Revisiting Modern Discounting of Risky Cash Flows: Imbedding Uncertainty Resolution Functions into Risk-Adjusted Discount Rates and Certainty Equivalent Factors (Revised)

Abstract

The paper revisits the prevailing RADR-based approach to cash flows valuation, indicates its restrictions and failures and suggests flexible methodology of discounting cash flows by means of certainty equivalents with implied instantaneous measure of risk consistent with specific uncertainty resolution function. The RADR discounting techniques do not allow estimate values of risky initial cash outflows or differed nonrecurring cash flows with known standard deviation or more sophisticated measure of risk, since they use periodic measure of risk according to CAPM, its modifications or ex-ante approach. However that is not an insolvable problem, but a restriction of RADR approach. The Certainty Equivalents approach to discounting is easily made consistent with the basic assumptions of RADR and may be fine-tuned to incorporate Equity Risk Premium as a basis for CEQ factors and when properly applied provides the same results as RADR-based discounting, meanwhile overcoming the above-mentioned shortcoming. However the most hard critics of RADR concern the fact that it implicitly produces rigid pattern of risk increasing over time, which is a special case without sufficient evidence to be overwhelming in practice. Particularly, the paper demonstrates that for a remote lottery cash flow there would be instantaneous standard deviation at the moment of payment, which is hardly possible to convert to periodic standard deviation and the discounting at a RADR is surely a mistake, since the risk of such cash flow is fixed over time. This is an extreme example of cash flow with specific uncertainty resolution pattern for which RADR-based discounting is inconsistent and provided strongly distorted results. The paper also presents the formula to calculate RADR-implied standard deviation and finds it being two times higher than the standard deviation of identical cash flows explicitly expressed by Wiener process, i.e. square root of time. Suggested methodology addresses using the arbitrary Uncertainty Resolution function to express risk at a given moment, for instance such as Wiener stochastic process. It’s shown how to embed the Uncertainty Resolution into Certainty Equivalents Method of discounting and thus have a possibility to evaluate cash flows according to their explicit risk pattern. The paper also proves the Propositions for discounting cash flows with different time-remote risk patterns. Specifically it substantiates the recursive algorithm to CE-based valuations and differentiates the natural risk of cash flows and risk of future revaluation of current estimates, which is jointly presented in CAPM.