1. Corporate Financial Policy Calculating an optimal capital structure
Professor André Farber
Solvay Business School
04/06/2019
Cofipo Merton Leland

2. References
Merton, R. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates Journal of Finance, 29 (May 1974)
Merton, R. Continuous-Time Finance Basil Blackwell 1990
Leland, H. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure Journal of Finance 44, 4 (September 1994) pp
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3. References
Altman, E., Resti, A. and Sironi, A., Analyzing and Explaining Default Recovery Rates, A Report Submitted to ISDA, December 2001
Bohn, J.R., A Survey of Contingent-Claims Approaches to Risky Debt Valuation, Journal of Risk Finance (Spring 2000) pp
Merton, R. On the Pricing of Corporate Debt: The Risk Structure of Interest Rate, Journal of Finance, 29 (May 1974)
Leland, H., Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, Journal of Finance, 44, 4 (September 1994) pp
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4. V Market value of comany
Merton (1974)
Limited liability: equity viewed as a call option on the company.
Mkt value of equity: E = V - D
Put-call parity (European options)
Call = Stock + Put - PV(Strike)
Conclusion: D = PV(Strike) - Put
E Market value of equity
Market value of risky debt
Market value of risk-free debt
Bankruptcy
Market value of put option (option to default)
F Face value of debt
V Market value of comany
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5. V Market value of company
Risky debt
Market value of debt
F Face value of debt
V Market value of company
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6. Merton Model: example using binomial option pricing
Data:
Market Value of Unlevered Firm: 100,000
Risk-free rate: 5% (cont.compounding)
Volatility: 40.55% u = 1.50 d = 0.667
Company issues 1-year zero-coupon
Face value = 70,000
Proceeds used to pay dividend
Binomial option pricing - Review
Up and down factors:
Risk neutral probability:
1-period valuation formula:
V = 150,000 E = 80,000 D = 70,000
V = 100,000 E = ? D = ?
V = 66,667 E = 0 D = 66,667
∆t = 1
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7. Binomial valuation
Risk neutral probability p =( )/( ) = 0.46
Market value of equity (a call option)
E =( 0.46 x 80, x 0)/1.051=35,048
Market value of debt:
B = V - E = 100, ,048 = 64,952
Market value of riskless debt = 66,586
Value of put option = 1,634
Borrowing rate = 70,000/64, = 7.77%
Spread = 2.77%
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8. Black-Scholes: Review
European call option: C = S N(d1) – PV(X) N(d2)
Put-Call Parity: P = C – S + PV(X)
European put option: P = S [N(d1)-1] + PV(X)[1-N(d2)]
P = - S N(-d1) +PV(X) N(-d2)
Risk-neutral probability of exercising the option = Proba(ST>X)
Delta of call option
Risk-neutral probability of exercising the option = Proba(ST<X)
Delta of put option
(Remember: 1-N(x) = N(-x))
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9. Merton Model: example using Black-Scholes
Data
Market value unlevered firm €100,000
Risk-free interest rate: 5%
Beta asset 1
Market risk premium 6%
Volatility unlevered 30%
Company issues 2-year zero-coupon
Face value = €60,000
Proceed used to buy back shares
Details of calculation:
PV(ExPrice) = 60,000/(1.05)²= 54,422
log[Price/PV(ExPrice)] = log(100,000/54,422) =
√t = 0.30 √ 2 =
d1 = log[Price/PV(ExPrice)]/ √ √ t
= / (0.5)(0.4243) =
d2 = d1 - √ t = =
N(d1) = 0.95
N(d2) = 0.89
C = N(d1) Price - N(d2) PV(ExPrice)
= 0.95 × 100, × 54,422
= 46,626
Using Black-Scholes formula
Price of underling asset 100,000
Exercise price 60,000
Volatility ó
Years to maturity 2
Interest rate 5%
Value of call option 46,626
Value of put option (using put-call parity)
C+PV(ExPrice)-Sprice 1,047
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10. Calculating borrowing cost
Initial situation
Balance sheet (market value)
Assets 100,000 Equity 100,000
Note: in this model, market value of company doesn’t change (Modigliani Miller 1958)
Final situation after: issue of zero-coupon shares buyback
Balance sheet (market value)
Assets 100,000 Equity 46,626
Debt 53,374
Decomposition of market value of debt:
Risk-free debt 54,422
- Put option 1,047
Yield to maturity on debt y:
D = FaceValue/(1+y)²
53,374 = 60,000/(1+y)²
y = 6.03%
Spread = 103 basis points (bp)
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11. Cost of equity (1) Start from WACC for unlevered company
As V does not change, WACC is unchanged
For unlevered company:
WACC = rA = rf + (rM - rf)βassets = 5%+6%× 1 = 11%
(2) Use WACC formula for levered company to find rE
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12. Beta of equity Remember : C = Deltacall × S - B
A call can be viewed as portfolio of the underlying asset combined with some borrowing B.
The fraction invested in the underlying asset is X = (Deltacall × S) / C
The beta of this portfolio is X βasset
When analyzing a levered company:
call option = equity
underlying asset = value of company
X = V/E = (1+D/E)
In example:
βasset = 1
Delta = 0.95
D/E = 1.14
βequity= 2.03
rE = 5% + 6% x 2.03
= 17.20%
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13. Beta of debt Remember : D = PV(FaceValue) - Put
Put = Deltaput × V + B !! Deltaput is negative
So : D = PV(FaceValue) - Deltaput × V - B
Fraction invested in underlying asset is X = - Deltaput × V/D
βput = - βasset Deltaput V/D
In example:
βasset = 1
Deltaput = -.05
V/D = 1.87
βD= 0.09
rD = 5% + 6% x 0.09
= 5.56%
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14. Agency costs Stockholders and bondholders have conflicting interests
Stockholders might pursue self-interest at the expense of creditors
Risk shifting
Underinvestment
Milking the property
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15. Risk shifting
The value of a call option is an increasing function of the value of the underlying asset
By increasing the risk, the stockholders might fool the existing bondholders by increasing the value of their stocks at the expense of the value of the bonds
Example (V = 100,000 – F = 60,000 – T = 2 years – r = 5%)
Volatility Equity Debt
30% 46, ,374
40% 48, ,494
+1, ,880
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16. Underinvestment
Levered company might decide not to undertake projects with positive NPV if financed with equity.
Example: F = 100,000, T = 5 years, r = 5%, σ = 30%
V = 100,000 E = 35,958 D = 64,042
Investment project: Investment 8,000 & NPV = 2,000
∆V = I + NPV
V = 110,000 E = 43,780 D = 66,220
∆ V = 10,000 ∆E = 7,822 ∆D = 2,178
Shareholders loose if project all-equity financed:
Invest 8,000
∆E 7,822
Loss = 178
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17. Milking the property
Suppose now that the shareholders decide to pay themselves a special dividend.
Example: F = 100,000, T = 5 years, r = 5%, σ = 30%
V = 100,000 E = 35,958 D = 64,042
Dividend = 10,000
∆V = - Dividend
V = 90,000 E = 28,600 D = 61,400
∆ V = -10,000 ∆E = -7,357 ∆D =- 2,642
Shareholders gain:
Dividend 10,000
∆E -7,357
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18. More on Merton
Market value of risky debt = Risk-free debt – Put Option
Exercise price = Face value of debt (F)
Underlying asset = Value of company (V)
D = PV(F) – [- V N(-d1) + PV(F) N(-d2)]
= PV(F) – N(-d2) [PV(F) – V N(-d1)/N(-d2)]
Default probability
Expected discounted loss given default
04/06/2019
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19. Another presentation D = PV(F) – N(-d2) [PV(F) – V N(-d1)/N(-d2)]
= e-rT F - N(-d2) [e-rT F – V N(-d1)/N(-d2)]
= e-rT {F - N(-d2) [F – V erT N(-d1)/N(-d2)]}
Discount factor
Face Value
Probability of default
Loss if no recovery
Expected Amount of recovery
Expected loss given default
04/06/2019
Cofipo Merton Leland

20. Expected amount of recovery
Recovery if default = VT
Expected recovery = E[VT|VT < F] (mean of truncated lognormal distribution)
We want to prove: E[VT|VT < F] = V erT N(-d1)/N(-d2)
To see why, notice that the value of the put option:
P = -V N(-d1) + e-rT F N(-d2)
can be written as
P = e-rT N(-d2)[- V erT N(-d1)/N(-d2) + F]
But, given default: VT = F – Put
So: E[VT|VT < F]=F - [- V erT N(-d1)/N(-d2) + F] = V erT N(-d1)/N(-d2)
Put F
Discount factor
Probability of default
Expected value of put given
Recovery F
Default VT
04/06/2019
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21. Example using Black-Scholes
Data Market value unlevered company € 100,000 Debt = 2-year zero coupon Face value € 60,000
Risk-free interest rate 5% Volatility unlevered company 30%
Using Black-Scholes formula
Value of risk-free debt € 60,000 x = 54,422
Probability of default N(-d2) = 1-N(d2) =
Expected recovery given default V erT N(-d1)/N(-d2) = (100,000 / ) (0.05/0.11) = 49,585
Expected recovery rate | default = 49,585 / 60,000 = 82.64%
Using Black-Scholes formula
Market value unlevered company € 100,000 Market value of equity € 46,626 Market value of debt € 53,374
Discount factor N(d1) N(d2)
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22. Leland 1994
Model giving the optimal debt level when taking into account:
limited liability
interest tax shield
cost of bankruptcy
Main assumptions:
the value of the unlevered firm (V) is known;
this value changes randomly through time according to a diffusion process with constant volatility  dV= µVdt + VdW;
the riskless interest rate r is constant;
bankruptcy takes place if the asset value reaches a threshold VB;
debt promises a perpetual coupon C;
if bankruptcy occurs, a fraction α of value is lost to bankruptcy costs.
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23. V Barrier VB Time Default point 04/06/2019
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24. Exogeneous level of bankruptcy
Market value of levered company VL = V + PVTS(V) - BC(V)
V: market value of unlevered company
PVTS(V): present value of tax benefits
BC(V): present value of bankruptcy costs
Closed form solution:
Define pB : present value of $1 contingent on future bankruptcy
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25. Tax shield if no default
Value of tax benefit
Tax shield if no default
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26. Present value of bankruptcy cost
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27. Value of debt
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28. Endogeneous bankruptcy level
If bankrupcy takes place when market value of equity equals 0:
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29. Example
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