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Lecture 6: Replication and Bankruptcy Analysis

Replication Approach​

Merton's Key Insight: The contingent claim approach shows not just payoffs, but how to price securities by replicating them with traded assets.

Law of One Price: If two securities have identical payoffs, they must have identical prices.

Replication Strategy: Express corporate securities in terms of:

  • Risk-free bonds (fixed payments)
  • Options (call and put options)
  • Other traded securities

Practical Application: Since we don't know how to price corporate bonds directly, we find other securities with the same payoff and use their prices.

Asset Value Dynamics​

Geometric Brownian Motion​

Model: Asset values follow a stochastic process with two components:

  1. Drift (μ): Constant upward trend
  2. Volatility (σ): Random fluctuations

Mathematical Representation:

dV=μV dt+σV dWdV = \mu V \, dt + \sigma V \, dW

Where:

  • V = Asset value
  • μ = Expected return (drift)
  • σ = Volatility
  • dW = Random shock (Brownian motion)

Visual Representation​

  • X-axis: Time
  • Y-axis: Asset value
  • Starting Point: V₀ (initial value)
  • Multiple Paths: Various possible future values
  • Default Threshold: F₁ + F₂ = $91.2M

Risk-Free vs Risky Debt​

Risk-Free Debt​

Characteristics:

  • Government bonds denominated in local currency
  • Fixed payment regardless of economic conditions
  • No credit risk (government can print money)
  • Example: US Treasury bills

Payoff: Always receives promised amount (e.g., $43.2M)

Risky Corporate Debt​

Characteristics:

  • Corporate bonds with credit risk
  • Payment depends on firm's asset value
  • Default risk when assets < liabilities
  • Example: Our S&P 500 investment firm

Payoff: min(V_T, F₁) - receives promised amount or all available assets

Key Difference​

Risk-free debt: Always pays $43.2M Risky debt: Pays $43.2M only if V_T ≥ $43.2M, otherwise pays V_T

Put Options and Credit Risk​

Mathematical Relationship​

Risky Bond = Risk-Free Bond - Put Option

Senior Debt Payoff:

min⁡(VT,F1)=F1−max⁡(F1−VT,0)\min(V_T, F_1) = F_1 - \max(F_1 - V_T, 0)

Where:

  • F₁ = Risk-free bond paying $43.2M
  • max(F₁ - V_T, 0) = Put option on firm's assets

Interpretation​

  • Risk-free bond: Always pays $43.2M
  • Put option: Pays (F₁ - V_T) when V_T < F₁, zero otherwise
  • Net result: Risky bond receives F₁ when V_T ≥ F₁, V_T when V_T < F₁

Market Trading​

  • Put options are traded on CBOE (Chicago Board Options Exchange)
  • Credit default swaps are similar instruments
  • Pricing: Risky bonds are cheaper than risk-free bonds

Absolute Priority Rule (APR)​

Definition​

Absolute Priority Rule: In bankruptcy, claims are paid in strict order of seniority.

Payment Order​

  1. Senior Debt: First priority
  2. Junior Debt: Second priority
  3. Equity: Residual claimant (last)

Example: Single Debt Case​

Scenario: Firm with $100M debt, $70M assets

  • According to APR: Debt holders get $70M, equity gets $0
  • Result: Debt holders become new owners, equity holders are wiped out

Key Principle​

Assets = Liabilities: The accounting identity must always hold.

Deviations from APR​

When Deviations Occur​

Real-world situations where APR is violated:

  1. Intangible Assets: High value in human capital, technology, relationships
  2. Bargaining Power: Equity holders can threaten to leave
  3. Bankruptcy Costs: Additional costs if firm is liquidated
  4. Going Concern Value: Firm worth more as ongoing business

Example: Tech Company​

Scenario: Firm owes $100M, has $70M assets, but founders control key technology

Threat: If equity holders leave, firm value drops to $30M Bargaining: Equity holders demand $10M to stay Result: Debt holders agree to reorganization to preserve value

Factors Affecting Deviations​

  • Asset Tangibility: More tangible assets → closer to APR
  • Human Capital: More intangible assets → larger deviations
  • Industry Type: Airlines (tangible) vs Tech (intangible)

Bankruptcy Procedures​

US Bankruptcy Code​

Two Main Procedures:

  1. Chapter 11: Reorganization

    • Firm continues operating
    • 90-180 days to negotiate
    • Debt and equity holders bargain
    • Judge can extend negotiation period

  2. Chapter 7: Liquidation

    • Firm ceases to exist
    • Assets sold off
    • Proceeds distributed according to APR
    • Judge can "cram down" if no agreement

Decision Process​

Step 1: Enter bankruptcy (Chapter 11) Step 2: Negotiate reorganization plan Step 3: If agreement reached → Reorganization Step 4: If no agreement → Chapter 7 liquidation

Key Players​

  • Debt Holders: Want maximum recovery
  • Equity Holders: Want to preserve some value
  • Judge: Can force decisions if no agreement
  • Employees: Affected by outcome

Time Value of Money​

Deferred Payments: Sometimes equivalent to reduced payments

  • Example: $70M now vs $70M in 15 years
  • Present Value: Deferred payment worth less today
  • Bargaining Tool: Debt holders may accept deferred payments

Key Takeaways​

  1. Replication Approach: Corporate securities can be replicated using traded assets (bonds, options)

  2. Law of One Price: Identical payoffs imply identical prices

  3. Asset Dynamics: Asset values follow stochastic processes with drift and volatility

  4. Risk-Free vs Risky: Government bonds are risk-free; corporate bonds have credit risk

  5. Put Options: Credit risk can be modeled as put options on firm's assets

  6. Absolute Priority Rule: Theoretical payment order in bankruptcy

  7. APR Deviations: Real-world deviations occur due to bargaining power and intangible assets

  8. Bankruptcy Procedures: Chapter 11 (reorganization) vs Chapter 7 (liquidation)

  9. Bargaining Power: Equity holders' power increases with intangible assets

  10. Time Value: Deferred payments can be equivalent to reduced payments

  11. Industry Differences: Tangible asset firms closer to APR than intangible asset firms

  12. Practical Applications: Understanding these concepts helps analyze real-world bankruptcy situations