The Merton Model Lecture Notes

Jul 9, 2024

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.