Below is an EditGrid spreadsheet that models subordinated debt under the Merton model (for FRM candidates, the assignment is Stulz Chapter 18. The last tab contains Stulz’s example on page 582).

Here are the basic mechanics:

• The firm has either high ($200) or low ($20) asset value (V)
• With senior debt (F) of $100 and subordinated debt (U) of $50
• Two call options are valued: a call option with strike equal to (F) and a call option with strike equal to (F+U).
• Firm value = Senior Debt + Subordinated Debt + Equity. Therefore,
• Subordinated Debt = Firm value – Senior Debt – Equity. Further, because Senior Debt = Firm Value – Call[asset value = V, strike price = F] and Equity = Call[asset value = V, strike = U+F]
• Subordinated Debt = Firm value - (Firm Value – Call[asset value = V, strike price = F]) - Call[asset value = V, strike = U+F], such that
• Subordinated Debt = Call[asset value = V, strike price = F] - Call[asset value = V, strike = U+F]. That’s how the subordinated debt is calculated below, as the difference between two call option where only the strike price is changed

Here are key points for the FRM candidate:

• As Stulz says, “an increase in firm volatility has an ambiguous effect on subordinated debt value.” That’s because subordinated debt value = Call[…,volatility,…] – Call[…, volatility,…]. The volatility increases one term but decreases the other.
  
• In regard to volatility versus subordinated debt value (see sheet below called Volatility_changing):
  
  If the firm is high value (unlikely to default), higher volatility associates with lower subordinated debt value. The subordinated debt is here behaving like senior debt.
  
  If the firm is low value (likely to default), higher volatility associates with higher subordinated debt value. The subordinated debt is here behaving like equity.
  
• In regard to maturity versus subordinated debt value (see sheet below called Maturity_changing):
  
  If the firm is high value (unlikely to default), longer maturity associates with lower subordinated debt value. The subordinated debt is again behaving like senior debt.
  
  If the firm is low value (likely to default), longer maturity associates with higher subordinated debt value. The subordinated debt is again behaving like equity. Stulz says, in this case, “there is a better chance [with longer maturity] that it will pay something.”

In summary, if the firm is high value, subordinated debt behaves like senior debt (i.e., increases in value with lower volatility and shorter maturity. If the firm is low value, subordinated debt behaves like equity (i.e., increases in value with higher volatility and longer maturity

Spreadsheet:

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