1. Advanced Finance 2007-2008 Risky debt (1)
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. Credit risk
Credit risk exist derives from the possibility for a borrower to default on its obligations to pay interest or to repay the principal amount.
Two determinants of credit risk:
Probability of default
Loss given default / Recovery rate
Consequence:
Cost of borrowing > Risk-free rate
Spread = Cost of borrowing – Risk-free rate
(usually expressed in basis points)
Function of a rating
Internal (for loans)
External: rating agencies (for bonds)
Advanced Finance Risky debt - binomial

3. Rating Agencies Investment-grades Speculative-grades
Moody’s (
Standard and Poors (
Fitch/IBCA (
Letter grades to reflect safety of bond issue
Very High Quality
High Quality
Speculative
Very Poor
S&P
AAA AA A
BBB BB B
CCC D
Moody’s
Aaa Aa
Baa Ba
Caa C
Reference
Bodie, Kane and Marcus, Investments, 4th ed. McGraw-Hill 1999
Moody’s, The Evolving Meaning of Moody’s Bond Rating – August 1999
Standard & Poors, European Default Rates Inch Closer to the Global Average, April 2002
Investment-grades
Speculative-grades
Advanced Finance Risky debt - binomial

4. Spread over Treasury for Industrial Bonds
This figure illustrate a pattern for the spread with different maturities and different credit ratings.
The spread increases as the rating declines. It also increases with maturity for investment-grades bonds. For speculative-grades bonds the relationship with maturity is more complex.
Advanced Finance Risky debt - binomial

5. Determinants of Bonds Safety
Key financial ratio used:
Coverage ratio: EBIT/(Interest + lease & sinking fund payments)
Leverage ratio
Liquidity ratios
Profitability ratios
Cash flow-to-debt ratio
Rating Classes and Median Financial Ratios,
Rating Category
Coverage Ratio
Cash Flow to Debt %
Return on Capital %
LT Debt to Capital %
AAA
17.5
55.4
28.2
15.2 AA
10.8
24.6
22.9
26.4 A
6.8
15.6
19.9
32.5
BBB
3.9
6.6
14.0
41.0 BB
2.3
1.9
11.7
55.8 B
1.0
(4.6)
7.2
70.7
Traditionally, rating agencies base their ratings on financial ratios. The key ratios used are:
Coverage ratios: EBIT/Interest
Cash flow to debt: Free operating cash flow/Total debt
Profitability ratio: Return on Capital
Leverage ratio: Total Debt / Capital
Source: Bodies, Kane, Marcus 2002 Table 14.3
Advanced Finance Risky debt - binomial

6. Moody’s:Average cumulative default rates 1920-1999 %3 4 5 10 15 20
Aaa
0.00
0.02
0.09
0.20
1.09
1.89
2.38 Aa
0.08
0.25
0.41
0.61
0.97
3.10
5.61
6.75 A
0.27
0.60
1.37
3.61
6.13
7.47
Baa
0.30
0.94
1.73
2.62
3.51
7.92
11.46
13.95
Inv. Grade
0.16
0.49
0.93
1.43
1.97
4.85
7.59
9.24 Ba
3.45
5.57
7.80
10.04
19.05
25.95
30.82 B
4.48
9.16
13.73
17.56
20.89
31.90
39.17
43.70
Spec. Grade
3.35
6.76
9.98
12.89
15.57
25.31
32.61
37.74
All Corp.
1.33
2.76
4.14
5.44
6.65
11.49
15.35
17.79
Advanced Finance Risky debt - binomial

7. Modeling credit risk 2 approaches:
Structural models (Black Scholes, Merton, Black & Cox, Leland..)
Utilize option theory
Diffusion process for the evolution of the firm value
Better at explaining than forecasting
Reduced form models (Jarrow, Lando & Turnbull, Duffie Singleton)
Assume Poisson process for probability default
Use observe credit spreads to calibrate the parameters
Better for forecasting than explaining
Advanced Finance Risky debt - binomial

8. Merton (1974)
Limited liability: equity viewed as a call option on the company.
D Market value of debt
E Market value of equity
Loss given default F
Bankruptcy
Why is debt risky? The key reason is limited liability: a company can default on its obligation to pay the interests or to repay the principal. From the stockholders’ perspective, going bankrupt means that they lose the money that they have invested in the company. But their loss is limited. They are not required to pay additional amounts.
Black and Scholes, in their seminal paper, discuss the implications of their option pricing model for the bond valuation. This insight was later expanded by Robert Merton in 1974 (hence the name “Merton Model” for risky debt valuation models based on an option pricing model).
We present here the problem in a simplified setting:
1. The value of a the firm (V) is given and is independent of the capital structure (the MM 58 proposition is satisfied)
2. The firm issues one single debt: a zero-coupon giving the holder the right to a fixed sum of money at maturity.
3. There are no cost of bankruptcy.
Two main conclusions appear is this setting.
1. The market value of equity is equal to the value of a call option
2. The market value of the debt is equal to the value of the riskless debt minus a put option.
F Face value of debt
V Market value of comany
F Face value of debt
V Market value of comany
Advanced Finance Risky debt - binomial

9. Put Call Parity (European options)
ST + Put = Call + K K
Call = Max(0,ST-K)
Put = Max(0,K-ST) K ST
Advanced Finance Risky debt - binomial

10. Stock = Call + PV(Strike) – Put
Using put-call parity
Market value of firm:
V = E + D
Put-call parity (European options)
Stock = Call + PV(Strike) – Put
In our setting:
V ↔Stock The company is the underlying asset
E↔Call Equity is a call option on the company
F↔Strike The strike price is the face value of the debt
→ D = PV(Strike) – Put
D = Risk-free debt - Put
The value of risky debt is equal to the value of a default-free debt minus a put option.
This put option appears because of limited liability. Stockholders have the insurance that the market value of equity will never become negative. If the value of the company is below the face value of the debt, they do not have to contribute to fund the difference. This insurance is provided by the bondholders.
There is no free lunch in finance. Stockholders have to pay for this guarantee that the value of their equity will not become negative. The price to pay is the value of the put option.
Advanced Finance Risky debt - binomial

11. Merton Model: example using binomial option pricing
Data:
Market Value of Unlevered Firm: 100,000
Risk-free rate per period: 5%
Volatility: 40%
Company issues 1-year zero-coupon
Face value = 70,000
Proceeds used to pay dividend or to buy back shares
Binomial option pricing: review Up and down factors:
V = 149,182 E = 79,182 D = 70,000
Risk neutral probability :
V = 100,000 E = 34,854 D = 65,146
V = 67,032 E = 0 D = 67,032
1-period valuation formula
We first use the binomial model to illustrate the Merton model. The calculation is a straightforward application of binomial option pricing. All securities are valued by discounting the risk-neutral expected value with the risk-free interest rate.
The analysis proceeds as follow:
1. Calculate the up and down factors
2. Calculate the risk neutral probabilities p and 1 – p
3. Calculate the value of the securities at maturity
4.Calculate the present value using the 1-period valuation formula.
Note that, in this example, the value of the risk-free debt would be:
PV(F) = 70,000 e-5% = 66,586
The difference between the value of the risk-free debt (66,586) and the value of the (risky) debt (65,146) is the value of the put option.
∆t = 1
Advanced Finance Risky debt - binomial

12. Calculating the cost of borrowing
Spread = Borrowing rate – Risk-free rate
Borrowing rate = Yield to maturity on risky debt
For a zero coupon (using annual compouding):
In our example:
y = 7.45%
Spread = 7.45% - 5% = 2.45% (245 basis points)
Once we know how to value risky debt, we can calculate the interest rate on the risky debt. This rate is defined the yield to maturity on the bond issue (or the discount rate required to set the value of the debt equal to the present value of the future cash flows).
In our simplified setting, the debt is a zero coupon.
Advanced Finance Risky debt - binomial

13. Decomposing the value of the risky debt
In our simplified model:
F: loss given default if no recovery
Vd : recovery if default
F – Vd : loss given default
(1 – p) : risk-neutral probability of default
Bank and financial institutions devote considerable resources to credit risk. They quantify this risk by estimating:
1. The probability that the firm will default
2. The loss given that a default occurs.
These two notions determine the crediworthiness of corporate debts.
Assume that Vd <F (otherwise, there would be no bankruptcy).
First, the value of the debt can be expressed is the risk neutral expected value discounted with the risk-free interest rate.
Rearranging the term, it is also be stated as:
Value of risk-free debt – (Probability of default) × (Loss given default)
The loss given default is equal to the difference between the face value of the debt and the recovery. In our simple 1-period binomial model, the recovery is equal to the value of the firm if default occurs.
The important point to note is the pricing of the risky debt is based on the risk-neutral probability of default and not the real-world probability.
Advanced Finance Risky debt - binomial

14. Weighted Average Cost of Capital
(1) Start from WACC for unlevered company
As V does not change, WACC is unchanged
Assume that the CAPM holds
WACC = rA = rf + (rM - rf)βA
Suppose: βA = 1 rM – rf = 6%
WACC = 5%+6%× 1 = 11%
(2) Use WACC formula for levered company to find rE
We now wish to understand the effect of leverage on the cost of equity and the cost of debt when debt is risky.
Our starting point in the Modigliani Miller (1958) propositions. The value of the firm is unaffected by the capital structure. Hence, the weighted average cost of capital of the levered firm is equal to the cost of capital of the unlevered firm. Similarly, the weighted average beta of equity and debt is equal to the asset beta.
Combining the option pricing model with the CAPM, we can work out either the values of rE and βE. The cost of debt and the beta of the debt can than be calculated using the WACC formula.
Advanced Finance Risky debt - binomial

15. Cost (beta) of equity Remember : C = Deltacall × S - B
A call can is as portfolio of the underlying asset combined with 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:
βA = 1
DeltaE = 0.96
V/E = 2.87
βE= 2.77
rE = 5% + 6% × 2.77
= 21.59%
To calculate the beta of equity, we proceed as follow.
1. The equity of the levered company is assimilated to a call option on the firm’value.
2. A call option is a portfolio composed of on a long position on delta shares combined with borrowing. The fraction of this portfolio invested in the shares is (delta V)/E.
3. The beta of a portfolio is equal to the weighted average of the beta of the underlying securities. In this setting, the betas of the underlying securities are the asset beta and the beta of borrowing in the replicating portfolio which is equal to zero.
4. Therefore βE = βA × Delta × (V/E)
This expression generalizes the formula that had been established when the debt is risk-free: βE = βA × (V/E)
Remember that 1≥Delta≥0. This implies that, for a given level of leverage, the equity is less risky with risky debt than when the debt is riskless.
Remember also that delta varies whenever V varies. As a consequence, the beta of equity will not be constant.
Advanced Finance Risky debt - binomial

16. Cost (beta) of debt Remember : D = PV(FaceValue) – Put
Put = Deltaput × V + B (!! Deltaput is negative: Deltaput=Deltacall – 1)
So : D = PV(FaceValue) - Deltaput × V - B
Fraction invested in underlying asset is X = - Deltaput × V/D
βD = - βA Deltaput V/D
In example:
βA = 1
DeltaD = 0.04
V/D = 1.54
βD= 0.06
rD = 5% + 6% × 0.09
= 5.33%
We now turn to the debt.
The beta of the debt is equal to minus the beta of the put option (remember that the beta of the put option is negative, therefore, the beta of the debt is positive).
An interesting point to note here is the difference between the cost of borrowing (y) calculated previously and the cost of debt (rD).
Remember that the cost of borrowing is defined as the yield to maturity on the debt whereas the cost of debt is equal to the expected return on the debt.
In our example, the cost of borrowing is 7.45% whereas the cost of debt is 5.33%.
To understand the difference, compare the formulas for y and rD.
where Π is the true probability of an up movement. This true probability can be calculated by solving the following equation:
rA = Π (u – 1) + (1 – Π) (d – 1)
In the example: Π = 0.535
The cost of borrowing calculation is based on the face value of the debt whereas the cost of debt calculation is based on the (true) expected future value.
Advanced Finance Risky debt - binomial

17. Multiperiod binomial valuation
Risk neutral proba
u4V p4
For European option,
(1) At maturity, calculate
- firm values;
- equity and debt values
- risk neutral probabilities
(2) Calculate the expected values in a neutral world
(3) Discount at the risk free rate
u3V
u²V
u3dV
4p3(1 – p) uV
u2dV V
udV
6p²(1 – p)²
u2d²V dV
ud²V Δt
ud3V
4p (1 – p)3
d²V
d3V
The price change in one period in independent of the price change in previous periods.
The probability of a path with k ups and n-k downs (where n is the total number of steps) is pk (1 – p)n-k
The number of path is given by the binomial coefficient:
The Excel function COMBIN(N,n) provides the desired result.
For small binomial tree, Pascal’s triangle is useful:
Each subsequent row is obtained by adding the two entries diagonally above.
This is known in China as the Yanghui triangle as it had been studied earlier by the Chinese mathematician Yanghui about 500 years earlier.
Reference mathworld.wolfram.com
d4V
(1 – p)4
Advanced Finance Risky debt - binomial

18. Multiperiod binomial valuation: example
Firm issues a 2-year zero-coupon Face value = 70,000 V = 100,000 Int.Rate = 5% (annually compounded) Volatility = 40% Beta Asset = 1
4-step binomial tree Δt = 0.50 u = 1.332, d = rf = 2.47% per period =(1.05)1/2-1 p = 0.471
We calculate the values using a binomial tree with 4 steps.
We first calculate the parameters of the binomial tree.
As the final maturity is 2 years, the length of one period is Δt =2/4=0.50
The up and dow factors are: and d = 1/1.332=0.751
The risk-free interest rate is 5% per annum with annual compounding. The risk-free interest per period in the binomial tree is thus rf = (1.05)1/2-1=2.47%
The risk-neutral probability of an up movement is:
p = ( – 0.751)/(1.332 – 0.751) = 0.471
The next step is to calculate the possible values of the firm at maturity and the associated probabilities.
We then calculate the values of equity and debt at maturity and their risk-neutral expected values
We get the present value by discounting the expected values at the risk-free interest rate.
Advanced Finance Risky debt - binomial

19. Multiperiod valuation: details
Binomial models are easier to implement in Excel using diagonal matrices instead of branching trees.
V uV u²V u3V …
dV udV u²dV …
d²V ud²V … …
The upper part of the slide shows the binomial evolution of the value of the company. The lower part shows the evolution of the value, the delta and the beta of the equity and of the debt.
For the equity, the delta increases (decreases) and the beta decreases (increases) when the value of the company goes up (down). The opposite applies for the debt.
Note that, as the value of the firm goes down, the debt becomes more and more similar to equity.
Advanced Finance Risky debt - binomial

20. Multiperiod binomial valuation: additional details
From the previous calculation, we can decompose D into:
Risk-free debt
Risk-neutral probability of default
Expected loss given default
Expected value at maturity:
Risk-free debt = 70,000
Default probability = 0.354
Expected loss given default = 18,552
Risky debt = 70,000 – × 18,552 = 63,427
Present value:
D = 63,427 / (1.05)² = 57,530
The value of the risky debt can also be expressed as follow:
D = {F – (RN probability of default) ×(Loss|Default)}×Discount factor
Default take place if the value of the firm at maturity is less than 70,000. Based on previous calculations, we have:
VT RN Proba Loss|Default
56, ,203
32, ,741
Based on these number, we get:
Probability of default = = 0.354
Expected loss given default = (0.277 ×13, ,741)/0.354
= 18,552
Advanced Finance Risky debt - binomial