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The Binomial Model

Winter 2006

• Binomial Pricing Model

• Risky Project

• Leverage-Increasing Recap

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The Binomial Pricing Model

Consider a firm that can take on two values at time T , i.e.

 V u with probability p,
T
VT =
 V d with probability 1 − p,
T

with VTu > VTd , where u stands for “up” and d stands for “down”.

Let

V ≡ Current value of the firm’s assets

D ≡ Current value of the firm’s debt
E ≡ Current value of the firm’s equity
X ≡ Payment promised to debtholders at time T
r f ≡ Risk-free rate of interest

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The Binomial Pricing Model

What are D and E?

At time T , the value of the firm’s debt is

 X if V ≥ X,
T
DT =
 VT if VT < X.

If VTd ≥ X, then VTu > X and thus debt is free of risk.

Risk-free assets are discounted at the risk-free rate, and thus
X
D = ,
(1 + r f )T
which gives
X
E = V −D = V − .
(1 + r f )T

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The Binomial Pricing Model: Risk-Free Debt

Using continuous discounting, this can be rewritten as

D = Xe−r f T
E = V − Xe−r f T .

The Binomial Pricing Model: Risk-Free Debt

Example
Consider a firm with present value V = \$400, future value

 V u = \$650 with probability p = 0.7,
3
V3 =
 V d = \$250 with probability 1 − p = 0.3.
3

in three years, and a pure-discount debt issue that pays X = \$200

in three years.
The risk-free interest rate is r f = 5%.

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The Binomial Pricing Model: Risk-Free Debt

Example
Note that the above values give us a return on assets of
Ã !1/3
u
pV3 + (1 − p)V3 d
rA = −1
V
µ ¶
.7 × 650 + .3 × 250 1/3
= −1
400
= 9.83%.

Note that rA ≡ ρ.

The Binomial Pricing Model: Risk-Free Debt

Example
What is the current value of debt?
Debt is risk-free and thus X can be discounted at the risk-free
rate to find D:
X 200
D = = = 173.
(1.05)3 (1.05)3
The current value of equity is then

E = V − D = 400 − 173 = 227.

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The Binomial Pricing Model: Risk-Free Debt

Example
The return on equity in this case is
µ ¶
.7 × 450 + .3 × 50 1/3
rLE = − 1 = 13.28%.
227

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The Binomial Pricing Model: Risk-Free Debt

Example
What happens to rLE if X increases to 250?
Debt is still risk-free and thus
250
D = = 216.
(1.05)3

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The Binomial Pricing Model: Risk-Free Debt

Example
The value of the firm has a whole remains V = 400 (M&M
Proposition I), we have

EL = 400 − 216 = 184,

which gives
µ ¶1/3
.7 × 400 + .3 × 0
rLE = − 1 = 15.02%.
184

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The Binomial Pricing Model: Risky Debt

Suppose now that debt is not risk-free. That is, suppose that

The value of debt at time T is then

 X if V = V u ,
T T
DT =
 V d if VT = V d .
T T

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The Binomial Pricing Model: Risky Debt

The value of equity at time T is

 V u − X if V = V u ,
T T T
ET =
 0 if VT = VTd .

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How can we find D and E in this case?

We have the risk-free discount rate and thus we can find the
present value of any risk-free asset.
Can we form a risk-free portfolio with V , D and E?

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The Binomial Pricing Model: Risky Debt

A risk-free portfolio is a portfolio that provides the same payoff

in each state of the world u and d.
How to make a portfolio that pays K, say, whether VT = VTu or
VT = VTd ?
What can K be?
What payoff can we guarantee with certainty?

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Consider a portfolio P, which involves the purchase of all of the

firm’s assets and the short sale of a fraction δ of the firm’s equity.
The payoff of portfolio P at time T is then

VTu − δETu in state u,

VTd − δETd in state d.

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The Binomial Pricing Model: Risky Debt

For portfolio P to be risk-free, we need

VTu −VTd
VTu − δETu = VTd − δETd ⇒ δ = u .
ET − ETd

VT − δET
V − δE = .
(1 + r f )T

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The Binomial Pricing Model: Risky Debt

VTu −VTd
With δ = ETu −ETd
, we have

VT − δET = VTu − δETu = (1 − δ)VTu + δX

= VTd − δETd = VTd

and thus
VTd
V − δE = .
(1 + r f )T

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The Binomial Pricing Model: Risky Debt

The current market value of equity is then

µ ¶
1 VTd
E = V −
δ (1 + r f )T
and the current market value of debt is

D = V − E.

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The Binomial Pricing Model: Risky Debt

Example
Consider a firm with present value V = \$400, future value

 V u = \$650 with probability p = 0.7,
3
V3 =
 V d = \$250 with probability 1 − p = 0.3.
3

in three years, and a pure-discount debt issue that pays X = \$400

in three years.
The risk-free interest rate is r f = 5%.

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The Binomial Pricing Model: Risky Debt

Example
Same firm as before, except that X = 400.
Debt is not default-free anymore.
What is the current value of debt and equity?
First thing to do: find δ in the portfolio V − δE such that the
latter be risk-free.

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The Binomial Pricing Model: Risky Debt

Example
To have V3u − δE3u = V3d − δE3d , we need

V3u −V3d
δ = u ,
E3 − E3d
where
E3u = max 0 , V3u − X = max 0 , 650 − 400 = 250
E3d = max 0 , V3u − X = max 0 , 250 − 400 = 0.

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The Binomial Pricing Model: Risky Debt

Example
This gives

V3u −V3d 650 − 250

δ = u = = 1.6
E3 − E3d 250 − 0

and thus
Ã ! µ ¶
1 V3d 1 250
E = V − = 400 − = 184
δ (1 + r f )3 1.6 (1.05)3

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The Binomial Pricing Model: Risky Debt

Example
µ ¶
.7 × 400 + .3 × 250 1/3
rD = − 1 = 18.0%.
216

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The Goal of Management

Does maximizing the market value of equity imply maximizing

the value of the firm?
Does maximizing the value of the firm imply maximizing the
market value of equity?
Can maximizing the market value of equity hurt other
stakeholders?

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with:

• Incentives to pay dividends instead of investing in profitable

projects.

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Risky Project such that V ↓ and E ↑

Consider a firm with present value V = \$400 and future value

 V u = \$650 with probability p = 0.7,
3
V3 =
 V d = \$250 with probability 1 − p = 0.3.
3

in three years, and a pure-discount debt issue that pays X = \$400

in three years.
The risk-free rate is r f = 5%.

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Risky Project such that V ↓ and E ↑

The firm has identified a project that pays off P3u = \$200 in the
“up” state or P3d = \$0 in the “down” state.
This project also requires an initial outlay of \$142.5.
Present value of the project at the firm’s cost fo capital of 9.7% is
.7 × 200 + .3 × 0
− 142.5 = − \$36.45
(1.097)3

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Risky Project such that V ↓ and E ↑

V n = 400 − 36.45 = \$363.55

V u,n = 650 − 142.5(1.05)3 + 200 = \$685.04
V d,n = 250 − 142.5(1.05)3 = \$85.04.

This gives

δ = 2.1050, E n = \$137.81, Dn = \$225.74

∆E = \$22.79, ∆D = −\$59.24, ∆E + ∆D = −\$36.45.

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What must the opportunity cost of the money invested in the

risky project be to prevent shareholders from investing in the
risky project?

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Shareholders and Bankruptcy Costs

bankruptcy.

interest rates.

• Direct bankruptcy costs can affect shareholders’ capacity to

expropriate wealth from bondholders.

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Firm’s present value: V = 100. Firm’s value in five years is

 V u = \$200 if boom (p = .7),
5
V5 =
 V d = \$50 if recession (1 − p = .3).
5

¡ .7×200+.3×50 ¢1/5
Note that 100 − 1 = 9.16%.
Do ≡ original issue of pure discount debt paying \$50 in five
50
years. If r f = 5%, then Do = 1.05 o
5 = \$39.18 and E = \$60.82.

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A Leverage-Increasing Recap, no Bankruptcy Costs

New issue of pure-discount debt that pays X n = \$40 at the end of

five years, same priority as the old debt on firm’s assets.
The proceeds from the issue are distributed to shareholders.

M&M Proposition I If future cash flows remain constant,

issuing more debt does not affect the value of the firm:

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This gives

Proceeds from the new issue:

4
× 55.40 = \$24.62.
9
Increase in shareholders’ wealth:

44.60 + 24.62 − 60.82 = \$8.40.

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A Leverage-Increasing Recap with Bankruptcy Costs

Suppose now that expected future bankruptcy costs are \$10.

Present value of bankruptcy costs:
10
= \$6.45 ⇒ V n = 100 − 6.45 = \$93.55
(1.0916)5

We now have
µ ¶
1 40
δ = 1.4545 ⇒ En = 93.55 − = \$42.77.
1.4545 (1.05)5

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Net proceeds from the new debt issue are then

4 4
× (V n − E n ) = × (93.55 − 42.77) = \$22.57.
9 9

42.77 + 22.57 − 60.82 = \$4.52.

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A Leverage-Increasing Recap with Bankruptcy Costs

What must expected future bankruptcy costs be to prevent

shareholders from issuing debt to pay themselves a dividend?

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Can the Cost of Debt Be Reduced?

Protective covenants:

• Limitations on amount of dividend the firm can pay

• The firm may not issue long-term debt with equal or higher
seniority

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