Dynamic capital structure modelling under alternative stochastic processes

Simon Seir Bjerrisgaard & Denis Fedoryaev

Student thesis: Master thesis


A coherent line of development of research focusing on modelling optimal capital structure is presented. We begin by revisiting the pricing model introduced in the seminal work of Merton (1974), which represents the fundamental building block for most capital structure models. The general partial differential equation is rederived, and it is demonstrated how it is applied to price debt and equity as contingent claims. The first stepping stone to our analysis is the static model along the lines of Leland (1994). We present comparative statics for optimally chosen everage, coupon, and default threshold as well as for tax advantage to debt, and discuss the drawbacks of the model in detail. It is evident that under our own base case parameters based on the most recent Danish data and empirical estimates which differ from unreasonably high values applied in Leland, the optimal leverage is too high compared to that observed in practice. We proceed by introducing dynamics in the model of capital structure; rather than focusing on any particular model, we outline a generalised framework to describe the overall family of existing dynamic capital structure models. The state variable is the operating income, and it is the value of a claim to the entire payout of the firm that is modelled as a stochastic process, which restores the no-arbitrage condition violated in the static model. As the leverage can be readjusted, the restructuring boundary is incorporated, stipulating when refinancing should take place, and is obtained from the smooth-pasting condition to ensure incentive compatibility. The key insight from the comparative statics analysis is that the optimal leverage drops substantially compared to that in the static model, and the default boundary is also lowered, reflecting the fact that the refinancing option, ceteris paribus, enhances the firm valuation. Further, we propose our own dynamic model which alters the cash flow process by assuming mean reversion. The vast majority of existing capital structure studies rests on the assumption that the state variable follows a geometric Brownian motion since this ensures mathematical tractability while modelling the capital structure readjustment. We argue that this underlying process due to its properties is not consistent with dynamics of the firm fundamental and that the assumption of mean reversion appears to be more suitable. Not only the latter describes the development of the real sector better, but is also reinforced by empirical evidence. Besides, by assuming that earnings follow the mean-reverting process, we obtain a better control over the process and can thus draw a much clearer distinction between the industries in terms of both profitability and stability through varying the long-term mean and the speed of mean reversion, respectively. As a matter of fact, optimal capital structure has never been modelled dynamically outside of the scope of a geometric Brownian motion, presumably due to the loss of homogeneity property. We assume the modified mean-reverting process with volatility being proportional to the current earnings level, and derive the ordinary differential equation used for pricing debt and equity. Further, using contingent claims analysis and state pricing, we prove that the new process still possesses the homogeneity property which implies that the mechanics of the model is unaltered in time. After that we describe what we call the optimal capital structure decision framework and formulate how the tuple that closes it is obtained, which fully determines the capital structure choice of the owner-manager. Extensive numerical simulations are conducted, with explicit focus on numerical algorithms and their shortcomings. We bring together the key optimal capital structure frameworks and carry out a cross-model comparison to uncover the implications that different assumptions have for the results. When benchmarking our model against the static mean-reverting model of Sarkar and Zapatero (2003), we find lower optimal leverage and optimal coupon. If compared to the conventional dynamic GBM-based model, our model suggests lower optimal leverage and higher restructuring frequency, stipulated by the difference in expectations of equity holders regarding the future development of cash flow under the two processes. This relationship is even more pronounced if we correct for the finiteness of the stationary variance of the mean-reverting process. Overall, given reasonable assumptions for base case parameters, the optimal leverage in our model is found to be closer to the empirical regularities than that in the existing models. All relationships are studied through the prism of mean reversion, and it is shown that the speed of earnings convergence and the long-term mean value of earnings are indeed important parameters as their impact on the key variables could be rather substantial. Finally, we extend the analysis by modifying the GBM-based model to take into account fixed operating costs, so that the resulting cash flow is not bounded at zero. We demonstrate that higher operating leverage implies lower financial leverage, and find that when the firm is allowed to have negative cash inflow, it will be less levered compared to an otherwise identical firm whose earnings are instead mean-reverting but always positive. Further, we incorporate a call premium in the dynamic model with mean reversion and show that assuming debt being callable at par may lead to somewhat understated leverage and too high restructuring frequency.

EducationsMSc in Advanced Economics and Finance, (Graduate Programme) Final Thesis
Publication date2011
Number of pages133