thanks to the shares 5%(profits – interests). Finally you will receive 5%interest + 5%(profits – interest) = 5% interest + 5% profits – 5%interests = 5% profits. So, as you can see, you are indifferent whether a firm is levered or not, you will receive 5% of its profits anyway. Because of this, the two different investment strategies have the same payoff. Thanks to the law of one price, you know that two investments having the same expected return have the same price. This means that: Va=Vb.

WHAT IS THE LAW OF CONSERVATION OF VALUE

Remember that, according to the principle of value additivity, when you have two cashflow to get their combined value you can simply sum their present values. If this is true, it must also work in reverse: we can split a big cashflow in multiple smaller cashflows. This property of cashflows is known as the law of conservation of value. The value of the funds you get are therefore independent on the way in which you get them. One application of this law can be found when firms can decide if it is better to raise a certain amount of money by issuing common stocks or preferred stocks. According to the law the choice among the two leaves the firm completely indifferent because as a whole the value of the equity increases by the same value.

RETURN AND RISK

When a firm uses debt, it increases the expected return on its stocks (because now it becomes riskier) but not their price. If we assume that the owner of a share will own it forever, we can calculate the expected return of the stock as the earnings to price ratio (because we are considering it as a perpetuity). Now, the expected returns on the company’s assets (ra) are:

When considering perfect capital markets, if the firm modifies its capital structure (by using debt for example), both the total market value and the value of the firm’s securities change. Because of that, it must also be true that, if a company decides to borrow money, the expected return on the assets of the firm must be unchanged. Because the expected return on a portfolio can be computed simply as the weighted average of the expected returns of the securities composing it, we can calculate the expected returns on the company’s assets using the expected returns on debt and equity and their proportions:

This expected return ra is the called the company’s cost of capital or the weighted average cost of capital (WACC). If you possess the company’s return on capital, the proportions of equity and debt and the return on debt you can calculate the expected returns on the equity’s stocks:

This is useful for example if you are asked to calculate how the stock returns change when the company shifts its capital structure. In this cases, you simply have to calculate the return on assets before the shift and then use the same return (because it does not change) to calculate the new return on equity. When the firm has no debt ra=re.

MORE ABOUT MODIGLIANI AND MILLER

Modigliani and Miller were the ones who theorized how the expected returns on the firm’s stocks increased as the debt to equity ratio (D/E) increases. Remember that we are always considering market and not book values. The rate at which the return on equity increases depends on the difference between the return on the assets and the return on debt (ra-rd). This happens because investors expect an higher return to compensate for the extra risk bore by a leveraged company. But remember that, as we said at the beginning of the lecture, the stockholders’ wealth is not influenced by how much debt the firm uses. The only thing that changes is the fact that stocks can provide higher returns but also higher losses when the firm is levered.

BETA AND CAPITAL STRUCTURE

Both the stockholders and the debtholders share the risks of the firm but the debtholders bear much less risk than stockholders. This is because on average the beta of debt is much lower than the beta of stocks. As we computed the expected return on the firm’s assets, we can compute the expected beta of the firm’s assets. We can do so simply by computing the weighted average of the betas of the firm’s debt and stocks.

If a company uses more debt, as we said, its stocks become riskier and therefore their beta will increase but the total beta, the beta of the company’s assets, will not change. You can calculate how the beta of the firm’s stocks changes by calculating the asset beta before the shift in capital structure and then, by using the same beta, find the equity beta. Suppose that the company has value composed 40% by debt and 60% by equity and that the corresponding betas are 0.05 and 1.5. The asset beta is:

Now, if the company wants to use more debt so that the its value its composed in equal parts by equity and debt, you can calculate how the equity beta changes:

MORE ABOUT THE WACC

Until now we assumed that bonds were always risk free and therefore their beta did not change as we increased the D/E ratio. This is not always the case in reality: it is true that the beta of debt does not change when firms use low debt, but, it is also true that, for large amounts of debt, also the debt increases in risk and therefore also in expected returns. As we said, the more the firm borrows the higher the expected return demanded by stockholders. But the expected return does not increase in a linear way: it starts sharply but then it slows down. This means that, when the firm has a lot of debt, increasing it by a little margin does not affect much the revenues asked by stockholders. If the value of the firm is not affected by its capital structure, than the combination of securities that minimizes the weighted average cost of capital and the one maximizes the market value are the same. But if the firm’s value changes depending on its choices of capital structure, then the combinations minimizing the cost of capital and maximizing the market value are the same only if the operating income is independent of the capital structure. If operating income is constant, when the firm’s value is increasing, the average cost of capital decreases. When the two objectives do not coincide, shareholders prefer when the firm pursues the highest possible value. Firm’s management does not have to fall into the trap of minimizing the weighted average cost of capital by increasing the proportion of debt because it is the cheapest source to obtain funds (because debtholders demand lower returns than shareholders). This is a trap because as debt increases, as we said, the returns demanded by stockholders already present increases thus making the return on assets constant.

AGAINST MODIGLIANI AND MILLER

Some argue that, when a firm uses some leverage, it is true that the expected return on equity increases but not as much as Modigliani and Miller predict. When instead the firm uses a lot of debt, the expected return on equity grows much faster than predicted. This implies that the cost of capital decreases when debt is small and increases as the debt becomes larger and larger. This, in reality, happens because shareholders do not demand higher returns when the debt used by the firms is so small. Therefore they accept a lower rate of return even if rationally they could expect an higher one. If markets are not perfect, it could happen that firms could use debt at a cheaper rate than single investors. If this happens it would be convenient for investors to buy the stocks of those companies and they would also be willing to pay a premium for them. Financial managers could also raise funds by finding an unsatisfied clientele: a group of investors willing to buy a certain kind of security which is not available in the market or that, it is available, but it is too expensive. So, financial managers can exploit a market imperfection and decide to offer those investors the kind of security they want and, by doing so, they can ask them for a premium. A market imperfection like this one creates for financial management an opportunity to make money. Sometimes the government is the one responsible of creating such imperfections thanks to regulations.

WACC WHEN YOU HAVE TAXES

When companies use debt, they have to pay interests on that. Those same interests in many countries like the USA can be deducted from the taxable income. When this is the case, the cost of debt is lower: it becomes rd(1-Tc) where Tc indicates the tax rate. When companies are aware of that, they will not use the normal WACC but the after tax weighted average cost of capital:

When companies can use the debt to reduce their taxable income, the opportunity cost of capital decreases as the proportion of debt increases.