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Stochastic Returns and Risk

Now allow for risk and uncertainty (strictly speaking, these are different concepts—see the discussion in Chapter 11). We could think of returns as being random, or stochastic, but it is more intuitively appealing to model returns as being state dependent. For example, suppose the dollar payoff to asset j, X_j, depends on which state of nature is observed. Assume there are six such states and the payoff will be one of the following , all with likelihood . Clearly, the payoff (and return) to X_j is random with uniform distribution in this case. We are interested in estimating the expected payoff, which, to us, is the mean value of X_j across all states. In general, the expected value E(X_j) is a weighted average of the possible outcomes. Formally, this is ; where X_i is a state value and f(x_i) denotes the probability of that state being realized; here, for all i outcomes. That is:

So, the expected payoff is 1.5. This statistic conveys a notion of average value across ...