The Merton Model. A Structural Approach to Default Prediction. Agenda. Idea. Merton Model. The iterative approach. Example: Enron

Transcription

1 The Merton Model A Structural Approach to Default Prediction Agenda Idea Merton Model The iterative approach Example: Enron A solution using equity values and equity volatility Example: Enron 2 1

2 Idea Consider the following company Following identity holds: Asset value = Value of equity + Value of liabilities 3 Consider the following scenario: Requirements to determine the PD: firm s liability from balance sheet specify the probability distribution of the asset value at T 4 2

3 Common assumption: log asset value in T follows normal distribution Besides, Merton assumes that follows a geometric Brownian motion, i.e

4 Conclusion: we can determine the PD if we know 7 Merton Model Problem: we can t observe Solution: use of option pricing theory Consider a publicly traded company as before Payoffs at time T: 8 4

5 9 If no dividends are paid, use the Black-Scholes call option formula 10 5

6 Implementation of the Merton Model Assumption: maturity T = one year Two different approaches: Iterative approach (1) Solution using equity values and equity volatilities (2) 11 The iterative approach (1) Rearranging the Black-Scholes formula, we get Going back in time for 260 trading days, we get a system of 261 equations in 261 unknowns σ can be estimated from a time series of V s 12 6

7 System of equations can be solved with an iterative procedure: Iteration 0: Set starting values for each a = 0,1,,260 and σ to the standard deviation of the log asset returns. For any further iteration k = 1,,end: Insert the V s and σ into the formula for and d 2, use Black-Scholes to compute d 1 new V s and compute a new σ. Go on until procedure converges 13 Example: Enron Default in December 2001 Biggest corporate default ever Implementation three months before its default Collect quarterly data on its liabilities from the SEC Edgar data base One-year US treasury serves as the risk-free rate of return Can obtain market value of equity from various data providers 14 7

8 15 Deriving an estimate of the drift rate of asset returns using CAPM 16 8

9 Implied default probability 17 A solution using equity values and equity volatilities (2) Use for the current date t only Introduce another equation We have two equations with two unknowns use numerical routine to solve it 18 9

10 19 Now we can solve the system of equations Need feasible initial values (i.e. > 0) for the two unknown variables Good choice: with assumption 20 10

11 Why does this assumption make sense? In general: Compare the two definitions It holds 21 Example: Enron Use same data and assumptions In addition: estimate of the equity volatility 22 11

12 23 Solving the equation system 24 12

13 Implied default probability Again we need the drift rate of assets Use CAPM as before 25 Comparing different approaches Key results Question: Why do the PDs differ that much if we have used the same one-year history of equity prices? Answer: The iterative approach models changes in leverage, the other one not! 26 13

14 Recall that we estimated from history of equity prices good way IF we think it is constant But equity risk varies if the asset-to-equity ratio varies equity risk varies with leverage E Conclusion: For data characterised by large changes in leverage one prefers the iterative approach 27 Thank you for your attention Thomas.Goswin@Bundesbank.de 28 14