THE THEORY BEHIND THE MERTON MODEL
The Merton Model was first developed by Robert Merton in 1974 and was the first of what are now called “structural models” of debt and default. Merton worked with Fisher Black and Myron Scholes to develop the Black-Scholes equation for option pricing, and the Merton Model is based on a similar understanding of price movements. The key point in understanding how structural models work is to consider the equity to be a contingent residual claim holder on the value of the assets of the firm. This is a complex idea that we will explain soon in more detail.
First, let us consider how companies default. In general, there are three types of ways that a company can default on its issued debt. One is that a company does not comply with the covenants it has agreed to in the loan documentation. This is sometimes referred to as a “technical default” because often a company will continue to pay its creditors, and the creditors may not take action against the debtor. From a quantitative point of view, while technical defaults may trigger certain legal repercussions or a renegotiation of terms, they are not considered economic defaults, and as a result they are not generally included in default analysis.
The other two general categories of defaults are generally considered “credit events,” which can impact all of a company's stakeholders. The first type of credit event is nonpayment, which occurs when a debtor does not pay principal or interest on an assigned date. Occasionally, these defaults are driven by liquidity: whether a company is able to raise cash to make a payment on time. In economies with well-developed financial systems, it is usually simple for a solvent company with capable management to avoid liquidity defaults. However, during financial turmoil such as banking crises, normal lines of liquidity may not be available, and otherwise solvent companies may enter default as a result. These defaults are important and can be extremely difficult to capture in simulation or quantitative forecasting, and the inability to account for these scenarios can be considered one of the shortcomings of classic simulation processes.
Finally, the third category of defaults (and the other category of credit event) is continued insolvency or strategic default. The vast majority of credit events occur when companies decide that they will no longer be able to support their debt or create profits. Companies then file for bankruptcy, restructure, or announce that they will not make payments on their debt. The key factor is that these defaults (and hence the majority of all defaults) are not exactly voluntary, but they are rational decisions based on the economic state of the debtor. In some industries such as banking and insurance, regulatory takeover exists as a fourth category of defaults, but these defaults can be considered similarly: The decision to reorganize, however, is made by regulators instead of the owners or their managers. In both cases, this decision is generally made when the value of the firm's assets is substantially less than the amount that needs to be paid back (i.e., it is unlikely that the firm will be able to make good on its liabilities).
The Model
The Merton Model is primarily focused on the relationship between the value of the firm, the value of the firm's assets, and the face value of the firm's debt. The value of the firm for public companies is generally available by adding the total value of the public stock to the total value of the debt (this is sometimes called the enterprise value). However, the true value of the firm's assets is not easily visible to us.
FIGURE 5.1 The simplified structure of a company's balance sheet. Debt holders have first claim on the proceeds of assets, but equity holders retain optionality.
Merton first simplified a company's balance sheet so that it only contained one class of debt and one class of stock. Consider the equity holders in such a situation. They control the assets of the company. One of the opportunities available to the equity holders of a company is the ability to liquidate the company's assets. This means the company closes shop or finds a buyer for the company as a whole. Of course, the decision to sell the company's assets does not mean they do not have to pay back money they have borrowed. Should an equity holder cause this sale, the owners would receive whatever proceeds from the sale remain after the debt holders are paid off. See Figure 5.1
To restate the equity holders' position: They have the right (but not the responsibility) to receive the excess value of the assets over a defined threshold. Does this sound familiar? It should; the equity is effectively a call option! However, instead of a traditional equity call option, which is an option upon a company's shares, the equity can be considered to be an option on a firm's assets, with a strike price equal to the face value of a company's debt.
Considered this way, the equity value of the firm can be related to the value of the firm's assets, as well as the volatility of the value of these assets, just like the value of an option can be related to the value and the volatility of an equity. The Merton Model starts with this construct in order to determine the volatility of the value of the assets of the firm, which is generally given in texts as σv and the actual market value of the assets, which is generally represented as V. The default barrier is represented by B and the period of time is T. So the Merton model can be written as (equation 5.1 through 5.3):
Where
With the exception of V and σv and potentially μ, the drift term that we will see coming up, all of the key components of the model are visible. Different academics and practitioners have come up with different methods for obtaining the asset value and the volatility of its returns. This chapter offers one version, but more complex, commercially available applications of structural models (the best known of these at the time of writing is Moody's-KMV) include large databases that contain vast amounts of historical data to improve the accuracy of the model's predictions. See Figure 5.2
FIGURE 5.2 Conceptually, structural models forecast a firm's default when the value of the firm's assets drops a fixed amount below the debt that needs to be repaid. In this situation the equity holders are sufficiently “under water” that they do not maintain the firm as a going concern.
The output of these models is the probability that a company will default. Since we make the assumption of normalcy when we use the Black-Scholes model, we can state the probability in terms of the number of standard deviations from the mean scenario we see a default. This is referred to as the “distance to default” or DD, and is typically calculated with equation 5.4:
If this is confusing, do not fret. We will walk through these calculations later in the chapter.