Time varying costs of capital and the expected present value of future cash flows

The use of an inter-temporally constant discount rate or cost of capital is a strong assumption in many ex ante models of finance and in applied procedures such as capital budgeting. We investigate how robust this assumption is by analysing the implications of allowing the cost of capital to vary stochastically over time. We use the Feynman-Kac functional to demonstrate how there will, in general, be systematic differences between present values computed on the assumption that the currently prevailing cost of capital will last indefinitely into the future and present values determined by discounting cash flows at the expected costs of capital that apply up until the point in time at which cash flows are to be received. Our analysis is based on three interpretations of the Feynman-Kac functional. The first assumes that the cost of capital evolves in terms of a state variable characterised by an Uhlenbeck and Ornstein ("On the Theory of the Brownian Motion." Physical Review 36(5): 823-841) process. The second and third interpretations of the Feynman-Kac functional are based on the continuous time branching process. The first of these assumes that the state variable tends to drift upwards over time, whilst the second assumes that there is no drift in the state variable. Our analysis shows that for all three stochastic processes there are significant differences between present values computed under the assumption that the currently prevailing cost of capital will last indefinitely into the future and present values determined by discounting cash flows at the expected costs of capital that apply up until the point in time at which cash flows are to be received. Comparisons are also made with the environmental economics literature where similar problems have been addressed by invoking a 'gamma discounting' methodology.