Black's Discounting Rule

Abstract:     
Black (1988) suggested a two-step rule for discounting uncertain cash flows: (1) form the expectation of the flow conditional on zero excess returns to traded securities in periods before the flow, and (2) discount the conditional expected value as if it were the amount of a certain payment. This paper explores formally the scope of Black?s rule and its use as a bound in cases where it does not yield correct present values. The scope is quite broad. Assuming that it is possible for securities to earn zero excess returns ex post, the rule correctly values the cash flow stream from any dynamically managed portfolio of nonderivative securities. For cash flow streams not generated or exactly duplicated by a portfolio the rule yields correct present values if the cash flow stream and its "least squares duplicate" among managed portfolios have the same mean conditional on zero excess security returns. The generic properties of least squares duplicates make this condition a plausible assumption except in cases of explicitly nonlinear derivative cash flows. Even in these exceptional cases, however, Black?s rule may provide useful information. For cash flows that are convex functions of contemporaneous security returns, Black?s rule yields the highest distribution-free lower bound on the flow?s present value. The problematic part of Black?s rule ? formation of the conditional cash flow forecasts ? requires no more (and generally less) knowledge about cash flows than is required to correctly apply "risk adjusted discount rate" present value rules.

Number of Pages in PDF File: 15

working papers series