# On cash flow valuation

Abstract: A fundamental fact in finance and economics is that moneyhas a time value, meaning that if we want to value an amount ofmoney we get at some future date we should discount the amountfrom the future date back to today. When facing a stream ofcash flows occurring at di®erent times we discount each ofthe cash flows using suitable discount rates and then sum thecontributions. This sum of discounted cash flows defines thevalue today of this stream. Most future cash flows that appearin models in finance and economics are assumed to be stochastic(nondefaultable bonds being a counter example). To be able tovalue these stochastic cash flows we also have to takeexpectations. In some cases even the discount rate should bemodelled as a stochastic object. The purpose of the two papersin this licentiate thesis (?On the Valuation of Cash Flows?Discrete Time Models?and?On the Valuation ofCash Flows?Continuous Time Models?) is to establishgeneral properties of the value process. As time passes twothings happen. Firstly, the cash flows that are realised are nomore parts of the value and secondly, the information we canuse to determine the expected cash flows and discount ratesincreases.The two papers consider discrete time models and continuoustime models respectively.Of course any continuous time modelis necessarily an idealisation. Thus one could argue from amodelling point of view that we should use discrete timemodels. The main reason for using continuous time models isthat we have the powerful machinery of stochastic calculus athand. Discrete time models are mostly used in practice whenvaluing a firm or a project, while the continuous time settingis more frequently used in thoretical approaches to valuation.Most of the results are parallelled in the two papers. Adi®erence is that we discuss some convergence results forthe value in discrete time which do not occur in the continuoustime paper. The reason for not including this in the continuoustime paper is because we find it a more important question indiscrete time. On the other hand the Brownian models incontinuous time, where the Martingale Representation Theorem isan important tool, make the analysis much more transparent.In both papers we first define the underlying objects: thediscount process and the cash flow process. We then define,using these two processes, the value process (i.e. the expecteddiscounted value of the cash flow stream). We show that thediscounted value tends to zero almost surely, and that thereare three equivalent ways of writing the value process, each ofwhich has its own merits. We also extend this result to thecase when the cash flow process and the value process areevaluated at a stopping time. The first paper, on discrete timemodels, then continues by showing examples from finance,economics, and insurance where the discounted value processplays an important role. Finally we present two propositionswith necessary conditions for the value process to convergealmost surely. The second paper, on continuous time models,discusses some properties of the local dynamics of the valueprocess and then continues with Brownian models. We show thatthe value process can equivalently be expressed as a solutionto a forward-backward stochastic di®erential equation.Finally we show that under some additional assumptions there isa one-to-one correspondance between the cash flow process andthe value process. We also investigate the inverse problem offinding a cash flow process generating a given value processand discuss applications to real options.

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