1. Advanced Finance 2006-2007 Risky debt (2)
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
Up to now, we have not explicitly model the risk associated with the debt. Risky debt require higher rates of return. In this lecture, we will use option pricing theory to estimate the cost of debt and the required rate of return on risky debt.

2. Toward Black Scholes formulas
Value
Increase the number to time steps for a fixed maturity
The probability distribution of the firm value at maturity is lognormal
Bankruptcy
What happens if we increase the number of steps of a given maturity?
The size of the binomial tree increases but the underlying logic remains unchanged. In a binomial tree with n steps (the length of each step is Δt=T/n), the firm can take n+1 different values at maturity depending on the number of ups (k) and downs (n – k):
Vk = ukdn – k k = 0, 1, …,n
The risk-neutral probability of Vk is:
Let fk be the value at maturity of a derivative if VT = Vk.
Risk-neutral pricing implies that the value at time 0 is:
f = e-rT (p0 f0 + p1 f1 + … + pk fk + … + pn+1 fn+1) [1]
As n→∞, the probability distribution of VT tends converges to a lognormal distribution.
ln(VT) ~ N[ln(V)+(r-0.5σ²)T, σ√T]
The Black-Scholes formulas is the continuous equivalent of [1] when the probability distribution of the underlying value at maturity is lognormal.
Today
Maturity
Time
Advanced Finance 2007 Risky debt - Merton

3. 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)
Delta of call option
Risk-neutral probability of exercising the option = Proba(ST>X)
The next step is to use the Black-Scholes formula to value the securities. This slide is a reminder of the formulas.
Delta of put option
Risk-neutral probability of exercising the option = Proba(ST<X)
(Remember: 1-N(x) = N(-x))
Advanced Finance 2007 Risky debt - Merton

4. Black-Scholes using Excel
Comments:
Stock price: for dividend paying stocks, the stock price should be reduced by the present value of the dividends paid before the option’s maturity.
Interest rate: the easiest way is to use the interest rate with continuous compounding. In that case, the t-year discount factor is exp(-rFt).
ln(S/PV(Strike)): this number is equal to minus the total excess return (with continuous compounding) in order for the stock price to reach the exercise price at maturity. To see this, note that:
PV(Strike) = S e-x → x=ln(S/PV(Strike)) and Strike = S e(rT – x)
Adjusted sigma: σ √T is the standard deviation of the total return until the option’s maturity.
Distance to exercice:
Advanced Finance 2007 Risky debt - Merton

5. Merton Model: example Data Market value unlevered firm €100,000
Risk-free interest rate (an.comp): 5%
Beta asset 1
Market risk premium 6%
Volatility unlevered 40%
Company issues 2-year zero-coupon
Face value = €70,000
Proceed used to buy back shares
Details of calculation:
PV(ExPrice) = 70,000/(1.05)²= 63,492
log[Price/PV(ExPrice)] = log(100,000/63,492) =
√t = 0.40 √ 2 =
d1 = log[Price/PV(ExPrice)]/ √ √ t = 1.086
d2 = d1 - √ t = = 0.520
N(d1) = 0.861
N(d2) = 0.699
C = N(d1) Price - N(d2) PV(ExPrice)
= × 100, × 63,492
= 41,772
Using Black-Scholes formula
Price of underling asset 100,000
Exercise price 70,000
Volatility s
Years to maturity 2
Interest rate 5%
Value of call option 41,772
Value of put option (using put-call parity)
C+PV(ExPrice)-Sprice 5,264
Using the Black-Scholes-Merton with a handheld calculator is tedious. Moreover, you need a table of the normal probability distribution to get N(d1) and N(d2).
The two important results in the calculation are:
- The delta of the call option (the equity) N(d1) = 0.86
- The risk neutral probability of no default N(d2) = 0.70
Keep in mind that this is a probability in a risk neutral world. As the stock doesn’t pay any dividend, the expected growth rate (equal to the expected return) of the value of the unlevered company is set equal to the risk-free interest rate (5% per annum with annual compounding). In the real world, the expected return would normally be higher if the beta of the unlevered firm is positive. As a consequence, the real probability of no default would be higher.
Advanced Finance 2007 Risky debt - Merton

6. Valuing the risky debt
Market value of risky debt = Risk-free debt – Put Option
D = e-rT F – {– V[1 – N(d1)] + e-rTF [1 – N(d2)]}
Rearrange:
D = e-rT F N(d2) + V [1 – N(d1)]
The value of the risky debt is equal to the value of the risk-free debt minus the value of the put option. It is easier to express the risk-free interest rate r with continuous compounding.The T-year discount factor is e-rT .
The value of the put option is equal to the value of the replicating portfolio.
The delta of the put option is N(d1) – 1 = –(1 – N(d1))
As N(d2) is the risk neutral (RN) probability that the option will be exercised. Therefore, 1 – N(d2) is the RN probability that the put option will be exercised. In the Merton model, this is the probability of default.
Rearranging, we show that risky debt is a mixture of risk-free debt and of the asset of the firm. When distinction between debt and equity blurs. If the probability of default is high, the risky debt looks more like equity than like debt.
Value of risk-free debt
Probability of no default
Discounted expected recovery given default
Probability of default × + ×
Advanced Finance 2007 Risky debt - Merton

7. Example (continued) D = V – E = 100,000 – 41,772 = 58,228
D = e-rT F – Put = 63,492 – 5,264 = 58,228
Advanced Finance 2007 Risky debt - Merton

8. Expected amount of recovery
We want to prove: E[VT|VT < F] = V erT[1 – N(d1)]/[1 – N(d2)]
Recovery if default = VT
Expected recovery given default = E[VT|VT < F]
(mean of truncated lognormal distribution)
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
Recovery
Discount factor
Probability of default
Expected value of put given F
This slide shows that V[1-N(d1)]/[1-N(d2)] is equal to the discounted expected recovery given default.
To understand this result, first note that if default take place (if VT<F), recovery is equal to F – Put (see figure).
Next, express the value of the put option as:
P = E(Put|default) ×(Probability of default)×(Disc.Factor)
After some manipulation, you get:
E(Put|default) = –V erT [1-N(d1)]/[1-N(d2)] + F
Therefore, the expected recovery given default is:
E(Recovery|Default) = F – E(Put|Default) = V erT [1-N(d1)]/[1-N(d2)]
Hence, the discounted expected recovery given default is:
E(Recovery|Default) ×(Disc. Factor) = V [1-N(d1)]/[1-N(d2)]
Default VT
Advanced Finance 2007 Risky debt - Merton

9. Another presentation Discount factor Face Value Probability of default
Loss if no recovery
Expected Amount of recovery given default
Expected loss given default
Advanced Finance 2007 Risky debt - Merton

10. 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)
Advanced Finance 2007 Risky debt - Merton

11. Calculating borrowing cost
Final situation after: issue of zero-coupon & shares buy back
Balance sheet (market value)
Assets 100,000 Equity 41,772
Debt 58,228
Yield to maturity on debt y:
D = FaceValue/(1+y)²
58,228 = 60,000/(1+y)²
y = 9.64%
Spread = 364 basis points (bp)
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)
Advanced Finance 2007 Risky debt - Merton

12. Determinant of the spreads
Quasi debt PV(F)/V
Volatility
Maturity
The spread is determined by 3 factors:
The quasi debt ratio defined as the ratio between the present value of the face value calculated with the risk-free interest rate;
The volatility of the firm’s value
The maturity
Advanced Finance 2007 Risky debt - Merton

13. Maturity and spread Proba of no default - Delta of put option
The most surprising result of the Merton model is the relationship between the spread and maturity.
Let d be the quasi debt ratio: d = F e-rT / V
The value of the debt can be written as:
D = F e-rT N(d2) + V N(-d1) = F e-rT [N(d2) + N(-d1)/d]
The cost of borrowing (with continuous compounding) is:
y = ln (F/D) / T = - ln(D/F) / T = r – ln[N(d2) + N(-d1)/d]/T
The spread s is the difference between the cost of borrowing and the risk-free rate:
s = y – r = – ln[N(d2) + N(-d1)/d]/T
For low levels of quasi debt ratio the spread is an increasing function of maturity. For high levels of quasi debt ratio (highly levered firms), the spread is a decreasing of maturity.
The explanation is the following:
N(d2), the probability that the company will not default at maturity, is a decreasing function of maturity.
However, for N(-d1), minus the delta of the put option, the relationship with maturity is more complex. For low values of d, N(-d1) is an increasing function of T. Therefore, the spread increases. But, for higher values of d, it decreases and the spread goes down as the maturity increases.
Advanced Finance 2007 Risky debt - Merton

14. Inside the relationship between spread and maturity
d = 0.6 d = 1.4
T = % 39.01%
T = % %
Probability of bankruptcy
d = 0.6 d = 1.4
T =
T =
Delta of put option
d = 0.6 d = 1.4
T =
T =
The relationship between the spread and maturity is complex. Two factors determine this relationship.
1. The probability of bankruptcy.
This probability increases sharply with maturity for low levels of leverage. However, this probability is much higher for high levels of leverage. Moreover, it doesn’t change a lot when the maturity changes. Therefore, the probability of bankruptcy is an important determinant of the relationship between the spread and maturity for low levels of leverage, but not for high levels. As it goes up, the spread goes up.
2. The delta of the put option.
Remember that delta is a measure of the sensitivity of the option to changes in the value of the underlying asset.
If the level of leverage is low, the delta of the put option is slightly negative and close to 0: the value of risky debt is not very sensitive to changes in the value of the company. The delta does not play an important role in the relationship between the spread and maturity.
If, on the other hand, the level of leverage is high, the delta of the put is highly negative and close to -1. The sensitivity of the risky debt to the value of the firm is high (and negative). The delta plays an important role. As it goes up, the spread goes down.
Advanced Finance 2007 Risky debt - Merton

15. Agency costs Stockholders and bondholders have conflicting interests
Stockholders might pursue self-interest at the expense of creditors
Risk shifting
Underinvestment
Milking the property
Advanced Finance 2007 Risky debt - Merton

16. 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
Advanced Finance 2007 Risky debt - Merton

17. 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
Advanced Finance 2007 Risky debt - Merton

18. 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
Advanced Finance 2007 Risky debt - Merton