^aPeter F. Drucker, The Essential Drucker (New York: Collins Business Essentials, 2001).

Example 9.1

My Finance Lab Solution Video

A firm is currently considering two investment opportunities. Two financial analysts, working independently of each other, are evaluating these opportunities. Assume the following information about investments A and B.

Investment A

The analyst studying this investment recalls that the company recently issued bonds paying a 6% rate of return. He reasons that because the investment project earns 7% while the firm can issue debt at 6%, the project must be worth doing, so he recommends that the company undertake this investment.

Investment B

Least costly financing source available

Equity = 14%

The analyst assigned to this project knows that the firm has common stock outstanding and that investors who hold the company’s stock expect a 14% return on their investment. The analyst decides that the firm should not undertake this investment because it only produces a 12% return while the company’s shareholders expect a 14% return.

In this example, each analyst is making a mistake by focusing on one source of financing rather than on the overall financing mix. What if instead the analysts used a combined cost of financing? By weighting the cost of each source of financing by its relative proportion in the firm’s target capital structure, the firm can obtain a weighted average cost of capital. Assuming that this firm desires a 50–50 mix of debt and equity, the weighted average cost here would be 10%[(0.50 × 6% debt) + (0.50 × 14% equity)]. With this average cost of financing, the firm should reject the first opportunity (7% expected return < 10% weighted average cost) and accept the second (12% expected return > 10% weighted average cost).

REVIEW QUESTIONS

net proceeds

Funds actually received by the firm from the sale of a security.

flotation costs

The total costs of issuing and selling a security.

Example 9.2

Duchess Corporation, a major hardware manufacturer, is contemplating selling $10 million worth of 20-year, 9% coupon (stated annual interest rate) bonds, each with a par value of $1,000. Because bonds with similar risk earn returns greater than 9%, the firm must sell the bonds for $980 to compensate for the lower coupon interest rate. The flotation costs are 2% of the par value of the bond (0.02 × $1,000), or $20. The net proceeds to the firm from the sale of each bond are therefore $960 ($980 minus $20).

Using Market Quotations

A relatively quick method for finding the before-tax cost of debt is to observe the yield to maturity (YTM) on the firm’s existing bonds or bonds of similar risk issued by other companies. The YTM of existing bonds reflects the rate of return required by the market. For example, if the market requires a YTM of 9.7 percent for a similar-risk bond, this value can be used as the before-tax cost of debt, r_d, for new bonds. Bond yields are widely reported by sources such as the Wall Street Journal.

Calculating the Cost

This approach finds the before-tax cost of debt by calculating the YTM generated by the bond’s cash flows, given the net proceeds that the firm receives when it issues the bonds. From the issuer’s point of view, this value is the cost to maturity of the cash flows associated with the debt. The YTM can be calculated by using a financial calculator or an electronic spreadsheet. It represents the annual before-tax percentage cost of the debt.

Example 9.3

In the preceding example, $960 were the net proceeds of a 20-year bond with a $1,000 par value and 9% coupon interest rate. The calculation of the annual cost is quite simple. The cash flow pattern associated with this bond’s sales consists of an initial inflow (the net proceeds) followed by a series of annual outlays (the interest payments). In the final year, when the debt is retired, an outlay representing the repayment of the principal also occurs. The cash flows associated with Duchess Corporation’s bond issue are as follows:

End of year(s)	Cash flow
0	$  960
1–20	−$   90
20	− $1,000

The initial $960 inflow is followed by annual interest outflows of $90 (9% coupon interest rate × $1,000 par value) over the 20-year life of the bond. In year 20, an outflow of $1,000 (the repayment of the principal) occurs. We can determine the cost of debt by finding the YTM, which is the discount rate that equates the present value of the bond outflows to the initial inflow.

My Finance Lab Financial Calculator

Calculator use (Note: Most calculators require either the present value [net proceeds] or the future value [annual interest payments and repayment of principal] to be input as negative numbers when we calculate yield to maturity. That approach is used here.) Using the calculator and the inputs shown at the left, you should find the before-tax cost of debt (yield to maturity) to be 9.452%.

Spreadsheet use The before-tax cost of debt on the Duchess Corporation bond can be calculated using an Excel spreadsheet. The following Excel spreadsheet shows that by referencing the cells containing the bond’s net proceeds, coupon payment, years to maturity, and par value as part of Excel’s RATE function, you can quickly determine that the appropriate before-tax cost of debt for Duchess Corporation’s bond is 9.452%.

Although you may not recognize it, both the calculator and the Excel function are using trial-and-error to find the bond’s YTM, they just do it faster than you can.

Approximating the Cost

Although not as precise as using a calculator, there is a method for quickly approximating the before-tax cost of debt. The before-tax cost of debt, r_d, for a bond with a $1,000 par value can be approximated by

(9.1)

Where

Example 9.4

Substituting the appropriate values from the Duchess Corporation example into the approximation formula given in Equation 9.1, we get

This approximate value of before-tax cost of debt is close to the 9.452%, but it lacks the precision of the value derived using the calculator or spreadsheet.

Example 9.5

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Duchess Corporation has a 40% tax rate. Using the 9.452% before-tax debt cost calculated above and applying Equation 9.2, we find an after-tax cost of debt of 5.67%[9.452% × (1 − 0.40)]. Typically, the cost of long-term debt for a given firm is less than the cost of preferred or common stock, partly because of the tax deductibility of interest.

Personal Finance Example 9.6

My Finance Lab Solution Video

Kait and Kasim Sullivan, a married couple in the 28% federal income-tax bracket, wish to borrow $60,000 to pay for a new luxury car. To finance the purchase, they can either borrow the $60,000 through the auto dealer at an annual interest rate of 6.0%, or they can take a $60,000 second mortgage on their home. The best annual rate they can get on the second mortgage is 7.2%. They already have qualified for both of the loans being considered.

If they borrow from the auto dealer, the interest on this “consumer loan” will not be deductible for federal tax purposes. However, the interest on the second mortgage would be tax deductible because the tax law allows individuals to deduct interest paid on a home mortgage. To choose the least-cost financing, the Sullivans calculated the after-tax cost of both sources of long-term debt. Because interest on the auto loan is not tax deductible, its after-tax cost equals its stated cost of 6.0%. Because the interest on the second mortgage is tax deductible, its after-tax cost can be found using Equation 9.2:

Because the 5.2% after-tax cost of the second mortgage is less than the 6.0% cost of the auto loan, the Sullivans may decide to use the second mortgage to finance the auto purchase.

REVIEW QUESTIONS

EXCEL REVIEW QUESTION

My Finance Lab

cost of preferred stock, r_p

The ratio of the preferred stock dividend to the firm’s net proceeds from the sale of preferred stock.

Example 9.7

Duchess Corporation is contemplating issuance of a 10% preferred stock that they expect to sell for $87 per share. The cost of issuing and selling the stock will be $5 per share. The first step in finding the cost of the stock is to calculate the dollar amount of the annual preferred dividend, which is $8.70 (0.10 × $87). The net proceeds per share from the proposed sale of stock equals the sale price minus the flotation costs ($87 − $5 = $82). Substituting the annual dividend, D_p, of $8.70 and the net proceeds, N_p, of $82 into Equation 9.3 gives the cost of preferred stock, 10.6% ($8.70 ÷ $82).

REVIEW QUESTION

cost of common stock equity, r_s

The rate at which investors discount the expected dividends of the firm to determine its share value.

Using the Constant-Growth Valuation (Gordon Growth) Model

In Chapter 7, we found the value of a share of stock to be equal to the present value of all future dividends, which in one model are assumed to grow at a constant annual rate over an infinite time horizon. This model, the constant-growth valuation model, is also known as the Gordon growth model. The key expression derived for this model, first presented as Equation 7.4, is

(9.4)

where

Solving Equation 9.4 for r_s results in the following expression for the cost of common stock equity:

(9.5)

Equation 9.5 indicates that the cost of common stock equity can be found by dividing the dividend expected at the end of year 1 by the current market price of the stock (the “dividend yield”) and adding the expected growth rate (the “capital gains yield”).

Example 9.8

Duchess Corporation wishes to determine its cost of common stock equity, r_s. The market price, P₀, of its common stock is $50 per share. The firm expects to pay a dividend, D₁, of $4 at the end of the coming year, 2016. The dividends paid on the outstanding stock over the past 6 years (2010 through 2015) were as follows:

Year	Dividend
2015	$3.80
2014	3.62
2013	3.47
2012	3.33
2011	3.12
2010	2.97

Using a financial calculator or electronic spreadsheet, in conjunction with the technique described for finding growth rates in Chapter 5, we can calculate the annual rate at which dividends have grown, g, from 2010 to 2015. It turns out to be approximately 5% (more precisely, it is 5.05%). Substituting D₁ = $4, P₀ = $50, and g = 5% into Equation 9.5 yields the cost of common stock equity:

The 13.0% cost of common stock equity represents the return required by existing shareholders on their investment. If the actual return is less than that, shareholders are likely to begin selling their stock.

Using the Capital Asset Pricing Model (CAPM)

Recall from Chapter 8 that the capital asset pricing model (CAPM) describes the relationship between the required return, r_s, and the nondiversifiable risk of the firm as measured by the beta coefficient, β. The basic CAPM is

where

Using the CAPM indicates that the cost of common stock equity is the return required by investors as compensation for the firm’s nondiversifiable risk, measured by beta.

Comparing Constant-Growth and CAPM Techniques

The CAPM technique differs from the constant-growth valuation model in that it directly considers the firm’s risk, as reflected by beta, in determining the required return or cost of common stock equity. The constant-growth model does not look at risk; it uses the market price, P₀, as a reflection of the expected risk–return preference of investors in the marketplace. The constant-growth valuation and CAPM techniques for finding r_s are theoretically equivalent, although in practice estimates from the two methods do not always agree. The two methods can produce different estimates because they require (as inputs) estimates of other quantities, such as the expected dividend growth rate or the firm’s beta.

Another difference is that when the constant-growth valuation model is used to find the cost of common stock equity, it can easily be adjusted for flotation costs to find the cost of new common stock; the CAPM does not provide a simple adjustment mechanism. The difficulty in adjusting the cost of common stock equity calculated by using the CAPM occurs because in its common form the model does not include the market price, P₀, a variable needed to make such an adjustment. Although the CAPM has a stronger theoretical foundation, the computational appeal of the traditional constant-growth valuation model justifies its use throughout this text to measure financing costs of common stock. As a practical matter, analysts might want to estimate the cost of equity using both approaches and then take an average of the results to arrive at a final estimate of the cost of equity.

cost of retained earnings, r_r

The same as the cost of an equivalent fully subscribed issue of additional common stock, which is equal to the cost of common stock equity, r_s.

Example 9.10

The cost of retained earnings for Duchess Corporation was actually calculated in the preceding examples: It is equal to the cost of common stock equity. Thus, r_r equals 13.0%. As we will show in the next section, the cost of retained earnings is always lower than the cost of a new issue of common stock because it entails no flotation costs.

Matter of fact

Retained Earnings, the Preferred Source of Financing

In the United States and most other countries, firms rely more heavily on retained earnings than any other financing source. For example, a 2013 survey of Chinese firms found that 64% of the companies surveyed listed retained earnings as one of their primary sources of funds. Bank loans were a distant second choice, mentioned as a primary source of funds by just 44% of the companies.¹

cost of a new issue of common stock, r_n

The cost of common stock, net of underpricing and associated flotation costs.

underpriced

Stock sold at a price below its current market price, P₀.

Example 9.11

In the constant-growth valuation example, we found Duchess Corporation’s cost of common stock equity, r_s, to be 13%, using the following values: an expected dividend, D₁, of $4; a current market price, P₀, of $50; and an expected growth rate of dividends, g, of 5%.

To determine its cost of new common stock, r_n, Duchess Corporation has estimated that on average, new shares can be sold for $47. The $3-per-share underpricing is due to the competitive nature of the market. A second cost associated with a new issue is flotation costs of $2.50 per share that would be paid to issue and sell the new shares. The total underpricing and flotation costs per share are therefore $5.50.

Subtracting the $5.50-per-share underpricing and flotation cost from the current $50 share price results in expected net proceeds of $44.50 per share ($50.00 minus $5.50). Substituting D₁ = $4, N_n = $44.50, and g = 5% into Equation 9.8 results in a cost of new common stock, r_n:

Duchess Corporation’s cost of new common stock is therefore 14.0%. That is the value to be used in subsequent calculations of the firm’s overall cost of capital.

REVIEW QUESTIONS

Example 9.12

In earlier examples, we found the costs of the various types of capital for Duchess Corporation to be as follows:

The company uses the following weights in calculating its weighted average cost of capital:

Because the firm expects to have a sizable amount of retained earnings available ($300,000), it plans to use its cost of retained earnings, r_r, as the cost of common stock equity. Duchess Corporation’s weighted average cost of capital is calculated in Table 9.1. The resulting weighted average cost of capital for Duchess is 9.8%. Assuming an unchanged risk level, the firm should accept all projects that will earn a return greater than 9.8%.

in practice focus on PRACTICE: Uncertain Times Make for an Uncertain Weighted Average Cost of Capital

As U.S. financial markets experienced and recovered from the 2008 financial crisis and 2009 “great recession,” firms struggled to keep track of their weighted average cost of capital. The individual component costs were moving rapidly in response to the financial market turmoil. Volatile financial markets can make otherwise manageable cost-of-capital calculations exceedingly complex and inherently error prone, possibly wreaking havoc with investment decisions. If a firm underestimates its cost of capital, it risks making investments that are not economically justified, and if a firm overestimates its financing costs, it risks foregoing value-maximizing investments.

Although the WACC computation does not change when markets become unstable, the uncertainty surrounding the components that comprise the WACC increases dramatically. The financial crisis pushed credit costs to a point where long-term debt was largely inaccessible, and the great recession saw Treasury bond yields fall to historic lows, making cost of equity projections appear unreasonably low. With these key components in flux, it is exceedingly difficult, if not impossible, for firms to get a handle on a cost of long-term capital.

According to CFO Magazine, at least one firm resorted to a two-pronged approach for determining its cost of capital during the uncertain times. Ron Domanico is the chief financial officer (CFO) at Caraustar Industries, Inc., and he reported that his company dealt with the cost-of-capital uncertainty by abandoning the conventional one-size-fits-all approach. “In the past, we had one cost of capital that we applied to all our investment decisions … today that’s not the case. We have a short-term cost of capital we apply to short-term opportunities, and a longer-term cost of capital we apply to longer-term opportunities … and the reality is that the longer-term cost is so high that it has forced us to focus only on those projects that have immediate returns,” Mr. Domanico is quoted saying.^a

Part of Caraustar’s motivation for implementing this two-pronged approach was to account for the excessively large spread between short- and long-term debt rates that emerged during the financial market crisis. Mr. Domanico reported that during the crisis Caraustar could borrow short-term funds at the lower of Prime plus 4 percent or LIBOR plus 5 percent, where either rate was reasonable for making short-term investment decisions. Alternatively, long-term investment decisions were being required to clear Caraustar’s long-term cost-of-capital calculation accounting for borrowing rates in excess of 12 percent.

Why don’t firms generally use both short- and long-run weighted average costs of capital?

Book Value versus Market Value

Book value weights use accounting values to measure the proportion of each type of capital in the firm’s financial structure. Market value weights measure the proportion of each type of capital at its market value. Market value weights are appealing because the market values of securities closely approximate the actual dollars to be received from their sale. Moreover, because firms calculate the costs of the various types of capital by using prevailing market prices, it seems reasonable to use market value weights. In addition, the long-term investment cash flows to which the cost of capital is applied are estimated in terms of current as well as future market values. Market value weights are clearly preferred over book value weights.

Historical versus Target Weights

Historical weights can be either book or market value weights based on actual capital structure proportions. For example, past or current book value proportions would constitute a form of historical weighting, as would past or current market value proportions. Such a weighting scheme would therefore be based on real—rather than desired—proportions.

Target weights, which can also be based on either book or market values, reflect the firm’s desired capital structure proportions. Firms using target weights establish such proportions on the basis of the “optimal” capital structure they wish to achieve. (The development of these proportions and the optimal structure are discussed in detail in Chapter 13.)

When one considers the somewhat approximate nature of the calculation of weighted average cost of capital, the choice of weights may not be critical. However, from a strictly theoretical point of view, the preferred weighting scheme is target market value proportions, and we assume this scheme throughout this chapter.

Personal Finance Example 9.13

Chuck Solis currently has three loans outstanding, all of which mature in exactly 6 years and can be repaid without penalty any time prior to maturity. The outstanding balances and annual interest rates on these loans are as follows:

Loan	Outstanding balance	Annual interest rate
1	$26,000	9.6%
2	9,000	10.6
3	45,000	7.4

After a thorough search, Chuck found a lender who would loan him $80,000 for 6 years at an annual interest rate of 9.2% on the condition that the loan proceeds be used to fully repay the three outstanding loans, which combined have an outstanding balance of $80,000 ($26,000 + $9,000 + $45,000).

Chuck wishes to choose the least costly alternative: (1) to do nothing or (2) to borrow the $80,000 and pay off all three loans. He calculates the weighted average cost of his current debt by weighting each debt’s annual interest cost by the proportion of the $80,000 total it represents and then summing the three weighted values as follows:

Given that the weighted average cost of the $80,000 of current debt of 8.5% is below the 9.2% cost of the new $80,000 loan, Chuck should do nothing and just continue to pay off the three loans as originally scheduled.