MF807

Prof. Thomas Chemmanur

Topic Note 8

1. The Cost of Capital

Now we turn to the problem of computing the rate of return to be used to discount the cashflows from a project. Conceptually, the answer to this question is clear: all cash flows of the project under consideration should be discounted at the opportunity cost of capital i.e., that rate of return which can be achieved by the investors in the firm in the best available alternative investment opportunity of the same riskiness as the project. Thus, this is the rate of return that is demanded by the investors in the firm from the project; If the project under consideration has a positive NPV at this discount rate, the value of the firm's shares will only go up if it undertakes the project; If the project has a negative NPV at this discount rate, the firm should not undertake this project since its stock price will go down in this case. Having said this, however, it can be rather difficult to identify this rate of return as a practical matter. We will discuss two different techniques that are commonly used.

2. The Weighted Average Cost of Capital Method

This method is applicable if (a) the project being considered is of the same level of riskiness as the existing projects of the firm, and further, (b) the capital structure of the firm will remain the same as before after the current project has been financed. (We will see later, when we study the financing decision, how to use this method even when the assumption (b) is violated.)

By capital structure, we mean the proportion of total firm value represented by the value of each security. For instance, in the case of a firm financed by debt, equity, and preferred stock, the total value of the firm, V is given by:

(1)

where E is the total value of all common shares issued by the firm and D the total market value of all bonds issued by the firm, and S is the total market value of preferred stock. Dividing (1) throughout by V,

(2)

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The ratio of the market value of the equity to the market value of the entire firm, E/V, the ratio of the market value of debt to the market value of the entire firm, D/V, and the ratio of the market value of preferred stock to total firm value, S/V, represent the "capital structure" proportions of the firm. The weighted average cost of capital is the weighted average of the rate of return required by equity holders, debt holders and preferred stock holders in the firm, the weights being these capital structure proportions.

Assumptions (a) and (b) are important because, if they hold, we can make use of available data on the market prices of the stocks and bonds and other securities of the firm in computing the cost of capital to be used in evaluating the project. Assumption (a) means that the riskiness of the securities issued by the firm will not change even after the firm undertakes the project: if the firm undertakes a project of much higher risk than the existing projects of the firm, the systematic risk of the firm's debt and equity will change, and we can no longer use past information about rates of return. Similarly, if the capital structure changes, again the systematic risk of the firm's debt and equity will change (we will see precisely how these change when we study the financing decision), and again we will not be able to use the available data (or at least we will have to modify it). Therefore, let us assume for the present that these assumptions do hold.

Now, if the firm's stock beta, denoted by ßE is known, and we know the parameters of the CAPM (ie., rF and rM), we can compute the rate of return demanded by equity holders, rE. This is usually referred to as the "cost of equity".

(3)

Another way we can estimate rE is from the constant dividend growth model we studied before. For this, we need to know the growth rate g in dividends expected by investors. Then, if PE is the price per share of the firm's equity, and D1 is the dividend the firm expects to pay in the next period,

(4)

(The choice of method here will depend on how good an estimate we think we have of g versus the CAPM parameters, and on our belief in the validity of the CAPM. Of course, if our estimates are completely accurate, either method should give the same answer).

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Now, we should compute the cost of debt, rD. This can be computed as the yield to maturity of the bonds issued by the company, which we learned how to compute when we studied bond valuation. We thus know that rD is that this is that value of r which satisfies the following equation:

(5)

where PD is the current price of each of the firm's debt (bonds), C is the amount of coupon payments per period, n is the number of periods to maturity and F is the face value of debt. rD is the rate of return expected by bond holders for the firm's debt.

If there is any preferred stock in the firm's capital structure, we can also compute the cost of preferred equity, which is the rate of return demanded by preferred equity holders, rPE:

(6)

where PPE is the price of a share of preferred stock in the firm, and DPE is the amount of the preferred dividend per share.

The weighted average cost of capital can now be computed as the weighted average of these rates of return rE, rD, and rPE, where the weights are the proportions of these securities in the capital structure of the firm. However, we also need to make an adjustment for the fact that interest payments by the company on any bonds issued by the company are deductible from the taxable income of the firm for corporate tax purposes. Making this adjustment to the required return on debt, the formula for the weighted average cost of capital r (for the case where the firm has sold only debt and equity to raise capital) is:

(7)

where Tc is the marginal corporate tax rate.

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Of course, if the firm there is no preferred stock in the firm's capital structure (as is the case with many firms), the last term in the above formula for the weighted average cost of capital can be omitted. (9) gives the return that investors expect from their investment in the firm's projects: in other words, it is the required rate of return, or the rate of return to be used in discounting project cash flows.

Example Problem to be worked out in class:

Nocil Inc. has $3 million worth of bonds and $7 million worth of stocks in its capital structure. The bonds have a 14% cost, and the stock is expected to pay $500,000 in dividends this year (for the entire equity). The growth rate of dividends is expected to be 11%. Find the cost of capital Nocil should use to evaluate a project of the same risk as the existing projects in the firm (Assume that Nocil's capital structure will remain unchanged even after the project has been financed and the marginal corporate tax rate of the firm is Tc = 40%)

3. The Asset Beta Method

In many cases, the two assumptions we have made in computing the weighted average cost of capital will not hold. In this case, it is useful to make use of the concept of "asset beta". The asset beta of a firm measures the systematic risk of the firm's assets or projects. In general, it is different from the equity beta, which is the beta of the firm's stock.

To understand what asset beta measures, it is useful to study the following example. Consider a firm having three different divisions: the first one is a division making food products; the second one is a division making electronics products; the third one is a division making chemicals. Clearly, the cash flows connected with each division will have different levels of systematic risk: in fact, it has been computed that the average beta of the food industry is 1, the average beta of projects in the electronics industry is 1.60, and the average beta of projects in the chemicals industry is 1.22. Let us assume these respective values for the betas of the three divisions of our firm and also that within a division, all projects have the same risk. Assume further that the food division constitutes 1/2 of the total value of our firm, the electronics division 1/4, and the chemicals division the remaining 1/4. To compute the asset beta of the firm, we can think of the firm as a portfolio of divisions. We know that the beta of a portfolio is the weighted average of the betas of the individual securities. Therefore, the asset beta, dentoed by ßA, is given by: Σi xi.ßi = (1/2)1 + (1/4)1.6 + (1/4)1.22 = 1.205, for the above firm.

Thus the asset beta, 1.205 in the above example, is a measure of the average business risk of the firm. Now, if the CAPM parameters are as follows: rF = 5% (say), and rM = 12%, the required return on the entire firm as a whole, denoted by rA, is given by, rA = rF + ßA (rM - rF) = 5 + 1.205 (12 - 5) = 13.435%.

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How much should be the rate of return that should be earned by the projects in each division if the firm as a whole is to make a return of rA = 13.45%? Projects in each division should have an expected return corresponding to their own riskiness (ie., corresponding to the project betas): Thus projects in the food division should have an expected return of r1 = 5 + 1(12 - 5)= 12%; similarly, projects in the electronics division should have an expected return of r2 = 5 + 1.60 (12 - 5) = 16.2% and projects in the chemicals division should have an expected return of r3 = 5 + 1.22 (12 - 5) = 13.54%. We can cross-check our results by computing the expected return on the entire firm from the individual divisions' expected returns: rA = (1/2)12 + (1/4)16.2 + (1/4)13.54 = 13.435%.

The above example illustrates an important principle to be used when different projects in a firm have different levels of risk: each project should be discounted at the rate of return corresponding to the riskiness of the individual project, as measured by the beta of the individual project. (A project's beta measures how sensitive the returns from the project are to changes in the market return). In the above example, we assumed that all projects in the same division have the same riskiness; if this is the case, we can use a single discounting rate for all projects in the same division: i.e., we can make use of a divisional cost of capital. It is, however, conceivable that even projects in the same division differ in their risk, in which case we should compute individual project betas (which is quite difficult to do!).

Let us now see how the asset beta of the firm, ßA is related to the equity beta, ßE. We know that the market value of the firm is the sum of market values of all the securities issued by the firm. We know that for a firm financed by only debt and equity,

V = E + D. This means that we can think of the firm value V as the value of a portfolio of debt and equity. Consequently, the beta of the firm must be the weighted average of the betas of the equity and debt betas, where the weights are the respective proportions of equity and debt in the capital structure of the firm (these are the "portfolio weights" of debt and equity, if we think of the firm as a portfolio):

(8)

We can simplify and re-write this as:

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(9)

The last expression gives us a way of predicting how the ßE of a firm changes in response to changes in its capital structure. If we assume that ßD is constant (which is usually the case; for debt with no default risk, ßD is zero; even for risky debt, it usually takes very low values except for debt of very high default risk), we can see from the above expression that the riskiness of equity, as measured by the equity beta of the firm, increases as the debt-equity ratio goes up. Remember that, irrespective of how the firm is financed, the asset beta of the firm is always a constant for a given set of projects undertaken by the firm, since the asset beta reflects only on the business risk of the firm. The asset beta of a firm will change only if the firm changes its portfolio of projects: i.e., only if the firm undertakes additional projects of systematic risk different from those of the existing projects of the firm.

The equity beta thus reflects the financial risk of the firm (which is additional risk coming from the way the firm is financed) as well as the business risk, while the asset beta reflects only the business risk. We can see that the asset beta will equal the equity beta only for the special case where the firm is 100% equity financed: i.e., for the case where E/V = 1, and D/V = 0. This gives us another way of thinking about the asset beta of a firm: it is the same as the equity beta of the firm if it were 100% equity financed. Further, it is clear that if the firm is 100% equity financed, the equity of the firm will have only the business risk of the firm, while introducing debt into the capital structure also subjects the equity holders to financial risk. We will have more to say about this when we discuss the theory of capital structure.

How can we use the concepts we have learned above to compute the cost of capital of a project which has a riskiness different from those of the existing projects in the firm, and which will also lead to changes in the firm's capital structure? The following steps are involved;

1. Compute the beta of the project. How do we do this? We often make use of the beta of the industry in which the project is undertaken as an approximate figure for the project beta. Another approach is to look at a firm similar to ours in the same industry as the project, compute its asset beta, and assume that the beta of the project we are evaluating is the same.

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2. Using the CAPM, compute the expected return corresponding to this project beta. Let us call this rate of return, rA.

3. Adjust the rate of return we got in step (2) for the tax deductibility of payments to debt holders, using the "target" proportion of debt to be used in financing the project, denoted by L. Thus, if Tc is the corporate tax rate, then the cost of capital r to be used to discount the project's cashflows is given by,