Mean-Variance Analysis In this chapter, we leave the certain world and introduce risk into our analysis. We start with a review of the expected utility hypothesis. We note the connection between risk-aversion and diminishing marginal income. We then introduce the concept of a risky asset, which is practically any asset that is available for investment (because the real world is always uncertain). Using EUH, we can compare different risky assets. Our next step is to study portfolios, which are baskets of assets. Using EUH, we can compare not only assets, but also portfolios. In this setting, the Markowitz theory of optimal portfolio tells us how the investor can conceptually choose the best portfolio, i.e. the portfolio that maximizes his/her EU. Finally, we compare the optimal portfolios of different investors. It is the remarkable insight of Markowitz that these optimal portfolios are actually very similar to each other. 1. Expected Utility HypothesisLet U(y) be a utility function which is defined over income. The income that an asset gives is equal to capital gain plus dividend. Both capital gain and dividend may be uncertain. To decide which asset we prefer to hold depends on the relative size of these capital gains and dividends. To enable to make comparisons among assets, we use the expected utility hypothesis. Suppose asset income (or equivalently, asset return) can take on ndifferent values, y1,...,yn, with probabilities p1,..., pnrespectively. Then the expected utility of holding the asset is EU = p1U(y1)+...+pnU(yn) To apply EUH, we calculate the EU for each asset and rank them, the most preferred asset being the one having the highest EU. If the random return can take on continuous values, the idea is the same, but instead of summing, we need to take the following integral over the entire range of return, EU = f(y)U(y)dywhere fis the (probability) density function. The function Ureflects the investor's attitude towards risk. The shape of U-- whether linear, concave, or convex -- corresponds to the three main types of investor preference -- risk-neutral, risk-averse, and risk-loving. Consider a chance game whose expected return is 0.(Also known as an actuarially fair bet). An example is the toss of a fair coin: head, you win $1; tail, you lose $1. If an investor starts out RN RARL y U(y) U(y) yyU(y)
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