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
.8  25
PV   15 
 £3.98.
1.054
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>:
Constant $
Amount
Just About
Impossible
Constant Proportion of
Firm Value
Really
Hard
Easy
Easy
APV
WACC
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:
D
E
WACC 
rE .
1  TC  rD 
D E
D+E
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 Industries
The 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
10
20
WACC =
.18  .136.
.6.08 
10+20
10  20
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.
WACC
Valuing 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.
WACC
Valuing 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
1. 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
2. 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:
 D  TC rD    D  TC rD  
rE = rA 1+ 1  -rD  1 .
 E  1+rD    E  1+rD  
How to Value Your Company
1. 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.
2. Calculate the average value of rA and assume
that is the asset return for your firm.
3. 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.
 D  TC rD  
 D  TC rD  
β E = β A 1+ 1  -β D  1 .
 E  1+rD  
 E  1+rD  
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:
 D D
rE = rA 1+  -rD ,
E
 E
and


 E =  A 1+
D
D

.
D

E
E
These are the equations for delevering absent taxes!
Valuing Your Company
continued
4. Using your estimate for the equity beta
calculate rE.
5. Now calculate the WACC.
6. Project out the firm’s all equity cash
flows.
7. 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
3. 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
1+rA
WACC = rA - LTC rD
.
1+rD
• 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
D
D
D/E
.75 3
L=


=
= .
EV D+E D/E+1 1.75 7
Solution Continued
• From the ME formula
3
1.2118
WACC = .2118 -  .3  .05
 .204381
7
1.05
• Fill in the perpetuity formula to get GP’s enterprise value
(EV).
EV =
1-.31.5
.204381-.01
= 5.40 .
• 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.
Asset to Equity Betas
Constant Amount of Debt Case


 D 1-TC  
E
rA = rE 
 +rD 
.
 E+D 1-TC  
 E+D 1-TC  


 D 1-TC  
E
βA = βE 
 +β D 
.
 E+D 1-TC  
 E+D 1-TC  
Adjusting the APV
Constant 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.
Year
Expected Debt Level
Expected Tax Shield
0
2.32
1
2.34
2
2.36
.0351
3
2.39
.0355
2.32×.05×.3=.0348
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
Year 1
Economy
Good
Bad
Year 1
Tax Shield 2.32×.05×.3 = .0348 .0348
Debt
2.32×1.22 = 2.83
2.32×.8 = 1.856
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
Year 1
Economy
Good
Good
Bad
Bad
Year 2
Economy
Good
Bad
Good
Bad
Year 2
Tax Shield .0425
.0425
.0278
.0278
Debt
2.264
2.264
1.485
3.453
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 debtequity 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.
Tracking the Debt Tax Shield
Year
Expected Debt Level
Expected Tax Shield
0
700,000
1
700,000
.05(.4)(700000)=14000
2
1.022×700000=728,280
.05(.4)(700000)=14000
3
1.022×700000=728,280
.05(.4)(728280)=14565.6
4
1.024×700000=757,702.51
.05(.4)(728280)=14565.6
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
14000 14000 1.022 14000 1.024
PV (Odd Years) 



2
4
1.05
1.05 1.17
1.05 1.17
14000
1 14000 14000 14000







1.05 1.05  1.316 1.3162 1.3163
 55,561.64.
To go from line one of the above equation to line two note
that 1.316 = (1.17/1.02)2.
14000 14000 1.022 14000 1.02 4
PV ( Even Years) 



2
2
2
2
4
1.05
1.05 1.17
1.05 1.17
14000
1 14000 14000 14000







1.052 1.052  1.316 1.316 2 1.3163
 52,915.85.
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:
 D  TC rD
rE = rA 1+ 1 E  1+rD
  D  TC rD 
  -rD 1.
E  1+rD 

Filling this in with the information we have:
rE

 .4  .05  
 .4  .05 
 .17 1  82.57 1 
   .05  82.57 1 

1.05
1.05





which implies rE = 10.72, that is NOT a percent. It is
1,072% due to the very high debt equity ratio!
The WACC
WACC
82.57
1

.05(1  .4) 
10.72
1  82.57
1  82.57
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 debtequity 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.