Lecture on Merton Model for Probability of Default

Introduction

• David Harper from Bionic Turtle presents a review of the Merton Model for probability of default (PD).
• Focus on credit risk models for Financial Risk Manager (FRM) candidates.
• Discussion on the expected loss of a credit or loan portfolio, which is the product of:
  • Adjusted exposure
  • Loss given default (LGD)
  • Expected default frequency (EDF) or probability of default (PD)

Approaches to Estimating Probability of Default

• Reduced Form Approach: Considers default as an exogenous process (not covered in this lecture).
• Structural Approach: Considers default as an endogenous process, based on the firm's balance sheet.
  • Popular approach: Merton Model.
  • Commercial application: Moody's KMV (similar concept, different application).

Merton Model Assumptions

• Firm's assets worth $11,000 (equity + debt).
• Default threshold: Point at which firm's assets drop below predicted default.
  • Rule: Short-term liabilities + 1/2 long-term liabilities.
• Predict the probability of default over one period (e.g. one year).
• Initial asset value ($11,000) 51% higher than the default point ($600).

Variables for Merton Model

• Volatility (Sigma): Standard deviation of firm’s assets (assume 25%).
• Expected Return: Growth rate of firm’s assets (assume 20%).

Prediction Calculation

• Continuous growth of assets: $11,000 growing continuously by 20% to $11,184.
• Dispersion considered: Standard deviation used to determine the probability of assets ending up below the default point.
• Difference (68%) between expected value and default point is standardized by dividing by volatility (25%).
  • Result: 2.72 standard deviations away from default.

Formula for Merton Model

• D2 (Distance to Default):
  • Formula: ln(Firm Value/Default Point) + (Expected Growth - (Volatility^2 / 2)) / Volatility
  • Converts difference to standard normal units.
• Apply inverse normal standard cumulative distribution to calculate PD.

Excel Example

• Assumptions entered: Firm value: $11,000, Default point: $600.
• Steps to calculate in Excel:
  • Numerator: ln($11,000/$600) + 20% - (25%^2 / 2) = 68%
  • Denominator: 25%
  • D2: 68% / 25% = 2.72 standard deviations
  • Cumulative distribution function (NormSdist): Probability of default = 0.33% (about 1/3 of 1%)

Conclusion

• Review of Merton Model: Calculates the probability of default based on firm's asset value, growth rate, and volatility.
• Structural approach offers an endogenous view of credit risk.

Note

• Option pricing theory is used initially to get the firm’s asset value and volatility, after which the process is mechanical.

-David Harper, Bionic Turtle