Cost of Capital and Project Valuation

1 Background

• Firm organization

– There are four types:

∗ sole proprietorships

∗ partnerships

∗ limited liability companies

∗ corporations

– Each organizational form has different legal and tax implications.

– In this handout, we focus only on the corporation.

• Corporate ownership

– Corporations have many owners; each owner owns only a fraction of the corporation.

– The ownership stake is divided into shares known as stock.

– The collection of all outstanding shares of stock is referred to as the equity of thecorporation.

– An owner of a share of stock is called a shareholder, stockholder or equityholder.

– Shareholders are entitled to dividend payments.

∗ Dividend payments are made at the discretion of the corporation to its equity-holders.

∗ The dividend share to each stockholder is typically in proportion to their own-ership stake.

– Ownership is distinct from control.

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• Stock market

– A public company is one whose shares trade on a stock market or stock exchange.

– In the US, three national stock exchanges are the New York Stock Exchange (NYSE),the American Stock Exchange (AMEX) and the National Association of SecurityDealers Automated Quotation (NASDAQ).

– Stock markets provide two useful features:

∗ They determine a market price for a company’s shares.

∗ They provide liquidity.A stockholder can quickly sell a portion of his/her ownership stake and for aprice very close to the price at which someone else can buy it.(The difference is called the bid-ask spread.)

• Corporate financing

– Corporations raise funds in two ways:

∗ Raise new equity by issuing new shares of stock and selling them to investors.The shares are sold on the primary market.Shares continue to trade in a secondary market between investors without in-volvement of the firm.

∗ Borrow funds from banks or by issuing debt (corporate bonds) via the financialmarkets.

– A firm’s capital structure refers to its relative proportions of equity and debt.

– A firm with no debt is said to be unlevered.Its equity is referred to as unlevered.

– A firm with debt is said to be levered.Its equity is referred to as levered.

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2 Valuation Definitions

• Firm value

– Defined to be the sum of its market value of equity and its market value of debt:

V L := D + E. (1)

– Debt is net of any cash and short-term investments and is referred to as net debt.

– V L represents the amount needed to acquire a firm — to buy out the firm’s share-holders and to pay off the firm’s bondholders.

– An unlevered firm’s value is denoted by the symbol V U .Here, V U = E as D = 0.

– Can be estimated as

V L = Present Value (expected Free Cash Flows (FCF)) @ rWACC (2)

rWACC = the firm’s weighted average cost of capital (3)

Free Cash Flow is the cash generated from a firm’s business operations.It is the cash flow generated if the firm were unlevered.

• Equity value

– Can be estimated as

E = Present Value (expected Free Cash Flows to Equity (FCFE)) @ rE (4)

rE = the firm’s equity cost of capital (5)

The equity cost of capital rE represents the risk-adjusted required rate of returndemanded by shareholders.

– For an unlevered firm, rE is denoted by rU , the firm’s unlevered or asset cost ofcapital.

– For public companies, it equals the company’s Market Capitalization (Market Cap).Market Cap is the product of the firm’s share price and the number of the firm’soutstanding shares of stock.

3

• Debt value

– Can be estimated as (excluding the cash and short-term investments)

D = Present Value (expected Cash Flows to Bondholders) @ rD (6)

rD = the firm’s debt cost of capital (7)

The debt cost of capital rD represents the risk-adjusted required rate of return de-manded by bondholders.

• Corporate debt

– Not risk free — defaults can happen.If so, a portion of the debt is repaid.

– Agencies like Moody’s and S&P rate corporate debt.Ratings indicate their assessment of the likelihood of default.

– Due to the possibility of default a bond’s yield to maturity (IRR) does not equal itsexpected rate of return.

– Many types of corporate debt.

– Difficult to determine market value.Standard practice is to use the book value of debt.

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3 Analysis of an Unlevered Firm

Market and firm data:

Market data:

• Risk-free rate rf = 2%

• Market premium R̄M − rF = 5%

ABC, Inc. data:

• Expected EBIT (Earnings Before Interest and Taxes) = 2,000 in perpetuity

• Capital Expenditures (CapExp) = Depreciation (Dep) each year

• Net Working Capital (NWC) = 0

• Corporate tax rate = 40%

• Equity beta = 1.2

• Number of outstanding shares = 1,000

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Analysis:

1. What is the cash flow to shareholders or Free Cash Flow to Equity (FCFE)?

• Profit before tax = 2,000

• Tax = 0.40(2,000) = 800

• Net income = 2,000 - 800 = 1,200

• FCFE = Net income + Dep - CapExp + Increase in NWC = 1,200

ABC Inc. generates a cash flow to shareholders of 1,200 in perpetuity.Since there is no debt the FCFE equals the FCF.

2. What is the ABC Inc.’s unlevered cost of capital?

We use the Security Market Line (SML):

rU = rf + β(R̄M − rf ) = 2 + 1.2(5) = 8%. (8)

3. What is ABC, Inc.’s market value?

V U = E = Present Value (FCFE cash flows) @ rU =1, 200

0.08= 15, 000. (9)

4. What is ABC, Inc.’s share price?

Total firm value is 15,000.There are 1,000 shares, so each share is worth 15,000/1,000 = $15.

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4 A Leverage Recapitalization

ABC, Inc. is an unlevered firm. Suppose it decides to change its capital structure by taking onsome debt. It announces it will undertake a leverage recapitalization. It will:

• Issue 7,500 of debt.

– The interest rate is 6%.

– The debt will be perpetual :

∗ Interest is repaid each year.

∗ No repayment of principal.

• Use proceeds to buy back shares of stock.

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Cash flow analysis:

1. What is the new cash flow to shareholders or FCFE?

• Interest expense = 0.06(7,500) = 450

• Profit before tax = 2,000 - 450 = 1,550

• Tax = 0.40(1,550) = 620

• Net income = 1,550 - 620 = 930

• FCFE = Net income + Dep - CapExp + Increase in NWC = 930

ABC Inc. generates a cash flow to shareholders of 930 in perpetuity.

2. What is the cash flow to bondholders?

Interest expense = 450.

3. What is the total cash flow or new Free Cash Flow (FCF) generated?

FCF = 930 + 450 = 1,380.

4. How does this new FCF compare to the FCF when there was no debt?

The old FCF = FCFE = 1,200 since there was no debt.The cash flow has increased by 1,380 - 1,200 = 180.

5. Why did the FCF increase?

Look at the taxes.Before debt, the firm paid 800; after the debt issue, the firm will pay only 620.Difference is 180. This is the interest tax shield.Notice that

180 = TCrDD = (0.40)(0.06)(7, 500). (10)

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Market value analysis:

6. What is the market value of the interest tax shield?

What is the value to the firm for generating an extra 180 in perpetuity? It depends onthe choice of discount rate. One acceptable choice is the firm’s unlevered cost of equitycapital, 8%. With this choice the answer is 180/0.08 = 2,250. Notice that

2, 250 =TCrDD

rU. (11)

7. What is the new value for ABC, Inc.?

When the announcement is made the market prices in the value of the interest tax shield.So, the new value is 15,000 + 2,250 = 17,250.Notice that

17, 250 = V L = V U +TCrDD

rU= 15, 000 + 2, 250. (12)

8. What is the market value of debt?

Debt issue = 7,500. This is the market value of debt, D.

9. What is the new value of equity?

V L = 17, 250 = D + E = 7, 500 + E =⇒ E = 9, 750.

10. What is the new share price?

There are still 1,000 shares after the announcement but before the buyback.The new share price = 17,250/1,000 = $17.25.

11. How many shares will be repurchased and how many shares will be left?

The total amount of the purchase is 7,500.At $17.25/share, 7, 500/17.25 = 435 shares will be repurchased.This leaves 1, 000− 435 = 565 shares.Note that at $17.25/share, E = 565(17.25) = 9, 750, which is the same answer as before.

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Cost of capital analysis:

12. What is the new weighted average cost of capital?

Firm value equals the present value of the FCF cash flow stream discounted at rWACC .So,

V L = 17, 250 =1, 200

rWACC=⇒ rWACC =

1, 200

17, 250= 6.96%. (13)

13. What is new levered cost of capital?

Equity value equals the present value of the FCFE cash flow stream discounted at rE . So,

E = 9, 750 =930

rE=⇒ rE =

930

9, 750= 9.54%. (14)

It can be shown that in this setting

rE = rU + (rU − rD)D

E(15)

= 8 + (8− 6)7, 500

9, 750= 9.54%. (16)

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5 Cost of Levered Equity Capital

ABC, Inc.’s cost of unlevered equity is 8%. After leverage, however, the cost of levered equityincreased to 8.86%. Since the equity cash flows are now “riskier,” equityholders should demanda higher expected rate of return to compensate for this risk.

To drive home this point, we consider the following simple stylized example in which thereis only one period and there are no corporate taxes.

Before leverage:

• Market value of the firm is 1,000.

• At the end of the period the firm’s free cash flow will either be

– 1,400 if the market is good, or

– 900 if the market is bad.

Each market state is equally likely.

• The return on unlevered equity is

– 40% if the market is good, or

– -10% if the market is bad.

• The expected return on unlevered equity is 0.5(40) + 0.5(−10) = 15%.

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After leverage:

• Firm issues debt of 500 @ 5%.

• With no taxes there is no interest tax shield. Firm value remains at 1,000.

• E = 500 since V L = 1, 000 = D + E = 500 + E.

• At the end of the period, the shareholders have to pay back the principal plus interest= (1.05)500 = 525.

• At the end of the period, the cash flow to the shareholders or Free Cash Flow to Equity(FCFE) will either be

– 1,400 - 525 = 875 if the market is good, or

– 900 - 525 = 375 if the market is bad.

• The return on levered equity is

– 875/500 - 1 = 75% if the market is good, or

– 375/500 - 1 = -25% if the market is bad.

• The expected return on the levered equity is 0.5(75) + 0.5(−25) = 25%.

Due to leverage, the return distribution is more “skewed” (risky).Investors demand a higher risk-adjusted rate of return to compensate.The levered cost of equity capital is now 25%.

Conclusion:

Taking on debt does, in fact, increase the expected return on equity.But this is as it should be: the equity is now riskier.That is, the increase is not for “free.”

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6 Weighted Average Cost of Capital Formula

The market value of a levered firm or project is the present value of a firm’s free cash flowdiscounted at the rWACC . This is very useful as it allows a project manager to focus onforecasting the project’s cash flows, letting the Chief Financial Officer (CFO) worry about howthe firm will finance the project.

It can be shown that the weighted average cost of capital is given by this formula:

rWACC :=

(E

D + E

)rE +

(D

D + E

)[(1− TC)rD]. (17)

Observations:

• rE denotes the firm’s cost of equity capital given the firm’s leverage or capital structure.Do not use the firm’s unlevered cost of equity capital, rU , in this formula.

• If the firm is unlevered, then D = 0, and so rWACC = rU .

Alternate formula:

It can be shown that in this setting

rWACC = rU − TCrD(D/V L). (18)

To use this formula you must use the unlevered cost of equity capital.Note that rU = rWACC when TC = 0.

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Example 1 XYZ, Inc. has 5 million shares outstanding at $10 per share. The market valueof its debt is $25 million. The cost of equity capital (given the riskiness of the company’s cashflow) is 12%. Its cost of debt is 7.5%. The company’s tax rate is 40%.

Given this data:

• Market value of the firm’s equity is 5($10) = $50 million.

• After-tax cost of debt is (1− 0.4)(7.5) = 4.5%.

• The company’s weighted average cost of capital is

rWACC =

(50

25 + 50

)12% +

(25

25 + 50

)4.5% = 9.5%. (19)

• The company’s cost of unlevered equity capital can be obtained from the formula

rE = rU + (rU − rD)(D

E) (20)

12 = rU + (rU − 7.5)(25

50) (21)

= 1.5rU − 3.75 =⇒ rU = 10.5%. (22)

• The company’s weighted average cost of capital can also be computed as

rWACC = rU − TCrD(D/V L)) (23)

= 10.5− 0.40(7.5)(25/75) = 9.5%. (24)

Alert! To use this formula for rWACC you have to use the unlevered cost of equity capital.If the company is levered, you will obtain the levered cost of equity capital. From thisquantity, you must unlever it to obtain rU .

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Example 2 Let’s return to ABC, Inc.’s leverage recapitalization. Recall that

• V L = 17, 250

• D = 7, 500

• E = 9, 750

• rD = 6%

• TC = 40%

• rE = 9.54%

The new weighted average cost of capital is

rWACC =

(9, 750

17, 250

)9.54% +

(7, 500

17, 250

)[(1− 0.40)6%] = 6.96%. (25)

This matches our earlier calculation using the market definition of rWACC .

Alert! When calculating rWACC under leverage you must use the new levered cost ofequity capital. A common mistake is to use the unlevered cost of equity capital in the formula,as in:

rWACC =

(9, 750

17, 250

)8% +

(7, 500

17, 250

)[(1− 0.40)6%] = 6.09%. (26)

Of course, 6.09% 6= 6.96%.

Given an unlevered cost of equity capital, the weighted average cost of capital can also becomputed as

rWACC = rU − TCrD(D/V L) (27)

= 8− 0.40(6)(7, 500/17, 250) = 6.96%. (28)

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7 Estimating the Weighed Average Cost of Capital

We illustrate one approach to estimating rWACC . The example firm is The Home Depot, Inc.Data was obtained from Yahoo! Finance on October 10, 2011.

Data:

• Key Statistics page.

– Market Cap = E = 54.22B.

– beta = 0.82.

• Balance Sheet Quarterly Data page.

– Short/Current Long Term Debt + Long Term Debt= 0.044B + 10.731B = 10.775B (July 31, 2011).

– Cash and Cash Equivalents = 2.551B (July 31, 2011).

– Short/Current Long Term Debt + Long Term Debt= 0.043B + 10.720B = 10.763B (May 1, 2011).

• Income Statement Quarterly Data page.

– Income Before Tax = 2.148B (July 31, 2011).

– Income Tax Expense = 0.785B (July 31, 2011).

– Interest Expense = 0.149B (July 31, 2011).

• Bonds Center page.

– US Treasury 2-yr bond rate yield = 0.30%.

– US Treasury 5-yr bond rate yield = 1.16%.

– US Treasury 10-yr bond rate yield = 2.21%.

– US Treasury 30-yr bond rate yield = 3.19%.

Assumptions:

• Market risk premium = 5.1%.Reference:Market Risk Premium used in 2010 by Analysts and Companies: a survey with 2,400answers. P. Fernandez and J. del Campo. Working paper, May 2010.

• Risk-free rate = 2.21%.

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Calculations:

• E = 54.22B.

• D = 10.78B − 2.55B = 8.23B. This is the net debt.

• rE = 2.21 + 0.82(5.1) = 6.39%.

• Average debt over the current quarter = (10.775B + 10.763B)/2 = 10.769B.

• For the current quarter, rD = 0.149B/10.769B = 1.3836%.

• Annual debt cost of capital rD = (1.013836)4 − 1 = 5.65%.

• TC = 0.785B/2.148B = 36.5%.

• After-tax cost of debt = (1 - 0.365)(5.65) = 3.64%.

• An estimate of the weighted average cost of capital is

rWACC =

(54.22B

54.22B + 8.23B

)6.39 +

(8.23B

54.22B + 8.23B

)3.64 = 6.03%. (29)

Remarks:

• Estimating beta is a non-trivial exercise.Several companies provide estimates (e.g. Bloomberg, Value Line, S&P).

• Highest quality corporate bond rates are sometimes used in lieu of treasury rates.

• Estimate of the cost of debt capital here uses the observed interest expense.Expected return on corporate debt is lower due to the possibility of default.An alternative approach is to estimate a firm’s debt beta and then use the SML.

• Estimate of corporate tax here uses the effective tax rate.An alternative approach is to use the marginal tax rate.

• In principle, estimates should be forward looking.

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8 Estimating Project Cost of Capital

Example 3 A firm is considering investing in a construction-related business whose businessrisk is similar to that of The Home Depot, Inc. The firm’s equity is valued at 30M and its debtis valued at 20M. Its debt cost of capital is 6% and its corporate tax rate is 30%.

What is a good estimate of a weighted average cost of capital to be used for valuation ofprojects within this business?

• Home Depot, Inc.’s rWACC = 6.03%.

• Calculate The Home Depot, Inc.’s rU from its rWACC :

rWACC = 6.03% (30)

= rU − TCrD(D/V L) (31)

= rU − 0.365(5.65)(8.23/62.45) =⇒ rU = 6.30%. (32)

• Calculate the firm’s rWACC from rU :

rWACC = 6.30− 0.3(6)(20/50)) = 5.58%. (33)

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Example 4 Let’s go back to mid-year 2009. A firm has decided to start a new division thatwill specialize in general retail goods such as can be found in large department stores. It wishesto estimate a suitable weighted average cost of capital it can use to valuing projects within thisdivision.

Data and Assumptions:

• Security Market Line will be used to estimate the cost of equity and debt.

• A market premium of 5.1% and a risk-free rate of 2.21% will be used.

• Firm’s tax rate is 36%.

• Firm’s target debt-to-value ratio D/V L = 0.40.

• Firm’s debt is rated BBB.

• Data from seven department stores has been collected — see Tables ?? and ??.

Table 1: Data for US Department Stores, mid-2009.Reference: Example 12.7, p. 394 in Corporate Finance by Berk and DeMarzo.

Company Ticker Equity Beta D/V Debt RatingDillard’s DDS 2.38 0.59 B

J.C. Penney Company JCP 1.60 0.17 BBKohl’s KSS 1.37 0.08 BBBMacy’s M 2.16 0.62 BB

Nordstrom JWN 1.94 0.35 BBBSaks SKS 1.85 0.50 CCC

Sears Holding SHLD 1.36 0.23 BB

Table 2: Average Debt Betas by Rating and Maturity.Reference: Table 12.3, p. 389 in Corporate Finance by Berk and DeMarzo.

By Rating A and above BBB BB B CCCAvg. Beta < 0.05 0.10 0.17 0.26 0.31

By Maturity (BBB and above) 1-5 year 5-10 year 10-15 year > 15 yearAvg. Beta 0.01 0.06 0.07 0.14

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Analysis:

• Table ?? shows the analysis.

• Corporate tax rates were obtained from the 2010 annual reports.

• The cost of unlevered equity in the last column can be inferred from the weighted averagecost of capital or the equity cost of capital:

rWACC = rU − TCrD(D/V L) or rE = rU + (rE − rU )(D/E). (34)

Calculating rU from rE does not require knowledge of each company’s corporate tax rate.

• rD = 2.21 + 0.10(5.1) = 2.72%.

• rWACC = 8.14− 0.36(2.72)(0.40)) = 7.75%.

Table 3: Cost of Capital for US Department Stores, mid-2009.

Company rE rD D/V L TC rWACC rUDillard’s 14.35% 3.54% 0.59 0.15 7.66% 7.97%

J.C. Penney Company 10.37% 3.08% 0.17 0.38 8.93% 9.13%Kohl’s 9.20% 2.72% 0.08 0.38 8.60% 8.68%Macy’s 13.23% 3.08% 0.62 0.35 6.27% 6.93%

Nordstrom 12.10% 2.72% 0.35 0.37 8.47% 8.82%Saks 11.65% 3.79% 0.50 0.28 7.19% 7.72%

Sears Holding 9.15% 3.08% 0.23 0.29 7.54% 7.75%Average 8.14%

Remark: Define a firm’s asset beta as

βU =E

V LβE +

D

V LβD. (35)

It can be shown that in this setting

rU = rf + βU (R̄M − rf ). (36)

To obtain an average unlevered cost of capital, ravgU , proceed, as follows:

• Calculate the asset beta for each department store, βiU .

• Take the average of these asset betas to obtain βavgU .

• Use this average beta in the Security Market Line to obtain ravgU .

You will obtain the same answer as above!

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Example 5 Let’s revisit the previous example with current data — see Table ??. Analysis isshown in Table ??.

Table 4: Data for US Department Stores, Oct 2011.

Company Ticker Equity Beta D/V L Debt RatingDillard’s DDS 2.72 0.23 BB-

J.C. Penney Company JCP 1.94 0.19 BB+Kohl’s KSS 0.84 0.16 BBB+Macy’s M 1.79 0.31 BBB-

Nordstrom JWN 1.83 0.14 A-Saks SKS 2.76 0.12 BB

Sears Holding SHLD 1.92 0.29 B+

Table 5: Cost of Capital for US Department Stores, Oct 2011.

Company rE rD D/V L TC rWACC rUDillard’s 16.08% 3.08% 0.23 0.31 12.87% 13.09%

J.C. Penney Company 12.10% 3.08% 0.19 0.35 10.18% 10.39%Kohl’s 6.49% 2.72% 0.16 0.37 5.73% 5.89%Macy’s 11.34% 2.72% 0.31 0.36 8.36% 8.67%

Nordstrom 11.54% 2.47% 0.14 0.38 10.14% 10.27%Saks 16.29% 3.08% 0.12 0.34 14.58% 14.70%

Sears Holding 12.00% 3.54% 0.29 0.35 9.19% 9.55%Average 10.37%

New rWACC = 10.37− 0.36(2.72)(0.40)) = 9.97%.

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9 Project Valuation

The Weighted Average Cost of Capital (WACC) method:

• Most common approach.

• Estimate the project’s free cash flows. (Ignore financing.)

• Estimate the project’s rWACC :

– Find comparable firm’s whose business risk is similar.

– Estimate the unlevered cost of equity capital, rU .

– Calculate rWACC from rU using a target or estimated D/V L ratio.

– Use firm’s rWACC if project risk is similar to firm’s overall risk.

– Not an exact science!

• Estimate project value by discounting free cash flows @ rWACC .

• Perform sensitivity analysis!

The Adjusted Present Value (APV) method:

• Most useful when

– Firm does not maintain a constant D/V L ratio.

– When debt is tied to project (e.g. real estate).

• It has been argued that a firm’s or project’s levered value is

V L = V U + PV (interest tax shield). (37)

• Estimate the project’s free cash flows. (Ignore financing.)

• Estimate the project’s rU :

– Find comparable firm’s whose business risk is similar.

– Not an exact science!

• Estimate project’s unlevered value V U by discounting free cash flows @ rU .

• Estimate PV(interest tax shield) using unlevered cost of equity capital.

• Estimate project value by adding PV(interest tax shield) to V U .

• Perform sensitivity analysis!

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10 Remarks

• Costs of leverage.

Benefit of debt is the value of the interest tax shield. But there are costs to financing.

– Banks that provide the loan or underwrite the sale of the securities charge fees.

– Higher debt levels increase the likelihood of financial distress, which can lower thefree cash flow of the firm.

∗ Potential loss of customers, suppliers, employees.

∗ Fire sales of assets to generate cash.

∗ Loss of receivables.

– Personal taxes were ignored.This effect will lower the benefits of the interest tax shield.

For these reasons some argue the costs offset the benefits.Firms do issue debt. It is typically easier to raise capital via debt than equity.

• Interest tax shield valuation.

– Most debt is not perpetual. One can still calculate the PV(interest tax shield).

– If the company issues permanent debt, then the debt cost of capital is an appropriatediscount rate to value the interest tax shield.Assuming perpetual debt, using rD to discount the interest tax shield cash flowstream will generate somewhat different formulae, as follows:

V L = V U + TCD (38)

rE = rU + (rU − rD)(1− TC)(D/E) (39)

rWACC = rU(1− TC(D/V L)

)(40)

=

(E

D + E

)rE +

(D

D + E

)[(1− TC)rD] as before (41)

• Hurdle rates.

– A firm’s target rate-of-return

– Higher than the theoretical risk-adjusted rate of return (cost of capital).

– Applicable in capital budgeting:

∗ Capital constraints impose limitations on funds.

∗ Implicitly values option to invest in future possibly more profitable projects.

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11 Key Formulae

• V L = V U + TCrDDrU

.

• rE = rU + (rU − rD)(D/E).

• rWACC =(

ED+E

)rE +

(D

D+E

)[(1− TC)rD].

• rWACC = rU − TCrD(D/V L)).

• ri = rf + βi(R̄M − rf ), i = E, U and D.

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