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UGBA 103 (Parlour, Spring 2015), Section 11
Capital Structure, APV, and Linear Regression
Raymond C. W. Leung
University of California, Berkeley
Haas School of Business, Department of Finance
Email: [email protected]
Website: faculty.haas.berkeley.edu/r_leung/ugba103
April 24, 2015
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §11
April 24, 2015
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Outline
1
Adjusted Present Value (APV)
APV Rule
2
(SEMI-OPTIONAL) Linear Regression
3
Examples
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Section 1
Adjusted Present Value (APV)
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Subsection 1
APV Rule
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The Adjusted Present Value (Rule)
Recall that we can write the value of a levered firm as the value of an otherwise identical
all-equity firm and the value of its financing decisions:
V = VU + NPV (financing decisions)
It is then obvious to define the APV of a project as the sum of its NPV to an all-equity firm and
the PV of the associated financing decisions,
APV = ∆V = ∆VU + ∆NPV (financing decisions)
That is to say,
APV = NPV (unlevered project) + NPV (change in financing decisions)
Separating the APV of a project into its NPV to an all-equity firm and the value of the
associated financing decisions hsould be generally useful for the financial manager.
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The Adjusted Cost of Capital Rule
While the APV approach is very flexible and takes into account different side effects of
accepting a project (e.g. issue costs, taxes), many firms use a simpler procedure that focuses
on the tax impact alone. This procedure discounts the project cash flows at an adjusted cost
of capital which reflects both the project’s risk and financing tax effects.
Unfortunately, there is no simple, universally correct method of calculating the adjusted cost
of capital, but only some short hand formulas. We will look at two of these formulas:
I
I
The WACC (“debt rebalanced”)
MM formula (“debt fixed”)
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The Adjusted Cost of Capital Formulas
The WACC formulas assumes the project maintains a constant D/V ratio over time:
WACC = (1 − tC )rD
D
V
+ rE
E
V
The MM formula assumes that the project supports a permanent additional debt ∆D issued
at rD .
∆D
∆D
∗
rMM
= rA (1 − tC L), where L =
=
APV + CF0
PV (project)
represents the project’s contribution to the firms’ debt capacity as a proportion of the project’s
market value. This is MM’s formula.
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Section 2
(SEMI-OPTIONAL) Linear Regression
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Linear regression — Excel punchline
Step by step guide
If you don’t want to know the theory and just want to get the results, the step-by-step guide to
linear regression is the following.
Step 1. Select your data. To select data, click the top left corner of your data set and then drag the
cursor down to the bottom right
Step 2. Select the “Insert” tab.
Step 3. Select “Scatter” in the Charts area and then click the first scatter graph (a simple scatter
graph).
Step 4. Click a data point on the graph then right click.
Step 5. Click “Add trendline” from the drop down menu.
Step 6. Scroll to the bottom of trendline options. Trendline Options is at the far right of the screen.
Step 7. Click the check boxes for Display R-squared value on chart and Display equation on chart.
R-squared and the regression equation will appear on the chart.
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(OPTIONAL) Linear regression — the smarter and faster way
Even with a 30-minute basic training in a proper statistical software (i.e. R, Stata, or even
statistical packages in programming languages like Python or MATLAB), you can do all those
tedious Excel steps much more efficiently and effectively. For example, to run a linear regression
in R:
# Multiple Linear Regression Example
fit <- lm(y ∼ x1 + x2 + x3, data=mydata)
summary(fit) # show results
See here for more details: http://www.statmethods.net/stats/regression.html.
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(SEMI-OPTIONAL) Linear regression
Linear model
I label this part as SEMI-OPTIONAL because:
1
2
You should have seen linear regression in your previous statistics / econometrics courses in
one form or another.
The level of mathematical detail in your previous statistics / econometrics course may vary.
That is, you may have only seen the application of linear regression, but not necessarily the
background mathematical and technical details.
I
I
If you haven’t seen linear regression model to this level of mathematical detail, then this is not the
appropriate course to talk about it.
I will only use a few lines to fix and introduce ideas — the subsequent formulas are not going to be
examinable.
For the purpose of this course:
Know how to run linear regression in any software package (say Excel). Period.
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Linear model
Suppose we have a linear model of the form, 1
Yi = α + Xi β + εi ,
i = 1, 2 . . . , n.
Suppose further that we have n observations {(Yi , Xi )}ni=1 of data. We need to estimate (α, β)
from the data.
1
We’re using one intercept and one regressor here. Clearly, we can use one intercept and K regressors,
Yi = α + X1i β1 + X2i β2 + . . . + XKi βK + εi
We’ll need linear algebra to handle this case, but linear algebra is not a course prerequisite to this course. . .
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OLS Estimator
ˆ of (α, β) are given by,
The ordinary least squares (OLS) estimator 2 (α,
ˆ β)
α
ˆ
n
= Pn
βˆ
i=1 Xi
−1 Pn
Pn
Pn i=1 Yi ,
Pni=1 X2i
i=1 Xi Yi
i=1 Xi
or more explicitly,
1
α
ˆ = Pn
P
n i=1 Xi2 − ( ni=1 Xi )2
"
1
Pn
2
2
i=1 Xi − ( i=1 Xi )
"
βˆ =
2
n
Pn
n
X
!
Xi2
i=1
n
n
X
i=1
n
X
!
−
Yi
i=1
Xi Yi −
n
X
i=1
n
X
n
X
!
Xi
i=1
!
Xi
n
X
!#
Xi Yi
i=1
!#
Yi
.
i=1
Also known as the least squares estimator.
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(OPTIONAL) More about econometrics and finance
More of my philosophy!
For the purpose of this course, the above estimation discussion will suffice to handle very
“basic and elementary” data work.
However, to really handle financial data, there are a lot more very complicated subtleties that
we need to handle, and that’s really far outside the scope of the course.
Essentially, financial data exhibit behavior that are complicated enough that the above
estimation methods will fail. That is to say, return data of stocks and other risky assets exhibit
behavior such as:
I
I
I
I
I
I
Stochastic volatility
Volatility clustering
Momentum and reversal
Leverage effect
Jumps
Factor effects
and many, many, many others that have been documented. To have a full treatment of these
econometric and statistical issues will take us far beyond the scope of an intro finance course.
And yet, if one can construct a very good statistical and econometric model that fits financial
data well (and consistently!), there is a fortune to be made. . . But I’m not sure if there exist a
model right now that can do everything.
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Section 3
Examples
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Problem 18-2
Suppose Caterpillar, Inc., has 665 million shares outstanding with a share price of $74.77, and
$25 billion in debt. If in three years, Caterpillar has 700 million shares outstanding trading for $83
per share, how much debt will Caterpillar have if it maintains a constant debt-equity ratio?
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Problem 18-2
Solutions
Initially, the equity value is E = 665 × $74.77 = $49.7bil and the debt value is $25bil. This implies,
D/E =
25
= 0.503.
49.7
After 3 years, the equity value becomes E 0 = 700 × $83.00 = $58.1bil. Maintaining the same
debt-to-equity ratio implies,
E 0 × D/E = $58.1 × 0.503 = $29.2bil.
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Problem 18-4
Suppose Goodyear Tire and Rubber Company is considering divesting one of its manufacturing
plants. The plant is expected to generate free cash flows of $1.5 million per year, growing at a rate
of 2.5% per year. Goodyear has an equity cost of capital of 8.5%, a debt cost of capital of 7%, a
marginal corporate tax rate of 35%, and a debt-equity ratio of 2.6. If the plant has average risk and
Goodyear plans to maintain a constant debt-equity ratio, what after-tax amount must it receive for
the plant for the divestiture to be profitable?
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Problem 18-4
Solutions
We can compute the levered value of the plant using the WACC method. The firm’s WACC is,
rWACC =
1
2.6
× 8.5% +
× 7% × (1 − 0.35) = 5.65%.
1 + 2.6
1 + 2.6
Therefore, the levered value of the firm at t = 0 is, via growing perpetuity,
V0,L =
1.5
FCF1
=
= $47.6mil.
rWACC − g
0.0565 − 0.025
Hence, a divestiture would be profitable if the firm received more than $47.6mil after tax.
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Problem 18-5
Suppose Lucent Technologies has an equity cost of capital of 10%, market capitalization of $10.8
billion, and an enterprise value of $14.4 billion. Suppose Lucent’s debt cost of capital is 6.1% and
its marginal tax rate is 35%.
(a) What is Lucents WACC?
(b) If Lucent maintains a constant debt-equity ratio, what is the value of a project with average risk
and the following expected free cash flows?
Year
FCF
0
-100
1
50
2
100
3
70
(c) If Lucent maintains its debt-equity ratio, what is the debt capacity of the project in part (b)?
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Problem 18-5
Solution
(a) We compute the WACC,
14.14 − 10.8
10.8
× 10% +
× 6.1% × (1 − 35%) = 8.49%.
14.4
14.4
(b) Using the WACC method, the levered value of the project at date 0 is,
WACC =
VL =
50
100
70
+
+
= $185.86.
1 + 0.0849
(1 + 0.0849)2
(1 + 0.0849)3
(c) The firm’s debt-to-value ratio is D/V = (14.4 − 10.8)/14.4 = 0.25. The project’s debt
capacity (i.e. how much debt D it wants to take on), is simply clearly,
D=
D
× VL = 0.25 × VL .
VL
But by simply discounting, we can quickly compute that the value of the levered firm VL,t at
time t is,
VL,0 = 185.86,
VL,2 = 64.52,
Thus, putting this back in table form,
Year
0
FCFt
-100
VL,t
185.86
Dt
46.47
VL,1 = 151.64
VL,3 = 0
1
50
151.64
37.91
2
100
64.52
16.13
3
70
0
0
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Problem 18-6
Acort Industries has 10 million shares outstanding and a current share price of $40 per share. It
also has long-term debt outstanding. This debt is risk free, is four years away from maturity, has
annual coupons with a coupon rate of 10%, and has a $100 million face value. The first of the
remaining coupon payments will be due in exactly one year. The riskless interest rates for all
maturities are constant at 6%. Acort has EBIT of $106 million, which is expected to remain
constant each year. New capital expenditures are expected to equal depreciation and equal $13
million per year, while no changes to net working capital are expected in the future. The corporate
tax rate is 40%, and Acort is expected to keep its debt-equity ratio constant in the future (by either
issuing additional new debt or buying back some debt as time goes on).
(a) Based on this information, estimate Acort’s WACC.
(b) What is Acort’s equity cost of capital?
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Problem 18-6
Solutions
(a) We don’t know Acort’s equity cost of capital, so we cannot calculate WACC directly. However,
we can compute it indirectly by estimating the discount rate that is consistent with Acort’s
market value. First, E = 10 × 40 = $400 million. The market value of Acort’s debt is (i.e.
growing perpetuity), and note the coupon dollar per year is $100mil × 10% = $10mil, and so,
1
1
100
D = $10mil ×
1−
+
= $113.86mil.
0.06
(1 + 0.06)4
(1 + 0.06)4
Therefore, Acort’s enterprise value is,
E + D = $400 + $113.86 = $513.86mil.
The firm’s free cash flows,
FCF = EBIT × (1 − τC ) + Depr − CAPEX − Inc in NWC
= $106 × (1 − 0.40) + $13 − $13 − $0
= $63.6mil.
Finally, since the firm is not expected to grow, it implies that the value of the levered firm is,
VL =
FCF
rWACC
=⇒
$513.86 =
63.6
,
rWACC
solving gives rWACC = 12.38%.
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Problem 18-6
Solutions
(b) Using,
rWACC =
D
E
rE +
rD (1 − τC ),
E +D
E +D
and substituting,
12.38% =
400
113.86
rE +
(6%)(1 − 0.40),
513.86
513.86
solving gives rE = 14.88%.
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Problem 18-7
Suppose Goodyear Tire and Rubber Company has an equity cost of capital of 8.5%, a debt cost of
capital of 7%, a marginal corporate tax rate of 35%, and a debt-equity ratio of 2.6. Suppose
Goodyear maintains a constant debt-equity ratio.
(a) What is Goodyear’s WACC?
(b) What is Goodyear’s unlevered cost of capital?
(c) Explain, intuitively, why Goodyear’s unlevered cost of capital is less than its equity cost of
capital and higher than its WACC.
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Problem 18-7
Solutions
(a) Compute,
2.6
1
× 8.5% +
× 7% × (1 − 0.35) = 5.65%.
1 + 2.6
1 + 2.6
rWACC =
(b) We want to compute rA . From MM Prop I and Prop II, we know that,
E
D
rE +
rD (1 − τC )
E +D
E +D
E
D
D
=
rE +
rD +
rD τC
E +D
E +D
E +D
|
{z
}
rWACC =
=rA
Thus, we can compute that,
rA =
2.6
1
× 8.5% +
× 7% = 7.42%.
1 + 2.6
1 + 2.6
(c) Goodyear’s equity cost of capital exceeds its unlevered cost of capital because leverage
makes equity riskier than the overall firm. Goodyear’s WACC is less than its unlevered cost of
capital because the WACC includes the benefit of the interest tax shield.
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Problem 18-9
Consider Lucent’s project in Problem 5.
(a) What is Lucent’s unlevered cost of capital?
(b) What is the unlevered value of the project?
(c) What are the interest tax shields from the project? What is their present value?
(d) Show that the APV of Lucents project matches the value computed using the WACC method.
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Problem 18-9
Solution
(a) The cost of capital of the unlevered firm,
rA =
14.4 − 10.8
10.8
× 10% +
× 6.1% = 9.025%.
14.14
14.4
(b) We discount using the unlevered cost of capital rA . That is,
VU =
50
100
70
+
+
= $184.01.
1 + 0.09025
(1 + 0.09025)2
(1 + 0.09025)3
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Problem 18-9
Solution
(c) Note that the interest expense Intt = rD × Dt , where here rD = 6.1%, and the interest tax
shield Interest tax shieldt = Intt × τ , where τ = 35% is the corporate tax shield.
Year
FCFt
VL,t
Dt
Intt
Int tax shieldt
0
-100
185.86
46.47
1
50
151.64
37.91
2.83
0.99
2
100
64.52
16.13
2.31
0.81
3
70
0
0
0.98
0.34
The present value of the interest tax shield is thus,
PV (Interest tax shield) =
0.99
0.81
0.34
+
+
= $1.85.
1 + 0.09025
(1 + 0.09025)2
(1 + 0.09025)3
(d) Recall that,
VL = APV = VU + PV (financing effects) = 184.01 + 1.85 = 185.86.
This matches the solution to Problem 18-5(c).
April 24, 2015
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Problem 18-11
In year 1, AMC will earn $2000 before interest and taxes. The market expects these earnings to
grow at a rate of 3% per year. The firm will make no net investments (i.e., capital expenditures will
equal depreciation) or changes to net working capital. Assume that the corporate tax rate equals
40%. Right now, the firm has $5000 in risk-free debt. It plans to keep a constant ratio of debt to
equity every year, so that on average the debt will also grow by 3% per year. Suppose the risk-free
rate equals 5%, and the expected return on the market equals 11%. The asset beta for this
industry is 1.11.
(a) If AMC were an all-equity (unlevered) firm, what would its market value be?
(b) Assuming the debt is fairly priced, what is the amount of interest AMC will pay next year? If
AMCs debt is expected to grow by 3% per year, at what rate are its interest payments
expected to grow?
(c) Even though AMCs debt is riskless (the firm will not default), the future growth of AMCs debt is
uncertain, so the exact amount of the future interest payments is risky. Assuming the future
interest payments have the same beta as AMCs assets, what is the present value of AMCs
interest tax shield?
(d) Using the APV method, what is AMCs total market value, VL ? What is the market value of
AMCs equity?
April 24, 2015
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Problem 18-11
(e) What is AMCs WACC? (Hint: Work backward from the FCF and VL .)
(f) Using the WACC method, what is the expected return for AMC equity?
(g) Show that the following holds for AMC:
βA =
D
E
βE +
βD .
D+E
D+E
(h) Assuming that the proceeds from any increases in debt are paid out to equity holders, what
cash flows do the equity holders expect to receive in one year? At what rate are those cash
flows expected to grow? Use that information plus your answer to part (f) to derive the market
value of equity using the FTE method. How does that compare to your answer in part (d)?
April 24, 2015
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Problem 18-11
Solution
(a) AMC has unlevered FCF of $2000 × (1 − 0.40) = $1200. From the CAPM, AMC’s unlevered
cost of capital is 5% + 1.11 × (11% − 5%) = 11.66%. Discounting the FCF as a growing
perpetuity tells us that the value of the firm, assuming growth of 3%, is:
V (all equity) =
1200
= $13857.
0.1166 − 0.03
(b) Since the debt is risk-free, the interest rate paid on it must equal the risk-free rate of 5% (or
else there would be an arbitrage opportunity). The firm has $5000 of debt next year. The
interest payment will be 5% of that, or $250. If the debt grows by 3% per year, so will the
interest payments.
(c) The expected value of next years tax shield will be $250 × 40% = $100, and it will grow (with
the growth of the debt) at a rate of 3%. But the exact amount of the tax shield is uncertain,
since AMC may add new debt or repay some debt during the year, depending on their cash
flows. This makes the actual amount of the tax shield risky (even though the debt itself is not).
Since the beta of the tax shield due to debt is 1.11, the appropriate discount rate is
5% + 1.11 × (11% − 5%) = 11.67%. We can now use the growing perpetuity formula and
conclude that
100
= $1155.
PV (Interest tax shields) =
0.1166 − 0.03
April 24, 2015
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Problem 18-11
Solution
(d) The APV tells us that the value of a firm with debt equals the sum of the value of an all equity
firm and the tax shield. From previous work (parts (a) and (c)), we get:
V (AMC) = 13857 + 1155 = $15012.
The market value of the equity is therefore V − D = 15012 − 5000 = $10012.
(e) Next year’s FCF is $2000 × 0.6 = $1200. It is expected to grow at 3%, so the WACC must
satisfy: 3
1200
= 15000.
V (AMC) =
rWACC − 0.03
Solving for the WACC, we get WACC = 11%.
(f) By definition,
E
D
rWACC =
× rE +
× rD × (1 − τC ).
V
V
The return on the debt is 5%; the value of the debt is $5000, the value of the firm is $15000
and therefore the value of the equity is $15000 − $5000 = $10000. Plugging into the above
expression, we get:
11% =
10000
5000
× rE +
× 5% × (1 − 0.4)
15000
15000
=⇒
rE = 15%.
(g) From the CAPM, βE must satisfy 15% = 5% + βE × (11% − 5%), so we conclude βE = 1.66.
The relationship holds since ($10, 000/$15, 000) × 1.66 = 1.11, and the beta of the debt
equals 0.
3
In the previous part we got $15,012 and in this part we have $15,000. Those are mere rounding errors.
April 24, 2015
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Problem 18-11
Solution
(h) The debt is expected to increase to $5000 × (1 + 0.03) = $5, 150, so the equity holders will
get $150 due to the increase in debt. These proceeds will increase by 3% annually. (The
second-year debt will be $5, 000 × (1 + 0.03)2 = $5, 304.5, with an increase in debt of
$154.5, 3% higher than the $150 proceeds of year 1.) The expected FCF to equity (FCFE) at
the end of the first year is therefore,
FCFE = EBIT − Interest − Taxes + Debt proceeds
= (2000 − 250) × (1.40) + 150 = $1200.
This cash flow is expected to grow at 3% per year. Thus, another way to compute the value of
equity is to discount these cash flows directly at the MCR for the equity of 15% (from (f)):
E=
1200
FCFE
=
= 10000.
rE − g
15% − 3%
This is the same value we computed in (d), using the APV.
April 24, 2015
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Slides - Faculty Directory | Berkeley-Haas

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