【正文】 tment with return .The functional , which maps distribution functions to the real numbers, is called a utility functional. Note that different investors in general have different utility functionals.There are many different forms of utility functionals. For simplicity, we assume that every investor has a utility functional of the form,where is a number that measures the investor’s level of risk aversion and is unique to each investor. (Here, and represent the mean and standard deviation of the return distribution .) There are good theoretical reasons for assuming a utility functional of this form. However, in the interest of brevity, we omit the details. Note that in assuming a utility functional of this form, we are implicitly assuming that among portfolios with the same expected return, less risk is preferable.The portfolio optimization problem for an investor with risk tolerance level can then 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。.)One of the Assets Is Risk Free　Suppose that is a riskfree asset (., ) and put , the riskfree rate of return. Further, let denote and write , in place of , . Then the efficient set is given by, .This is a line in riskreward space with slope and intercept (see Figure ). Portfolios of Two Risky Securities and a RiskFree AssetSuppose now that we are to construct a portfolio from two risky securities and a riskfree asset. This corresponds to the problem of allocating assets among stocks, bonds, and shortterm moneymarket securities. Let , denote the returns on the risky securities and suppose that and . Further, let denote the riskfree rate.The Efficient SetFrom our discussion in 167。, we know that the portfolios consisting only of the two risky securities , must lie on a hyperbola of the type illustrated in Figure .We claim that when a riskfree asset is also available, the efficient set consists of the portfolios on the tangent line through (0,) (Figure ). Note that in this figure is the intercept of the tangent line through .FIGURE Portfolio Opportunity Set for Two SecuritiesFIGURE Efficient Set as a Tangent Line