A NOTE ON DISCOUNTING THE COMPONENTS OF AN INCOME STREAM

Abstract

IN Modigliani and Miller's correction [3] to their cost of capital article [2] they introduce the idea of breaking up the expected cash flow stream into component streams and discounting each component stream by the rate appropriate to that stream viewed in isolation. In particular, they break up the firm's cash flow stream into its uncertain and certain parts and discount the expected value of the former by a rate appropriate to that stream while capitalizing the certain portion of the stream by the interest rate. This procedure enabled them to derive formulae for the firm's average and marginal costs of capital as a function of the firm's leverage. Other theoretical applications may arise for this particular form of discounting, and it therefore seems appropriate to ask under what conditions may a stream be broken up into component parts and each discounted by the rate appropriate to that stream if it were viewed in isolation. Assuming the individual is an expected utility maximizer, we first seek the maximum price, p, that the individual would be willing to pay for the asset. For simplicity, we concentrate on assets that have a one period life. Although the market for the asset that one is interested in purchasing may be imperfect, we assume that the market for funds is not; thus there exists a single riskless rate, i, at which the investor may borrow or lend. Denote the utility function of total wealth at the beginning of the period by u. Let wo, x, x/(1 + i), and x represent initial wealth at the beginning of the period, the investment's uncertain gross dollar return, the investment's uncertain contribution to total wealth at the beginning of the period, and the expected gross return, respectively.' Now the maximum price p is obtained by determining the amount of initial wealth that the individual would be willing to part with to purchase the asset and still be as well off in utility terms as he was before making the purchase. Formally, p is the solution to,