DISCOUNTING THE COMPONENTS OF AN INCOME STREAM: REPLY

Abstract

THE PURPOSE OF [1] was to define conditions, consistent with the expected utility maxim, under which a project's present value could be calculated by application of the income-component discounting notion first introduced in [7]. The anlaysis focused on an individual decision-maker and unfortunately did not consider the corporation with stock trading in a perfectly competitive market. Before turning to the implications of market environment for my results, let us first consider Turnbull's certain world counter-example from which he reasons that my additivity condition does not necessarily hold. One can see that if Turnbull were true to his analysis or to my assumption, that any individual may borrow or lend at a riskless rate without affecting that rate (see [1], p. 995, second paragraph), the U(C1, C2) preference function and his corresponding analysis are irrelevant to this discussion. For if one does allow borrowing and lending at a single interest rate in a certain world-the perfect market assumption, then clearly for any preference function that gives rise to convex indifference curves in (C1, C2) then Pz = Px + Py. To see this draw, the Fisher diagram (Figure 1) where the absolute slope of the market opportunity line is one plus the interest rate, i. Then starting at (C*, C*) it is clear that if the individual is promised an additional Z dollars in period 2 then the maximum price, Pz, that he would be willing to pay in period 1 is Z/(l + i), for he could obtain that trade-off in the market place. Identical reasoning leads to Px= X/(l + i) and Py = Y/(l + i), consequently Pz = Px + Py. My interpretation of the remainder of Turnbull's comment is that the calculation of a project's net present value should simply yield the additions to a firm's future equilibrium stock price and if the market values of the stock's risky component income streams are additive, as they are when equilibrium is represented by a complete set of Arrow-Debreu securities, or in a mean-variance setting with a riskless asset and homogeneous expectations, then additivity holds for the present values or prices of the project's risky income streams. These observations by Turnbull are clearly valid. The additivity principle in the context of a competitive stock market also holds true for heterogeneous expectations, given, of course, equal and costless access to all information by all potential investors. (c.f. [5], and [6].) The common thread connecting these results is that in perfectly competitive security markets all participants achieve efficient portfolio diversification, and therefore equilibrium prices will consequently reflect all possible gains from risk reduction. Since I was negligent in considering the impact of market conditions in deriving my conclusions, some statement should have been made regarding the implications for my results of that other type of market, an imperfect one. If a merger affects