A closed‐form expression for the net present value of a time‐power cash‐flow function

Abstract

Previous work has shown how Laplace Transform Theory and z-Transform Theory may be. used in Net Present Value Analysis. A closed-form expression is easily derived for a continuouscash flow tk, where t denotes time and k is a positive integer. In the corresponding discrete case, explicit expressions have only been provided for k < 10. In this article we derive a closed-form formula for the missing general case when k is an arbitrary non-negative integer. As a byproduct, we also obtain a closed-form expression for the z-transform of such a power series.

Variance amplification and the golden ratio in production and inventory control

“… minimising expected costs in the next period by using the variance ratios in probability density functions and assigning costs to inventory holding and backlogs,  maximising the Net Present Value of the economic consequences of the cash flows through time created by an ordering system. There is a long history of work emanating from Linköping Institute of Technology (Grubbström 1967 and1991) that has observed and exploited the fact that replacing the complex frequency (s or z) with the discount rate (or one plus the discount rate in discrete time) in the transfer function of the cash flow yields the Net Present Value of the cash flow.  incorporating customer service measures such as availability (or the probability of inventory being available from the shelf at the end of each period) and fill rates (percentage of demand shipped on time) into the bullwhip and inventory variance trade-off.…”