Lecture 4: Modigliani-Miller Proposition 2
ECON6015: Finance
Yidi Sun
University of Southampton
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Recap
Suppose a firm increases its debt
This increases the risk of holding/buying the firm’s equity.
Higher debt repayment expenses
Vulnerable to downturns
Less funds for growth and development
In the event of bankruptcy, creditors have 1st claim on assets
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This lecture: Modigliani-Miller Proposition 2 (MMP2)
If a firm increases its debt, then the return on equity must increase
to compensate for increased risk.
Main ideas:
1 Return on equity varies +vely and linearly with leverage
2 The increase in equity’s risk is compensated by an increase in
equity’s return
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Notation
D: market value of outstanding debt
E : market value of outstanding equity
rD : (expected) return on a unit of debt (or cost of debt capital)
rE : (expected) return on a unit of equity (or cost of equity capital)
rA: (expected) return on assets (or weighted average cost of
capital)
pi: (expected) profit
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(Expected) Return on assets
Definition
rA =
pi
V
=
pi
D + E
Assumption 1
All profit is paid out to creditors and stockholders, i.e.
pi = rDD + rEE
Implication of Assumption 1
rA =
rDD + rEE
D + E
=⇒ rE = rA + D
E
(
rA − rD
)
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Further assumptions
Assumption 2
MMP1 holds. Specifically, changing capital structure
does not change V
does not change pi
does not change rA
Only investment decisions can change these things.
Assumption 3
rD is constant (for simplicity and by exogeneity, but not necessary)
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Comparative statics
rE = rA +
D
E
(
rA − rD
)
D
E
x
By Assumption 2, rA is constant
By Assumption 3, rD is constant
Result: rE ↑
The rate of increase in rE is (rA − rD)
Explanation
Shareholders require a greater return on equity as compensation
for the greater risk associated with holding equity.
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MMP2
Modigliani-Miller Proposition 2
Theorem
The expected return on equity of a leveraged firm increases in
proportion to the debt-to-equity ratio, and the rate of increase
depends on (rA − rD).
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Numerical illustration 1
Consider an unleveraged firm with an expected profit of £1,500.
The firm has 1,000 shares outstanding with a share price of £10.
The firm’s return on equity is
rE =
Suppose the firm decides to become leveraged and have equal
proportions of debt and equity. It issues £5,000 in debt at an
interest rate of rD = 0.10 and uses the proceeds to buy back 500
shares. The return on equity becomes
rE =
Conclusion:
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Useful extension
We can break Assumption 3 (rD is constant)
Justification
By logic similar to equity, if the firm holds more debt then creditors
need compensation for the greater risk they face
Warnings
Breaking Assumption 3 may not change results
Changing rD is random
rD is exogenously determined
requires modelling to endogenize and avoid randomness
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Weighted average cost of capital (WACC)
rA =
rDD + rEE
D + E
can be written as
rA =
(
D
D + E
)
rD +
(
E
D + E
)
rE
or more compactly as
rA = xDrD + xE rE
where xD =
D
D + E
=
D
V
and xE =
E
D + E
=
E
V
are weights of
debt and equity in the firm’s capital structure
Interpretation
(Expected) return on assets is a weighted sum of (expected)
returns on debt and equity
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Comparative statics: more debt
rA =
(
D
D + E
)
rD +
(
E
D + E
)
rE
Assumptions (MMP1)
1 rA is constant
2 V is constant
Alternatively, for simplicity, we can assume E is constant
without changing results
D ↑
E ↓ such that D + E is unchanged, and therefore E
D + E
y
D
D + E
x
Result: rE ↑
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Numerical illustration 2
Consider a leveraged firm with £30 in debts, £70 in equity outstanding,
and offering returns of 7.5% and 15% on debt and equity, respectively.
The expected return on assets is
rA = 7.5
(
30
30 + 70
)
+ 15
(
70
30 + 70
)
= 12.75%
The firm is considering issuing £10 more debt and using the cash to buy
back its equity. It will therefore have £40 in debts and £60 in equity. By
MM Proposition 1, the return on assets is unchanged. With more risk,
the return on debt must increase, say, to 7.875%. Then
rA = 12.75 =⇒ 7.875
(
40
40 + 60
)
+ rE
(
60
40 + 60
)
= 12.75%
=⇒ rE = 16
Conclusion: increasing leverage increases the risk of equity; shareholders
require an increase in the return on equity (from 15% to 16%); similarly,
investors get an increase in the return on debt (from 7.5% to 7.875%).
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β: a reminder
Definition
β is a measure of risk that arises from movements in the market:
β =
covariance between stock and market returns
variance of market returns
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β and capital structure
Consider an investor holding a portfolio of the firms’ debt and
equity, and no other assets
The beta of the portfolio is equal to the beta of all the firm’s
assets (βA), which is a weighted sum of betas on debt and equity:
βA =
(
D
D + E
)
βD +
(
E
D + E
)
βE
which can be re-structured as
βE = βA +
D
E
(
βA − βD
)
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Comparative statics
βE = βA +
D
E
(
βA − βD
)
Assumptions (MMP1)
1 βA is constant
2 βD is exogenously determined and constant (for simplicity)
Then
D
E
x =⇒ βE ↑
Interpretation
The risk of equity increases with leverage, at a rate of (βA − βD)
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Summary
Assumptions
1.
2.
3.
Main results
Implication for capital structure
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Closing Remarks
MMP1 and MMP2 warn us that higher leverage increases
both equity risk and expected returns on equity, but does not
change shareholder value.
However you must watch out for hidden changes in leverage,
e.g. a decision to lease new equipment, or an underfunded
pension scheme; both imply new debt. You might observe an
increase in expected equity returns (which is the result of this
increase in debt) and interpret this as an increase in
shareholder value, rather than the compensation for increased
risk which is what it really is!
In Lecture 5, we will explore the Modigliani-Miller model in the
presence of taxes.