Financing and Valuation

In Chapters 5 and 6, we showed how to value a capital investment project by a four-step procedure:

1. Forecast after-tax cash flows, assuming all-equity financing.

2. Assess the project’s risk.

3. Estimate the opportunity cost of capital.

4. Calculate NPV, using the opportunity cost of capital as the discount rate.

There’s nothing wrong with this procedure, but now we’re going to extend it to include value contributed by financing decisions. There are two ways to do this:

1. Adjust the discount rate. To take account of the interest tax shields, companies usually adjust the discount rate downward. They do this by calculating an after-tax weighted average cost of capital (WACC). We introduced the after-tax WACC in Chapters 9 and 17, but here we provide a lot more guidance on how it is calculated and used.

2. Adjust the present value. In this case, you start by estimating the base-case value of the firm or project, assuming it is all-equity-financed, and then adjust this base-case value to account for financing.

Adjusted present value (APV)

= base-case value + value of financing side effects

Once you identify and value the financing side effects, calculating APV is no more than addition or subtraction.

This is a how-to-do-it chapter. In the first section, we explain and derive the after-tax WACC and use it to value a project and business. Then in Section 19-2, we work through a more complex and realistic valuation problem. Section 19-3 covers some tricks of the trade: helpful hints on how to estimate inputs and on how to adjust WACC when business risk or capital structure changes. Section 19-4 turns to the APV method. The idea behind APV is simple enough, but tracing through all the financing side effects can be tricky. We conclude the chapter with a question-and-answer section designed to clarify points that managers and students often find confusing. The Appendix covers an important special case—namely, the after-tax valuation of safe cash flows.

This chapter contains several extended numerical examples. To keep things simple, we will follow current U.S. tax law in all of them, using the 21% tax rate and 100% bonus depreciation implemented in 2018. You can explore the effects of different tax rates and depreciation schedules when you tackle the end-of-chapter problems.

19-1The After-Tax Weighted-Average Cost of Capital

We first addressed problems of valuation and capital budgeting in Chapters 5 and 6. In those early chapters, we said hardly a word about financing decisions. We separated investment from financing decisions. If the investment project was positive-NPV, we assumed that the firm would go ahead, without asking whether financing the project would add or subtract additional value. We were really assuming a Modigliani–Miller (MM) world in which all financing decisions are irrelevant. In a strict MM world, firms can analyze real investments as if they are all-equity-financed; the actual financing plan is a mere detail to be worked out later.

Under MM assumptions, decisions to spend money can be separated from decisions to raise money. Now we reconsider the capital budgeting decision when investment and financing decisions interact and cannot be wholly separated.

One reason financing and investment decisions interact is taxes. Interest is a tax-deductible expense. Think back to Chapters 9 and 17, where we introduced the after-tax weighted-average cost of capital:

Here D and E are the market values of the firm’s debt and equity, V = D + E is the total market value of the firm, r_D and r_E are the costs of debt and equity, respectively, and T_c is the marginal corporate tax rate.

Notice that the WACC formula uses the after-tax cost of debt r_D(1 – T_c). That is how the after-tax WACC captures the value of interest tax shields. Notice too that all the variables in the WACC formula refer to the firm as a whole. As a result, the formula gives the right discount rate only for projects that are just like the firm undertaking them. The formula works for the “average” project. It is incorrect for projects that are safer or riskier than the average of the firm’s existing assets. It is incorrect for projects whose acceptance would lead to an increase or decrease in the firm’s target debt ratio.

The WACC is based on the firm’s current characteristics, but managers use it to discount future cash flows. That’s fine as long as the firm’s business risk and debt ratio are expected to remain constant, but when the business risk and debt ratio are expected to change, discounting cash flows by the WACC is just approximately correct.

Many firms set a single, companywide WACC and update it only if there are major changes in risk and interest rates. The WACC is a common reference point that avoids divisional squabbles about discount rates.¹ But all financial managers need to know how to adjust WACC when business risks and financing assumptions change. We show how to make these adjustments later in this chapter.

EXAMPLE 19.1 Calculating Sangria’s WACC

Sangria is a U.S.-based company whose products aim to promote happy, low-stress lifestyles. Let’s calculate Sangria’s WACC. Its book and market-value balance sheets are

Sangria Corporation (Book Values, $ millions)
Asset value	$1,000			$ 500	Debt
				500	Equity
	$1,000			$ 1,000

Sangria Corporation (Market Values, $ millions)
Asset value	$1,250			$ 500	Debt
				750	Equity
	$1,250			$ 1,250

We calculated the market value of equity on Sangria’s balance sheet by multiplying its current stock price ($7.50) by 100 million, the number of its outstanding shares. The company’s future prospects are good, so the stock is trading above book value ($7.50 vs. $5.00 per share). However, interest rates have been stable since the firm’s debt was issued, so the book and market values of debt are, in this case, equal.

Sangria’s cost of debt (the market interest rate on its existing debt and on any new borrowing)² is 6%. Its cost of equity (the expected rate of return demanded by investors in Sangria’s stock) is 12.5%.

The market-value balance sheet shows assets worth $1,250 million. Of course, we can’t observe this value directly because the assets themselves are not traded. But we know what they are worth to debt and equity investors ($500 + 750 = $1,250 million). This value is entered on the left of the market-value balance sheet.

Why did we show the book balance sheet? Only so you could draw a big X through it. Do so now.

Think of the WACC as the expected rate of return on a portfolio of the firm’s outstanding debt and equity. The portfolio weights depend on market values. The expected rate of return on the market-value portfolio reveals the expected rate of return demanded by investors for committing their hard-earned money to the firm’s assets and operations.

When estimating the weighted-average cost of capital, you are not interested in past investments but in current values and expectations for the future. Sangria’s true debt ratio is not 50%, the book ratio, but 40% because its assets are worth $1,250 million. The cost of equity, r_E = .125, is the expected rate of return from purchase of stock at $7.50 per share, the current market price. It is not the return on book value per share. You can’t buy shares in Sangria for $5 anymore.

Sangria is consistently profitable and pays taxes at the marginal rate of 21%.³ This tax rate is the final input for Sangria’s WACC. The inputs are summarized here:

Cost of debt (rD)	0.06
Cost of equity (rE)	0.125
Marginal tax rate (Tc)	0.21
Debt ratio (D/V)	500/1,250 = 0.4
Equity ratio (E/V)	750/1,250 = 0.6

The company’s after-tax WACC is

WACC = .06 × (1 − .21) × .4 + .125 × .6 = .094, or 9.4%

That’s how you calculate the weighted-average cost of capital. Now let’s see how Sangria would use it.

EXAMPLE 19.2 Using Sangria’s WACC to Value a Project

Sangria’s enologists have proposed investing $12.5 million in the construction of a perpetual crushing machine, which (conveniently for us) never depreciates and generates a perpetual stream of earnings and cash flow of $1.487 million per year pretax. The project is average risk, so we can use WACC. The after-tax cash flow is:

Pretax cash flow	$1.487 million
Tax at 21%	0.312
After-tax cash flow	C = $1.175 million

Note: This after-tax cash flow takes no account of interest tax shields on debt supported by the perpetual crusher project. As we explained in Chapter 6, standard capital budgeting practice separates investment from financing decisions and calculates after-tax cash flows as if the project were all-equity-financed. However, the interest tax shields will not be ignored: We are about to discount the project’s cash flows by Sangria’s WACC, in which the cost of debt is entered after tax. The value of interest tax shields is picked up not as higher after-tax cash flows, but in a lower discount rate.

The crusher generates a perpetual after-tax cash flow of C = $1.175 million, so NPV is

NPV = 0 means a barely acceptable investment. The annual cash flow of $1.175 million per year amounts to a 9.4% rate of return on investment (1.175/12.5 = .094), exactly equal to Sangria’s WACC.

If project NPV is exactly zero, the return to equity investors must exactly equal the cost of equity, 12.5%. Let’s confirm that Sangria shareholders can actually look forward to a 12.5% return on their investment in the perpetual crusher project.

Suppose Sangria sets up this project as a mini-firm. Its market-value balance sheet looks like this:

Perpetual Crusher (Market Values, $ millions)
Asset value	$12.5			$ 5.0	Debt
				7.5	Equity
	$12.5			$12.5

Calculate the expected dollar return to shareholders:

After-tax interest = r_D(1 − T_c )D = .06 × (1 − .21) × 5 = .237

Expected equity income = C − r_D(1 − T_c )D = 1.175 − .237 = .938

The project’s earnings are level and perpetual, so the expected rate of return on equity is equal to the expected equity income divided by the equity value:

The expected return on equity equals the cost of equity, so it makes sense that the project’s NPV is zero.

Review of Assumptions

It is appropriate to discount the perpetual crusher’s cash flows at Sangria’s WACC only if

· The project’s business risks are the same as those of Sangria’s other assets and remain so for the life of the project.

· Throughout its life, the project supports the same fraction of debt to value as in Sangria’s overall capital structure.

You can see the importance of these two assumptions: If the perpetual crusher had greater business risk than Sangria’s other assets, or if the acceptance of the project would lead to a permanent, material change in Sangria’s debt ratio, then Sangria’s shareholders would not be content with a 12.5% expected return on their equity investment in the project.

But users of WACC need not worry about small or temporary fluctuations in debt ratios. Nor should they be misled by the immediate source of financing. Suppose that Sangria decides to borrow $12.5 million to get a quick start on construction of the crusher. This does not necessarily change Sangria’s long-term financing targets. The crusher’s debt capacity is only $5 million. If Sangria decides for convenience to borrow $12.5 million for the crusher, then sooner or later it will have to borrow $12.5 – $5 = $7.5 million less for other projects.

We have illustrated the WACC formula only for a project offering perpetual cash flows. But the formula works for any cash-flow pattern as long as the firm adjusts its borrowing to maintain a constant debt ratio over time.⁴ When the firm departs from this borrowing policy, WACC is only approximately correct.

Mistakes People Make in Using the Weighted-Average Formula

The weighted-average formula is very useful, but it is also dangerous. It tempts people to make logical errors. For example, manager Q, who is campaigning for a pet project, might look at the formula

and think, “Aha! My firm has a good credit rating. It could borrow, say, 90% of the project’s cost if it likes. That means D/V = .9 and E/V = .1. My firm’s borrowing rate r_D is 8%, and the required return on equity, r_E, is 15%. The tax rate is now 21%. Therefore,

WACC = .08(1 − .21)(.9) + .15(.1) = .072

or 7.2%. When I discount at that rate, my project looks great.”

Manager Q is wrong on several counts. First, the weighted-average formula works only for projects that are carbon copies of the firm. The firm isn’t 90% debt-financed.

Second, the immediate source of funds for a project has no necessary connection with the hurdle rate for the project. What matters is the project’s overall contribution to the firm’s borrowing power. A dollar invested in Q’s pet project will not increase the firm’s debt capacity by $.90. If the firm borrows 90% of the project’s cost, it is really borrowing in part against its existing assets. Any advantage from financing the new project with more debt than normal should be attributed to the old projects, not to the new one.

Third, even if the firm were willing and able to lever up to 90% debt, its cost of capital would not decline to 7.2%, as Q’s naive calculation predicts. You cannot increase the debt ratio without creating financial risk for stockholders and thereby increasing r_E, the expected rate of return they demand from the firm’s common stock. Going to 90% debt would certainly increase the borrowing rate, too.

19-2Valuing Businesses

On most workdays, the financial manager concentrates on valuing projects, arranging financing, and helping to run the firm more effectively. The valuation of the business as a whole is left to investors and financial markets. But on some days, the financial manager has to take a stand on what an entire business is worth. When this happens, a big decision is typically in the offing. For example:

· If firm A is about to make a takeover offer for firm B, then A’s financial managers have to decide how much the combined business A + B is worth under A’s management. This task is particularly difficult if B is a private company with no observable share price.

· If firm C is considering the sale of one of its divisions, it has to decide what the division is worth in order to negotiate with potential buyers.

· When a firm goes public, the investment bank must evaluate how much the firm is worth in order to set the issue price.

· If a mutual fund owns shares in a company that is not traded, then the fund’s directors are obliged to estimate a fair value for those shares. If the directors do a sloppy job of coming up with a value, they are liable to find themselves in court.

In addition, thousands of analysts in stockbrokers’ offices and investment firms spend every workday burrowing away in the hope of finding undervalued firms. Many of these analysts use the valuation tools we are about to cover.

In Chapter 4, we took a first pass at valuing free cash flows from an entire business. We assumed then that the business was financed solely by equity. Now we will show how WACC can be used to value a company that is financed by a mixture of debt and equity. You just treat the company as if it were one big project. You forecast the company’s free cash flows (the hardest part of the exercise) and discount back to present value. But be sure to remember three important points:

1. If you discount at WACC, cash flows have to be projected just as you would for a capital investment project. Do not deduct interest. Calculate taxes as if the company were all-equity-financed. (The value of interest tax shields is not ignored, because the after-tax cost of debt is used in the WACC formula.)

2. Unlike most projects, companies are potentially immortal. But that does not mean that you need to forecast every year’s cash flow from now to eternity. Financial managers usually forecast to a medium-term horizon and add a terminal value to the cash flows in the horizon year. The terminal value is the present value at the horizon of all subsequent cash flows. Estimating the terminal value requires careful attention because it often accounts for the majority of the company’s value.

3. Discounting at WACC values the assets and operations of the company. If the object is to value the company’s equity, that is, its common stock, don’t forget to subtract the value of the company’s outstanding debt.

Here’s an example.

Valuing Rio Corporation

Sangria is tempted to acquire the Rio Corporation, which is also in the business of promoting relaxed, happy lifestyles. Rio has developed a special weight-loss program called the Brazil Diet, based on barbecues, red wine, and sunshine. The firm guarantees that within three months you will have a figure that will allow you to fit right in at Ipanema or Copacabana beach in Rio de Janeiro. But before you head for the beach, you’ve got the job of working out how much Sangria should pay for Rio.

Rio is a U.S. company. It is privately held, so Sangria has no stock market price to rely on. Rio is in the same line of business as Sangria, so we will assume that it has the same business risk as Sangria, and, like Sangria, its debt capacity is 40% of firm value. Therefore, we can use Sangria’s WACC.

Your first task is to forecast Rio’s free cash flow (FCF). Free cash flow is the amount of cash that the firm can pay out to investors after making all investments necessary for growth. Free cash flow is calculated assuming the firm is all-equity-financed. Discounting the free cash flows at the after-tax WACC gives the total value of Rio (debt plus equity). To find the value of its equity, you will need to subtract the 40% of the firm that can be financed with debt.

We will forecast each year’s free cash flow out to a valuation horizon (H) and predict the business’s value at that horizon (PV_H). The cash flows and horizon value are then discounted back to the present:

Of course, the business will continue after the horizon, but it’s not practical to forecast free cash flow year by year to infinity. PV_H stands in for the value in year H of free cash flow in periods H + 1, H + 2, etc.

Free cash flow and net income are not the same. They differ in several important ways:

· Income is the return to shareholders, calculated after interest expense. Free cash flow is calculated before interest.

· Income is calculated after various noncash expenses, including depreciation. Therefore, we will add back depreciation when we calculate free cash flow.

· Capital expenditures and investments in working capital do not appear as expenses on the income statement, but they do reduce free cash flow.

Free cash flow can be negative for rapidly growing firms, even if the firms are profitable, because investment exceeds cash flow from operations. Negative free cash flow is normally temporary, fortunately for the firm and its stockholders. Free cash flow turns positive as growth slows down and the payoffs from prior investments start to roll in.

Table 19.1 sets out the information that you need to forecast Rio’s free cash flows. We will follow common practice and start with a projection of sales. In the year just ended, Rio had sales of $83.6 million. In recent years, sales have grown between 5% and 8% a year. You forecast that sales will grow about 7% a year for the next three years. Growth will then slow to 4% for years 4 to 6 and to 3% starting in year 7.

TABLE 19.1 Free-cash-flow projections and company value for Rio Corporation ($ millions)

Note: Columns may not add exactly because of rounding.

The other components of cash flow in Table 19.1 are driven by these sales forecasts. For example, you can see that costs are forecasted at 74% of sales in the first year with a gradual increase to 76.5% of sales in later years, reflecting increased marketing costs as Rio’s competitors gradually catch up.

Increasing sales are likely to require further investment in fixed assets and working capital. Rio’s net fixed assets are currently about $.79 for each dollar of sales. Unless Rio has surplus capacity or can squeeze more output from its existing plant and equipment, its investment in fixed assets will need to grow along with sales. Therefore, we assume that every dollar of sales growth requires an increase of $.79 in net fixed assets. We also assume that working capital grows in proportion to sales.

BEYOND THE PAGE

Try It! Rio’s spreadsheet

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Rio’s free cash flow is calculated in Table 19.1 as profit after tax, plus depreciation,⁵ minus investment. Investment is the change in the stock of (gross) fixed assets and working capital from the previous year. For example, in year 1:

Estimating Horizon Value

We will forecast cash flows for each of the first six years. After that, Rio’s sales are expected to settle down to stable, long-term growth starting in year 7. To find the present value of the cash flows in years 1 to 6, we discount at the 9.4% WACC:

Now we need to find the value of the cash flows from year 7 onward. In Chapter 4, we looked at several ways to estimate horizon value. Here we will use the constant-growth DCF formula. This requires a forecast of the free cash flow for year 7, which we have worked out in the final column of Table 19.1, assuming a long-run growth rate of 3% per year.⁶ The free cash flow is $8.5 million, so

We now have all we need to value the business:

This is the total value of Rio. To find the value of the equity, we simply subtract the 40% of firm value that will be financed with debt:

Value of debt = .40 × 106.4 = $42.6 million

Total value of equity = $106.4 − 42.6 = $63.8 million

If Rio has 1.5 million shares outstanding, its value per share is:

Value per share = 63.8/1.5 = $42.53

Thus, Sangria could afford to pay up to about $42 per share for Rio.

You now have an estimate of the value of Rio Corporation. But how confident can you be in this figure? Notice that only about a quarter of Rio’s value comes from cash flows in the first six years. The rest comes from the horizon value. Moreover, this horizon value can change in response to minor changes in assumptions. For example, if the long-run growth rate is 4% rather than 3%, firm value increases from $106.4 million to $110.5 million.

Thus, faster growth increases Rio’s horizon value and PV(company). At this point, we must check the two warnings from the concatenator valuation example in Chapter 4. Did we account for the extra investment required to support the faster long-run growth? Yes. Growth at 4% instead of 3% increases year 7’s investment in fixed assets from $15.9 to $16.9 million and investment in working capital from $0.4 to $0.6 million. (To confirm this, go to the Rio Spreadsheet Beyond the Page and change the long-run growth rate.) Did we assume that Rio can earn more that its cost of capital in perpetuity? Yes, because the increased investment in year 7 and after added NPV. In other words, the horizon value contains positive PVGO, the present value of growth opportunities.

Will competition eliminate the PVGO? The financial manager will have to think hard about the competitive landscape. Perhaps he or she will decide that the long-run cost forecast at 76.5% of sales is too optimistic.

The financial manager will probably also look at the values that investors place on comparable listed companies. For example, suppose that similar lifestyle companies commonly trade at a ratio of company value to EBITDA of 4.8. Then Sangria’s manager might judge that Rio’s horizon value is 4.8 × $27.9 million = $133.9 million in year 6 and $78.1 million discounted to year 0. This would suggest that Rio is currently worth $29.0 + 78.1 = $107.1 million, marginally higher than our initial DCF estimate. The manager might also look at the market-to-book ratio for comparable businesses and calculate what Rio would be worth if it sold at a similar ratio.

Financial managers should also check whether a business is worth more dead than alive. Sometimes a company’s liquidation value exceeds its value as a going concern. Sometimes financial analysts can ferret out idle or underexploited assets that would be worth much more if sold to someone else. Such assets would be valued at their likely sale price and the rest of the business valued without them.

WACC vs. the Flow-to-Equity Method

When valuing Rio, we forecasted the cash flows assuming all-equity financing and we used the WACC to discount these cash flows. The WACC formula picked up the value of the interest tax shields. Then, to find the equity value, we subtracted the value of debt from the total value of the firm.

If our task is to value a firm’s equity, there’s an obvious alternative to discounting the total cash flows at the firm’s WACC: Discount cash flows to equity after interest and after taxes at the cost of equity capital. This is called the flow-to-equity method. If the company’s debt ratio is constant over time, the flow-to-equity method should give the same answer as discounting total cash flows at the WACC and then subtracting the value of the debt.

Suppose that you are asked to value Rio by the flow-to-equity method, assuming that the company adjusts its debt each year to maintain a constant debt ratio. You are given as a starter an estimate of Rio’s horizon value at the end of year 6. Perhaps this value was obtained by discounting subsequent cash flows by Rio’s WACC or perhaps it was estimated by looking at how investors value comparable, publicly traded companies. You decide to expand the spreadsheet in Table 19.1 by calculating each year’s interest payments and issues or repayments of debt. You recompute taxes, recognizing that the interest payments are a tax-deductible expense. Finally, you discount the free cash flow to equity at the cost of equity, which in our example is r_E = 12.5%.

It sounds straightforward, but in practice, it can be tricky to do it right. The problem arises because each year’s interest payment depends on the amount of debt at the start of the year, and this depends in turn on Rio’s value at the start of the year (remember Rio’s debt is a constant proportion of value). So you seem to have a catch-22 situation in which you first need to know Rio’s value each year before you can go on to calculate and discount the cash flows to equity. Fortunately, a simple formula allows you to solve simultaneously for the company’s value and the cash flow in each year. We won’t get into that here, but if you would like to see how the flow-to-equity method can be used to value Rio, click on the nearby Beyond the Page feature to access the worked example.

BEYOND THE PAGE

Try It! Cash- flow-to-equity model

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19-3Using WACC in Practice

Some Tricks of the Trade

Sangria had just one asset and two sources of financing. A real company’s market-value balance sheet has many more entries, for example:⁷

Current assets, including cash, inventory, and accounts receivable			Current liabilities, including accounts payable and short-term debt
Property, plant, and equipment			Long-term debt (D)
			Preferred stock (P)
Growth opportunities			Equity (E)
Total assets			Total liabilities plus equity

Several questions immediately arise:

How does the formula change when there are more than two sources of financing? Easy: There is one cost for each element. The weight for each element is proportional to its market value. For example, if the capital structure includes both preferred and common shares,

where r_P is investors’ expected rate of return on the preferred stock, P is the amount of preferred stock outstanding, and V = D + P + E.

What about short-term debt? Many companies consider only long-term financing when calculating WACC. They leave out the cost of short-term debt. In principle, this is incorrect. The lenders who hold short-term debt are investors who can claim their share of operating earnings. A company that ignores this claim will misstate the required return on capital investments.

But “zeroing out” short-term debt is not a serious error if the debt is only temporary, seasonal, or incidental financing or if it is offset by holdings of cash and marketable securities. Suppose, for example, that one of your foreign subsidiaries takes out a six-month loan to finance its inventory and accounts receivable. The dollar equivalent of this loan will show up as a short-term debt. At the same time, headquarters may be lending money by investing surplus dollars in short-term securities. If this lending and borrowing offset, there is no point in including the cost of short-term debt in the weighted-average cost of capital, because the company is not a net short-term borrower.

What about other current liabilities? Current liabilities are usually “netted out” by subtracting them from current assets. The difference is entered as net working capital on the left-hand side of the balance sheet. The sum of long-term financing on the right is called total capitalization.

Net working capital = current assets – current liabilities			Long-term debt (D) Preferred stock (P)
Property, plant, and equipment
Growth opportunities			Equity (E )
Total assets			Total capitalization (V )

When net working capital is treated as an asset, forecasts of cash flows for capital investment projects must treat increases in net working capital as a cash outflow and decreases as an inflow. This is standard practice, which we followed in Section 6-2. We also did so when we estimated the future investments that Rio would need to make in working capital.

Because current liabilities include short-term debt, netting them out against current assets excludes the cost of short-term debt from the weighted-average cost of capital. We have just explained why this can be an acceptable approximation. But when short-term debt is an important, permanent source of financing—as is common for small firms and firms outside the United States—it should be shown explicitly on the right-hand side of the balance sheet, not netted out against current assets.⁸ The interest cost of short-term debt is then one element of the weighted-average cost of capital.

How are the costs of financing calculated? You can often use stock market data to get an estimate of r_E, the expected rate of return demanded by investors in the company’s stock. With that estimate, WACC is not too hard to calculate because the borrowing rate r_D and the debt and equity ratios D/V and E/V can be directly observed or estimated without too much trouble.⁹ Estimating the value and required return for preferred shares is likewise usually not too complicated.

Estimating the required return on other security types can be troublesome. Convertible debt, where the investors’ return comes partly from an option to exchange the debt for the company’s stock, is one example. We leave convertibles to Chapter 24.

Junk debt, where the risk of default is high, is likewise difficult. The higher the odds of default, the lower the market price of the debt, and the higher is the promised rate of interest. But the weighted-average cost of capital is an expected—that is, average—rate of return, not a promised one. For example, as we write this in 2018, the three-year bonds issued by Bon-Ton Department Stores are priced at 12.5%% of face value and offer a promised yield of 110%, about 108 percentage points above yields on the highest-quality debt issues maturing at the same time. The price and yield on the Bon-Ton bond demonstrated investors’ concern about the company’s chronic financial ill-health. But the 110% yield was not an expected return because it did not average in the losses to be incurred if the company were to default. Including 110% as a “cost of debt” in a calculation of WACC would therefore have overstated Bon-Ton’s true cost of capital.

This is bad news: There is no easy way of estimating the expected rate of return on most junk debt issues. The good news is that for most debt, the odds of default are small. That means the promised and expected rates of return are close, and the promised rate can be used as an approximation in calculating the weighted-average cost of capital.

Should I use a company or industry WACC? Of course, you want to know what your company’s WACC is. Yet industry WACCs are sometimes more useful. Here’s an example. Kansas City Southern used to be a portfolio of (1) the Kansas City Southern Railroad, with operations running from the U.S. Midwest south to Texas and Mexico, and (2) Stillwell Financial, an investment-management business that included the Janus mutual funds. It’s hard to think of two more dissimilar businesses. Kansas City Southern’s overall WACC was not right for either of them. The company would have been well advised to use a railroad industry WACC for its railroad operations and an investment management WACC for Stillwell.¹⁰

Kansas City Southern spun off Stillwell in 2000 and is now a pure-play railroad. But even now, the company would be wise to check its WACC against a railroad industry WACC. Industry WACCs are less exposed to random noise and estimation errors. Fortunately for Kansas City Southern, there are five large, pure-play railroads (including Canadian Pacific and Canadian National Railway) that the company could use to calculate an industry WACC. Of course, use of an industry WACC for a particular company’s investments assumes that the company and industry have approximately the same business risk. Industry WACCs also have to be adjusted (by the three-step procedure given below) if industry-average debt ratios differ from the target debt ratio for the project to be valued.

Don’t use industry WACCs blindly. Railroad companies are relatively homogenous, and therefore it may be helpful to look at a railroad WACC. But an industry-average WACC for Miscellaneous Consumer Goods would be almost useless as a guide to the WACC of an individual company.¹¹

What tax rate should I use? Taxes are complicated. Corporations can often reduce average tax rates by taking advantage of special provisions (some might say “loopholes”) in the tax code. But the WACC formula calls for the marginal tax rate—that is, the cash taxes paid as a percentage of each dollar of additional income generated by a capital-investment project.

The examples in this chapter use 21%, the current U.S. space corporate rate. In practice, U. S. corporations add three or four percentage points to cover state taxation. Thus, a corporation operating nationwide—and paying income taxes in most states—might use a 24% or 25% rate to calculate WACC.

What if the company can’t use all its interest tax shields? So far, we have assumed that the company is consistently profitable and will pay taxes at the full statutory rate, which in the United States is now 21%. Thus, the company can still use a new project’s depreciation tax shields even if a project endures a period of start-up losses. The project’s depreciation expense shields some of the company’s overall taxable income. Interest tax shields on debt supported by the new project can be captured in the same way.

Sometimes interest tax shields from new debt cannot be captured immediately because (1) the company is suffering losses overall or (2) its total interest payments exceed 30% of EBITDA.¹² Should the company change its WACC if it finds itself in one or both of these unfortunate states?

Probably not, if the losses or constraint are temporary. Tax losses and nondeductible interest can be carried forward and used to shield future income. (The tax rate in the WACC formula could be reduced to account for delay in using tax shields.) But if the wait to use interest tax shields from additional borrowing is long enough, it may be best to use the APV method, which we explain in the next section.

Adjusting WACC When Debt Ratios and Business Risks Differ

The WACC formula assumes that the project or business to be valued will be financed in the same debt–equity proportions as the company (or industry) as a whole. What if that is not true? For example, what if Sangria’s perpetual crusher project supports only 20% debt, versus 40% for Sangria overall?

Moving from 40% to 20% debt may change all the inputs to the WACC formula.¹³ Obviously, the financing weights change. But the cost of equity r_E is less because financial risk is reduced. The cost of debt may be lower, too.

Take a look at Figure 19.1, which plots Sangria’s WACC and the costs of debt and equity as a function of its debt–equity ratio. The flat line is r, the opportunity cost of capital. Remember, this is the expected rate of return that investors would want from the company if it were all-equity-financed. The opportunity cost of capital depends only on business risk and is the natural reference point.

FIGURE 19.1 This plot shows WACC for the Sangria Corporation at debt-toequity ratios of 25% and 67%. The corresponding debt-to-value ratios are 20% and 40%.

Suppose Sangria or the perpetual crusher project were all-equity-financed (D/V = 0). At that point, WACC and the opportunity cost of capital are identical. Start from that point in Figure 19.1. As the debt ratio increases, the cost of equity increases because of financial risk, but notice that WACC declines. The decline is not caused by use of “cheap” debt in place of “expensive” equity. It falls because of the tax shields on debt interest payments. If there were no corporate income taxes, the weighted-average cost of capital would be constant and equal to the opportunity cost of capital at all debt ratios. We showed this in Chapter 17.

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Try it! Figure 19.1: Sangria’s WACC

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Figure 19.1 shows the shape of the relationship between financing and WACC, but initially we have numbers only for Sangria’s current 40% debt ratio. We want to recalculate WACC at a 20% ratio.

Here is the simplest way to do it. There are three steps.

Step 1 Calculate the opportunity cost of capital. In other words, calculate WACC and the cost of equity at zero debt. This step is called unlevering the WACC. The simplest unlevering formula is

Opportunity cost of capital = r = r_D D/V + r_E E/V

This formula comes directly from Modigliani and Miller’s proposition 1 (see Section 17-1). If taxes are left out, the weighted-average cost of capital equals the opportunity cost of capital and is independent of leverage.

Step 2 Estimate the cost of debt, r_D, at the new debt ratio, and calculate the new cost of equity.

r_E = r + (r − r_D )D / E

This formula is Modigliani and Miller’s proposition 2 (see Section 17-2). It calls for D/E, the ratio of debt to equity, not debt to value.

Step 3 Recalculate the weighted-average cost of capital at the new financing weights.

Let’s do the numbers for Sangria at D/V = .20, or 20%.

Step 1. Sangria’s current debt ratio is D/V = .4. So the opportunity cost of capital is:

r = .06(.4) + .125(.6) = .099, or 9.9%

Step 2. We will assume that the debt cost stays at 6% when the debt ratio is 20%. Then

r_E = .099 + (.099 − .06)(.25) = .109, or 10.9%

Note that the debt–equity ratio is .2/.8 = .25.

Step 3. Recalculate WACC.

WACC = .06(1 − .21)(.2) + .109(.8) = .097, or 9.7%

Figure 19.1 enters these numbers on the plot of WACC versus the debt–equity ratio.

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WACC and changing debt ratios

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Unlevering and Relevering Betas

Our three-step procedure (1) unlevers and then (2) relevers the cost of equity. Some financial managers find it convenient to (1) unlever and then (2) relever the equity beta. Given the beta of equity at the new debt ratio, the cost of equity can be calculated from the capital asset pricing model. We can then compute the WACC at the new debt ratio.

Suppose Sangria’s debt and equity betas in our example are β_D = .135 and β_E = 1.07.¹⁴ If the risk-free rate is 5%, and the market risk premium is 7.0%, then Sangria’s cost of equity is

r_E = r_f + ( r_m − r_f ) β_E = .05 + (.07)1.07 = .125, or 12.5%

This matches the cost of equity in our example at a 40/60 debt–equity ratio.

To find Sangria’s weighted average cost of capital at a 20% debt ratio, we can follow a three-step procedure that pretty much matches the procedure that we used earlier.

Step 1 Unlever beta. The unlevered beta is the beta of the equity if the company had zero debt. The formula for unlevering beta was given in Section 17-2.

β_A = β_D(D/V) + β_E(E/V)

This equation says that the beta of a firm’s assets (β_A) is equal to the beta of a portfolio of all of the firm’s outstanding debt and equity securities. An investor who bought such a portfolio would own the assets free and clear and absorb only business risks. For Sangria,

β_A = β_D(D/V) + β_E(E/V) = .135(.4) + 1.07(.6) = .696

Step 2 Estimate the betas of the debt and equity at the new debt ratio. The formula for relevering beta closely resembles MM’s proposition 2, except that betas are substituted for rates of return:

β_E = β_A + ( β_A − β_D )D/E

Use this formula to recalculate β_E when D/E changes. If the beta of Sangria’s debt stays at .135 when it moves to a debt-equity ratio of .2/.8 =.25, then

β_E = β_A + ( β_A − β_D )D/E = .696 + (.696 − .135).25 = .836

Step 3 Recalculate the cost of equity and the WACC at the new financing weights:

r_E = r_f + ( r_m − r_f ) β_E = .05 + .07(.836) = .109, or 10.9%

WACC = .06 ( 1 – .21 ) ( .2 ) + .8 ( .109 ) = .097, or 9.7%

This corresponds to the figure that we calculated above and plotted in Figure 19.1.

The Importance of Rebalancing

The formulas for WACC and for unlevering and relevering expected returns are simple, but we must be careful to remember the underlying assumptions. The most important point is rebalancing.

Calculating WACC for a company at its existing capital structure requires that the capital structure won’t change; in other words, the company must rebalance its capital structure to maintain the same market-value debt ratio for the relevant future. Take Sangria Corporation as an example. It starts with a debt-to-value ratio of 40% and a market value of $1,250 million. Suppose that Sangria’s products do unexpectedly well in the marketplace and that market value increases to $1,500 million. Rebalancing means that it will then increase debt to .4 × 1,500 = $600 million, thus regaining a 40% ratio. The proceeds of the additional borrowing could be used to finance other investments or it could be paid out to the stockholders. If market value instead falls, Sangria would have to pay down debt proportionally.

Of course, real companies do not rebalance capital structure in such a mechanical and compulsive way. For practical purposes, it’s sufficient to assume gradual but steady adjustment toward a long-run target.¹⁵ But if the firm plans significant changes in capital structure (for example, if it plans to pay off its debt), the WACC formula won’t work. In such cases, you should turn to the APV method, which we describe in the next section.

Our three-step procedure for recalculating WACC with a different debt ratio makes a similar rebalancing assumption.¹⁶ Whatever the starting debt ratio, the firm is assumed to rebalance to maintain that ratio in the future.¹⁷

The Modigliani–Miller Formula, Plus Some Final Advice

What if the firm does not rebalance to keep its debt ratio constant? In this case, the only general approach is adjusted present value, which we cover in the next section. But sometimes financial managers turn to other discount-rate formulas, including one derived by Modigliani and Miller (MM). MM considered a company or project generating a level, perpetual stream of cash flows financed with fixed, perpetual debt. There is then a simple relationship between the after-tax discount rate (r_MM) and the opportunity cost of capital (r):¹⁸

r_MM = r(1 − T_c D/V)

Here it’s easy to unlever: just set the debt-capacity parameter (D/V) equal to zero.¹⁹

MM’s formula is still used in practice, but it is exact only in the special case where there is a level, perpetual stream of cash flows and fixed, perpetual debt. However, the formula is not a bad approximation for projects that are not perpetual as long as debt is issued in a fixed amount.²⁰

So which team do you want to play with, the fixed-debt team or the rebalancers? If you join the fixed-debt team you will be outnumbered. Most financial managers use the plain, after-tax WACC, which assumes constant market-value debt ratios and therefore assumes rebalancing. That makes sense, because the debt capacity of a firm or project must depend on its future value, which will fluctuate.

At the same time, we must admit that the typical financial manager doesn’t care much if his or her firm’s debt ratio drifts up or down within a reasonable range of moderate financial leverage. The typical financial manager acts as if a plot of WACC against the debt ratio is “flat” (constant) over this range. This too makes sense, if we just remember that interest tax shields are the only reason why the after-tax WACC declines in Figures 17.4 and 19.1. The WACC formula doesn’t explicitly capture costs of financial distress or any of the other non-tax complications discussed in Chapter 18.²¹ All these complications may roughly cancel the value added by interest tax shields (within a range of moderate leverage). If so, the financial manager is wise to focus on the firm’s operating and investment decisions, rather than on fine-tuning its debt ratio.

19-4Adjusted Present Value

We now turn to an alternative way to take account of financing decisions. This is to calculate an adjusted present value or APV. The idea behind APV is to divide and conquer. Instead of capturing the effects of financing by adjusting the discount rate, APV makes a series of present value calculations. The first calculation establishes a base-case value for the project or firm: its value as a separate, all-equity-financed venture. The discount rate for the base-case value is just the opportunity cost of capital. Once the base-case value is set, then each financing side effect is traced out, and the present value of its cost or benefit to the firm is calculated. Finally, all the present values are added together to estimate the project’s total contribution to the value of the firm:

APV = base-case NPV + sum of PVs of financing side effects²²

The most important financing side effect is the interest tax shield on the debt supported by the project (a plus). Other possible side effects are the issue costs of securities (a minus) or financing packages subsidized by a supplier or government (a plus).

APV gives the financial manager an explicit view of the factors that are adding or subtracting value. APV can prompt the manager to ask the right follow-up questions. For example, suppose that base-case NPV is positive but less than the costs of issuing shares to finance the project. That should prompt the manager to look around to see if the project can be rescued by an alternative financing plan.

APV for the Perpetual Crusher

APV is easiest to understand in simple numerical examples. Let’s apply it to Sangria’s perpetual crusher project. We start by showing that APV is equivalent to discounting at WACC if we make the same assumptions about debt policy.

We used Sangria’s WACC (9.4%) as the discount rate for the crusher’s projected cash flows. The WACC calculation assumed that debt will be maintained at a constant 40% of the future value of the project or firm. In this case, the risk of interest tax shields is the same as the risk of the project.²³ Therefore, we can discount the tax shields at the opportunity cost of capital (r). We calculated the opportunity cost of capital in the last section by unlevering Sangria’s WACC to obtain r = 9.9%.

The first step is to calculate base-case NPV. This is the project’s NPV with all-equity financing. To find it, we discount after-tax project cash flows of $1.175 million at the opportunity cost of capital of 9.9% and subtract the $12.5 million outlay. The cash flows are perpetual, so

Thus, the project would not be worthwhile with all-equity financing. But it actually supports debt of $5 million. At a 6% borrowing rate (r_D = .06) and a 21% tax rate (T_c = .21), annual tax shields are .21 × .06 × 5 = .063, or $63,000.

What are those tax shields worth? If the firm is constantly rebalancing its debt, we discount at r = 9.9%:

APV is the sum of base-case value and PV(interest tax shields):

APV = −0.63 million + 0.63 million = 0

This is exactly the same as we obtained by one-step discounting with WACC. The perpetual crusher is a break-even project by either valuation method.²⁴

But with APV, we don’t have to hold debt at a constant proportion of value. Suppose Sangria plans to keep project debt fixed at $5 million. In this case, we assume the risk of the tax shields is the same as the risk of the debt and we discount at the 6% rate on debt:

Now the project is more attractive. With fixed debt, the interest tax shields are safe and therefore worth more. (Whether the fixed debt is safer for Sangria is another matter. If the perpetual crusher project fails, the $5 million of fixed debt may end up as a burden on Sangria’s other assets.)

Other Financing Side Effects

Suppose Sangria has to finance the perpetual crusher by issuing debt and equity. It issues $7.5 million of equity with issue costs of 7% ($.53 million) and $5 million of debt with issue costs of 2% ($.10 million). Assume the debt is fixed once issued, so that interest tax shields are worth $1.05 million. Now we can recalculate APV, taking care to subtract the issue costs:

APV = −0.63 + 1.05 − .53 − .10 = −.21 million, or − $210,000

The issue costs would result in a negative APV.

Sometimes there are favorable financing side effects that have nothing to do with taxes. For example, suppose that a potential manufacturer of crusher machinery offers to sweeten the deal by leasing it to Sangria on favorable terms. Then to calculate APV you would need to add in the NPV of the lease. Or suppose that a local government offers to lend Sangria $5 million at a very low interest rate if the crusher is built and operated locally. The NPV of the subsidized loan could be added in to APV. (We cover leases in Chapter 25 and subsidized loans in the Appendix to this chapter.)

APV for Entire Businesses

APV can also be used to value entire businesses. Let’s take another look at the valuation of Rio. In Table 19.1, we assumed a constant 40% debt ratio and discounted free cash flow at Sangria’s WACC. Table 19.2 runs the same analysis, but with a fixed debt schedule.

TABLE 19.2 APV valuation of Rio Corporation ($ millions)

We’ll suppose that Sangria has decided to make an offer for Rio. If successful, it plans to finance the purchase with $62 million of debt. It intends to pay down the debt to $53 million in year 6. Recall Rio’s horizon value of $132.7 million, which is calculated in Table 19.1 and shown again in Table 19.2. The debt ratio at the horizon is therefore projected at 53/132.7= .40, or 40%. Thus, Sangria plans to take Rio back to a normal 40% debt ratio at the horizon.²⁵ But Rio will be carrying a heavier debt load before the horizon. For example, the $62 million of initial debt is about 56% of company value as calculated in Table 19.1.

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Try it! Rio’s spreadsheet

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Let’s see how Rio’s APV is affected by this more aggressive borrowing schedule. Table 19.2 shows projections of free cash flows from Table 19.1.²⁶ Now we need Rio’s base-case value. This is its value with all-equity financing, so we discount these flows at the opportunity cost of capital (9.9%), not at WACC. The resulting base-case value for Rio is $28.5 + 75.3 =103.8 million. Table 19.2 also projects debt levels, interest payments, and interest tax shields. If the debt levels are taken as fixed, then the tax shields should be discounted back at the 6% borrowing rate. The resulting PV of interest tax shields is $3.6 million. Thus,

an increase of $1.0 million from NPV in Table 19.1. The increase can be traced to the higher early debt levels and to the assumption that the debt levels and interest tax shields are fixed and safe.²⁷

Now a difference of $1.0 million is not a big deal, considering all the lurking risks and pitfalls in forecasting Rio’s free cash flows. But you can see the advantage of the flexibility that APV provides. The APV spreadsheet allows you to explore the implications of different financing strategies without locking into a fixed debt ratio or having to calculate a new WACC for every scenario.

APV is particularly useful when the debt for a project or business is tied to book value or has to be repaid on a fixed schedule. For example, Kaplan and Ruback used APV to analyze the prices paid for a sample of leveraged buyouts (LBOs). LBOs are takeovers, typically of mature companies, heavily debt-financed. However, the new debt is not intended to be permanent. LBO business plans call for generating extra cash by selling assets, shaving costs, and improving profit margins. The extra cash is used to pay down the LBO debt. Therefore, you can’t use WACC as a discount rate to evaluate an LBO because its debt ratio will not be constant.

APV works fine for LBOs. The company is first evaluated as if it were all-equity-financed. That means that cash flows are projected after tax, but without any interest tax shields generated by the LBO’s debt. The tax shields are then valued separately and added to the all-equity value. Any other financing side effects are added also. The result is an APV valuation for the company.²⁸ Kaplan and Ruback found that APV did a pretty good job explaining prices paid in these hotly contested takeovers, considering that not all the information available to bidders had percolated into the public domain. Kaplan and Ruback were restricted to publicly available data.

APV and Limits on Interest Deductions

The United States now limits the amount of interest that can be deducted for tax to 30% of each year’s EBITDA (or 30% of EBIT starting in 2022). Germany has a similar restriction, and the European Commission has proposed an EU-wide limit.

Most companies will not be caught by these rules. But what about the few that are caught? How should a financial manager take limits on interest-expense deductions into account?

Suppose the 30% constraint is and will be binding. Assume the firm is profitable and paying taxes. Then the future interest tax shields generated by a new investment project are proportional to its future EBITDA. The financial manager should forecast EBITDA and the associated tax shields and discount at a rate depending on the risk of EBITDA.²⁹ The APV formula is the same as before:

APV = base-case NPV + PV(interest tax shields)

but PV(interest tax shields) now depends on the project’s forecasted EBITDA.

Those projects that generate plenty of EBITDA will be especially valuable to tax-paying firms that are subject to the 30% constraint. The EBITDA of the project can relax the constraint for the firm as a whole, thus unlocking interest tax shields on the firm’s existing debt.

The APV of an entire business or company subject to the 30% constraint should also include the PV of interest tax shields generated by its expected future EBITDA. If the 30% limit on interest deductions is temporary—in one or two low-profit years, for example—then the unused tax shields are not lost but can be carried forward indefinitely and may therefore be merely delayed. The financial manager could assign the tax shields to future years, discount to PV and include them in APV.

APV for International Investments

APV is most useful when financing side effects are numerous and important. This is frequently the case for large international investments, which may have custom-tailored project financing and special contracts with suppliers, customers, and governments. Here are a few examples of financing side effects resulting from the financing of a project.

We explain project finance in Chapter 24. It typically means very high debt ratios to start, with most or all of a project’s early cash flows committed to debt service. Equity investors have to wait. Since the debt ratio will not be constant, you have to turn to APV.

Project financing may include debt available at favorable interest rates. Most governments subsidize exports by making special financing packages available, and manufacturers of industrial equipment may stand ready to lend money to help close a sale. Suppose, for example, that your project requires construction of an on-site electricity generating plant. You solicit bids from suppliers in various countries. Don’t be surprised if the competing suppliers sweeten their bids with offers of low interest rate project loans or if they offer to lease the plant on favorable terms. You should then calculate the NPVs of these loans or leases and include them in your project analysis.

Sometimes international projects are supported by contracts with suppliers or customers. Suppose a manufacturer wants to line up a reliable supply of a crucial raw material—powdered magnoosium, say. The manufacturer could subsidize a new magnoosium smelter by agreeing to buy 75% of production and guaranteeing a minimum purchase price. The guarantee is clearly a valuable addition to the smelter’s APV: If the world price of powdered magnoosium falls below the minimum, the project doesn’t suffer. You would calculate the value of this guarantee (by the methods explained in Chapters 20 to 22) and add it to APV.

Sometimes local governments impose costs or restrictions on investment or disinvestment. For example, Chile, in an attempt to slow down a flood of short-term capital inflows in the 1990s, required investors to “park” part of their incoming money in non-interest-bearing accounts for a period of two years. An investor in Chile during this period could have calculated the cost of this requirement and subtracted it from APV.³⁰

19-5Your Questions Answered

Question: All these cost of capital formulas—which ones do financial managers actually use?

Answer: The after-tax weighted-average cost of capital, most of the time. WACC is estimated for the company, or sometimes for an industry. We recommend industry WACCs when data are available for firms with similar assets, operations, business risks, and growth opportunities.

Of course, conglomerate companies, with divisions operating in two or more unrelated industries, should not use a single company or industry WACC. Such firms should try to estimate a different industry WACC for each operating division.

Question: But WACC is the correct discount rate only for “average” projects. What if the project’s financing differs from the company’s or industry’s?

Answer: Remember, investment projects are usually not separately financed. Even when they are, you should focus on the project’s contribution to the firm’s overall debt capacity, not on its immediate financing. (Suppose it’s convenient to raise all the money for a particular project with a bank loan. That doesn’t mean the project itself supports 100% debt financing. The company is borrowing against its existing assets as well as the project.)

But if the project’s debt capacity is materially different from the company’s existing assets, or if the company’s overall debt policy changes, WACC should be adjusted. The adjustment can be done by the three-step procedure explained in Section 19-3.

Question: Could we do one more numerical example?

Answer: Sure. Suppose that WACC has been estimated as follows at a 30% debt ratio:

What is the correct discount rate at a 50% debt ratio?

Step 1. Calculate the opportunity cost of capital.

Step 2. Calculate the new costs of debt and equity. The cost of debt will be higher at 50% debt than 30%. Say it is r_D = .095. The new cost of equity is

Step 3. Recalculate WACC.

Question: How do I use the capital asset pricing model to calculate the after-tax weighted-average cost of capital?

Answer: First plug the equity beta into the capital asset pricing formula to calculate r_E, the expected return to equity. Then use this figure, along with the after-tax cost of debt and the debt-to-value and equity-to-value ratios, in the WACC formula.

Of course, the CAPM is not the only way to estimate the cost of equity. For example, you might be able to use the dividend discount model (see Section 4-3).

Question: But suppose I do use the CAPM? What if I have to recalculate the equity beta for a different debt ratio?

Answer: The formula for the equity beta is

β_E = β_A + ( β_A − β_D )D/E

where β_E is the equity beta, β_A is the asset beta, and β_D is the beta of the company’s debt. The asset beta is a weighted average of the debt and equity betas:

β_A = β_D (D / V) + β_E (E/V)

Suppose you needed the opportunity cost of capital r. You could calculate β_A and then r from the capital asset pricing model.

Question: I think I understand how to adjust for differences in debt capacity or debt policy. How about differences in business risk?

Answer: If business risk is different, then r, the opportunity cost of capital, is different.

Figuring out the right r for an unusually safe or risky project is never easy. Sometimes the financial manager can use estimates of risk and expected return for companies similar to the project. Suppose, for example, that a traditional pharmaceutical company is considering a major commitment to biotech research. The financial manager could pick a sample of biotech companies, estimate their average beta and cost of capital, and use these estimates as benchmarks for the biotech investment.

But in many cases, it’s difficult to find a good sample of matching companies for an unusually safe or risky project. Then the financial manager has to adjust the opportunity cost of capital by judgment. Section 9-3 may be helpful in such cases.

Question: When do I need adjusted present value (APV)?

Answer: The WACC formula picks up only one financing side effect: the value of interest tax shields on debt supported by a project. If there are other side effects—subsidized financing tied to a project, for example—you should use APV.

You can also use APV to break out the value of interest tax shields:

APV = base-case NPV+PV(tax shield)

Suppose, for example, that you are analyzing a company just after a leveraged buyout. The company has a very high initial debt level but plans to pay down the debt as rapidly as possible. APV could be used to obtain an accurate valuation.

Question: When should personal taxes be incorporated into the analysis?

Answer: Always use T_c, the marginal corporate tax rate, when calculating WACC as a weighted average of the costs of debt and equity. The discount rate is adjusted only for corporate taxes.

In principle, APV can be adjusted for personal taxes by replacing the marginal corporate rate T_c with an effective tax rate that combines corporate and personal taxes and reflects the net tax advantage per dollar of interest paid by the firm. We provided back-of-the-envelope calculations of this advantage in Section 18-2. The effective tax rate is almost surely less than T_c, but it is very difficult to pin down the numerical difference. Therefore, in practice T_c is almost always used as an approximation.

Question: Are taxes really that important? Do financial managers really fine-tune the debt ratio to minimize WACC?

Answer: As we saw in Chapter 18, financing decisions reflect many forces beyond taxes, including costs of financial distress, differences in information, and incentives for managers. There may not be a sharply defined optimal capital structure. Therefore most financial managers don’t fine-tune their companies’ debt ratios, and they don’t rebalance financing to keep debt ratios strictly constant. In effect, they assume that a plot of WACC for different debt ratios is “flat” over a reasonable range of moderate leverage.

SUMMARY

In this chapter, we considered how financing can be incorporated into the valuation of projects and ongoing businesses. There are two ways to take financing into account. The first is to calculate NPV by discounting at an adjusted discount rate, usually the after-tax weighted-average cost of capital (WACC). The second approach discounts at the opportunity cost of capital and then adds or subtracts the present values of financing side effects. The second approach is called adjusted present value, or APV.

The formula for the after-tax WACC is

where r_D and r_E are the expected rates of return demanded by investors in the firm’s debt and equity securities, D and E are the current market values of debt and equity, and V is the total market value of the firm (V = D + E). Of course, the WACC formula expands if there are other sources of financing, for example, preferred stock.

Strictly speaking, discounting at WACC works only for projects that are carbon copies of the existing firm—projects with the same business risk that will be financed to maintain the firm’s current ratio of debt to market value. But firms can use WACC as a benchmark rate to be adjusted for differences in business risk or financing. We gave a three-step procedure for adjusting WACC for different debt ratios.

Discounting cash flows at the WACC assumes that debt is rebalanced to keep a constant ratio of debt to market value. The amount of debt supported by a project is assumed to rise or fall with the project’s after-the-fact success or failure. The WACC formula also assumes that financing matters only because of interest tax shields. When this or other assumptions are violated, only APV will give an absolutely correct answer.

APV is, in concept at least, simple. First calculate the base-case NPV of the project or business on the assumption that financing doesn’t matter. (The discount rate is not WACC, but the opportunity cost of capital.) Then calculate the present values of any relevant financing side effects and add or subtract from base-case value. A capital investment project is worthwhile if

APV = base-case NPV + PV(financing side effects)

is positive. Common financing side effects include interest tax shields, issue costs, and special financing packages offered by suppliers or governments.

For firms or going-concern businesses, value depends on free cash flow. Free cash flow is the amount of cash that can be paid out to all investors, debt as well as equity, after deducting cash needed for new investment or increases in working capital. Free cash flow does not include the interest tax shields. The WACC formula accounts for interest tax shields by using the after-tax cost of debt. APV adds PV(interest tax shields) to base-case value.

Businesses are usually valued in two steps. First, free cash flow is forecasted out to a valuation horizon assuming all-equity financing and is then discounted back to present value using WACC. Then a horizon value is calculated and also discounted back. Be particularly careful to avoid unrealistically high horizon values. By the time the horizon arrives, competitors will have had several years to catch up. Also, when you are done valuing the business, don’t forget to subtract its debt to get the value of the firm’s equity.

All of this chapter’s examples reflect assumptions about the amount of debt supported by a project or business. Remember not to confuse “supported by” with the immediate source of funds for investment. For example, a firm might, as a matter of convenience, borrow $1 million for a $1 million research program. But the research is unlikely to contribute $1 million in debt capacity; a large part of the $1 million new debt would be supported by the firm’s other assets.

Also remember that debt capacity is not meant to imply an absolute limit on how much the firm can borrow. The phrase refers to how much it chooses to borrow against a project or ongoing business.

FURTHER READING

The Harvard Business Review has published a popular account of APV:

T. A. Luehrman, “Using APV: A Better Tool for Valuing Operations,” Harvard Business Review 75 (May–June 1997), pp. 145–154.

There have been dozens of articles on the weighted-average cost of capital and other issues discussed in this chapter. Here are three:

J. Miles and R. Ezzell, “The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification,” Journal of Financial and Quantitative Analysis 15 (September 1980), pp. 719–730.

R. A. Taggart Jr., “Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes,” Financial Management 20 (Autumn 1991), pp. 8–20.

R. S. Ruback, “Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows,” Financial Management 31 (Summer 2002), pp. 85–103.

The valuation rule for safe, nominal cash flows is developed in:

R. S. Ruback, “Calculating the Market Value of Riskless Cash Flows,” Journal of Financial Economics 15 (March 1986), pp. 323–339.

PROBLEM SETS

Select problems are available in McGraw-Hill’s Connect. Please see the preface for more information.

1. WACC True or false? Use of the WACC formula assumes

a. A project supports a fixed amount of debt over the project’s economic life.

b. The ratio of the debt supported by a project to project value is constant over the project’s economic life.

c. The firm rebalances debt each period, keeping the debt-to-value ratio constant.

2. WACC The WACC formula seems to imply that debt is “cheaper” than equity—that is, that a firm with more debt could use a lower discount rate. Does this make sense? Explain briefly.

3. WACC* Calculate the weighted-average cost of capital (WACC) for Federated Junkyards of America, using the following information:

· Debt: $75,000,000 book value outstanding. The debt is trading at 90% of book value. The yield to maturity is 9%.

· Equity: 2,500,000 shares selling at $42 per share. Assume the expected rate of return on Federated’s stock is 18%.

· Taxes: Federated’s marginal tax rate is T_c = .21.

4. WACC* Suppose Federated Junkyards decides to move to a more conservative debt policy. A year later, its debt ratio is down to 15% (D/V = .15). The interest rate has dropped to 8.6%. Recalculate Federated’s WACC under these new assumptions. The company’s business risk, opportunity cost of capital, and tax rate have not changed. Use the three-step procedure explained in Section 19-3.

5. WACC Whispering Pines Inc. is all-equity-financed. The expected rate of return on the company’s shares is 12%.

a. What is the opportunity cost of capital for an average-risk Whispering Pines investment?

b. Suppose the company issues debt, repurchases shares, and moves to a 30% debt-to-value ratio (D/V = .30). What will be the company’s weighted-average cost of capital at the new capital structure? The borrowing rate is 7.5% and the tax rate is 21%.

6. WACC Table 19.3 shows a book balance sheet for the Wishing Well Motel chain. The company’s long-term debt is secured by its real estate assets, but it also uses short-term bank loans as a permanent source of financing. It pays 10% interest on the bank debt and 9% interest on the secured debt. Wishing Well has 10 million shares of stock outstanding, trading at $90 per share. The expected return on Wishing Well’s common stock is 18%.

Calculate Wishing Well’s WACC. Assume that the book and market values of Wishing Well’s debt are the same. The marginal tax rate is 21%.

Cash and marketable securities	100	Bank loan	280
Accounts receivable	200	Accounts payable	120
Inventory	50	Current liabilities	400
Current assets	350
Real estate	2,100	Long-term debt	1,800
Other assets	150	Equity	400
Total	2,600	Total	2,600

7. TABLE 19.3 Book balance sheet for Wishing Well Inc. (figures in $ millions)

8. WACC Table 19.4 shows a simplified balance sheet for the Dutch manufacturer Rensselaer Felt. Calculate this company’s weighted-average cost of capital. The debt has just been refinanced at an interest rate of 6% (short term) and 8% (long term). The expected rate of return on the company’s shares is 15%. There are 7.46 million shares outstanding, and the shares are trading at €46. The tax rate is 25%.

Cash and marketable securities	1,500	Short-term debt	75,600
Accounts receivable	120,000	Accounts payable	62,000
Inventory	125,000	Current liabilities	137,600
Current assets	246,500
Property, plant, and equipment	212,000	Long-term debt	208,600
Deferred taxes	45,000
Other assets	89,000	Shareholders’ equity	246,300
Total	592,500	Total	592,500

9. TABLE 19.4 Simplified book balance sheet for Rensselaer Felt (figures in € thousands)

10. WACC See Problem 7. How will Rensselaer’s WACC and cost of equity change if it issues €50 million in new equity and uses the proceeds to retire long-term debt? Assume the company’s borrowing rates are unchanged. Use the three-step procedure from Section 19-3.

11. WACC Nevada Hydro is 40% debt-financed and has a weighted-average cost of capital of 10.2%:

WACC = ( 1 − Tc ) rD D / V + rE E / V = ( 1 − .21 ) ( .085 ) ( .40 ) + .125 ( .60 ) = .102

Goldensacks Company is advising Nevada Hydro to issue $75 million of preferred stock at a dividend yield of 9%. The proceeds would be used to repurchase and retire common stock. The preferred issue would account for 10% of the pre-issue market value of the firm.

Goldensacks argues that these transactions would reduce Nevada Hydro’s WACC to 9.84%:

Do you agree with this calculation? Explain.

12. Forecasting cash flow Suppose Wishing Well (see Problem 6) is evaluating a new motel and resort on a romantic site in Madison County, Wisconsin. Explain how you would forecast the after-tax cash flows for this project. (Hints: How would you treat taxes? Interest expense? Changes in working capital?)

13. Flow-to-equity valuation What is meant by the flow-to-equity valuation method? What discount rate is used in this method? What assumptions are necessary for this method to give an accurate valuation?

14. APV True or false? The APV method

a. Starts with a base-case value for the project.

b. Calculates the base-case value by discounting project cash flows, forecasted assuming all-equity financing, at the WACC for the project.

c. Is especially useful when debt is to be paid down on a fixed schedule.

15. APV* A project costs $1 million and has a base-case NPV of exactly zero (NPV = 0). What is the project’s APV in the following cases?

a. If the firm invests, it has to raise $500,000 by a stock issue. Issue costs are 15% of net proceeds.

b. If the firm invests, there are no issue costs, but its debt capacity increases by $500,000. The present value of interest tax shields on this debt is $76,000.

16. APV Consider a project lasting one year only. The initial outlay is $1,000, and the expected inflow is $1,200. The opportunity cost of capital is r = .20. The borrowing rate is r_D = .10, and the tax shield per dollar of interest is T_c = .21.

a. What is the project’s base-case NPV?

b. What is its APV if the firm borrows 30% of the project’s required investment?

17. APV To finance the Madison County project (see Problem 10), Wishing Well needs to arrange an additional $80 million of long-term debt and make a $20 million equity issue. Underwriting fees, spreads, and other costs of this financing will total $4 million. How would you take this into account in valuing the proposed investment?

18. APV Digital Organics (DO) has the opportunity to invest $1 million now (t = 0) and expects after-tax returns of $600,000 in t = 1 and $700,000 in t = 2. The project will last for two years only. The appropriate cost of capital is 12% with all-equity financing, the borrowing rate is 8%, and DO will borrow $300,000 against the project. This debt must be repaid in two equal installments of $150,000 each. Assume debt tax shields have a net value of $.30 per dollar of interest paid. Calculate the project’s APV using the procedure followed in Table 19.2.

19. APV Consider another perpetual project like the crusher described in Section 19-1. Its initial investment is $1,000,000, and the expected cash inflow is $95,000 a year in perpetuity. The opportunity cost of capital with all-equity financing is 10%, and the project allows the firm to borrow at 7%. The tax rate is 21%.

Use APV to calculate the project’s value.

a. Assume first that the project will be partly financed with $400,000 of debt and that the debt amount is to be fixed and perpetual.

b. Then assume that the initial borrowing will be increased or reduced in proportion to changes in the market value of this project.

Explain the difference between your answers to (a) and (b).

20. Opportunity cost of capital Suppose the project described in Problem 17 is to be undertaken by a university. Funds for the project will be withdrawn from the university’s endowment, which is invested in a widely diversified portfolio of stocks and bonds. However, the university can also borrow at 7%. The university is tax exempt.

The university treasurer proposes to finance the project by issuing $400,000 of perpetual bonds at 7% and by selling $600,000 worth of common stocks from the endowment. The expected return on the common stocks is 10%. He therefore proposes to evaluate the project by discounting at a weighted-average cost of capital, calculated as

What’s right or wrong with the treasurer’s approach? Should the university invest? Should it borrow? Would the project’s value to the university change if the treasurer financed the project entirely by selling common stocks from the endowment?

21. APV Consider a project to produce solar water heaters. It requires a $10 million investment and offers a level after-tax cash flow of $1.75 million per year for 10 years. The opportunity cost of capital is 12%, which reflects the project’s business risk.

a. Suppose the project is financed with $5 million of debt and $5 million of equity. The interest rate is 8% and the marginal tax rate is 21%. An equal amount of the debt will be repaid in each year of the project’s life. Calculate APV.

b. How does APV change if the firm incurs issue costs of $400,000 to raise the $5 million of required equity?

22. APV and debt capacity Suppose KCS Corp. buys out Patagonia Trucking, a privately owned business, for $50 million. KCS has only $5 million cash in hand, so it arranges a $45 million bank loan. A normal debt-to-value ratio for a trucking company would be 50% at most, but the bank is satisfied with KCS’s credit rating.

Suppose you were valuing Patagonia by APV in the same format as Table 19.2. How much debt would you include? Explain briefly.

23. APV and issue costs The Bunsen Chemical Company is currently at its target debt ratio of 40%. It is contemplating a $1 million expansion of its existing business. This expansion is expected to produce a cash inflow of $130,000 a year in perpetuity.

The company is uncertain whether to undertake this expansion and how to finance it. The two options are a $1 million issue of common stock or a $1 million issue of 20-year debt. The flotation costs of a stock issue would be around 5% of the amount raised, and the flotation costs of a debt issue would be around 1½%.

Bunsen’s financial manager, Polly Ethylene, estimates that the required return on the company’s equity is 14%, but she argues that the flotation costs increase the cost of new equity to 19%. On this basis, the project does not appear viable. On the other hand, she points out that the company can raise new debt on a 7% yield, which would make the cost of new debt 8½%. She therefore recommends that Bunsen should go ahead with the project and finance it with an issue of long-term debt.

Is Ms. Ethylene right? How would you evaluate the project?

24. APV and limits on interest tax shields Take another look at the APV calculation for the perpetual crusher project in Section 19-4. This time assume that the corporation investing in the project has hit the 30% constraint on interest deductions as a percentage of EBITDA. How does the constraint change the project’s APV?

Notice that the crusher’s pretax cash flow of $1.487 million a year is also its EBITDA and EBIT. The project is perpetual, so there is no depreciation or amortization. Assume for simplicity that the constraint is permanently binding, but that the firm will continue to pay tax at the 21% statutory rate.

BEYOND THE PAGE

Try It! Rio’s spreadsheet

mhhe.com/brealey13e

25. WACC and APV Take another look at the valuations of Rio in Tables 19.1 and 19.2. Now use the live spreadsheets in Connect to show how Rio’s value depends on:

a. The forecasted long-term growth rate.

b. The required amounts of investment in fixed assets and working capital.

c. The opportunity cost of capital. (Note: You can also vary the opportunity cost of capital in Table 19.1.)

d. Profitability—that is, cost of goods sold as a percentage of sales.

e. The assumed amount of debt financing.

26. Company valuation Chiara Company’s management has made the projections shown in Table 19.5. Use this table as a starting point to value the company as a whole. The WACC for Chiara is 12%, and the forecast long-run growth rate after year 5 is 4%. The company, which is located in South Africa, has ZAR 5 million of debt and 865,000 shares outstanding. What is the value per share?

TABLE 19.5 Cash flow projections for Chiara Corp. (ZAR thousands)

CHALLENGE

25. Miles-Ezzell formula In footnote 17, we referred to the Miles–Ezzell discount rate formula, which assumes that debt is not rebalanced continuously, but at one-year intervals. Derive this formula. Then use it to unlever Sangria’s WACC and calculate Sangria’s opportunity cost of capital. Your answer will be slightly different from the opportunity cost that we calculated in Section 19-3. Can you explain why?

26. Rebalancing The WACC formula assumes that debt is rebalanced to maintain a constant debt ratio D/V. Rebalancing ties the level of future interest tax shields to the future value of the company. This makes the tax shields risky. Does that mean that fixed debt levels (no rebalancing) are better for stockholders?

27. Horizon value Modify Table 19.1 on the assumption that competition eliminates any opportunities to earn more than WACC on new investment after year 7 (PVGO = 0). How does the valuation of Rio change?

FINANCE ON THE WEB

Table 19.6 is a simplified book balance sheet for Nike in November 2017. Here is some further information:

Number of outstanding shares (N)	1.32billion
Price per share (P)	$60
Beta	0.55
Treasury bill rate	0.8%
20-year Treasury bond rate	2.7%
Cost of debt (rD)	3.8%
Marginal tax rate (from 2018)	21%

a. Calculate Nike’s WACC. Use the capital asset pricing model and the additional information given above. Make additional assumptions and approximations as necessary.

b. What is Nike’s opportunity cost of capital?

c. Finally, go to finance.yahoo.com and update your answers to parts (a) and (b).

Current assets	$16,582	Current liabilities	$ 6,750
Net property, plant, and equipment	4,117	Long-term debt	3,472
Investments and other assets	3,356	Other liabilities	2,075
		Shareholders’ equity	11,758
Total	$24,055	Total	$24,055

d. TABLE 19.6 Simplified book balance sheet for Nike, November 2017 (figures in $ millions)

APPENDIX

Discounting Safe, Nominal Cash Flows

Suppose you’re considering purchase of a $100,000 machine. The manufacturer sweetens the deal by offering to finance the purchase by lending you $100,000 for five years, with annual interest payments of 5%. You would have to pay 13% to borrow from a bank. Your marginal tax rate is 21% (T_c = .21).

How much is this loan worth? If you take it, the cash flows, in thousands of dollars, are

What is the right discount rate?

Here you are discounting safe, nominal cash flows—safe because your company must commit to pay if it takes the loan,³¹ and nominal because the payments would be fixed regardless of future inflation. Now, the correct discount rate for safe, nominal cash flows is your company’s after-tax, unsubsidized borrowing rate,³² which is r_D(1 – T_c) = .13(1 – .21) = .1027. Therefore,

The manufacturer has effectively cut the machine’s purchase price from $100,000 to $100,000 – $23,790 = $76,210. You can now go back and recalculate the machine’s NPV using this fire-sale price, or you can use the NPV of the subsidized loan as one element of the machine’s adjusted present value.

A General Rule

Clearly, we owe an explanation of why r_D(1 – T_c) is the right discount rate for safe, nominal cash flows. It’s no surprise that the rate depends on r_D, the unsubsidized borrowing rate, for that is investors’ opportunity cost of capital, the rate they would demand from your company’s debt. But why should r_D be converted to an after-tax figure?

Let’s simplify by taking a one-year subsidized loan of $100,000 at 5%. The cash flows, in thousands of dollars, are

	Period 0	Period 1
Cash flow	100	−105
Tax shield on interest of $5		+1.05
After-tax cash flow	100	−103.95

Now ask, What is the maximum amount X that could be borrowed for one year through regular channels if $103,950 is set aside to service the loan?

“Regular channels” means borrowing at 13% pretax and 10.27% after tax. Therefore, you will need 110.27% of the amount borrowed to pay back principal plus after-tax interest charges. If 1.1027X = 103,950, then X = 94,269. Now if you can borrow $100,000 by a subsidized loan, but only $94,269 through normal channels, the difference ($5,731) is money in the bank. Therefore, it must also be the NPV of this one-period subsidized loan.

When you discount a safe, nominal cash flow at an after-tax borrowing rate, you are implicitly calculating the equivalent loan, the amount you could borrow through normal channels, using the cash flow as debt service. Note that

In some cases, it may be easier to think of taking the lender’s side of the equivalent loan rather than the borrower’s. For example, you could ask: How much would my company have to invest today to cover next year’s debt service on the subsidized loan? The answer is $94,269: If you lend that amount at 13%, you will earn 10.27% after tax, and therefore have 94,269(1.1027) = $103,950. By this transaction, you can in effect cancel, or “zero out,” the future obligation. If you can borrow $100,000 and then set aside only $94,269 to cover all the required debt service, you clearly have $5,731 to spend as you please. That amount is the NPV of the subsidized loan.

Therefore, regardless of whether it’s easier to think of borrowing or lending, the correct discount rate for safe, nominal cash flows is an after-tax interest rate.³³

In some ways, this is an obvious result once you think about it. Companies are free to borrow or lend money. If they lend, they receive the after-tax interest rate on their investment; if they borrow in the capital market, they pay the after-tax interest rate. Thus, the opportunity cost to companies of investing in debt-equivalent cash flows is the after-tax interest rate. This is the adjusted cost of capital for debt-equivalent cash flows.³⁴

Some Further Examples

Here are some further examples of debt-equivalent cash flows.

Payout Fixed by Contract

Suppose you sign a maintenance contract with a truck leasing firm, which agrees to keep your leased trucks in good working order for the next two years in exchange for 24 fixed monthly payments. These payments are debt-equivalent flows.

Prejudgment Interest Awards

Court cases involving the award of damages are often complex, and by the time the decision has been reached, many years may have elapsed since the time of the original harm. To compensate for the delay in payment, courts customarily award “prejudgment interest.” In other words, they add on an additional award for the return that the claimant could have earned over the period since the offense. This increment is often larger than the amount of the original damage. For example, when GM was held to have infringed a company’s patent, it was ordered to pay $8.8 million in royalties and $11 million in prejudgment interest. A company that has to wait for compensation for damages until long after the damages were incurred has effectively made a debt-equivalent loan to the offender. The award should therefore be increased by the company’s after-tax interest rate.³⁵

Depreciation Tax Shields

Since 2018, U.S. companies have been able to immediately write off most expenditure on capital equipment. In other countries, capital expenditures must generally be written off over their likely life. These deductions generate a depreciation tax shield.

Capital projects are normally valued by discounting the total after-tax cash flows they are expected to generate. Depreciation tax shields contribute to project cash flow, but they are not valued separately; they are just folded into project cash flows along with dozens, or hundreds, of other specific inflows and outflows. The project’s opportunity cost of capital reflects the average risk of the resulting aggregate.

However, suppose we ask what depreciation tax shields are worth by themselves. For a firm that’s sure to pay taxes, depreciation tax shields are a safe, nominal flow. Therefore, they should be discounted at the firm’s after-tax borrowing rate.

Perhaps you are CFO of a Polish company that proposes to buy an asset for 500,000 zloty,³⁶ which can be depreciated straight-line over five years. The corporate tax rate in Poland is 19%. Therefore, the resulting depreciation tax shields are

If the pretax borrowing rate is 10%, the after-tax discount rate is r_D(1 – T_c) = .10(1 – .19) = .081. The present value of these shields is

A Consistency Check

You may have wondered whether our procedure for valuing debt-equivalent cash flows is consistent with the WACC and APV approaches presented earlier in this chapter. Yes, it is consistent, as we will now illustrate.

Let’s look at another very simple numerical example. You are asked to value a $1 million payment to be received from a blue-chip company one year hence. After taxes at 21%, the cash inflow is $790,000. The payment is fixed by contract.

Because the contract generates a debt-equivalent flow, the opportunity cost of capital is the rate investors would demand on a one-year note issued by the blue-chip company, which happens to be 8%. For simplicity, we’ll assume this is your company’s borrowing rate too. Our valuation rule for debt-equivalent flows is therefore to discount at r_D(1 – T_c) = .08(1 – .21) = .0632:

What is the debt capacity of this $650,000 payment? Exactly $611,362. Your company could borrow that amount and pay off the loan completely—principal and after-tax interest—with the $650,000 cash inflow. The debt capacity is 100% of the PV of the debt-equivalent cash flow.

If you think of it that way, our discount rate r_D(1 – T_c) is just a special case of WACC with a 100% debt ratio (D/V = 1).

Now let’s try an APV calculation. This is a two-part valuation. First, the $650,000 inflow is discounted at the opportunity cost of capital, 8%. Second, we add the present value of interest tax shields on debt supported by the project. Because the firm can borrow 100% of the cash flow’s value, the tax shield is r_D T_c APV, and APV is:

Solving for APV, we get $611,362, the same answer we obtained by discounting at the after-tax borrowing rate. Thus, our valuation rule for debt-equivalent flows is a special case of APV.

QUESTIONS

1. The U.S. government has settled a dispute with your company for $16 million. The government is committed to pay this amount in exactly 12 months. However, your company will have to pay tax on the award at a marginal tax rate of 21%. What is the award worth? The one-year Treasury rate is 5.5%.

2. You are considering a five-year lease of office space for R&D personnel. Once signed, the lease cannot be canceled. It would commit your firm to six annual $100,000 payments, with the first payment due immediately. What is the present value of the lease if your company’s borrowing rate is 9% and its tax rate is 21%? The lease payments would be tax-deductible.

¹See Section 9-1 under “Perfect Pitch and the Cost of Capital.”

²Always use an up-to-date interest rate (yield to maturity), not the interest rate when the firm’s debt was first issued and not the coupon rate on the debt’s book value.

³This is the U.S. corporate tax rate starting in 2018. Most U.S. corporations add a few percentage points to the tax rate to cover state income taxes, depending on how their sales, assets, and income are distributed across states.

⁴We can prove this statement as follows. Denote expected after-tax cash flows (assuming all-equity financing) as C₁, C₂, . . . , C_T. With all-equity financing, these flows would be discounted at the opportunity cost of capital r. But we need to value the cash flows for a firm that is financed partly with debt. Start with value in the next to last period: V_{T– 1} = D_{T– 1} + E_{T– 1}. The total cash payoff to debt and equity investors is the cash flow plus the interest tax shield. The expected total return to debt and equity investors is

(1)

(2)

Assume the debt ratio is constant at L = D/V. Equate (1) and (2) and solve for V_{T– 1}:

The logic repeats for V_{T– 2}. Note that the next period’s payoff includes V_{T– 1}: Expected cash payoff in T − 1 = C_{T– 1} + T_cr_DD_{T– 2} + V_{T– 1}

We can continue all the way back to date 0:

⁵For simplicity we have tied depreciation to growth in sales. We have not tracked bonus depreciation on each year’s new investments.

⁶Notice that expected free cash flow increases by about 4% from year 6 to year 7 because the transition from 4% to 3% sales growth reduces required investment. But sales, investment, and free cash flow will all increase at 3% once the company settles into stable growth. Recall that the first cash flow in the constant-growth DCF formula occurs in the next year, year 7 in this case. Growth progresses at a steady-state 3% from year 7 onward. Therefore, it’s OK to use the 3% growth rate in the horizon-value formula.

⁷This balance sheet is for exposition and should not be confused with a real company’s books. It includes the value of growth opportunities, which accountants do not recognize, though investors do. It excludes certain accounting entries, for example, deferred taxes.

Deferred taxes arise when a company uses faster depreciation for tax purposes than it uses in reports to investors. That means the company reports more in taxes than it pays. The difference is accumulated as a liability for deferred taxes. In a sense, there is a liability because the Internal Revenue Service “catches up,” collecting extra taxes as assets age. But this is irrelevant in capital investment analysis, which focuses on actual after-tax cash flows and uses accelerated tax depreciation.

Deferred taxes should not be regarded as a source of financing or an element of the weighted-average cost of capital formula. The liability for deferred taxes is not a security held by investors. It is a balance sheet entry created for accounting purposes.

Deferred taxes can be important in regulated industries, however. Regulators take deferred taxes into account in calculating allowed rates of return and the time patterns of revenues and consumer prices.

⁸Financial practitioners have rules of thumb for deciding whether short-term debt is worth including in WACC. One rule checks whether short-term debt is at least 10% of total liabilities and net working capital is negative. If so, then short-term debt is almost surely being used to finance long-term assets and is explicitly included in WACC.

⁹Most corporate debt is not actively traded, so its market value cannot be observed directly. But you can usually value a nontraded debt security by looking to securities that are traded and that have approximately the same default risk and maturity. See Chapter 23.

For healthy firms the market value of debt is usually not too far from book value, so many managers and analysts use book value for D in the weighted-average cost of capital formula. However, be sure to use market, not book, values for E.

¹⁰We noted the difficulty of estimating expected rates of return on junk debt. This problem largely disappears for industry WACCs, provided that most or all of the companies in the industry sample are not relying on junk-debt financing.

¹¹Levi and Welch argue against using industry-average betas to predict betas for individual companies in Y. Levi and I. Welch, “Best Practice for Cost-of-Capital Estimates,” Journal of Financial and Quantitative Analysis 52 (April 2017), pp. 427–463.

¹²See the summary of the U.S. Tax Cuts and Jobs Act in Chapter 6. The limit on interest deductions changes to 30% of EBIT in 2022.

¹³Even the tax rate could change. For example, Sangria might have enough taxable income to cover interest payments at 20% debt but not at 40% debt. In that case, the effective marginal tax rate would be higher at 20% than 40% debt.

¹⁴Debt betas are generally small, and many managers simplify and assume β_D = 0. Junk-debt betas can be well above zero, however.

¹⁵Here’s another way to interpret the WACC formula’s assumption of a constant debt ratio. Assume that the debt capacity of a project is a constant fraction of the project’s value. (“Capacity” does not mean the maximum amount that could be borrowed against the project, but the amount that managers would optimally choose to borrow.) Discounting at WACC gives the project credit for interest tax shields on the project’s debt capacity, even if the firm does not rebalance its capital structure and ends up borrowing more or less than the total capacity of all its projects.

¹⁶Similar, but not identical. The basic WACC formula is correct whether rebalancing occurs at the end of each period or continuously. The unlevering and relevering formulas used in steps 1 and 2 of our three-step procedure are exact only if rebalancing is continuous so that the debt ratio stays constant day-to-day and week-to-week. However, the errors introduced from annual rebalancing are very small and can be ignored for practical purposes.

¹⁷Here’s why the formulas work with continuous rebalancing. Think of a market-value balance sheet with assets and interest tax shields on the left and debt and equity on the right, with D + E = PV(assets) + PV(tax shield). The total risk (beta) of the firm’s debt and equity equals the blended risk of PV(assets) and PV(tax shield):

(1)

where α is the proportion of the total firm value from its assets and 1 – α is the proportion from interest tax shields. If the firm readjusts its capital structure to keep D/V constant, then the beta of the tax shield must be the same as the beta of the assets. With rebalancing, an x% change in firm value V changes debt D by x%. So the interest tax shield T_cr_DD will change by x% as well. Thus the risk of the tax shield must be the same as the risk of the firm as a whole:

(2)

This is our unlevering formula expressed in terms of beta. Since expected returns depend on beta:

(3)

Rearrange formulas (2) and (3) to get the relevering formulas for β_E and r_E. (Notice that the tax rate T_c has dropped out.)

All this assumes continuous rebalancing. Suppose instead that the firm rebalances once a year, so that the next year’s interest tax shield, which depends on this year’s debt, is known. Then you can use a formula developed by Miles and Ezzell:

See J. Miles and J. Ezzell, “The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification,” Journal of Financial and Quantitative Analysis 15 (September 1980), pp. 719–730.

¹⁸The formula first appeared in F. Modigliani and M. H. Miller, “Corporate Income Taxes and the Cost of Capital: A Correction,” American Economic Review 53 (June 1963), pp. 433–443. It is explained more fully in M. H. Miller and F. Modigliani: “Some Estimates of the Cost of Capital to the Electric Utility Industry, 1954–1957,” American Economic Review 56 (June 1966), pp. 333–391. Given perpetual fixed debt,

¹⁹In this case the relevering formula for the cost of equity is

r_E = r_A + (1 − T_c )( r_A − r_D )D / E

The unlevering and relevering formulas for betas are

and

See R. Hamada, “The Effect of a Firm’s Capital Structure on the Systematic Risk of Common Stocks,” Journal of Finance 27 (May 1972), pp. 435–452.

²⁰See S. C. Myers, “Interactions of Corporate Financing and Investment Decisions—Implications for Capital Budgeting,” Journal of Finance 29 (March 1974), pp. 1–25.

²¹Costs of financial distress can show up as rapidly increasing costs of debt and equity, especially at high debt ratios. The costs of financial distress could “flatten out” the WACC curve in Figures 17.4 and 19.1 and finally increase WACC as leverage climbs. Thus some practitioners calculate an industry WACC and take it as constant, at least within the range of debt ratios observed for healthy companies in the industry.

Personal taxes could also generate a flatter curve for after-tax WACC as a function of leverage. See Section 18-2.

²²The adjusted-present-value rule was developed in S. C. Myers, “Interactions of Corporate Financing and Investment Decisions—Implications for Capital Budgeting,” Journal of Finance 29 (March 1974), pp. 1–25.

²³That is, β_A = βtax shields. See footnote 17.

²⁴Calculating the present value of the tax shields is straightforward when the project is a perpetuity. When it is not, the expected value of the project changes as time passes and so does the expected tax shield. With a finite project and debt that is a constant proportion of project value, we would need to calculate the expected project value at each future date before calculating the present value of the tax shields. Therefore, whenever the debt ratio is constant, managers use WACC to account for the interest tax shield, and they save APV for times when debt is repaid on a fixed schedule.

²⁵Therefore, we still calculate the horizon value in year 6 by discounting subsequent free cash flows at WACC. The horizon value in year 6 is discounted back to year 0 at the opportunity cost of capital, however.

²⁶Many of the assumptions and calculations in Table 19.1 have been hidden in Table 19.2. The hidden rows can be recalled in the Beyond the Page spreadsheets for Tables 19.1 and 19.2.

²⁷But will Rio really support debt at the levels shown in Table 19.2? If not, then the debt must be partly supported by Sangria’s other assets, and only part of the $3.6 million in PV(interest tax shields) can be attributed to Rio itself.

²⁸Kaplan and Ruback actually used “compressed” APV, in which all cash flows, including interest tax shields, are discounted at the opportunity cost of capital. S. N. Kaplan and R. S. Ruback, “The Valuation of Cash Flow Forecasts: An Empirical Analysis,” Journal of Finance 50 (September 1995), pp. 1059–1093.

²⁹In most cases the risk of EBITDA will be similar to the risk of the project’s overall cash flows. If so the interest tax shields generated by the project’s EBITDA can be discounted at the same opportunity cost of capital used to calculate base-case NPV.

³⁰Such capital controls have been described as financial roach motels: Money can get in, but it can’t get out.

³¹In theory, safe means literally “risk-free,” like the cash returns on a Treasury bond. In practice, it means that the risk of not paying or receiving a cash flow is small.

³²In Section 13-1, we calculated the NPV of subsidized financing using the pretax borrowing rate. Now you can see that was a mistake. Using the pretax rate implicitly defines the loan in terms of its pretax cash flows, violating a rule promulgated way back in Section 6-1: Always estimate cash flows on an after-tax basis.

³³Borrowing and lending rates should not differ by much if the cash flows are truly safe—that is, if the chance of default is small. Usually your decision will not hinge on the rate used. If it does, ask which offsetting transaction—borrowing or lending—seems most natural and reasonable for the problem at hand. Then use the corresponding interest rate.

³⁴All the examples in this section are forward-looking; they call for the value today of a stream of future debt-equivalent cash flows. But similar issues arise in legal and contractual disputes when a past cash flow has to be brought forward in time to a present value today. Suppose it’s determined that company A should have paid B $1 million 10 years ago. B clearly deserves more than $1 million today because it has lost the time value of money. The time value of money should be expressed as an after-tax borrowing or lending rate or, if no risk enters, as the after-tax risk-free rate. The time value of money is not equal to B’s overall cost of capital. Allowing B to “earn” its overall cost of capital on the payment allows it to earn a risk premium without bearing risk. For a broader discussion of these issues, see F. Fisher and R. C. Romaine, “Janis Joplin’s Yearbook and the Theory of Damages,” Journal of Accounting, Auditing & Finance 5 (Winter/Spring 1990), pp. 145–157.

³⁵In practice, courts use a variety of methods for calculating prejudgment interest. For a discussion of the issue see Fisher and Romaine, op. cit.

³⁶Zloty, the Polish currency, often abbreviated as PLN.