Merton Debt Payoff as a Risk-Free Bond Minus a Put

Summary

The document asks how the Merton model’s debt payoff can be written both as debt face value minus a put payoff and as the lesser of face value and asset value. At maturity, debt holders receive the firm’s assets when those assets fall short of the promised amount, and otherwise receive the promised amount. The put represents the shortfall borne by debt holders when assets are insufficient.

The two expressions are algebraically equivalent: subtracting the positive part of face value minus assets leaves the asset value below the debt threshold and the full face value above it. The discussion frames this as a payoff identity, rather than a derivation of the debt’s value before maturity. It does not cover default timing, recovery assumptions beyond the stated payoff, or the valuation of the bond and option components.

Key ideas

• At maturity, debt holders receive the lesser of the firm’s asset value and the debt face value.
• The payoff can be represented as face value minus the positive shortfall of assets relative to debt.
• The short put on firm assets captures the loss to debt holders when asset value is below the promised debt amount.
• The payoff identity describes maturity cash flows and does not by itself give the debt’s value before maturity.

Tags

# Future value of the debt under Merton model


# Future value of the debt under Merton model












The author Malz states the future value of the firm's debt under the Merton model can be found from:

$$ D_{t} = D - \max(D - A_{t} , 0) $$

(where $D$ is the par value of the debt, $A_{t}$ is the current value of the firm's assets)

I know that according to the model, the value of the debt can be modeled as a simultaneous position in a risk-less bond with face value of the risky debt discounted using the risk free rate and a short put on the firm's assets with a strike of the value of the debt. If the asset value is below the value of the debt at maturity, the payoff is max(asset value, 0).

How then do we get $D_{t} = D - \max(D - A_{t}, 0)$ ?

I can see $D_{t} = \min(D, \max(A_{t}, 0))$ but not $D_{t} = D - \max(D - A_{t}, 0)$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.