In Chapter 1 we introduced the elements of a decision problem under certainty. The same elements are present when we recognize the existence of risk; however, their formulation becomes more complex. In the next two chapters we explore the nature of the opportunity set under risk. Before we begin the analysis we present a brief summary or road map of where we are going. The existence of risk means that the investor can no longer associate a single number or payoff with investment in any asset. The payoff must be described by a set of outcomes and each of their associated probabilities of occurrence, called a frequency function or return distribution. In this chapter we start by examining the two most frequently employed attributes of such a distribution: a measure of central tendency, called the expected return, and a measure of risk or dispersion around the mean, called the standard deviation. Investors should not and, in fact, do not hold single assets; they hold groups or portfolios of assets. Thus a large part of this chapter is concerned with how one can compute the expected return and risk of a portfolio of assets given the attributes of the individual assets. One important aspect of this analysis is that the risk on a portfolio is more complex than a simple average of the risk on individual assets. It depends on whether the returns on individual assets tend to move together or whether some assets give good returns when ...

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