Structural And Reduced Form Models

Structural Models

Structural models of corporate credit risk are based on the structure of a company’s balance sheet and rely on insights provided by option pricing theory.

Option Analogy

Consider a hypothetical company with assets that are financed by equity and a single issue of zero coupon debt. The value of the assets at any point in time is the sum of the value of equity and the value of debt.

Due to the limited liability nature of corporate equity, the shareholders effectively have a European call option on the company’s assets with a strike price equal to the face value of debt. If at the maturity of the debt, the value of the company’s assets is higher than the face value of debt, shareholders will exercise their call option to acquire the assets (and then pay off the debt and keep the residual). On the other hand, if the value of the company’s assets is less than the face value of debt, the shareholders will let the option expire worthless (i.e., default on the debt), leaving the company’s assets to the debt investors.

Hence, at time T, (corresponding to the maturity of debt):

which means that value of $deb{t}_T$ = min ($A_T$, K)

An alternate, but related interpretation considers equity investors as long the net assets of the company (with a time T value of $A_T$ − K) and long a European put option, allowing them to sell the assets at an exercise price of K. Default is then synonymous to exercising the put option.

Under the put option analogy, the investors in risky debt can be construed to have a long position in risk free debt and a short position in that put option.

From our earlier discussion on CVA, recall that:

Therefore, the value of the put option = CVA.

Figure 26.2 shows the distribution of asset values at time T. If the value of assets falls below the default barrier K, the company defaults. The probability of default is indicated by the region in the left tail below the default barrier of K.

Figure 26.2: Distribution of Asset Value at Time T

Advantages of structural models:

1. Structural models provide an economic rationale for default (i.e., $A_T$ < K) and explain why default occurs.
2. Structural models utilize option pricing models to value risky debt.

Disadvantages of structural models:

1. Because structural models assume a simple balance sheet structure, complex balance sheets cannot be modeled. Additionally, when companies have off balance sheet debt, the default barrier under structural models (K) would be inaccurate and hence the estimated outputs of the model will be inaccurate.
2. One of the key assumptions of the structural model is that the assets of the company are traded in the market. This restrictive assumption makes the structural model impractical.

Reduced Form Models

Reduced form (RF) models do not rely on the structure of a company’s balance sheet and therefore do not assume that the assets of the company trade. Unlike the structural model, reduced form models do not explain why default occurs. Instead, they statistically model when default occurs.

Default under the RF model is a randomly occurring exogenous variable. As such, the RF model imposes assumptions on the output of the structural model (asset values, recovery rates, etc.).

A key input into the RF model is the default intensity, which is the probability of default over the next (small) time period. Default intensity can be estimated using regression models. These regression models employ several independent variables including company specific variables (e.g., leverage, beta, interest coverage ratio) as well as macro economic variables.

Advantages of reduced form models:

1. RF models do not assume that the assets of a company trade.
2. Default intensity is allowed to vary as company fundamentals change, as well as when the state of the economy changes.

Disadvantages of reduced form models:

1. RF models do not explain why default occurs.
2. Under RF models, default is treated as a random event (i.e., a surprise), but in reality, default is rarely a surprise (it is often preceded by several downgrades).