Behavior of discount rates in present value calculation

The purpose of this paper is to examine the behavior of the required return (discount rate) within the present value method. This method is a foundational tool in capital budgeting and investment planning for evaluating the favorability of various decisions. Additionally, it is commonly used in the valuation of companies through discounted cash flows. In both practice and academic literature, discount rates are often assumed to be constant and deterministic, without explicitly stating the assumptions that justify such behavior. We demonstrate that the appropriate discount rate is inherently linked to the stochastic process governing the cash flow stream — even when the pricing kernel is held constant. Specifically, discount rates remain deterministic and time-invariant only when cash-flow streams are generated by a simple nonlinear and nondisrupted autoregressive process of the first order. In contrast, cash flows generated by trend-stationary or disrupted processes typically require time-varying discount rates. For these processes, we show how the present value formula can be adapted accordingly. To our knowledge, this relationship has not been formally developed or illustrated in existing NPV literature. More complex cash flow dynamics generally require stochastic discount rates, making the traditional present value method problematic.