# 15.1 The Capital Structure Question and the Pie Theory

How should a firm choose its debt–equity ratio? We call our approach to the capital structure question the pie model. If you are wondering why we chose this name, just take a look at Fig. 15.1. The pie in question is the sum of the financial claims of the firm - debt and equity in this case. We define the value of the firm to be this sum. Hence the value of the firm, V, is

where B is the market value of the debt and S is the market value of the equity. Figure 15.1 presents two possible ways of slicing this pie between equity and debt: 40 per cent/60 per cent and 60 per cent/40 per cent. If the goal of a firm’s management is to make the firm as valuable as possible, then the firm should pick the debt–equity ratio that makes the pie – the total value – as big as possible.
This discussion begs two important questions:
1. Why should the shareholders in the firm care about maximizing the value of the entire firm? After all, the value of the firm is, by definition, the sum of both the debt and the equity. Instead, why should the shareholders not prefer the strategy that maximizes their interests only?
2. What ratio of debt to equity maximizes the shareholders’ interests?
Let us examine each of the two questions in turn.
 Figure 15.1 Two pie models of capital structure

# 15.2 Maximizing Firm Value versus Maximizing Shareholder Interests

The following example shows that the capital structure that maximizes the value of the firm is the one that financial managers should choose for the shareholders.
EXAMPLE 15.1Debt and Firm Value
Suppose the market value of J.J. Sprint plc is £1,000. The company currently has no debt, and each of J.J. Sprint’s 100 shares sells for £10. A company such as J.J. Sprint with no debt is called an unlevered company. Further suppose that J.J. Sprint plans to borrow £500 and pay the £500 proceeds to shareholders as an extra cash dividend of £5 per share. After the issuance of debt, the firm becomes levered. The investments of the firm will not change as a result of this trans-action. What will the value of the firm be after the proposed restructuring?
Management recognizes that, by definition, only one of three outcomes can occur from restructuring. Firm value after restructuring can be: (a) greater than the original firm value of £1,000; (b) equal to £1,000; or (c) less than £1,000. After consulting with investment bankers, management believes that restructuring will not change firm value more than £250 in either direction. Thus it views firm values of £1,250, £1,000 and £750 as the relevant range. The original capital structure and these three possibilities under the new capital structure are presented next:

Note that the value of equity is below £1,000 under any of the three possibilities. This can be explained in one of two ways. First, the table shows the value of the equity after the extra cash dividend is paid. Because cash is paid out, a dividend represents a partial liquidation of the firm. Consequently there is less value in the firm for the equity-holders after the dividend payment. Second, in the event of a future liquidation, shareholders will be paid only after bondholders have been paid in full. Thus the debt is an encumbrance of the firm, reducing the value of the equity.
Of course, management recognizes that there are infinite possible outcomes. These three are to be viewed as representative outcomes only. We can now determine the pay-off to shareholders under the three possibilities:

No one can be sure ahead of time which of the three outcomes will occur. However, imagine that managers believe that outcome (a) is most likely. They should definitely restructure the firm, because the shareholders would gain £250. That is, although the price of the shares declines by £250 to £750, they receive £500 in dividends. Their net gain is £250 = -£250 + £500. Also, notice that the value of the firm would rise by £250 = £1,250 - £1,000.
Alternatively, imagine that managers believe that outcome (c) is most likely. In this case they should not restructure the firm, because the shareholders would expect a £250 loss. That is, the equity falls by £750 to £250, and they receive £500 in dividends. Their net loss is -£250 = -£750 + £500. Also, notice that the value of the firm would change by -£250 = £750 - £1,000.
Finally, imagine that the managers believe that outcome (b) is most likely. Restructuring would not affect the shareholders’ interest, because the net gain to shareholders in this case is zero. Also notice that the value of the firm is unchanged if outcome (b) occurs.

This example explains why managers should attempt to maximize the value of the firm. In other words, it answers question 1 in Section 15.1. We find in this example the following wisdom:
Changes in capital structure benefit the shareholders if and only if the value of the firm increases.
Conversely, these changes hurt the shareholders if and only if the value of the firm decreases. This result holds true for capital structure changes of many different types.1 As a corollary, we can say the following:
Managers should choose the capital structure that they believe will have the highest firm value because this capital structure will be most beneficial to the firm’s shareholders.
Note, however, that this example does not tell us which of the three outcomes is most likely to occur. Thus it does not tell us whether debt should be added to J.J. Sprint’s capital structure. In other words, it does not answer question 2 in Section 15.1. This second question is treated in the next section.

# 15.3 Financial Leverage and Firm Value: An Example

#### Leverage and Returns to Shareholders

The previous section shows that the capital structure producing the highest firm value is the one that maximizes shareholder wealth. In this section we wish to determine that optimal capital structure. We begin by illustrating the effect of capital structure on returns to shareholders. We shall use a detailed example, which we encourage students to study carefully. Once we fully understand this example, we shall be ready to determine the optimal capital structure.
Autoveloce SpA currently has no debt in its capital structure. The firm is considering issuing debt to buy back some of its equity. Both its current and proposed capital structures are presented in Table 15.1. The firm’s assets are €8,000. There are 400 shares of the all-equity firm, implying a market value per share of €20. The proposed debt issue is for €4,000, leaving €4,000 in equity. The interest rate is 10 per cent. Assume in all our examples that debt is issued at par.
 Table 15.1 Financial structure of Autoveloce SpA

The effect of economic conditions on earnings per share is shown in Table 15.2 for the current capital structure (all equity). Consider first the middle column, where earnings are expected to be €1,200. Because assets are €8,000, the return on assets (ROA) is 15 per cent (= €1,200/€8,000). Assets equal equity for this all-equity firm, so return on equity (ROE) is also 15 per cent. Earnings per share (EPS) is €3.00 (= €1,200/400). Similar calculations yield EPS of €1.00 and €5.00 in the cases of recession and expansion, respectively.
 Table 15.2 Autoveloce’s current capital structure: no debt

 Table 15.3 Autoveloce’s proposed capital structure

The case of leverage is presented in Table 15.3. ROA in the three economic states is iden-tical in Tables 15.2 and 15.3, because this ratio is calculated before interest is considered. Debt is €4,000 here, so interest is €400 (= 0.10 × €4,000). Thus earnings after interest are €800 (= €1,200 − €400) in the middle (expected) case. Because equity is €4,000, ROE is 20 per cent (= €800/€4,000). Earnings per share are €4.00 (= €800/200). Similar calculations yield earnings of €0 and €8.00 for recession and expansion, respectively.
Tables 15.2 and 15.3 show that the effect of financial leverage depends on the company’s earnings before interest. If earnings before interest are equal to €1,200, the return on equity (ROE) is higher under the proposed structure. If earnings before interest are equal to €400, the ROE is higher under the current structure.
This idea is represented in Fig. 15.2. The solid line represents the case of no leverage. The line begins at the origin, indicating that earnings per share (EPS) would be zero if earnings before interest (EBI) were zero. The EPS rise in tandem with a rise in EBI.
 Figure 15.2 Financial leverage: EPS and EBI for Autoveloce SpA

The dotted line represents the case of €4,000 of debt. Here EPS are negative if EBI are zero. This follows because €400 of interest must be paid, regardless of the firm’s profits.
Now consider the slopes of the two lines. The slope of the dotted line (the line with debt) is higher than the slope of the solid line. This occurs because the levered firm has fewer shares of equity outstanding than the unlevered firm. Therefore any increase in EBI leads to a greater rise in EPS for the levered firm, because the earnings increase is distributed over fewer shares of equity.
Because the dotted line has a lower intercept but a higher slope, the two lines must intersect. The break-even point occurs at €800 of EBI. Were earnings before interest to be €800, both firms would produce €2 of earnings per share (EPS). Because €800 is break-even, earnings above €800 lead to greater EPS for the levered firm. Earnings below €800 lead to greater EPS for the unlevered firm.

#### The Choice between Debt and Equity

Tables 15.2 and 15.3 and Fig. 15.2 are important, because they show the effect of leverage on earnings per share. Students should study the tables and figure until they feel comfortable with the calculation of each number in them. However, we have not yet presented the punch line. That is, we have not yet stated which capital structure is better for Autoveloce.
At this point many students believe that leverage is beneficial, because EPS are expected to be €4.00 with leverage and only €3.00 without leverage. However, leverage also creates risk. Note that in a recession EPS are higher (€1.00 versus €0) for the unlevered firm. Thus a risk-averse investor might prefer the all-equity firm, whereas a risk-neutral (or less risk-averse) investor might prefer leverage. Given this ambiguity, which capital structure is better?
Modigliani and Miller (MM or M & M) have a convincing argument that a firm cannot change the total value of its outstanding securities by changing the proportions of its capital structure. In other words, the value of the firm is always the same under different capital structures. In still other words, no capital structure is any better or worse than any other capital structure for the firm’s shareholders. This rather pessimistic result is the famous MM Proposition I.2
Their argument compares a simple strategy, which we call strategy A, with a two-part strategy, which we call strategy B. Both of these strategies for shareholders of Autoveloce are illustrated in Table 15.4. Let us now examine the first strategy.
 Table 15.4 Pay-off and cost to shareholders of Autoveloce SpA under the proposed structure and under the current structure with homemade leverage

Strategy A: Buy 100 shares of the levered equity
The first line in the top panel of Table 15.4 shows EPS for the proposed levered equity in the three economic states. The second line shows the earnings in the three states for an individual buying 100 shares. The next line shows that the cost of these 100 shares is €2,000.
Let us now consider the second strategy, which has two parts to it.
1. Borrow €2,000 from either a bank or, more likely, a brokerage house. (If the brokerage house is the lender, we say that this activity is going on margin.)
2. Use the borrowed proceeds plus your own investment of €2,000 (a total of €4,000) to buy 200 shares of the current unlevered equity at €20 per share.
The bottom panel of Table 15.4 shows pay-offs under strategy B, which we call the homemade leverage strategy. First observe the middle column, which indicates that 200 shares of the unlevered equity are expected to generate €600 of earnings. Assuming that the €2,000 is borrowed at a 10 per cent interest rate, the interest expense is €200 (= 0.10 × €2,000). Thus the net earnings are expected to be €400. A similar calculation generates net earnings of either €0 or €800 in recession or expansion, respectively.
Now let us compare these two strategies, both in terms of earnings per year and in terms of initial cost. The top panel of the table shows that strategy A generates earnings of €0, €400 and €800 in the three states. The bottom panel of the table shows that strategy B generates the same net earnings in the three states.
The top panel of the table shows that strategy A involves an initial cost of €2,000. Similarly, the bottom panel shows an identical net cost of €2,000 for strategy B.
This shows a very important result. Both the cost and the pay-off from the two strategies are the same. Thus we must conclude that Autoveloce is neither helping nor hurting its shareholders by restructuring. In other words, an investor is not receiving anything from corporate leverage that she could not receive on her own.
Note that, as shown in Table 15.1, the equity of the unlevered firm is valued at €8,000. Because the equity of the levered firm is €4,000 and its debt is €4,000, the value of the levered firm is also €8,000. Now suppose that, for whatever reason, the value of the levered firm were actually greater than the value of the unlevered firm. Here strategy A would cost more than strategy B. In this case an investor would prefer to borrow on his own account and invest in the equity of the unlevered firm. He would get the same net earnings each year as if he had invested in the equity of the levered firm. However, his cost would be less. The strategy would not be unique to our investor. Given the higher value of the levered firm, no rational investor would invest in the shares of the levered firm. Anyone desiring shares in the levered firm would get the same euro return more cheaply by borrowing to finance a purchase of the unlevered firm’s shares. The equilibrium result would be, of course, that the value of the levered firm would fall and the value of the unlevered firm would rise until they became equal. At this point individuals would be indifferent between strategy A and strategy B.
This example illustrates the basic result of Modigliani–Miller (MM) and is, as we have noted, commonly called their Proposition I. We restate this proposition as follows:
MM Proposition I (no taxes): The value of the levered firm is the same as the value of the unlevered firm.
This is perhaps the most important result in all of corporate finance. In fact, it is generally considered the beginning point of modern corporate finance. Before MM, the effect of leverage on the value of the firm was considered complex and convoluted. Modigliani and Miller showed a blindingly simple result: if levered firms are priced too high, rational investors will simply borrow on their personal accounts to buy shares in unlevered firms. This substitution is oftentimes called homemade leverage. As long as individuals borrow (and lend) on the same terms as the firms, they can duplicate the effects of corporate leverage on their own.
The example of Autoveloce SpA shows that leverage does not affect the value of the firm. Because we showed earlier that shareholders’ welfare is directly related to the firm’s value, the example also indicates that changes in capital structure cannot affect the shareholders’ welfare.

#### A Key Assumption

The MM result hinges on the assumption that individuals can borrow as cheaply as corporations. If, alternatively, individuals can borrow only at a higher rate, we can easily show that corporations can increase firm value by borrowing.
Is this assumption of equal borrowing costs a good one? Individuals who want to buy shares and borrow can do so by establishing a margin account with a broker. Under this arrangement the broker lends the individual a portion of the purchase price. For example, the individual might buy €10,000 of equity by investing €6,000 of her own funds and borrowing €4,000 from the broker. Should the shares be worth €9,000 on the next day, the individual’s net worth or equity in the account would be €5,000 = €9,000 - €4,000.3
The broker fears that a sudden price drop will cause the equity in the individual’s account to be negative, implying that the broker may not get her loan repaid in full. To guard against this possibility, stock exchange rules require that the individual make additional cash contributions (replenish her margin account) as the share price falls. Because (a) the procedures for replenishing the account have developed over many years and (b) the broker holds the equity as collateral, there is little default risk to the broker. In particular, if margin contributions are not made on time, the broker can sell the shares to satisfy her loan. Therefore brokers generally charge low interest, with many rates being only slightly above the risk-free rate.
By contrast, corporations frequently borrow using illiquid assets (e.g. plant and equipment) as collateral. The costs to the lender of initial negotiation and ongoing supervision, as well as of working out arrangements in the event of financial distress, can be quite substantial. Thus it is difficult to argue that individuals must borrow at higher rates than corporations.

# 15.4 Modigliani and Miller: Proposition II (No Taxes)

#### Risk to Equity-holders Rises with Leverage

At an Autoveloce board meeting a director said, “Well, maybe it does not matter whether the corporation or the individual levers - as long as some leverage takes place. Leverage benefits investors. After all, an investor’s expected return rises with the amount of the leverage present.” He then pointed out that, as shown in Tables 15.2 and 15.3, the expected return on unlevered equity is 15 per cent, whereas the expected return on levered equity is 20 per cent.
However, another director replied, “Not necessarily. Though the expected return rises with leverage, the risk rises as well.” This point can be seen from an examination of Tables 15.2 and 15.3. With earnings before interest (EBI) varying between €400 and €2,000, earnings per share (EPS) for the shareholders of the unlevered firm vary between €1.00 and €5.00. EPS for the shareholders of the levered firm vary between €0 and €8.00. This greater range for the EPS of the levered firm implies greater risk for the levered firm’s shareholders. In other words, levered shareholders have better returns in good times than do unlevered shareholders, but have worse returns in bad times. The two tables also show greater range for the ROE of the levered firm’s shareholders. The earlier interpretation concerning risk applies here as well.
The same insight can be taken from Fig. 15.2. The slope of the line for the levered firm is greater than the slope of the line for the unlevered firm. This means that the levered shareholders have better returns in good times than do unlevered shareholders, but have worse returns in bad times, implying greater risk with leverage. In other words, the slope of the line measures the risk to shareholders, because the slope indicates the responsiveness of ROE to changes in firm performance (earnings before interest).

Because levered equity has greater risk, it should have a greater expected return as compensation. In our example, the market requires only a 15 per cent expected return for the unlevered equity, but it requires a 20 per cent expected return for the levered equity.
This type of reasoning allows us to develop MM Proposition II. Here MM argue that the expected return on equity is positively related to leverage, because the risk to equity-holders increases with leverage.
To develop this position, recall that the firm’s weighted average cost of capital, RWACC, can be written as4

where RB is the cost of debt; RS is the expected return on equity, also called the cost of equity or the required return on equity; RWACC is the firm’s weighted average cost of capital; B is the market value of the firm’s debt or bonds; and S is the market value of the firm’s shares or equity.
Equation (15.2) is quite intuitive. It simply says that a firm’s weighted average cost of capital is a weighted average of its cost of debt and its cost of equity. The weight applied to debt is the proportion of debt in the capital structure, and the weight applied to equity is the proportion of equity in the capital structure. Calculations of RWACC from Eq. (15.2) for both the unlevered and the levered firm are presented in Table 15.5.
An implication of MM Proposition I is that RWACC is a constant for a given firm, regardless of the capital structure.5 For example, Table 15.5 shows that RWACC for Autoveloce is 15 per cent, with or without leverage.
Let us now define R0 to be the cost of capital for an all-equity firm. For Autoveloce SpA, R0 is calculated as

As can be seen from Table 15.5, RWACC is equal to R0 for Autoveloce. In fact, RWACC must always equal R0 in a world without corporate taxes.
Proposition II states the expected return on equity, RS, in terms of leverage. The exact relationship, derived by setting RWACC = R0 and then rearranging Eq. (15.2), is6
MM Proposition II (no taxes):

 Table 15.5 Cost of capital calculations for Autoveloce

 Figure 15.3 The cost of equity, the cost of debt, and the weighted average cost of capital: MM Proposition II with no corporate taxes

Equation (15.3) implies that the required return on equity is a linear function of the firm’s debt–equity ratio. Examining Eq. (15.3), we see that if R0 exceeds the cost of debt, RB, then the cost of equity rises with increases in the debt–equity ratio, B/S. Normally R0 should exceed RB. That is, because even unlevered equity is risky, it should have an expected return greater than that of riskless debt. Note that Eq. (15.3) holds for Autoveloce in its levered state:

Figure 15.3 graphs Eq. (15.3). As you can see, we have plotted the relation between the cost of equity, RS, and the debt–equity ratio, B/S, as a straight line. What we witness in Eq. (15.3) and illustrate in Fig. 15.3 is the effect of leverage on the cost of equity. As the firm raises the debt–equity ratio, each euro of equity is levered with additional debt. This raises the risk of equity and therefore the required return, RS, on the equity.
Figure 15.3 also shows that RWACC is unaffected by leverage, a point we have already made. (It is important for students to realize that R0, the cost of capital for an all-equity firm, is represented by a single dot on the graph. By contrast, RWACC is an entire line.)
EXAMPLE 15.2MM Propositions I and II
Canary Motors, an all-equity firm, has expected earnings of £10 million per year in perpetuity. The firm pays all of its earnings out as dividends, so the £10 million may also be viewed as the shareholders’ expected cash flow. There are 10 million shares outstanding, implying expected annual cash flow of £1 per share. The cost of capital for this unlevered firm is 10 per cent. In addition, the firm will soon build a new plant for £4 million. The plant is expected to generate additional cash flow of £1 million per year. These figures can be described as follows:

The project’s net present value is

assuming that the project is discounted at the same rate as the firm as a whole. Before the market knows of the project, the market value balance sheet of the firm is this:

The value of the firm is £100 million because the cash flow of £10 million per year is capitalized (discounted) at 10 per cent. A share of equity sells for £10 (= £100 million/10 million) because there are 10 million shares outstanding.
The firm will issue £4 million of either equity or debt. Let us consider the effect of equity and debt financing in turn.
Equity Financing Imagine that the firm announces that in the near future it will raise £4 million in equity to build a new plant. The share price, and therefore the value of the firm, will rise to reflect the positive net present value of the plant. According to efficient markets, the increase occurs immediately. That is, the rise occurs on the day of the announcement, not on the date of either the onset of construc-tion of the plant or the forthcoming equity offering. The market value balance sheet becomes this:

Note that the NPV of the plant is included in the market value balance sheet. Because the new shares have not yet been issued, the number of outstanding shares remains 10 million. The price per share has now risen to £10.60 (= £106 million/10 million) to reflect news concerning the plant.
Shortly thereafter, £4 million of equity is issued, or floated. Because the shares are selling at £10.60 per share, 377,358 (= £4 million/£10.60) shares are issued. Imagine that funds are put in the bank temporarily before being used to build the plant. The market value balance sheet becomes this:

The number of shares outstanding is now 10,377,358, because 377,358 new shares were issued. The share price is £10.60 (= £110,000,000/10,377,358). Note that the price has not changed. This is consistent with efficient capital markets, because the share price should move as a result of new information only.
Of course, the funds are placed in the bank only temporarily. Shortly after the new issue, the £4 million is given to a contractor, who builds the plant. To avoid problems in discounting, we assume that the plant is built immediately. The market value balance sheet then looks like this:

Though total assets do not change, the composition of the assets does change. The bank account has been emptied to pay the contractor. The present value of cash flows of £1 million a year from the plant is reflected as an asset worth £10 million. Because the building expenditures of £4 million have already been paid, they no longer represent a future cost. Hence they no longer reduce the value of the plant. According to efficient capital markets, the share price remains £10.60.
Expected yearly cash flow from the firm is £11 million, £10 million of which comes from the old assets and £1 million from the new. The expected return to shareholders is

Because the firm is all equity, RS = R0 = 0.10.
Debt Financing Alternatively, imagine the firm announces that in the near future it will borrow £4 million at 6 per cent to build a new plant. This implies yearly interest payments of £240,000 (= £4,000,000 × 6%). Again, the share price rises immediately to reflect the positive net present value of the plant. Thus we have the following:

The value of the firm is the same as in the equity financing case, because (a) the same plant is to be built, and (b) MM proved that debt financing is neither better nor worse than equity financing.
At some point £4 million of debt is issued. As before, the funds are placed in the bank temporarily. The market value balance sheet becomes this:

Note that debt appears on the right side of the market value balance sheet. The share price is still £10.60, in accordance with our discussion of efficient capital markets.
Finally the contractor receives £4 million and builds the plant. The market value balance sheet turns into this:

The only change here is that the bank account has been depleted to pay the contractor. The shareholders expect yearly cash flow after interest of

The shareholders expect to earn a return of

This return of 10.15 per cent for levered shareholders is higher than the 10 per cent return for the unlevered shareholders. This result is sensible because, as we argued earlier, levered equity is riskier. In fact, the return of 10.15 per cent should be exactly what MM Proposition II predicts. This prediction can be verified by plugging values into

We obtain

This example was useful for two reasons. First, we wanted to introduce the concept of market value balance sheets, a tool that will prove useful elsewhere in the text. Among other things, this technique allows us to calculate the share price of a new issue of shares. Second, the example illustrates three aspects of Modigliani and Miller:
1. The example is consistent with MM Proposition I, because the value of the firm is £110 million after either equity or debt financing.
2. Students are often more interested in share price than in firm value. We show that the share price is always £10.60, regardless of whether debt or equity financing is used.
3. The example is consistent with MM Proposition II. The expected return to shareholders rises from 10 to 10.15 per cent, just as Eq. (15.3) states. This rise occurs because the shareholders of a levered firm face more risk than do the shareholders of an unlevered firm.

#### MM: An Interpretation

The Modigliani–Miller results indicate that managers cannot change the value of a firm by repackaging the firm’s securities. Though this idea was considered revolutionary when it was originally proposed in the late 1950s, the MM approach and proof have since met with wide acclaim.7
MM argue that the firm’s overall cost of capital cannot be reduced as debt is substituted for equity, even though debt appears to be cheaper than equity. The reason for this is that, as the firm adds debt, the remaining equity becomes more risky. As this risk rises, the cost of equity capital rises as a result. The increase in the cost of the remaining equity capital offsets the higher proportion of the firm financed by low-cost debt. In fact, MM prove that the two effects exactly offset each other, so that both the value of the firm and the firm’s overall cost of capital are invariant to leverage.
 In Their Own Words In Professor Miller’s Words . . . The Modigliani–Miller results are not easy to understand fully. This point is related in a story told by Merton Miller.* How difficult it is to summarize briefly the contribution of the [Modigliani–Miller] papers was brought home to me very clearly last October after Franco Modigliani was awarded the Nobel Prize in Economics in part - but, of course, only in part - for the work in finance. The television camera crews from our local stations in Chicago immediately descended upon me. “We understand,” they said, “that you worked with Modigliani some years back in developing these M and M theorems and we wonder if you could explain them briefly to our television viewers.” “How briefly?” I asked. “Oh, take ten seconds,” was the reply. Ten seconds to explain the work of a lifetime! Ten seconds to describe two carefully reasoned articles, each running to more than thirty printed pages and each with sixty or so long footnotes! When they saw the look of dismay on my face, they said, “You don’t have to go into details. Just give us the main points in simple, commonsense terms.” The main point of the first or cost-of-capital article was, in principle at least, simple enough to make. It said that in an economist’s ideal world of complete and perfect capital markets and with full and symmetric information among all market participants, the total market value of all the securities issued by a firm was governed by the earning power and risk of its underlying real assets and was independent of how the mix of securities issued to finance it was divided between debt instruments and equity capital . . . Such a summary, however, uses too many shorthanded terms and concepts, like perfect capital markets, that are rich in connotations to economists but hardly so to the general public. So I thought, instead, of an analogy that we ourselves had invoked in the original paper . . . “Think of the firm,” I said, “as a gigantic tub of whole milk. The farmer can sell the whole milk as is. Or he can separate out the cream and sell it at a considerably higher price than the whole milk would bring. (That’s the analogy of a firm selling low-yield and hence high-priced debt securities.) But, of course, what the farmer would have left would be skim milk with low butterfat content and that would sell for much less than whole milk. That corresponds to the levered equity. The M and M proposition says that if there were no costs of separation (and, of course, no government dairy support programmes), the cream plus the skim milk would bring the same price as the whole milk.” The television people conferred among themselves and came back to inform me that it was too long, too complicated, and too academic. “Don’t you have anything simpler?” they asked. I thought of another way that the M and M proposition is presented these days, which emphasizes the notion of market completeness and stresses the role of securities as devices for “partitioning” a firm’s pay-offs in each possible state of the world among the group of its capital suppliers. “Think of the firm,” I said, “as a gigantic pizza, divided into quarters. If now you cut each quarter in half into eighths, the M and M proposition says that you will have more pieces but not more pizza.” Again there was a whispered conference among the camera crew, and the director came back and said: “Professor, we understand from the press releases that there were two M and M propositions. Can we try the other one?” [Professor Miller tried valiantly to explain the second proposition, though this was apparently even more difficult to get across. After his attempt:] Once again there was a whispered conversation. They shut the lights off. They folded up their equipment. They thanked me for giving them the time. They said that they’d get back to me. But I knew that I had somehow lost my chance to start a new career as a packager of economic wisdom for TV viewers in convenient ten-second bites. Some have the talent for it . . . and some just don’t. *Taken from GSB Chicago, University of Chicago (Autumn 1986).

Food found its way into this chapter earlier when we viewed the firm as a pie. MM argue that the size of the pie does not change, no matter how shareholders and bondholders divide it. MM say that a firm’s capital structure is irrelevant; it is what it is by some historical accid-ent. The theory implies that firms’ debt–equity ratios could be anything. They are what they are because of whimsical and random managerial decisions about how much to borrow and how much equity to issue.
 Summary of Modigliani–Miller Propositions without Taxes Assumptions No taxes. No transaction costs. Individuals and corporations borrow at same rate. Results Proposition I: VL = VU (Value of levered firm equals value of unlevered firm) Proposition II: Intuition Proposition I: Through homemade leverage individuals can either duplicate or undo the effects of corporate leverage. Proposition II: The cost of equity rises with leverage because the risk to equity rises with leverage.

Although scholars are always fascinated with far-reaching theories, students are perhaps more concerned with real-world applications. Do real-world managers follow MM by treating capital structure decisions with indifference? Unfortunately for the theory, virtually all companies in certain industries, such as banking, choose high debt–equity ratios. Conversely, companies in other industries, such as pharmaceuticals, choose low debt–equity ratios. In fact, almost any industry has a debt–equity ratio to which companies in that industry tend to adhere. Thus companies do not appear to be selecting their degree of leverage in a frivolous or random manner. Because of this, financial economists (including MM themselves) have argued that real-world factors may have been left out of the theory.
Though many of our students have argued that individuals can borrow only at rates above the corporate borrowing rate, we disagreed with this argument earlier in the chapter. But when we look elsewhere for unrealistic assumptions in the theory, we find two:8
1. Taxes were ignored.
2. Bankruptcy costs and other agency costs were not considered.
We turn to taxes in the next section. Bankruptcy costs and other agency costs will be treated in the next chapter. A summary of the main Modigliani–Miller results without taxes is presented in the nearby boxed section.

# 15.5 Corporate Taxes

#### The Basic Insight

The previous part of this chapter showed that firm value is unrelated to debt in a world without taxes. We now show that in the presence of corporate taxes the firm’s value is positively related to its debt. The basic intuition can be seen from a pie chart, such as the one in Fig. 15.4. Consider the all-equity firm on the left. Here both shareholders and tax authorities have claims on the firm. The value of the all-equity firm is, of course, that part of the pie owned by the shareholders. The proportion going to taxes is simply a cost.
The pie on the right for the levered firm shows three claims: shareholders, debtholders, and taxes. The value of the levered firm is the sum of the value of the debt and the value of the equity. In selecting between the two capital structures in the picture, a financial manager should select the one with the higher value. Assuming that the total area is the same for both pies,9 value is maximized for the capital structure paying the least in taxes. In other words, the manager should choose the capital structure that the government hates the most.
We shall show that, because of a quirk in corporate tax law, the proportion of the pie allocated to taxes is less for the levered firm than it is for the unlevered firm. Thus managers should select high leverage.
 Figure 15.4 Two pie models of capital structure under corporate taxes

EXAMPLE 15.3Taxes and Cash Flow
Wasserprodukte GmbH has a corporate tax rate, tC, of 35 per cent and expected earnings before interest and taxes (EBIT) of €1 million each year. Its entire earnings after taxes are paid out as dividends.
The firm is considering two alternative capital structures. Under Plan I Wasserprodukte would have no debt in its capital structure. Under Plan II the company would have €4,000,000 of debt, B. The cost of debt, RB, is 10 per cent.
The chief financial officer for Wasserprodukte makes the following calculations:

The most relevant numbers for our purposes are the two on the bottom line. Dividends, which are equal to earnings after taxes in this example, are the cash flow to shareholders, and interest is the cash flow to bondholders. Here we see that more cash flow reaches the owners of the firm (both shareholders and bondholders) under Plan II. The difference is €140,000 = €790,000 − €650,000. It does not take us long to realize the source of this difference. The government receives less tax under Plan II (€210,000) than it does under Plan I (€350,000). The difference here is €140,000 = €350,000 - €210,000.
This difference occurs because the way governments treat interest is different from the way they treat earnings going to shareholders.10 Interest totally escapes corporate taxation, whereas earnings after interest but before corporate taxes (EBT) are taxed at the corporate tax rate.

#### Present Value of the Tax Shield

The previous discussion shows a tax advantage to debt or, equivalently, a tax disadvantage to equity. We now want to value this advantage. The interest in monetary terms is

This interest is €400,000 (= 10% × €4,000,000) for Wasserprodukte. All this interest is tax-deductible. That is, whatever the taxable income of Wasserprodukte would have been without the debt, the taxable income is now €400,000 less with the debt.
Because the corporate tax rate is 0.35 in our example, the reduction in corporate taxes is €140,000 (= 0.35 × €400,000). This number is identical to the reduction in corporate taxes calculated previously.
Algebraically, the reduction in corporate taxes is

That is, whatever the taxes that a firm would pay each year without debt, the firm will pay tCRBB less with the debt of B. Expression (15.4) is often called the tax shield from debt. Note that it is an annual amount.
As long as the firm expects to be paying tax, we can assume that the cash flow in Expression (15.4) has the same risk as the interest on the debt. Thus its value can be determined by discounting at the cost of debt, RB. Assuming that the cash flows are perpetual, the present value of the tax shield is

#### Value of the Levered Firm

We have just calculated the present value of the tax shield from debt. Our next step is to calculate the value of the levered firm. The annual after-tax cash flow of an unlevered firm is

where EBIT is earnings before interest and taxes. The value of an unlevered firm (that is, a firm with no debt) is the present value of EBIT × (1 - tC):

Here VU is the present value of an unlevered firm; EBIT × (1 - tC) represents firm cash flows after corporate taxes; tC is the corporate tax rate; and R0 is the cost of capital to an all-equity firm. As can be seen from the formula, R0 now discounts after-tax cash flows.
As shown previously, leverage increases the value of the firm by the tax shield, which is tCB for perpetual debt. Thus we merely add this tax shield to the value of the unlevered firm to get the value of the levered firm.
We can write this algebraically as follows:11
MM Proposition I (corporate taxes):

Equation (15.5) is MM Proposition I under corporate taxes. The first term in Eq. (15.5) is the value of the cash flows of the firm with no debt tax shield. In other words, this term is equal to VU, the value of the all-equity firm. The value of the levered firm is the value of an all-equity firm plus tCB, the tax rate times the value of the debt. tCB is the present value of the tax shield in the case of perpetual cash flows.12 Because the tax shield increases with the amount of debt, the firm can raise its total cash flow and its value by substituting debt for equity.
EXAMPLE 15.4MM with Corporate Taxes
Divided Airlines is currently an unlevered firm. The company expects to generate €153.85 in earnings before interest and taxes (EBIT) in perpetuity. The corporate tax rate is 35 per cent, implying after-tax earnings of €100. All earnings after tax are paid out as dividends.
The firm is considering a capital restructuring to allow €200 of debt. Its cost of debt capital is 10 per cent. Unlevered firms in the same industry have a cost of equity capital of 20 per cent. What will the new value of Divided Airlines be?
The value of Divided Airlines will be equal to

The value of the levered firm is €570, which is greater than the unlevered value of €500. Because VL = B + S, the value of levered equity, S, is equal to €570 - €200 = €370. The value of Divided Airlines as a function of leverage is illustrated in Fig. 15.5.
 Figure 15.5 The effect of financial leverage on firm value: MM with corporate taxes in the case of Divided Airlines

#### Expected Return and Leverage under Corporate Taxes

MM Proposition II under no taxes proposes a positive relationship between the expected return on equity and leverage. This result occurs because the risk of equity increases with leverage. The same intuition also holds in a world of corporate taxes. The exact formula in a world of corporate taxes is this:13
MM Proposition II (corporate taxes):

Applying the formula to Divided Airlines, we get

This calculation is illustrated in Fig. 15.6.
Whenever R0 > RB, RS increases with leverage, a result that we also found in the no-tax case. As stated earlier in this chapter, R0 should exceed R0B. That is, because equity (even un-levered equity) is risky, it should have an expected return greater than that on the less risky debt.
 Figure 15.6 The effect of financial leverage on the cost of debt and equity capital

Let’s check our calculations by determining the value of the levered equity in another way. The algebraic formula for the value of levered equity is

The numerator is the expected cash flow to levered equity after interest and taxes. The denominator is the rate at which the cash flow to equity is discounted.
For Divided Airlines we get

which is the same result we obtained earlier (ignoring a small rounding error).

#### The Weighted Average Cost of Capital, RWACC, and Corporate Taxes

In Chapter 12 we defined the weighted average cost of capital (with corporate taxes) as follows (note that VL = S + B):

Note that the cost of debt capital, RB, is multiplied by (1 - tC), because interest is tax-deductible at the corporate level. However, the cost of equity, RS, is not multiplied by this factor, because dividends are not deductible. In the no-tax case RWACC is not affected by leverage. This result is reflected in Fig. 15.3, which we discussed earlier. However, because debt is tax-advantaged relative to equity, it can be shown that RWACC declines with leverage in a world with corporate taxes. This result can be seen in Fig. 15.6.
For Divided Airlines, RWACC is equal to

Divided Airlines has reduced its RWACC from 0.20 (with no debt) to 0.1754 with reliance on debt. This result is intuitively pleasing, because it suggests that when a firm lowers its RWACC, the firm’s value will increase. Using the RWACC approach, we can confirm that the value of Divided Airlines is €570:

#### Share Prices and Leverage under Corporate Taxes

At this point students often believe the numbers - or at least are too intimidated to dispute them. However, they sometimes think we have asked the wrong question. “Why are we choosing to maximize the value of the firm?” they will say. “If managers are looking out for the shareholders’ interests, why aren’t they trying to maximize share price?” If this question occurred to you, you have come to the right section.
Our response is twofold. First, we showed in the first section of this chapter that the capital structure that maximizes firm value is also the one that most benefits the interests of the shareholders.
However, that general explanation is not always convincing to students. As a second procedure, we calculate the share price of Divided Airlines both before and after the exchange of debt for equity. We do this by presenting a set of market value balance sheets. The market value balance sheet for the company in its all-equity form can be represented as follows:

Assuming that there are 100 shares outstanding, each share is worth €5 = €500/100.
Next imagine the company announces that in the near future it will issue €200 of debt to buy back €200 of equity. We know from our previous discussion that the value of the firm will rise to reflect the tax shield of debt. If we assume that capital markets price securities efficiently, the increase occurs immediately. That is, the rise occurs on the day of the announcement, not on the date of the debt-for-equity exchange. The market value balance sheet now becomes this:

Note that the debt has not yet been issued. Therefore only equity appears on the right side of the balance sheet. Each share is now worth €570/100 = €5.70, implying that the shareholders have benefited by €70. The shareholders gain because they are the owners of a firm that has improved its financial policy.
The introduction of the tax shield to the balance sheet is perplexing to many students. Although physical assets are tangible, the ethereal nature of the tax shield bothers these students. However, remember that an asset is any item with value. The tax shield has value because it reduces the stream of future taxes. The fact that one cannot touch the shield in the way that one can touch a physical asset is a philosophical, not a financial, consideration.
At some point the exchange of debt for equity occurs. Debt of €200 is issued, and the proceeds are used to buy back shares. How many shares are repurchased? Because shares are now selling at €5.70 each, the number of shares that the firm acquires is €200/€5.70 = 35.09. This leaves 64.91 (= 100 − 35.09) shares outstanding. The market value balance sheet is now this:

Each share is worth €370/64.91 = €5.70 after the exchange. Notice that the share price does not change on the exchange date. As we mentioned, the share price moves on the date of the announcement only. Because the shareholders participating in the exchange receive a price equal to the market price per share after the exchange, they do not care whether they exchange their equity.
This example was provided for two reasons. First, it shows that an increase in the value of the firm from debt financing leads to an increase in the price of the shares. In fact, the shareholders capture the entire €70 tax shield. Second, we wanted to provide more work with market value balance sheets.
A summary of the main results of Modigliani–Miller with corporate taxes is presented in the following boxed section:
 Summary of Modigliani–Miller Propositions with Corporate Taxes Assumptions Corporations are taxed at the rate tC, on earnings after interest. No transaction costs. Individuals and corporations borrow at same rate. Results Proposition I: VL = VU + tCB (for a firm with perpetual debt) Proposition II: Intuition Proposition I: Because corporations can deduct interest payments but not dividend payments, corporate leverage lowers tax payments. Proposition II: The cost of equity rises with leverage because the risk to equity rises with leverage.

# 15.6 Personal Taxes

So far in this chapter we have considered corporate taxes only. Because interest on debt is tax-deductible, whereas dividends on equity are not deductible, we argued that the tax system gives firms an incentive to issue debt. But corporations are not the only ones paying taxes; individuals must pay taxes on both the dividends and the interest that they receive. We cannot fully understand the effect of taxes on capital structure until all taxes, both corporate and personal, are considered.

#### The Basics of Personal Taxes

Let’s begin by examining an all-equity firm that receives €1 of pre-tax earnings. If the corporate tax rate is tc, the firm pays taxes tc, leaving itself with earnings after taxes of 1 - tC. Let’s assume that this entire amount is distributed to the shareholders as dividends. If the personal tax rate on share dividends is tS, the shareholders pay taxes of (1 - tC) × tS, leaving them with (1 - tC) × (1 - tS) after taxes.
Alternatively, imagine that the firm is financed with debt. Here, the entire €1 of earnings will be paid out as interest, because interest is deductible at the corporate level. If the personal tax rate on interest is tB, the bondholders pay taxes of tB, leaving them with 1 - tB after taxes.

#### The Effect of Personal Taxes on Capital Structure

To explore the effect of personal taxes on capital structure, let’s consider three questions:
1. Ignoring costs of financial distress, what is the firm’s optimal capital structure if dividends and interest are taxed at the same personal rate: that is, tS = tB?
The firm should select the capital structure that gets the most cash into the hands of its investors. This is tantamount to selecting a capital structure that minimizes the total amount of taxes at both the corporate and personal levels.
As we have said, beginning with €1 of pre-tax corporate earnings, shareholders receive (1 - tC) × (1 - tS), and bondholders receive 1 - tB. We can see that if tS = tB, bondholders receive more than shareholders. Thus the firm should issue debt, not equity, in this situation. Intuitively, income is taxed twice - once at the corporate level and once at the personal level - if it is paid to shareholders. Conversely, income is taxed only at the personal level if it is paid to bondholders.
Note that the assumption of no personal taxes, which we used in the previous chapter, is a special case of the assumption that both interest and dividends are taxed at the same rate. Without personal taxes, the shareholders receive 1 - tC while the bondholders receive €1. Thus, as we stated in a previous section, firms should issue debt in a world without personal taxes.
2. Under what conditions will the firm be indifferent between issuing equity or debt?
The firm will be indifferent if the cash flow to shareholders equals the cash flow to bondholders. That is, the firm is indifferent when
3. What should companies do in the real world?
Although this is clearly an important question, it is, unfortunately, a hard one - perhaps too hard to answer definitively. Nevertheless, let’s begin by working with the highest tax rates for a specific country, the UK. As of 2009, the corporate tax rate was 28 per cent. For investors in the highest tax bracket, interest income was taxed at 40 per cent. Investors in this highest bracket faced an effective 25 per cent tax rate on dividends.
At these rates, the left side of Eq. (15.7) becomes (1 − 0.28) × (1 − 0.25), which equals 0.54. The right side of the equation becomes 1 − 0.28, which equals 0.72. Because any rational firm would rather get £0.72 instead of £0.54 into its investors’ hands, it appears at first glance that firms should prefer debt over equity, just as we argued earlier.
Does anything else in the real world alter this conclusion? Perhaps: our discussion on equity income is not yet complete. Firms can repurchase shares with excess cash instead of paying a dividend. Although capital gains in the UK are taxed at 18 per cent, the shareholder pays a capital gains tax only on the gain from sale, not on the entire proceeds from the repurchase. Thus the effective tax rate on capital gains is actually lower than 18 per cent. Because firms both pay dividends and repurchase shares, the effective personal tax rate on share distributions must be below 18 per cent.
This lower effective tax rate makes equity issuance less burdensome, but the lower rate will not induce any firm to choose shares over bonds. For example, suppose that the effective tax rate on share distributions is 10 per cent. From every pound of pre-tax corporate income, shareholders in the UK receive (1 − 0.28) × (1 − 0.10), which equals £0.648. This amount is less than the £0.72 that bondholders receive. In fact, as long as the effective tax rate on equity income is positive, bondholders will still receive more than shareholders from a pound of pre-tax corporate income. And we have assumed that all bondholders face a tax rate of 40 per cent on interest income. In reality, plenty of bondholders are in lower tax brackets, further tipping the scales towards bond financing.