Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

8.2 The Role of Time – Cash Flows and Associated Risk Measures

8.2.1 Costs Over a Time Period – the Cash Flow Model

In Section 8.1, costs were introduced in a ‘lumped’ view as output variables of interest occurring once at the future time of interest. Most economic models would more generally consider a variable of interest that accumulates costs occurring over a number of future time periods. Considering the standard engineering economics total cost, this would result in the following:

(8.28)

Investment cost is often concentrated at the beginning of the period, while the uncertain costs occur at the varying time periods, depending typically on the time realisations of the technical performance z_t(t). For instance, the annual floods may generate costs or may not (when overspill is negligible).

In finance or insurance, one would consider more generally an income statement or cash flow, with events distributed over the successive time periods generating either positive or negative terms (meaning cost vs. revenues or the reverse), cf. Figure 8.4. A generalised cash flow may be viewed as a vector (t_k, c_k)_k = _{1 . . . K} representing the times t_k and amounts c_k in- or out-flowing, where any of the components may be random: fixed times but partially uncertain payments, or known payments at uncertain times, random time lengths (as in life insurance) and so on. Similar to a natural alea (riverflow, ...