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How to calculate the cost of capital for a firm using the weighted average cost of capital (WACC) method. It covers the cost of debt, preferred stock, retained earnings, and new equity, including the calculation of after-tax costs and the impact of flotation costs.
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The major theme of the last few sections of notes has been valuation—that is, the time value of money concepts provides you with the computations to determine the value of any asset; the valuation section showed you how to determine the value of financial assets, such as bonds and stocks; and the capital budgeting section showed how to determine whether an asset is an acceptable investment for the firm, which entails comparing the value of the asset to its cost (initial investment). To compute the value of an asset, you need to know the rate of return, r, that is required to invest in the asset. To this point, we have not discussed how the required rate of return is determined. The purpose of this section is to show you how the required rate of return for a firm (or for an individual for that matter) is determined and what the value for the required rate of return means. As you read this section, keep in mind that the reason a firm has to earn a particular rate of return on its assets is because investors who provide funds to the firm demand to receive a return on their funds. As a result, the firm must earn enough on its investments to provide the return investors demand—that is, r, or the required rate of return. The concept is the same as if you borrow money for the purposes of investing in the stock market. If your investments don’t earn a return that covers the interest you are paying on the loan, which is your required rate of return in this case, then you lose money—that is, the net present value (NPV) of the investments is negative, which means you reduce your wealth.
d d d
After-tax component (^) = Bondholders' required- Tax savings cost of debt rate of return associated with debt = r - r ×T = r (1-T)
where rd is the required rate of return of investors who hold the firm’s bonds and T is the marginal tax rate of the firm. Remember from the notes titled “Risk and Rates of Return” that rd is the rate of return that investors demand (require) to invest in the firm’s bonds. This rate is also referred to as the market return or the yield to maturity (YTM) on the bond. Thus, effectively, investors who are the participants in the financial markets determine the firm’s cost of debt. This is also the case for the other component costs of capital—that is, the cost of equity.
Example: A firm is considering issuing a bond to raise funds to finance its assets. The bond will have a face value equal to $1,000, a coupon rate of interest equal to 11 percent with interest paid semiannually, and a 10-year life. The firm expects the market price of each bond in the issue to be $942.65. If the firm’s marginal tax rate is 40 percent, what is rdT for this bond? To determine the component cost of debt, first consider the fact that we know the following situation must exist:
1 2 20 942 65^55 55 1 055 1 d^2 1 d^2 1 d 2
$. $^ $^ $ , r r r
= + + + (^) + (^) + (^) +
Solving for rd gives us the YTM for this bond. Remember that you can solve for the YTM by using: (1) trial-and-error—that is, plug in different values for rd until the right side of the equation equals $942.65; (2) the TVM keys on your calculator—input N = 20 (20 interest payments), PV = –942.65 (market value), PMT = 55 (semiannual interest payment), and FV = 1,000 (maturity value), and then solve for I; or (3) a spreadsheet (see the section of the notes titled “Risk and Rates of Return” for an explanation). Whichever method you us, you should find rd /2 in the previous equation equals 6 percent. But, because this result represents a six-month rate of return, it must be annualized to be useful. In other words, the YTM = 12% = 6% × 2 = rd.
Because rd = YTM = 12% is computed without considering the fact that the interest the firm pays its bondholders is a tax deductible expense to the firm, rd represents a before-tax amount and it needs to be adjusted so as to represent the actual cost to the firm—that is, the cost of the bond to the firm isn’t really 12 percent. As an example of the tax deductibility of interest, consider the fact that when you finance your primary residence, the interest paid on the mortgage is tax deductible. Suppose the interest rate on a $100,000 mortgage is 10 percent and the mortgage calls for only one payment at the end of each of the next 10 years. The mortgage payment each year will equal $16,274.54 (you should be able to compute this value using TVM techniques discussed in earlier sections). In the first year, the portion of this payment that represents interest is $10,000 = 0.10 x $100,000; thus the remainder, $6,274.54, represents repayment of the principal amount of the
From the notes titled “Risk and Rates of Return,” we know that we can estimate the cost of retained earnings, rs, using the following relationships:
1 s RF M RF s s 0
of return of return Dˆ r = r + (r -r )β = + g = rˆ P
where the variables are as defined previously: rRF is the risk-free rate of return, rM is the return on the market, βs is the beta coefficient associated with the firm’s common stock, (^) Dˆ^1 is the dividend the firm expects to pay next year assuming constant growth exists, P 0 is the current market price of the stock, and g is the constant rate at which the firm is expected to grow in the future. Remember that in equilibrium rs = rˆs^. So, we can apply these two concepts to determine the cost of retained earnings. In addition, we can get a “rough guess” of rs by using the “bond-yield-plus-risk-premium approach.” o The CAPM Approach—using the CAPM, which is given in the above relationship, the cost of retained earnings is stated as follows:
rs = rRF + (rM - rRF)βs
Thus, if the risk-free rate of return is 6 percent, the market risk premium is expected to be 8 percent, and the firm’s beta coefficient is 1.5, then the cost of retained earnings is 18% = 6% + (8%)1.5.
One of the assumptions associated with the CAPM is that the firm’s stockholders are very well diversified. If this is not the case, then beta probably is not the appropriate measure of risk for determining the firm’s cost of retained earnings. Also, we usually associate the risk-free return with Treasury securities. Because at any point in time there are many different rates for Treasuries that have different terms to maturity, the result of the CAPM computation depends on which Treasury security is used—that is, if different people use Treasury securities with different maturities, then the results will differ somewhat. o Bond-yield-plus-risk-premium approach—studies have shown that the return on equity for a particular firm is approximately 3 to 5 percentage points higher than the return on its debt. Thus, as a general rule of thumb, firms often compute the YTM for their bonds and then add 3 to 5 percent to the result. For example, earlier we found that the before-tax return (YTM) on
our illustrative firm is 12 percent. Thus, as a rough estimate, we might say the cost of retained earnings is 16% = 12% + 4%. o Discounted cash flow (DCF) approach—if the firm is expected to grow at a constant rate, then we have the following relationship:
1 1 0 s s s 0
Dˆ^ Dˆ^ Dividend Capital P = or r =r =ˆ +g= + r -g P yield gain
Thus, if our illustrative firm expects the next dividend payment to be $3.30 per share, the price of its stock currently is $22, and it is estimated that the firm will grow at a constant rate of 5 percent, then this information suggests that the cost of retained earnings should be:
rs = + = =
For our illustrative firm, the three approaches we used to determine the cost of retained earnings give three different results. This is not unexpected, because the three approaches are based on different assumptions. As was mentioned, the CAPM approach assumes investors are extremely well diversified; the DFC approach assumes the firm grows at a constant rate; and the bond-yield-plus-risk-premium approach assumes that the return on equity is related to the return on the firm’s debt. In an ideal situation (the perfect world), all three approaches should give the same result. But, when we find different values and we don’t have any reason to discard any of them, generally we simply average the results. In this case, the average is 18% = (18% + 16% + 20%) ÷ 3. Like preferred stock, this figure is not adjusted for taxes because dividends, regardless of the type, are not tax deductible expenses for the firm.
As you can tell from the discussion to this point, determining the cost of retained earnings requires some judgment—there is no one simple rule that can be used when computing the cost of retained earnings.
g NP
g P( 1 F)
r 0
1 0
1 e + = + −
Using the values we computed earlier, if the firm does not have to issue new common stock, then its weighted average cost of capital, WACC, is
WACC = 0.3(7.20%) + 0.1(13.56%) + 0.6(18.00%) = 14.32%
According to the information given here, each dollar of funding that is attributed to debt costs the firm 7.2 percent, each dollar raised using preferred stock costs the firm 13.56 percent, and each dollar of retained earnings that is used to finance the firm costs 18 percent. But, because the firm does not use each type of financing in equal proportions, the average cost is not a simple average of these individual costs. If you think about it, the information in the above table indicates that for every dollar of financing the firm has, 30¢ is in the form of debt, 10¢ is preferred stock, and 60¢ is common equity. Thus, the average cost of the firm’s financing, its WACC, is 14.32 percent, which represents the average rate of return the firm must earn on its existing investments to ensure the value of the firm does not decrease—that is, WACC = r = required rate of return. In general terms, WACC is calculated as follows:
Proportion (^) × After-tax (^) + Proportion of (^) × Cost of (^) + Proportion of (^) × Cost of of debt cost of debt preferred stock preferred stock common equity common equity WACC=
=
(^) (^) (^) ^ ^ ^ ^ ^ ^ ^ ^
w rd dT + w (^) ps psr + w (^) s (r (^) s or re )
As you can see from the graph, the are points, which we call break points , where the WACC increases—that is, at $5 million the WACC increases from 14.32 percent to 14.86 percent and at $20 million the WACC increases from 14.86 percent to 17.26 percent. What causes these break points? Look at the equation given at the end of the last section—if any of the component costs of capital change, then so does the WACC. For example, if the after-tax cost of debt for our illustrative firm increases from 7.2 percent to 9 percent at $5 million of new funds ($1. million of new debt because debt represents 30 percent of total capital), then WACC = 14.86% = 0.3(9%) + 0.1(13.56%) + 0.6(18%). Then if the cost of equity increases from 18 percent to 22 percent when the firm raises $20 million of new funds ($12 million of new common stock because common equity represents 60 percent of total capital), WACC = 17.26% = 0.3(9%) + 0.1(13.56%) + 0.6(22%). Consequently, as you can see, the WACC changes whenever any of the component costs of capital change.
If we know the level at which a particular type of financing will increase, the break point stated in terms of total new funds can be written as follows:
point Proportion of this type of capital in the capital structure
In our example, the after-tax cost of debt is 7.2 percent if the firm raises from $1 (one dollar) to $1.5 million and then it increases to 9 percent for every dollar of new debt greater than $1. million, the break point caused by an increase in the cost of debt is:
$ 5 , 000 , 000
$1,500, Break point= =
Notice that the $5 million is the amount of total new funds the firm can raise before WACC increases from 14.32 percent to 14.86 percent. The reason for the increase in WACC is because above $5 million in total financing the amount of debt funding that is needed is greater than
20 Dollars of New Funds Raised (millions)
New WACC = MCC (%)
Break Points
14.86 percent, which is less than IRR (^) D = 15.5%. Just like we did in the “Capital Budgeting” section of the notes, the decision to accept projects is based on the relationship of the project’s IRR to the firm’s required rate of return, which we have now identified as its WACC. Thus, as long as IRR > WACC = r, a project is acceptable.
WACC Versus Required Rates of Return
Investor’s Required Rate of Return/Firm’s Cost of Capital:
Financial Asset Financial Asset’s Market Value Return to Investors Cost to Firms
Debt, rd YTM = r (^) d = return investors require r (^) d = YTM = before-tax cost of debt to purchase the firm’s debt r (^) dT = r (^) d(1-T) = after-tax cost of debt
Preferred Stock, rps = return investors require = cost of preferred stock to purchase the firm’s preferred stock
Common Equity, ; (constant growth firm) = return investors require = cost of retained rs (internal) or to purchase the firm’s earnings (internal) re (external) common stock = cost of new common equity (external)
Variable Definitions:
r (^) RF = nominal risk-free rate of return YTM = yield to maturity P 0 = market value of the financial asset T = the firm’s marginal tax rate INT = dollar interest payment D (^) ps = preferred stock dividend M = maturity (face) value Dˆ^1 = next period’s dividend N = number of remaining interest payments F = cost of issuing new stock (in decimal form) g = constant growth rate of the firm
0
ps ps (^) P
r = P( 1 F)
r 0
ps ps = −
g P
r 0
1 s =^ + P g
r 0
1 s = +
g P( 1 F)
r 0
1 e = − +
ps
ps (^0) r
r g
s
1 0 = −
costof capital
Firm'scomponent r r premiumRisk r,r ,orr rateofreturn
Investor' srequired RF (^) = d ps s=