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### Cost of Capital

What Number Goes in the Denominator?

Discount Rate(a.k.a. Cost of Capital, or the Interest Rate)

- Rate at which you can move money of similar risk through time.
- An individual moves money into the future by buying securities such as stock, bonds, and depositing funds into a savings, checking, or other bank account.
- An individual moves money from the future to today by borrowing, or selling securities.

Discount Rate and Companies

- A company moves money through time by buying and selling securities as well.
- It takes funds from the future and sells them for cash today. This moves money backwards in time (closer to today).
- It funds from today and moves them into the future (forwards in time) by buying real assets: think plant and equipment. These assets then (hopefully) produce future cash flows.

Discount Rates and Projects

- The discount rate for a particular set of cash flows is determined by the market risk of those cash flows.
- This means a single firm with multiple projects may need to use multiple discount rates.
- One for each project depending on that project’s market risk.

Practical Points About Determining the Discount Rate

- Companies often use higher discount rates for investments involving new products.
- Bad idea! The idiosyncratic (a.k.a. firm specific) risk associated with a new product is NOT market risk. It alters the expected cash flows (numerator of the present value equation) not the discount rate (the denominator of the present value equation).

Example

During the middle ages, In a previous life, you ran the Tail of Fairies Caviar Import Company. Among your best customers are the besieged aristocrats in the great and wonderful town of Nottingham.

Alas, in the forests lies the evil communist Robin Hood! As a result, there is a 20% chance any shipment of yours will be intercepted by him and lost.

Our Story Continues

It takes you four months to transport caviar from Russia to Nottingham. Assume you can purchase the caviar for £10. Shipping, whether everything goes well or not comes to £5.

In Nottingham the sale price of the caviar is, on average, £25, but the actual sales price depends on how well the local economy is doing. Thus, the price has a market beta of 1.5. Assume the annual risk free rate (rf) equals 2% and the market premium (rm-rf) for the same period equals 10%. Should you proceed to ship caviar to Nottingham?

Solution

- Question 1: Is Robin Hood risk market risk?
- NO! The covariance between the market return and having Robin Hood intercept your shipment is (according to the story) zero. Thus, it should not alter your discount rate.

Solution Continued

- Question 2: What is the correct discount rate?
- r = rf+β(rm-rf)
- rf for a year is 2%, and rm-rf is 10%. So the annual discount rate is rannual = .02+1.5(.1) = .17.
- Four months (time from purchase to sale) is a third of a year. So the discount rate 1.171/3 or 5.4%.

The Present Value of Caviar Shipping

Notice that the chance Robin Hood intercepts your shipment goes into the NUMERATOR. It reduces your expected payoff. Remember the numerator in the present value equation contains the expected cash flows from the project.

PV – More on the Practical Issues

- In some cases the investment calculations are difficult to carry out.
- Employee safety equipment investments.
- Investments for environmental reasons
- “Solution”
- If you must make an investment to meet regulations, or satisfy a labor agreement there is no need to calculate anything. You just have to do it.
- Otherwise, the standard PV calculations apply. You need to estimate productivity gains, legal costs, etcetera.
- The correct discount rate is likely to be the risk free rate as the cash flows are probably uncorrelated with the market.

WACC and APV

I am assuming future debt levels are a <blank> and I want to make my life <blank>:

Just About Impossible

Really Hard

Easy

Easy

Weighted Average Cost of Capital(WACC)

- Ninety-nine times out of a hundred people will use a firm's WACC as the discount rate for any project a firm undertakes. While this may be tempting, it only works if two important assumptions hold:
- The firm maintains a constant debt/equity ratio over its lifetime.
- The project has the same risk as the rest of the firm.

WACC Defined

WACC is defined by:

- where:
- D = dollar value of the outstanding debt
- E = dollar value of the outstanding equity
- rD = expected return on the firm’s debt
- rE = expected return on the firm’s equity
- TC = firm’s tax rate.

Using the WACC

- The WACC makes it easy to value marginal projects.
- Use market values to find E and D.
- Use past market returns to estimate rE and rD.
- Use the accounting statement to get TC.
- Once you have the WACC
- Calculate the firm’s cash flows under the assumption it is 100% equity financed.
- Discount these cash flows at the WACC to get the PV.

Example: Marginal Industries

Marginal Industries currently provides outfits for NFL backup players. It is considering the initiation of marketing project 5 at a cost of 1 million. The firm hopes that project 5 will increase sales by convincing the NCAA to purchase its clothing line. Assume sales to the NCAA will have the same level of market risk as the firm’s current business.

Marginal IndustriesThe Status Quo

- Current market value of:
- Equity = 20 million
- Debt = 10 million
- Current return on MI’s securities:
- rE = 18%
- rD = 8%
- Firm’s tax rate:
- TC = 40%

MI – 5 Project

- MI expects that the MI-5 project will generate revenues of 5 million a year and costs of 3 million per year starting in year 1.
- MI expects these cash flows to continue in perpetuity, but grow at a rate of 2% per year.
- Question: How much is the MI-5 project worth?

Valuing the MI-5 Project

- First calculate the cash flows under the assumption the firm is all equity financed.
- All Equity After Tax Cash Flow =

(1-TC)(Revenues-Costs) = .6(2) = 1.2.

- Second calculate the WACC

Discount MI-5 with the WACC

- And the answer is:
- PV = After tax investment cost + PV of growing profit perpetuity.
- PV = -1(.6) + 1.2/(.136-.02) = 9.74.

WACCValuing an Entire Firm

- Good news:
- You can get Tc from financial statements.
- Debt values are often close to their book values. If this is true for your firm you now have D.
- rD can be obtained by looking at the historical return to debt from firms with similar ratings.
- Lazy persons solution use the coupon rate. This will over estimate the cost of debt. Investors expect some debt issues to go “belly up” and not pay off in full. Their expected return is overall average return from those firms that pay in full and those that do not.
- If you are not sure what a “rating” is we will cover it later on this semester.

WACCValuing an Entire Firm

- Bad news:
- You need E.
- The I am not thinking solution: use the market value of the equity.
- But . . . you (for some reason) are trying to value the firm on your own. If you are going to use the market value why go through all this work? Here you are projecting out the cash flows, finding the discount rate and getting a PV for what? Just use the market value.
- You need rE
- Here you at least have some hope if you can figure out how to calculate the equity beta without using the market value of the equity.
- If there are several firms in the industry and all have similar capital structures you might try using the average return on the industry’s equity over the past several years.

Typical Solutions

- Go with the “I am not thinking method” of analysis.
- Get E from the market price of the equity.
- Get rE by using the historical beta on the firm’s equity.
- Project the cash flows and discount with the WACC.

Solutions Continued

- Use market data from comparable firms.
- This can be combined with method 1.
- Under this method you use debt and equity values from comparable firms and then apply the results to the firm you are interested in.
- If the firms differ in their capital structure you will need to adjust the estimated betas to account for this.

Adjusting returns for Leverage

- Definition: A company’s asset return (rA) is the return associated with the cash flows it would generate under all equity financing.
- If the firm has a constant D/E ratio then:

How to Value Your Company

- For each comparable firm estimate rA using the previous formula (just rearrange for rA).
- For each firm use its E, D, rD, rE, and TC.
- Calculate the average value of rA and assume that is the asset return for your firm.
- Calculate the equity return using the previous equation.

If You Like Betas Better

The formula for beta is identical to the return formula except that the r’s are now replaced with β’s.

Do You Believe in the WACC?

- If you really believe, why not also believe that firms adjust their debt-equity ratio to the target level 24 hours a day seven days a week?
- Why? Makes it really easy to calculate asset betas!

Unlevering β With When Debt-Equity Ratio is Adjust Continuously

- As the time between adjustments goes to zero, the per period risk free rate rD goes to zero.
- This causes the rD/(1+rD) terms in the equations relating equity betas and returns to asset and debt betas and returns to go to zero. Leaving:

and

These are the equations for delevering absent taxes!

Valuing Your Companycontinued

- Using your estimate for the equity beta calculate rE.
- Now calculate the WACC.
- Project out the firm’s all equity cash flows.
- Discount at the WACC and declare victory!

Problems with the Comparable Firm WACC

- For some reason you do not believe the market is correctly valuing the firm you are looking at.
- But at the same time you seem to think the market has correctly valued every other firm in the industry.
- Very handy pair of coincidences, no?

Yet Another Solution

- You know what they say the third time is the charm!
- If you believe the firm will target its D/(D+E) ratio to some number L, and that eventually the market will get the firm’s value right then you can use the Miles and Ezzell formula for the WACC.
- For use later on represent the Enterprise Value as EV = D+E. In this notation L = D/EV.

Miles and Ezzell (ME) WACC Formula

- Looks easy but . . .
- Where are you going to get rA from?
- Back to the comparable firm method?
- As an aside this is the formula used to derive the relationship between rA, rD, and rE developed before.

Some Questions

- Based upon the ME formula setting L = 1 is optimal since the WACC declines as L goes up!
- Why?
- What stops this from happening in the real world?
- What has the formula ignored as you change L?

WACC Example

- Golden Products (GP) has stated its long term goal is to maintain a D/E ratio of .75.
- Assume it has an asset beta of 1.618.
- The firm will produce pre-tax profits of $1.50 million next year. You expect this to grow at a rate of 1% per year forever.
- The firm’s tax rate is 30%
- Assume rf = 5%, and rm = 15%.
- How much is a share of GP worth?

Solution

- Since GP will maintain a constant D/E ratio we can use the WACC to value its cash flows.
- rA = .05 + 1.618(.15-.05) = .2118

Solution Continued

- From the ME formula

- Fill in the perpetuity formula to get GP’s enterprise value (EV).

- To get the equity value subtract the value of the debt from the enterprise value.
- E = EV – (D/EV)xEV = 5.4 – (3/7) ×5.4 = 3.087.

Target Amount of Debt

- So far every problem has assumed the firm maintains a target D/E ratio. Not every firm does this.
- Highly levered firms may be looking to reduce their outstanding debt.
- Firms with little debt may seek to increase it substantially to finance a new project and then pay it off.
- If the ratio is not constant you cannot use WACC to get the present value of the cash flows.

Adjusted Present Value(APV)

- APV = PV(All Equity Cash Flows) + PV(Debt Tax Shield).
- Advantages
- Easy to use when a firm has a target dollar value of debt.
- Can be adjusted (with some work) for cases where a firm’s dollar value of debt varies over time.
- Can handle (again with some work) even the case where the firm maintains a constant D/E ratio.

Constant $ Amount of Debt Example

- The Perpetual Rest Lounge Company (PRLC) has $12 million in outstanding debt, and it does not plan to either add to or subtract from this.
- The firm pays interest on the debt at a rate of 5% per annum. Assume the debt is risk free.
- Starting next year the firm expects to generate earnings before interest and taxes (EBIT) of $2 million, and that this figure will grow at a rate of 1% per year.

PRLC Continued

- Assume
- rf = 5%
- rm = 15%
- βA = .8
- Tc = .4
- What is PRLC’s enterprise value?
- What is the value of PRLC’s equity?

PRLC Solution

- First calculate PRLC’s value as an all equity firm. In this case it produces after tax profits of (1-.4)2 = 1.2 in year 1 and this grows at 1% per year.
- Since the asset beta is .8, its

rA = .05+.8(.15-.05) = .13.

- PV(All Equity) = 1.2/(.13-.01) = 10.

PRLC Continued

- Now calculate the PV of the debt tax shield.
- The company pays 12×.05 = 0.6 in interest each year. This generates a tax shield of 0.6×.4 = .24 per year. Because this amount never varies and is risk free

PV(Debt Tax Shield) = .24/.05 = 4.8.

- EV of PRLC =

PV(All Equity Cash Flows) + PV(Debt Tax Shield) =

10 + 4.8 = 14.8.

- Note: Since PRLC is growing its D/E ratio declines over time! It does not maintain a constant D/E ratio so the WACC will not generate the correct EV for the firm.

Adjusting the APVConstant D/E Ratio Case

- The APV can be adjusted to handle the very case the WACC is designed for: the constant D/E case.
- The key to understanding how is through the famous question, “Just how much debt will you have and when will you know it?”
- An example will explain why.

Valuing Golden Products with the APV

- The first part is relatively easy
- Calculate the value of the firm without any debt financing.
- After tax cash flows start at 1.5×.7 = 1.05 and grow at 1% per year.
- Have rA equal to .2118.

All Equity Value

- All Equity Value =

1.05/(.2118-.01) =5.203171

- Yes, a lot of decimal places but we want to show the APV gives the same exact answer as the WACC when both are calculated correctly.

Debt Tax Shield

- Calculating the PV of the debt tax shield is somewhat tricky until you get the hang of it.
- The problem is the amount of debt the firm will have outstanding depends on how well it does over time.
- Good years → more debt → higher tax shield.
- Bad years → less debt → lower tax shield.
- Tax shield is pro-cyclical with the economy as so has a positive beta!
- From the original problem D starts at 2.32. Use that to begin the debt tax shield calculation.

The Tax Shield: What do You Know and When?

- The following table describes the expected debt tax shield. Arrows indicate which debt amounts generate which tax shields.

Expectations vs. Realizations

- The year one debt level produces the year two tax shield.
- In year one you will know with certainty what the year two tax shield will be.
- Prior to that you are not certain.
- The firm is expected to grow at a rate of 1% per year. In actuality it may grow faster or slower. That will impact the amount of debt it will eventually have.

Numerical Example

- Each year the firm’s value goes up by either 22% or down by 20% with equal probability. Notice that on average it grows at 1%.
- Year 0: No debt tax shield

Year 1 Debt Tax Shield

Today you know for sure what next period’s tax shield will be. Therefore you get its present value by discounting at the risk free rate of .05, assuming its debt is risk free.

PV(Year 1 tax shield) = .0348/1.05.

Year 2 Tax Shield Discounted

- Today you do not know with certainty what the period 2 tax shield will equal.
- Depends upon whether year 1 is good or bad.
- Does not depend on whether year 2 is good or bad. I
- Impacted by just one year of market risk.
- Year 2 tax shield is known for sure when we reach year 1.
- Discount the expected tax shield back from year 2 to year 1 using the risk free rate of 5%.
- Between today and year 1, the amount of debt (and hence the year 2 tax shield) will increase or decrease in proportion to the firm’s value. It has the same risk as the enterprise as a whole.
- Use the same discount rate as for the firm’s assets, 21.18%, to discount back from year 1 to the present.
- PV(Year 2 tax shield) = .0348/(1.05×1.2118).

Year t Tax Shield

- This is determined by the amount of debt outstanding in year t-1, which goes up or down in proportion to the value of the firm for t-1 years.
- Then is known for sure as of year t-1.
- Discount back from year t to t-1 using the risk free rate, 5%.
- From year t-1 to today using the appropriate rate for the firm’s assets, 21.18%.
- PV(Year t tax shield) = .0348×1.01t-1/(1.05×1.2118t-1).

PV of the Debt Tax Shield

PV (tax shields) =

.0348/1.05 +

.0348×1.01/(1.05×1.2118) +

.0348×1.012/(1.05 × 1.21182) + …

=.0348/1.05 +

[.0348/(.2118-.01)][1.01/1.05]

= .199

APV Solution

- APV = PV(All Equity Firm) + PV(Debt Tax Shield).
- APV = 5.203171+.199 = 5.40.

APV Your Friend in Tough Times

- Many people use the WACC to value the cash flows for any project any firm undertakes.
- WACC only accurate if the firm maintains a constant debt to equity ratio.
- What if the firm only adjusts its debt to maintain a constant debt to equity ratio once and a while rather than every period? How many firms issue debt every quarter or even every year?
- How poorly will WACC perform?
- How do you discount the cash flows with the APV to get the exact answer?

Example: Only a Somewhat Constant Debt to Equity Ratio

- Sam’s Toasters and Appliances (STA)
- STA (known in the industry by the less flattering nickname Slow To Adjust) has decided to expand into the market for stereo equipment.
- STA believes that the expansion will produce an additional $400,000 in profits per year starting in year one and that these profits will grow at a rate of 2% per year.
- The expansion will cost STA $1,000,000.
- Initial financing $700,000 in risk free debt. The rest with equity.
- This is in line with the firm’s current debt equity policy.
- As in the past, the firm will adjust its debt equity ratio every other year in order to restore the debt-equity ratio to its current value.

Overall Assumptions

- Beta of the profits equal 1.2.
- rf = .05
- rm = .15
- Tc = .4
- How much is the expansion worth?

All Equity Value

- The discount rate equals .05 + 1.2(.15-.05) = .17.
- With all equity financing the firm keeps (1−.4)400000 = 240,000 per year starting in year 1 with an expected growth rate of 2%.
- All equity value of the project equals the initial cost plus the present value of the profits or:

PV(All Equity Financed) =

−1000000 + 240000/(.17-.02) = 600,000.

Deciphering the Next Table

- Diagonal arrows indicate which debt amounts produce which tax shields.
- Arrows pointing straight down indicate when one debt amount will automatically equals the amount of debt in the previous year.
- The problem states that the firm only adjusts its debt-equity ratio every other year. Thus, once you know the amount of debt taken out in year 0 you know how much debt the firm will have in year 1.
- Similarly when you know how much debt the firm has in year 2 you will also know how much debt it has outstanding in year 3.

PV of the Debt Tax Shield: Year 1

- You discount the first year’s tax shield by the risk free rate since you know with certainty what it will be as of today. There is no market risk.

PV(Year 1 Tax Shield) = 14,000/1.05.

PV of the Debt Tax Shield: Year 2

- You twice discount the second year’s tax shield by the risk free rate.
- The firm only adjusts its debt-equity ratio every other year. Year 0 debt value fixes the year 2 debt tax shield. There is no uncertainty.
- Since it arrives two periods hence you discount it twice at the risk free rate.

PV(Year 2 Tax Shield) = 14,000/1.052.

PV of the Debt Tax Shield: Year 3

- You know the exact value of the year 3 tax shield in year 2.
- There are 2 periods of market risk associated with its value.
- Discount twice at the asset’s discount rate of 17%.
- Once year 2 arrives you know with certainty the year 3's tax shield.
- One year should be discounted at the risk free rate.

PV(Year 3 Tax Shield) = (14000×1.022)/(1.172×1.05).

PV of the Debt Tax Shield: Year 4

- You know the exact value of the year 4 tax shield in year 2.
- There are 2 periods of market risk associated with its value.
- Discount twice at the 17% rate.
- Once year 2 arrives you will know with certainty the year 4 tax shield.
- Discount twice at the risk free rate of 5%.

PV(Year 4 Tax Shield) = (14000×1.022)/(1.172×1.052).

PV Tax Shields

To go from line one of the above equation to line two note that 1.316 = (1.17/1.02)2.

Total Value of the Debt Tax Shield

- PV(Total Tax Shield) = 55,561.64 + 52,915.85 = 108,477.49.
- Therefore the APV of the project equals 600,000 + 108,477.49 = 708,477.49.
- While that was a lot of work it does show how flexible the APV is!

WACC: How Far Off?

- How far off would the calculation have been if the WACC had been applied instead?
- In order to apply the WACC we need both the initial debt to equity ratio, and the return on equity.
- The calculations that follow are not right! They assume that the firm will maintain a constant debt-equity ratio and it does not!
- Our goal is to find out how far off we would have been by mindlessly applying the WACC.

WACC Calculations

- The entire project is really worth 708,477.49, and presumably the market knows this. Thus, if you looked up the value of the equity for the project in year 0 it would equal
- E = 708,477.49 - 700,000 = 8,477.49.
- This is the value of the project minus the value of the debt issued to finance it. So it would appear the firm uses a debt-equity ratio of 700000/8,477.49 = 82.57!

WACC: rE Calculation

- From before the formula relating the equity, debt, and asset returns under the WACC:

Filling this in with the information we have:

which implies rE = 10.72, that is NOT a percent. It is 1,072% due to the very high debt equity ratio!

The WACC

- so WACC = .1579.
- The value you obtain via the mindless application of the WACC is
- −1000000 + 400,000(1-.4)/(.1579-.02) = 740,391.58.
- This is an overestimate of 31,914.10 or 4.5% of the project’s value.

Where Does the WACC Go Wrong?

- The WACC assumes the firm readjusts its debt-equity ratio every year.
- In reality this firm only does so every other year.
- Among other errors:
- Every other year when the firm has not adjusted its D/E ratio, the WACC assumes it did.
- This causes the WACC to use too large a value for the expected tax shield in those years.
- Of course, the error is even larger for firms that adjusts their debt-equity ratio even less frequently.

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