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Stochastic Continuous-Time Cash Flows

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Stochastic Continuous-Time Cash Flows STOCHASTIC CONTINUOUS-TIME CASH FLOWS A COUPLED LINEAR-QUADRATIC MODEL BY JOHANNUS GERARDUS JOSEPHUS VAN DER BURG A thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy Victoria University of Wellington 2018 ii iii Synopsis Substantiated by theoretical considerations and empirical evidence, this study corroborates the suitability of a stochastic differential equation specification with a linear drift function and a quadratic diffusion function to model continuous-time cash flows. Ignoramus et ignorabimus iv v ACKNOWLEDGEMENTS Soon after completing my Masters, a considerable time ago, I embarked on a PhD that I unfortunately had to abandon prematurely. The decision, not taken lightly, was forced upon me by a rapidly developing and very time-consuming career in (financial) management. However, a completed PhD was something that always lingered in my mind, and a few years into a second career in academia, this opportunity arose. I took it with both hands and invested every spare moment until reaching the finish line. In hindsight, I must admit that there are benefits in doing a PhD later in life. Blessed with the rich professional and personal life experience that I was able to bring this into this dissertation, I have likely avoided some challenges and pitfalls that I may have encountered earlier in life. The process of writing this dissertation proved to be overall a satisfying and enjoyable experience. Yes, there were a few frustrations and desperate moments on my journey but, fortunately, never a true low. It felt like playing chess though: finding solutions to a problem usually required a series of well-considered moves. At each step, there was the anticipation to discover if the selected approach was going to work. More often than not, this was not the case. In chess-parlance: every time I made a move, my imaginary opponent came up with an unexpected, equally brilliant counter-move. That forced me to reformulate and rethink the problem, sometimes fundamentally, in order to develop a new solution strategy. However, never did I belief ‘the other player’ would deliver a checkmate: somewhere, there was always a solution waiting to be discovered, frequently in unanticipated, hidden places. Despite being an activity of solitude, undertaking a PhD is never a solitary process. Here is the place to express my gratitude to all those people who have supported me to realise my academic dream. First, my sincere thanks and appreciation goes out to my supervisors, professors Mark Tippett and Tony van Zijl. They stimulated me intellectually to go the extra mile and showed me new horizons to improve the quality of my work. I truly enjoyed the weekly Skype-conversations with Mark: beside this dissertation, our discussion topics ranged from last week’s achievements of the All Blacks and Wallabies, monetary policy and free markets, to econometric models based on nonsense correlations. Tony had always a listening ear, taught me how to improve my academic writing style and make publications vi more accessible, and, importantly, gently guided me through the intricacies of a university’s bureaucracy. Patiently accepting my mental absence for over two years, my wife Angela deserves the ultimate gratitude, respect and appreciation. Without her loving support and deep involvement, this dissertation would never have been completed. Thank you, my dearest Angie. A special word of thanks to my family: my Dad, almost ninety years, and Corine, Andries and Niels, who are all immensely proud. Sadly, my Mum passed away at the end of last year, too early to see this work completed, but she undoubtedly would have considered the PhD as a major achievement. Last but not least, thanks go out to my employer Nelson Marlborough Institute of Technology for supporting this PhD-dissertation, not only financially but also by generously allocating research time. To my NMIT colleagues, some of whom are on a path to completing a PhD too: I wish them best of luck. I hope that this study will make a contribution, however modestly, to a strongly growing, often challenging but always exciting discipline within the field of quantitative finance. John Van der Burg, March 2018. Not everything that counts can be counted – and not everything that is counted, truly counts. Albert Einstein vii STATEMENT OF ORIGINAL AUTHORSHIP I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of materials which have been accepted for the award of any other degree or diploma at Victoria University of Wellington or any other educational institutions, except where due acknowledgement is made in this thesis. Any contribution made to the research by others, with whom I have worked at Victoria University of Wellington or elsewhere, is explicitly acknowledged in this thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged. Signature: Date: viii ix ABSTRACT 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) x 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
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