• If the firm's value $V_T$ is greater than or equal to the face value $F$, the debt holders receive $F$, and the term $\max(F - V_T, 0)$ becomes zero.
• If the firm's value $V_T$ is less than $F$, the debt holders receive $V_T$, and the term $\max(F - V_T, 0)$ represents the shortfall $F - V_T$.

Thus, the formula $F - \max(F - V_T, 0)$ correctly represents the amount debt holders receive at maturity based on the firm’s value.

Things to Remember

• This model captures the essence of debt as a financial instrument: debt holders have a capped upside (up to $F$) but face the risk of receiving less if $V < F$.
• The formula emphasizes the seniority of debt over equity; in default scenarios, equity holders are wiped out before debt holders incur losses.
• The Merton model helps in quantifying the risk of default based on the current value of the firm relative to its debt obligations.
• This framework is particularly useful in credit risk analysis, where assessing the likelihood and impact of default is key.