Mathematical Credit Analysis

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.

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

General Payoff Graphs from Holding Investments with Future Uncertain Returns

Payoff Graphs from Call Option – Payoffs when Conditions Improve

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

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

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)

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.

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).

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

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

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

Merton Model and Credit Spreads

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.

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

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).

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.

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

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

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

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

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.

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].

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

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

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

Defaults in 2005