On The Use Of Probability Distribution Of Risky Investments
Authors: Akindele Femi M. , Shapiro Robert A. 
Published: December 2007

Abstract
An investment risk analysis by the probability distribution technique was proposed and published by Hillier in a 1963 issue of Management Science. His work was quite theoretical. A simplified form of the method was presented in petroleum literature by Davidson and Cooper in 1975. However, their work did not extend to using the method in practical decision making. Both of these works critically discussed why this approach is preferable to earlier methods. In as much as the main value of such an innovation and the ultimate need for its publication is its application, discussion of the practical aspects of the method is important. The purpose of this paper is to fulfill this need.

Introduction
Regardless of what criterion an investor uses to determine the profitability of his venture, the amount of risk involved must be considered for a sound, final decision. As such, many techniques have evolved over the years for quantifying risk. Among the older techniques which have been used are:Payback Period - The length of time required to Payback Period - The length of time required to recover an investment through the net cash flows from the project.Expected Monetary Value - Sum of the products of investment outcomes and their respective products of investment outcomes and their respective chances of occurrence.The Certainity Equivalent - A method of discounting cash flow by a risk-free interest rate.Sensitivity Analysis - Determining the profitability measure by varying an uncertain parameter.
For reasons well discussed in parameter. For reasons well discussed in literature, all of these techniques are weak and are inadequate for proper risk portrayal. The more recent and certainly more sophisticated method of risk analysis is the probability distribution of the profitability index. One form of this is the Monte profitability index. One form of this is the Monte Carlo simulation which uses randomly selected parameters from appropriate ranges of values to generate parameters from appropriate ranges of values to generate a series of possible outcomes. While this technique is a quite reliable tool, it does not lend itself to routine application because of the computer work involved. Another form is the, analytical technique proposed by Fredrick Hillier. This approach adopts proposed by Fredrick Hillier. This approach adopts well know statistical theories and assumes the distribution of the profitability index to derive the probability distribution. Hillier's work, though probability distribution. Hillier's work, though amenable to hand calculators, was quite theoretical. A simplified form of the method was presented in petroleum literature by Davidson and presented in petroleum literature by Davidson and Cooper in 1975. However, their work did not extend to using the method in practical decision making.
This paper discusses the practical aspects of Hillier's approach. Although Davidson and Cooper suggested log-normal distribution assumption for analysis of petroleum investments, this work is based on the validity of normal distribution when the random variable is the cash flow. Both assumptions are compared in the discussion. Arriving at a decision to accept or reject an investment proposal on the basis of the method requires the use of utility theory. Therefore, the concept of the theory is reviewed. Its use in conjunction with the probability distribution of net present value is then shown. Finally, a decision present value is then shown. Finally, a decision concerning two or more risky investments may be influenced by the skewness of normal distribution. Hence, the semi-variance equation for testing the normality or magnitude and direction of skewness is presented and its application discussed.

DERIVATION OF PROBABILITY DISTRIBUTION
In his work, Hillier assumed that under conditions of uncertainity, cash flows from an investment are random variables which can be characterized by their means and variances. Therefore, by the central limit theorem, the distribution of these means will be approximately normal. Moreover, their sum will approach a normal distribution. Based on these, he derived the probability distributions of net present values and internal rate of return for three possible cases of cash flows from an investment. These cases include (1) statistically independent cash flows (2) perfectly correlated cash flows and, (3) combined independent and correlated cash flows. This paper considers the second case only.