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Lecture 5: Contingent Claim Approach

Introduction to Merton's Model​

Background: Developed by Robert Merton (Nobel Prize winner) in 1974

Purpose: The contingent claim approach shows not just the payoff of different securities, but also how to price them using options (call and put options).

Focus for Today: We will focus on payoff functions and replication equations that express the value of securities at future time, not current pricing.

Connection to Previous Session: Builds directly on the simple corporation example from Lecture 4 with S&P 500 investment financed by three securities.

Model Setup and Assumptions​

Basic Setup​

  • Single Asset: Company invests in one asset with current market value V₀
  • Our Example: V₀ = $100 million (S&P 500 investment)
  • Maturity: T = 1 year
  • Asset Value at Maturity: V_T (stochastic variable)

Key Assumptions​

  1. No Taxes: No tax implications
  2. No Transaction Costs: No bid-ask spreads
  3. No Early Default: Default can only occur at maturity
  4. Risk-Free Rate: Continuous compounding (not crucial for our calculations)
  5. Asset Dynamics: Asset value changes according to drift and volatility

Payoff Functions for Different Securities​

Senior Debt Payoff (F₁)​

  • Face Value: $40 million
  • Coupon Payment: $3.2 million
  • Total Promised Payment: F₁ = $43.2 million

Junior Debt Payoff (F₂)​

  • Face Value: $40 million
  • Coupon Payment: $8 million
  • Total Promised Payment: F₂ = $48 million

Total Debt Obligation​

  • Total Owed: F₁ + F₂ = $91.2 million
  • Payment Waterfall: Senior first, then junior, then equity (residual)

Senior Debt Payoff Analysis​

Payoff Function​

Senior Debt Payoff=min⁡(VT,F1)\text{Senior Debt Payoff} = \min(V_T, F_1)

Where:

  • V_T = Asset value at maturity
  • F₁ = $43.2 million (promised payment)

Two Scenarios​

  1. V_T < F₁: Senior debt receives V_T (partial payment)
  2. V_T ≥ F₁: Senior debt receives F₁ (full payment)

Alternative Expression​

Senior Debt Payoff=VT−max⁡(VT−F1,0)\text{Senior Debt Payoff} = V_T - \max(V_T - F_1, 0)

This shows that senior debt is equivalent to:

  • Risk-free bond paying F₁
  • Minus a put option on V with strike price F₁

Equity Payoff Analysis​

Payoff Function​

Equity Payoff=max⁡(VT−(F1+F2),0)\text{Equity Payoff} = \max(V_T - (F_1 + F_2), 0)

Where:

  • F₁ + F₂ = $91.2 million (total debt obligation)

Interpretation​

  • Below $91.2M: Equity receives $0 (limited liability)
  • Above $91.2M: Equity receives residual (V_T - $91.2M)

Key Characteristics​

  • Unlimited upside potential
  • Limited downside (cannot go below zero)
  • Call option-like payoff on the firm's assets

Junior Debt Payoff Analysis​

Direct Calculation Method​

Junior Debt=VT−Senior Debt−Equity\text{Junior Debt} = V_T - \text{Senior Debt} - \text{Equity}

This uses the accounting identity: Assets = Liabilities

Complex Payoff Function​

Junior Debt=min⁡(F2,max⁡(VT−F1,0))\text{Junior Debt} = \min(F_2, \max(V_T - F_1, 0))

Where:

  • F₂ = $48 million (promised payment)
  • F₁ = $43.2 million (senior debt payment)

Three Segments​

  1. V_T < F₁: Junior debt = $0
  2. F₁ ≤ V_T < F₁ + F₂: Junior debt = V_T - F₁ (linear increase)
  3. V_T ≥ F₁ + F₂: Junior debt = F₂ (capped at promised amount)

Three-State Analysis Framework​

State 1: V_T < F₁ (Below $43.2M)​

Example: V_T = $30M

  • Senior Debt: Receives $30M (all available assets)
  • Junior Debt: Receives $0
  • Equity: Receives $0

State 2: F₁ ≤ V_T < F₁ + F₂ (Between $43.2M and $91.2M)​

Example: V_T = $70M

  • Senior Debt: Receives $43.2M (full promised payment)
  • Junior Debt: Receives $26.8M (V_T - F₁ = $70M - $43.2M)
  • Equity: Receives $0

State 3: V_T ≥ F₁ + F₂ (Above $91.2M)​

Example: V_T = $1000M

  • Senior Debt: Receives $43.2M (full promised payment)
  • Junior Debt: Receives $48M (full promised payment)
  • Equity: Receives $908.8M (V_T - F₁ - F₂ = $1000M - $91.2M)

Key Takeaways​

  1. Merton's Model: Nobel Prize-winning framework for analyzing corporate securities as contingent claims

  2. Payoff Functions: Each security has a mathematical payoff function based on asset value at maturity

  3. Senior Debt: min(V_T, F₁) - receives promised payment or all available assets, whichever is less

  4. Equity: max(V_T - (F₁ + F₂), 0) - call option on firm's assets with strike price equal to total debt

  5. Junior Debt: Can be calculated directly using accounting identity or complex payoff function

  6. Three-State Framework: Systematic way to analyze payoffs across different asset value scenarios

  7. Limited Liability: Equity holders cannot lose more than their initial investment

  8. Payment Priority: Senior debt → Junior debt → Equity (residual claimant)

  9. Risk-Return Trade-off: Higher priority securities have lower risk and lower return potential

  10. Mathematical Approach: Provides precise framework for understanding security payoffs and relationships