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en be stated as follows:Maximize: Subject to: .This is a simple constrained optimization problem that can be solved by substituting the condition into the objective function and then using standard optimization techniques from single variable calculus. Alternatively, this optimization problem can be solved using the Lagrange multiplier method from multivariable calculus.Graphically, the maximum value of is the number such that the parabola is tangent to the hyperbola . (See Figure . The optimal portfolio in this figure is denoted by .) Clearly, the optimal portfolio depends on the value of , which specifies the investor’s level of risk aversion. Portfolio with Greatest UtilityCarrying out the details of the optimization, we find that when and are both risky securities (. and ), the riskreward coordinates of the optimal portfolio are,.Since , it follows that the portion of the portfolio that should be invested in is.Comment We have assumed that short selling without margin posting is possible (., we have assumed that can assume any real value, including values outside the interval[0,1]). In the more realistic case, where short selling is restricted, the optimal portfolio may differ from the one just determined.EXAMPLE 1: The return on a bond fund has expected value 5% and standard deviation 12%, while the return on a stock fund has expected value 10% and standard deviation 20%. The correlation between the returns is . Suppose that an investor’s utility functional is of the form . Determine the investor’s optimal allocation between stocks and bonds assuming short selling without margin posting is possible.It is customary in problems of this type to assume that the utility functional is calibrated using percentages. Hence, if , represent the returns on the bond and stock funds, respectively, then,.Note that such a calibration can always be achieved by proper selection of .From the formulas that have been developed, the expected return on the optimal portfolio is,where , and. Hence, the portion of the portfolio that should be invested in bonds is .Thus, for a portfolio of $1000, it is optimal to sell short $ worth of bonds and invest $ in stocks. ■Special Cases of the Portfolio Opportunity SetWe conclude this section by high lighting the form of the portfolio opportunity set in some special cases. Throughout, we assume that and are securities such that and .(The situation where and is not interesting since then is always preferable to .) We also assume that no short positions are allowed.Assets Are Perfectly Positively Correlated Suppose that (. and are perfectly positively correlated). Then the set of possible portfolios is a straight line, as illustrated in Figure .Assets Are Perfectly Negatively Correlated Suppose that (, and are perfectly negatively correlated). Then the set of possible portfolios is as illustrated in Figure . Note that, in this case, it is possible to construct a perfectly hedged portfolio (., portfolio with ).a. b. c. d. One of the Assets Is Risk FreeFIGURE Special Cases of the Portfolio Opportunity SetAssets Are Uncorrelated Suppose that . Then the portfolio opportunity set has the form illustrated in Figure . From this picture, it is clear that starting from a portfolio consisting only of the lowrisk security , it is possible to decrease risk and increase expected return simultaneously by adding a portion of the highrisk security to the portfolio. Hence, even investors with a low level of risk tolerance should have a portion of their portfolios invested in the highrisk security . (See also the discussion on the standard deviation of a sum in 167。P 500 and do not attempt to “beat the market”) can be traced to a surprising consequence of the Markowitz model: that every investor, regardless of risk tolerance, should hold the same portfolio of risky securities. This result has called into question the conventional wisdom that it is possible to beat the market with the “right” investment manager and in so doing has revolutionized the investment industry.Our presentation of the Markowitz model is organized in the following way. We begin by considering portfolios of two securities.