# Merton model for Probability of Default - What liabilities?

In Merton structural model for credit risk (74), the company's Assets and Liabilities are used to imply the default probability of the firm. At the end, we don't need to know the assets value, and just use the equity's price process, and use the liabilities as a strike.

My question is, what information will you use for the liabilities? is it the total debt value divided by the outstanding shares?

The Kealhoffer-Merton-Vasicek (KMV) model is derivative of Merton. Essentially, it codifies the calibration process and extends the framework to empirical distributions.

The following entry, Modeling Default Risk, contains one such passage regarding KMV’s parameterization of liabilities:

Oldrich Vasicek and Stephen Kealhofer have extended the Black-Scholes-Merton framework to produce a model of default probability known as the Vasicek-Kealhofer (VK) model. This model assumes the firm’s equity is a perpetual option with the default point acting as the absorbing barrier for the firm’s asset value. When the asset value hits the default point, the firm is assumed to default. Multiple classes of liabilities are modeled: short-term liabilities, long-term liabilities, convertible debt, preferred equity, and common equity. When the firm’s asset value becomes very large, the convertible securities are assumed to convert and dilute the existing equity. In addition, cash payouts such as dividends are explicitly used in the VK model. A default database is used to derive an empirical distribution relating the distance-to-default to a default probability. In this way, the relationship between asset value and liabilities can be captured without resorting to a substantially more complex model characterizing a firm’s liability process.

Personally, I find that attempts to systemically implement KMV in the real world often fall short by attempting to introduce more precision than is practical, especially when one wishes to retain analytically desirable properties of plain vanilla Merton. When partitioning various types of assets and liabilities, for example, there may be different default probabilities assigned to different debt maturities and different classes of claims with or without maturity. Rather than introduce excessive complexity, I usually make simplifying assumptions such as grouping all non-current debt by average YTM and maturity and retaining assumptions regarding the log-normality of the processes.

In spite of its complexities, I believe that KMV’s canonical approach to structuring liabilities is instructive as to how one might decide to structure a similar model.

My question is, what information will you use for the liabilities? is it the total debt value divided by the outstanding shares?

Common choices for the liabilities is the short term debt plus 50% of the long term debt. The equity value is the taken to be the share price times the number of shares. See e.g., my notes here and the references therein.