1. Mathematical Credit Analysis

2. Probability of Default
This chart shows rating migrations and the probability of default for alternative loans. Note the increase in default probability with longer loans.

3. Mathematical Credit Analysis
Debt as a sold put Option
Merton Model
Monte Carlo Simulation

4. General Payoff Graphs from Holding Investments with Future Uncertain Returns

5. Payoff Graphs from Call Option – Payoffs when Conditions Improve

6. Payoff Graphs from Buying Put Option – Returns are realized to buyer when the value declines

7. Payoff Graphs from Selling Put Option – Value Changes with Value Decreases

8. Payoff to claimholders Equity F Debt V(T)
At maturity date T, the debt-holders receive face value of bond F as long as the value of the firm V(T) exceeds F and V(T) otherwise.
They get F - Max[F - V(T), 0]: The payoff of riskless debt minus the payoff of a put on V(T) with exercise price F.
Equity holders get Max[V(T) - F, 0], the payoff of a call on the firm.
Value of the company and changes in value to equity and debt investors
Equity F
Debt
V(T)

9. The payoffs to the bond holders are limited to the amount lent B
Payoff to
debt holders
Credit spread is the payoff from selling a put option A1 A2
Assets B
The payoffs to the bond holders are limited to the amount lent B
at best.

10. The Black-Scholes/Merton Approach
Consider a firm with equity and one debt issue.
The debt issue matures at date T and has principal F.
It is a zero coupon bond for simplicity.
Value of the firm is V(t).
Value of equity is E(t).
Current value of debt is D(t).

11. Merton’s Model
Merton’s model regards the equity as an option on the assets of the firm
In a simple situation the equity value is
max(VT -D, 0)
where VT is the value of the firm and D is the debt repayment required

12. Merton’s Model Assumptions
Markets are frictionless, there is no difference between borrowing and lending rates
Market value of the assets of a company follow Brownian Motion Process with constant volatility
No cash flow payouts during the life of the debt contract – no debt re-payments and no dividend payments
APR is not violated

13. Merton‘s Structural Model (1974)
Assumes a simple capital structure with all debt represented by one zero coupon bond – problem in project finance because of amortization of bonds.
We will derive the loss rates endogenously, together with the default probability
Risky asset V, equity S, one zero bond B maturing at T and face value (incl. Accrued interest) F
Default risk on the loan to the firm is tantamount to the firm‘s assets VT falling below the obligations to the debt holders F
Credit risk exists as long as probability (V<F)>0
This naturally implies that at t=0, B0<Fe-rT; yT>rf, where πT=yT-rf is the default spread which compensates the bond holder for taking the default risk

14. Merton Model and Credit Spreads

15. Merton Model Propositions
Face value of zero coupon debt is strike price
Can use the Black-Scholes model with equity as a call or debt as a put option to directly measure the value of risky debt
Can use to compute the required yield on a risky bond:
PV of Debt = Face x (1+y)^t or
(1+y)^t = PV/Face
(1+y) = (PV/Face)^(1/t)
y = (PV/Face)^(1/t) – 1
With continual compounding = - Ln(PV/Face)/t
Computation of the yield allows computation of the required credit spread and computation of debt value
Borrower always holds a valuable default or repayment option. If things go well repayment takes place, borrower pays interest and principal keeps the remaining upside, If things go bad, limited liability allows the borrower to default and walk away losing his/her equity.

16. Default Occurs at Maturity of Debt if V(T)<F
Asset Value VT V0 F
Probability of default T
Time

17. Black-Scholes Assumptions with Respect to Firm Value
Firm value is lognormal; constant volatility; deterministic interest rate; no frictions.
E(t) = Call[V(t),F,T]
D(t) = Exp[-r(T-t)]*F – Put[V(t),F,T]
Put-call parity implies also:
D(t) = V(t) – Call[V(t),F,T]
Firm value is simply sum of equity and debt:
V(t) = E(t) + D(t).

18. Equity Price Method
Following the Merton’s model, the fair value of the Put is
The annual protection fee will be the cost of Put divided by the number of years.

19. Applying Option Valuation Model
Merton showed value of a risky loan
F(t) = Be-it[(1/d)N(h1) +N(h2)]
Written as a yield spread
k(t) - i = (-1/t)ln[N(h2) +(1/d)N(h1)]
where k(t) = Required yield on risky debt
ln = Natural logarithm
i = Risk-free rate on debt of equivalent maturity

20. Merton Model and Recovery Rate
If Merton’s model applies, the probability of default can be computed.
It is the probability that firm value will exceeds debt face value at T.
Debt Value = Face at Risk Free Rate less Value of Put Option
Put Option Value = Probability of Default x Cost When Default
Black-Scholes formulation allows one to divide the value of the put option into probability of default and recovery rate

21. Enron EDF and Bond Ratings – KMV Model Default Probability in Blue, Implied Default Probability from Bond Ratings in Red
Enron remained investment grade until three months before bankruptcy

22. Subordinated Debt Payoff
Payoff to
claimholders
Equity
F+U
Subordinated
debt F
Debt
V(T)

23. Subordinated Debt
Let the firm have subordinated debt with face value of U.
The subordinated debt holders receive U if firm value exceeds U + F.
If firm value is less than U + F but less than F, they receive nothing.
If firm value is less than U + F but more than F, they get what is left.

24. Payoff and Pricing of Subordinated Debt
Subordinated debt with face value of U pays :
Max[V - F,0] – Max[V - (F+U), 0]
If V > F+U, the value of the firm is greater than the value of senior plus junior:
the payoff is V - F - (V - (F+U)) = U.
If F < V < F+U, the value of the firm is greater than the senior debt, but less than junior:
the payoff is V - F.
If V < F, the firm value is less than the face value of the senior debt:
the payoff is 0.
The value of subordinated debt is
Call[V,F,T] – Call[V,F+U,T].

25. Merton‘s Model With Subordinated Debt
So the joint probability of both obligors defaulting would be:
N2 is the standard normal distribution with correlation ρ among the 2 variables

26. Class 4: Model Assessment
Various ratios that some use to assess management performance are most appropriate for testing the validity of a model. Examples of these ratios include:
EBITDA/Revenue
Working Capital – Activity Ratios
Inventory Turnover
AR/Revenues;
AP/Expenses
Income Tax Payable/Income Taxes
Depreciation Rate – Depreciation Expense/Gross PP&L
Depreciation Expense to Cap Exp
Average Interest Rate – Interest Expense/Average Debt
Capital Expenditure/Capacity

27. Other Ratios in Financial Modeling
Other ratios should be computed in financial models to test the validity of assumptions and the structure of the model. For these ratios, the historic levels can be used to make forecasts of relevant assumptions.
Examples include average interest rate, depreciation rate, working capital ratios, dividend payout ratio, EBITDA/Sales

28. Defaults in 2005