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UGBA 103 (Parlour, Spring 2015), Section 10
Capital Structure (With and Without Taxes)
Raymond C. W. Leung
University of California, Berkeley
Haas School of Business, Department of Finance
Email: r_leung@haas.berkeley.edu
Website: faculty.haas.berkeley.edu/r_leung/ugba103
April 17, 2015
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §10
April 17, 2015
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Outline
1
Modigliani-Miller Theorems
MM Theorems (no tax)
MM Theorems (with taxes)
2
(OPTIONAL) Viewing equity and debt as financial derivatives
3
Examples
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Section 1
Modigliani-Miller Theorems
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Subsection 1
MM Theorems (no tax)
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WACC
The expected return of the firm’s assets — expected return on a portfolio of all the company’s
securities or weighted average cost of capital:
D
E
WACC =
rD + rE ,
V
V
where D is the market value of debt, E is the market value of equity, and V = D + E is the total
value of the firm; and rD is the cost of debt, and rE is the cost of equity.
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Modigliani-Miller Theorems (no taxes)
Proposition (MM I (no tax))
The levered value of the firm VL is equal to the unlevered value of the firm VU ; that is,
VL = VU
Proposition (MM II (no tax))
The cost of equity,
rE = rA +
D
E
(rA − rD ).
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Modigliani-Miller Theorems (no taxes)
Notes
Just to make it more transparent: MM I is discussing the values of the firm; and MM II is
discussing the cost of capital of the firm.
Once we use MM I to deduce that rA = WACC, and using the definition of WACC, we can just
simply rearrange terms to get MM II:
rA = WACC =
D
E
rD +
rE
D+E
D+E
=⇒
=⇒
=⇒
(D + E)rA = DrD + ErE
D
D
+ 1 rA = rD + rE
E
E
D
rE = rA + (rA − rD ).
E
Recall also how asset, equity and debt betas are related to each other.
βE = βA +
D
E
(βA − βD ).
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Subsection 2
MM Theorems (with taxes)
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Modigliani-Miller theorems (with taxes)
Firstly, with taxes, the expected returns of the firm’s assets are,
D
E
rA =
rD + rE .
V
V
Most notably, here with taxes, that rA 6= WACC!
Proposition (MM I (with tax))
The levered value of the firm is equal to the unlevered value of the firm plus the present value of its
tax shield. That is,
VL = VU + tC DL ,
where tC is the corporate tax rate.
Proposition (MM II (with tax))
The cost of equity of the levered firm is equal to,
rE = rA +
D
E
(1 − tC )(rA − rD )
Thus, the weighted average cost of capital (with taxes) is equal to,
D
E
D
WACC = (1 − tC )
rD +
rE = rA 1 − tC
.
VL
VL
VL
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Personal Taxes
Suppose there are different tax rates. That is, suppose that the personal income tax rate for
interest income is tD and the personal income tax rate for equity income is tE . When $1 in pre-tax
is paid to a:
Shareholder: he/she receives (1 − tE )(1 − tC );
Bondholder: he/she receives (1 − tD ) (because tax deductible).
Thus, the tax advantage of debt in the presence of both corporate and personal taxes become,
t∗ = 1 −
(1 − tE )(1 − tC )
.
1 − tD
Thus, we rewrite MM I (with tax) in the presence of corporate and personal taxes become,
VL = VU + t ∗ DL .
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Bankruptcy and Agency costs
Bankruptcy costs: Financial distress due to increased leverage.
Agency costs: Risk shifting due to the presence of leverage (i.e. bondholders are paid off
first in terms of interest and debt principal, before the shareholders get their dividends.)
The levered value of the firm that take into account bankruptcy and agency costs:
VL = VU + PV (tax shield) − PV (bankruptcy costs) − PV (agency costs)
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Section 2
(OPTIONAL) Viewing equity and debt as financial derivatives
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Financial Options
Call option
A (European) call option with strike price K with maturity T on an asset with price ST at the
maturity date has the time T payoff of,
max{ST − K , 0}.
Payoff
0
Strike price
K
Ending stock
price ST
The call option contract gives the holder the right but not the obligation to buy the stock at
price K at maturity date T .
The call option is effectively a bullish gamble on the stock price. The cost of this call option is
substantially cheaper than buying the stock outright.
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Financial Options
Put option
A (European) put option with strike price K with maturity T on an asset with price ST at the
maturity date has the time T payoff of,
max{K − ST , 0}.
Payoff
0
Strike price
K
Ending stock
price ST
The put option contract gives the holder the right but not the obligation to sell the stock at
price K at maturity date T .
The call option is effectively a bearish gamble on the stock price. The cost of this put option is
substantially cheaper than buying the stock outright.
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Equity and Debt via financial derivatives
Suppose the following:
A — the payoff of the unlevered firm / assets
E — payoff of the levered equity
D — payoff of debt with a face value of F
Since debt payments have highest priority in claimants to the corporate cash flows and assets, and
equity holders are a residual claimant, this implies the payoff of equity is linked through this way:
(
A − F , if A ≥ F (solvent firm)
E=
0,
if A < F (defaulted firm).
= max{A − F , 0}.
This is the perspective of an individual holding a unit of stock.
Relating back to the previous discussion, we see that the payoff of equity E can be viewed as
a long position on a call option, where the underlying security is the unlevered value (asset
value) of the firm A, and the strike price is the face value F of debt.
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Equity and Debt via financial derivatives
What about the perspective of the bond holder? The payoff of the bond holder D then becomes,
(
F , if A ≥ F (solvent firm)
D=
A, if A < F (defaulted firm)
= min{A, F }
= F − max{F − A, 0}.
Relating back to the previous discussion, we see that the payoff of debt D can be viewed as a
long position on cash in the amount of F , and a short position on a put option, where the
underlying security is the unlevered value (asset value) of the firm A, and the strike price is
the face value F of debt.
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Real world relevance?
The realization that debt and equity can be viewed as financial derivative contracts was a
significant breakthrough. This led to the Nobel winning Merton (1974) paper, which we know
of today as the Black-Scholes-Merton model.
The realization that (defaultable corporate) debt can be viewed as a put option meant that we
can use option pricing methodologies (i.e. Black-Scholes-Merton equation) to compute not
only the value of debt, but also compute a “probability of default (PD)” number and a “distance
to default (DD)” number. Both of these things are critical inputs for corporate bond investors
and risk managers.
The Merton (1974) model (and along with Leland (1994) and others) form the backbone of
Moody’s Analytics (their KMV model), S&P Capital IQ Credit Analytics, and many other credit
analytics methods used in practice.
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Section 3
Examples
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Problem 15-1
Pelamed Pharmaceuticals has EBIT of $325 million in 2006. In addition, Pelamed has interest
expenses of $125 million and a corporate tax rate of 40%.
(a) What is Pelamed’s 2006 net income?
(b) What is the total of Pelamed’s 2006 net income and interest payments?
(c) If Pelamed had no interest expenses, what would its 2006 net income be? How does it
compare to your answer in part b?
(d) What is the amount of Pelamed’s interest tax shield in 2006?
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Problem 15-1
Solutions
(a) Compute,
Net Income = EBIT − Interest − Taxes = ($325 − $125) × (1 − 0.40) = $120mil.
(b) Compute,
Net Income + Interest = $120 + $125 = $245mil.
(c) Compute,
Net Income = EBIT − Taxes = $325 × (1 − 0.40) = $195mil.
This is $245 − $195 = $50mil lower than part (b).
(d) Compute,
Interest tax shield = $125 × 0.40 = $50mil.
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Problem 15-2
Grommit Engineering expects to have net income next year of $20.75 million and free cash flow of
$22.15 million. Grommit’s marginal corporate tax rate is 35%.
(a) If Grommit increases leverage so that its interest expense rises by $1 million, how will its net
income change?
(b) For the same increase in interest expense, how will free cash flow change?
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Problem 15-2
Solutions
(a) Net income will fall by the after-tax interest expense to million.
(b) Free cash flow is not affected by interest expenses.
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Problem 15-3
Suppose the corporate tax rate is 40%. Consider a firm that earns $1000 before interest and taxes
each year with no risk. The firm’s capital expenditures equal its depreciation expenses each year,
and it will have no changes to its net working capital. The risk-free interest rate is 5%.
(a) Suppose the firm has no debt and pays out its net income as a dividend each year. What is
the value of the firm’s equity?
(b) Suppose instead the firm makes interest payments of $500 per year. What is the value of
equity? What is the value of debt?
(c) What is the difference between the total value of the firm with leverage and without leverage?
(d) The difference in part (c) is equal to what percentage of the value of the debt?
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Problem 15-3
Solutions
(a) Compute,
Net income = $1000 × (1 − 0.40) = $600.
Thus, equity holders receive dividends of $600 per year with no risk. This implies the value of
equity is,
$600
= $12, 000.
E=
0.05
(b) Compute,
Net income = ($1000 − $500) × (1 − 0.40) = $300.
This implies the value of equity,
E=
$300
= $6000.
0.05
The debt holders receive interest of $500 per year. This implies the value of debt is
D = $10, 000.
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Problem 15-3
Solutions
(c) Observe that the total value of the firm,
With leverage = $6, 000 + $10, 000 = $16, 000
Without leverage = $16, 000,
and so,
Difference in value = $16, 000 − $12, 000 = $4, 000.
(d) Observe that,
$16, 000 − $12, 000
= 40%,
$10, 000
which is exactly the corporate tax rate.
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Problem 15-6
Arnell Industries has just issued $10 million in debt (at par). The firm will pay interest only on this
debt. Arnell’s marginal tax rate is expected to be 35% for the foreseeable future.
(a) Suppose Arnell pays interest of 6% per year on its debt. What is its annual interest tax shield?
(b) What is the present value of the interest tax shield, assuming its risk is the same as the loan?
(c) Suppose instead that the interest rate on the debt is 5%. What is the present value of the
interest tax shield in this case?
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Problem 15-6
Solutions
(a) Compute,
Interest tax shield = $10mil × 6% × 35% = $0.21mil
(b) Compute,
PV (Interest tax shield) =
$0.21mil
= $3.5mil
0.06
(c) Compute,
Interest tax shield = $10mil × 5% × 35% = $0.175mil,
and so
PV (Interest tax shield) =
$0.175mil
= $3.5mil.
0.05
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Problem 15-8
Bay Transport Systems (BTS) currently has $30 million in debt outstanding. In addition to 6.5%
interest, it plans to repay 5% of the remaining balance each year. If BTS has a marginal corporate
tax rate of 40%, and if the interest tax shields have the same risk as the loan, what is the present
value of the interest tax shield from the debt?
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Problem 15-8
Solutions
Interest tax shield in year 1 = $30 × 6.5% × 40% = $0.78 million. As the outstanding balance
declines, so will the interest tax shield. Therefore, we can value the interest tax shield as a growing
perpetuity with a growth rate of g = −5% and r = 6.5%:
PV =
$0.78
= $6.78mil.
0.065 − (−0.05)
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Problem 15-20
Suppose the corporate tax rate is 40%, and investors pay a tax rate of 15% on income from
dividends or capital gains and a tax rate of 33.3% on interest income. Your firm decides to add
debt so it will pay an additional $15 million in interest each year. It will pay this interest expense by
cutting its dividend.
(a) How much will debt holders receive after paying taxes on the interest they earn?
(b) By how much will the firm need to cut its dividend each year to pay this interest expense?
(c) By how much will this cut in the dividend reduce equity holders annual after-tax income?
(d) How much less will the government receive in total tax revenues each year?
(e) What is the effective tax advantage of debt τ ∗ ?
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Problem 15-20
Solutions
(a) $15 × (1 − 0.333) = $10 million each year
(b) Given a corporate tax rate of 40%, an interest expense of $15 million per year reduces net
income by 15(1 − 0.4) = $9 million after corporate taxes.
(c) $9 million dividend cut =⇒ $9 × (1 − 0.15) = $7.65 million per year.
(d) Compute,
Govt tax revenue changes = Interest taxes − Corporate taxes − Dividend taxes
= (0.333 × $15) − (0.40 × $15) − (0.15 × $9)
= $5mil − $6mil − $1.35mil
= $2.35mil.
Note that this equals part (a) less part (c).
(e) Here,
τ∗ = 1 −
(1 − 0.40)(1 − 0.15)
= 23.5%.
1 − 0.333
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Problem 15-25
With its current leverage, Impi Corporation will have net income next year of $4.5 million. If Impi’s
corporate tax rate is 35% and it pays 8% interest on its debt, how much additional debt can Impi
issue this year and still receive the benefit of the interest tax shield next year?
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Problem 15-25
Solutions
The net income of $4.5mil implies we have,
$4.5mil
= $6.923mil
1 − 0.035
in taxable income. Therefore, the firm can increase its interest expenses by $6.923mil, which
corresponds to debt of,
$6.923
= $86.5mil.
D=
0.08
April 17, 2015
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