A PORTFOLIO MODEL OF CAPITAL BUDGETING UNDER RISK*

Abstract

IT HAS LONG BEEN RECOGNIZED that the capital investment decision has two aspects: (1) it requires a commitment of wealth at one point in time in exchange for the opportunity to obtain income at one or more future points in time, and (2) the future income to be obtained is uncertain. A subproblem under (2) is the uncertainty associated with selecting combinations of projects where the returns from the several projects are statistically interdependent. This study develops a mathematical model for analyzing and selecting an optimal portfolio of capital projects when returns from project opportunities and from the firm's existing capital assets are not known with certainty. The model combines the problem of uncertainty of future cash flows from the firm's project opportunities and existing capital assets with the problem of selecting combinations of projects promising the highest return for a given level of risk. Recognizing that a decision to commit resources to current projects may affect resource availabilities and capital investment decisions in future periods, the model also emphasizes the time dimension of capital planning, explicitly treating the intertemporal relationship between current commitments and project opportunities available at various future points of time. The model in the study is developed in three stages. The first stage focuses on the decision maker's subjective probability distribution of projects' annual net cash flow. No restrictive assumptions regarding the shape of the probability distributions are necessary. Specific cash-flow values from the distributions are identified by Monte Carlo simulation. In the second stage of the model, the decision maker is asked to formulate his risk-preference function and to specify his financial resources. Funds available for capital investment may be limited or unlimited, and the relevant capital markets be perfect or imperfect. The final stage of the model uses mathematical programming to identify optimal combinations of new projects and existing capital assets. The type of programming algorithm used-quadratic or integer-depends on the acceptability of fractional projects in the optimal portfolio. Weingartner's basic horizon model is used with the Monte Carlo-simulated cash flows and Markowitz's variance-covariance measure of risk to find a distribution of optimal portfolio values. The project portfolio maximizing the decision maker's preference function is found, within the framework of a payoff table, by identifying the portfolio with the highest expected utility. Extensions of the portfolio model include detailed study of the dual to the quadraticprogramming problem of project selection and a computer test of the model using artificial data. Through the dual problem, the firm's annual opportunity cost of funds (discount rate) is shown to be a function of its annual liquidity position and available project opportunities and, therefore, to change from year to year with changes in the firm's liquidity and investment alternatives. The marginal utility of any project in the project opportunity set is also found through the dual problem. An individual project's marginal utility is shown to depend on the absolute level of its cash flows, the effect of its cash flows on the annual liquidity position of the firm as well as the liquidity