【正文】 ubject to: and then determining by . Indeed, will still be minimized because the required , will be the same in both optimization problems.The simpler optimization problem can be solved using the Lagrange multiplier method. In general, we will have and ,where and is the Lagrange multiplier. The letter is generally reserved in investment theory for the rewardtovariability ratio and, hence, will not be used to represent a Lagrange multiplier here. Performing the required differentiation, we obtain,or equivalently,where, .Note that the Lagrange multiplier will depend in general on .Now the tangency portfolio lies on the efficient set and has the property that (., no portion of the tangency portfolio is invested in the riskfree asset). Hence, the values of and for the tangency portfolio are given by, ,where (,) is the unique solution of the preceding matrix equation. Indeed, since lies on the efficient set, we must have (,)=(,), and since , we must have . The riskreward coordinates (,) for the tangency portfolio are then determined using the equations,where , are the fractions just calculated.中文譯文：第十章：Markowitz投資組合選擇模型這本書前面九個(gè)章節(jié)提出了保險(xiǎn)和投資任一名學(xué)生應(yīng)該熟悉的基本的概率理論。, 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 LineFIGURE Portfolios Containing the Tangency Portfolio Dominate All OthersTo see why this is so, consider a portfolio consisting only of and and let be the tangency portfolio (., the portfolio which is on both the hyperbola and the tangent line). From our discussion in 167。 in other words, “no pain, no gain.” The portfolios on the lower branch of Set of Possible Portfolios consisting of and the hyperbola, while theoretically possible, will never be selected risk level , the portfolio on the upper branch with standard deviation will always have higher expected return (., higher reward) than the portfolio on the lower branch with standard deviation and, hence, will always be preferred to the portfolio on the lower branch. Consequently, the only portfolios that need be considered further are the ones on the upper branch. These portfolios are referred to as efficient portfolios. In general, an efficient portfolio is one that provides the highest reward for a given level of risk.Determining the Optimal PortfolioNow let’s consider which portfolio in the efficient set is best. To do this, we need to consider the investor’s tolerance for risk. Since different investors in general have different risk tolerances, we should expect each investor to have a different optimal portfolio. We will soon see that this is indeed the case.Let’s consider one particular investor and let’s suppose that this investor is able to assign a number to each possible investment return distribution with the following properties:1. if and only if the investor prefers the investment with return to the investment with return .2. if and only if the investor is indifferent to choosing between the investment with return and the investment 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 th