Risk, Ruin, and Investment Analysis: A Comment

In their provocative article that discusses risk as the probability of an investment's worth falling below some specified minimal value, Machol and Lerner observe that by this definition investments may be risky over a short time horizon but not over a long one [5, p. 484], and that a person who could invest in the stock market over a relatively long period of time without needing to withdraw capital during the period could invest with “relatively little worry†[5, p. 488]. The purpose of this comment is to examine the foregoing position rather more closely insofar as the time path of investment values is concerned. To this end, we model the value of an investment in the New York Stock Exchange Index, relative to its initial value, as a Markov chain. We assume that no part of the initial investment or dividends received on it is withdrawn before termination of the process, at which point the entire amount accumulated (which may be less than the initial investment) is realized. Values taken from a record of annual percentage changes in the New York Stock Exchange Index over the period 1940–1968 are then used to define a representative matrix of transition probabilities which describes the manner in which investment values can change from one period to the next. The probability distributions of relative investment values over differing lengths of time for which the investment may be held are then investigated using the Markov chain model.