Cross-Ownership as a Structural Explanation for Over- and Underestimation of Default Probability

TL;DR

This paper extends Merton's asset valuation model to include cross-ownership of firms, revealing that traditional lognormal assumptions can misestimate default probabilities, either overestimating or underestimating actual risks.

Contribution

It generalizes the Merton model to account for cross-ownership, demonstrating its impact on default probability estimation both theoretically and through simulations.

Findings

Lognormal model can overestimate or underestimate default probabilities due to cross-ownership.

Maximum cross-ownership levels lead to theoretical limits on default probability estimates.

Lognormal assumptions restrict the range of possible default probabilities, unlike actual probabilities which span 0 to 1.

Abstract

Based on the work of Suzuki (2002), we consider a generalization of Merton's asset valuation approach (Merton, 1974) in which two firms are linked by cross-ownership of equity and liabilities. Suzuki's results then provide no arbitrage prices of firm values, which are derivatives of exogenous asset values. In contrast to the Merton model, the assumption of lognormally distributed assets does not result in lognormally distributed firm values, which also affects the corresponding probabilities of default. In a simulation study we see that, depending on the type of cross-ownership, the lognormal model can lead to both, over- and underestimation of the actual probability of default of a firm under cross-ownership. In the limit, i.e. if the levels of cross-ownership tend to their maximum possible value, these findings can be shown theoretically as well. Furthermore, we consider the default probability of a firm in general, i.e. without a distributional assumption, and show that the lognormal model is often able to yield only a limited range of probabilities of default, while the actual probabilities may take any value between 0 and 1.