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

• Maybe this helps: $\min(x, y) = x-\max(x-y, 0)$ – Slug Pue Aug 15 '15 at 14:45
• Slug , if you post this as an answer, I can accept it. – AfterWorkGuinness Nov 17 '15 at 22:51