The Merton Model Lecture Notes

Introduction

• Speaker: MJ, Fellow Actuary
• Topic: Merton Model

History and Background

• Creator: Robert Cox Merton
  • Part of Black-Scholes Model
  • Won Nobel Prize
  • Involved in Long-Term Capital Management (LTCM)
  • LTCM had +40% returns before catastrophic losses
• Full Name: Black-Scholes-Merton Model
• Student: Robert Jarrow
  • Developed the Jarrow-Turnbull Model
  • Uses Markov Jump processes to determine probability of default
• KMV Extension
  • Adopted by Moody's (credit rating agency)
  • Used in proprietary models: LossCalc and RiskCalc
• Foundation: Key in current financial models

Merton Model Basics

• Company Finance: Financed by equity and debt
• Model Type: Structured model
  • Uses share price (equity) to calculate probability of default
• Assumptions:
  • Equity pays no dividends
  • Debt is a zero-coupon bond (interest combined in final redemption)

Key Formulae

• Company Value at Time 0:
  • V(0) = E(0) + D(0)
  • Where V = Value, E = Equity, D = Debt, and T = Future Date
• Debt at Future Time (T):
  • Minimum(D(T), V(T))
• Equity at Future Time (T):
  • Maximum(V(T) - D(T), 0)

Examples and Scenarios

• Company Performs Well:
  • Equity: V(T) - D(T)
  • Debt: D(0) * (1 + interest rate)
• Company Performs Poorly:
  • Equity: 0
  • Debt: Minimum(D(T), V(T))

Payoff Diagrams

• Equity: Acts like a long call option
  • Above debt: Equity holders are "in the money"
  • Below debt: Liquidation occurs, equity holders receive nothing
• Debt: Acts like a short put option
  • Minimum payoff: Minimum(D(T), V(T))

Merton Model and Black-Scholes Link

• Black-Scholes Model Used: Values options
  • Equity premium: Long call option
  • Debt interest repayment: Put option premium
• Translating Black-Scholes to Merton:
  • E(0): Premium of a call option
  • D(T): Strike price
  • D(0) * (1+interest rate): Put option premium
  • V(T): Future share price of the company
  • V(0): Current share price of the company

Challenges and Considerations

• Assumption Issues:
  • Constant observable asset value is unrealistic
  • Volatility estimation is complex and error-prone

Key Formulas and Application

• Probability of Default (P(default)):

  • P(S(T) < K) = N(-d2), where d2 is from the Black-Scholes formula
  • P(default) = N(-d2)

• Credit Spread Calculation:

  • Use both risk-free rate (r) and corporate bond rate (B)
  • Calculate implied credit spread: B - r

Conclusion

• Powerful model for determining:

  • Probability of default
  • Company's debt at various times
  • Credit spread

• Recommendation: Review call/put options, payoff diagrams, and the Black-Scholes model for better understanding.

• Utility: Using share prices to determine probability of default, debt, and credit spreads.