Stochastic continuous-time cash flows: A coupled linear-quadratic model
Authors:   , Van der Burg Johannus Gerardus Josephus 
Published: December 2021

<p>The focal point of this dissertation is stochastic continuous-time cash flow models. These models, as underpinned by the results of this study, prove to be useful to describe the rich and diverse nature of trends and fluctuations in cash flow randomness. Firstly, this study considers an important preliminary question: can cash flows be fully described in continuous time? Theoretical and empirical evidence (e.g. testing for jumps) show that under some not too stringent regularities, operating cash flow processes can be well approximated by a diffusion equation, whilst investing processes -preferably- will first need to be rescaled by a system-size variable. Validated by this finding and supported by a multitude of theoretical considerations and statistical tests, the main conclusion of this dissertation is that an equation consisting of a linear drift function and a complete quadratic diffusion function (hereafter: “the linear-quadratic model”) is a specification preferred to other specifications frequently found in the literature. These so-called benchmark processes are: the geometric and arithmetic Brownian motions, the mean-reverting Vasicek and Cox, Ingersoll and Ross processes, and the modified Square Root process. Those specifications can all be considered particular cases of the generic linear-quadratic model. The linear-quadratic model is classified as a hybrid model since it is shown to be constructed from the combination of geometric and arithmetic Brownian motions. The linear-quadratic specification is described by a fundamental model, rooted in well-studied and generally accepted business and financial assumptions, consisting of two coupled, recursive relationships between operating and investing cash flows. The fundamental model explains the positive feedback mechanism assumed to exist between the two types of cash flows. In a stochastic environment, it is demonstrated that the linear-quadratic model can be derived from the principles of the fundamental model. There is no (known) general closed-form solution to the hybrid linear-quadratic cash flow specification. Nevertheless, three particular and three approximated exact solutions are derived under not too stringent parameter restrictions and cash flow domain limitations. Weak solutions are described by (forward or backward) Fokker-Planck- Kolmogorov equations. This study shows that since the process is converging in time (that is, approximating a stable probability distribution), (uncoupled) investing cash flows can be described by a Pearson diffusion process approaching a stationary Person-IV probability density function, more appropriately a Student diffusion process. In contrast, (uncoupled) operating cash flow processes are diverging in time, that is exploding with no stable probability density function, a dynamic analysis in a bounded cash flow domain is required. A suggested solution method normalises a general hypergeometric differential equation, after separation of variables, which is then transformed into a Sturm-Liouville specification, followed by a choice of three separate second transformations. These second transformations are the Jacobi, the Hermitian and the Schrödinger, each yielding a homonymous equation. Only the Jacobi transformation provides an exact solution, the other two transformations lead to approximated closed-form general solutions. It turns out that a space-time density function of operating cash flow processes can be construed as the multiplication of two (independent) time-variant probability distributions: a stationary family of distributions akin to Pearson’s case 2, and the evolution of a standard normal distribution. The fundamental model and the linear-quadratic specification are empirically validated by three different statistical tests. The first test provides evidence that the fundamental model is statistically significant. Parameter values support the conclusion that operating and investing processes are converging to overall long-term stable values, albeit with significant stochastic variation of individual firms around averages. The second test pertains to direct estimation from approximated SDE solutions. Parameter values found, are not only plausible but agree with theoretical considerations and empirical observations elaborated in this study. The third test relates to an approximated density function and its associated approximated maximum likelihood estimator. The Ait-Sahalia- method, in this study adapted to derive the Fourier coefficients (of the Hermite expansion) from a (closed) system of moment ODEs, is considered a superior technique to derive an approximated density function associated with the linear-quadratic model. The maximum likelihood technique employed, proper for high-parametrised estimations, includes re-parametrisation (based on the extended invariance principle) and stepwise maximisation. Reported estimation results support the hypothesised superiority of the linear-quadratic cash flow model, either in complete (five-parameter form) or in a reduced-parameter form, in comparison to the examined five benchmark processes.</p>