外文翻译---Markowitz投资组合选择模型

1、.英文原文：10The Markowitz Investment Portfolio Selection ModelThe first nine chapters of this book presented the basic probability theory with which any student of insurance and investments should be familiar. In this final chapter, we discuss an important application of the basic theory: the Nobel Priz

2、e winning investment portfolio selection model due to Harry Markowitz. This material is not discussed in other probability texts of this level; however, it is a nice application of the basic theory and it is very accessible.The Markowitz portfolio selection model has a profound effect on the investm

3、ent industry. Indeed, the popularity of index funds (mutual funds that track the performance of an index such as the S&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 t

4、he 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 follow

5、ing way. We begin by considering portfolios of two securities. An important example of a portfolio of this type is one consisting of a stock mutual fund and a bond mutual fund. Seen from this perspective, the portfolio selection problem with two securities is equivalent to the problem of asset alloc

6、ation between stocks and bonds. We then consider portfolios of two risky securities and a risk-free asset, the prototype being a portfolio of a stock mutual fund, a bond mutual fund, and a money-market fund. Finally, we consider portfolio selection when an unlimited number of securities is available

7、 for inclusion in the portfolio.We conclude this chapter by briefly discussing an important consequence of the Markowitz model, namely, the Nobel Prize winning capital asset pricing model due to William Sharpe. The CAPM, as it is referred to, gives a formula for the fair return on a risky security w

8、hen the overall market is in equilibrium. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice.10.1Portfolios of Two SecuritiesIn this section, we consider portfolios consisting of only two securities, and . These two securities could be a stock mutual fun

9、d and a bond mutual fund, in which case the portfolio selection problem amounts to asset allocation, or they could be something else. Our objective is to determine the “best mix” of and in the portfolio.Portfolio Opportunity SetLet's begin by describing the set of possible portfolios that can be

10、 constructed from and . Suppose that the current value of our portfolio is dollars and let and be the dollar amounts invested in and , respectively. Let and be the simple returns on and over a future time period that begins now and ends at a fixed future point in time and let be the corresponding si

11、mple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideration,.Hence, the return on the portfolio over the given time period is,where is the fraction of the portfolio currently invested in . Consequently, by varying , we can change the re

12、turn characteristics of the portfolio.Now if and are risky securities, as we will assume throughout this section, then , , and are all random variables. Suppose that and are both normally distributed and their joint distribution has a bivariate normal distribution. This may appear to be a strong ass

13、umption. However, data on stock price returns suggest that, as a first approximation, it is not unreasonable. Then, from the properties of the normal distribution, it follows that is normally distributed and that the distributions of , , and are completely characterized by their respective means and

14、 standard deviations. Hence, since is a linear combination of and , the set of possible investment portfolios consisting of and can be described by a curve in the plane.To see this more clearly, note that from the identity and the properties of means and variances, we have,where is the correlation b

15、etween and , Eliminating from these two equations by substituting , which we obtain from the equation for , into the equation for , we obtain,which describes a curve in the plane as claimed.Notice that and change with , while , , , , remain fixed. To emphasize the fact that and are variables, lets d

16、rop the subscript from now on. Then, the preceding equation for can be written as,where , , are parameters depending only on and with and . Indeed,(the inequality holding since ), and(again since ). Further,.Consequently, the possible portfolios lie on the curve,，which we recognize as being the righ

17、t half of a hyperbola with vertex at . (Figure 10.1). Notice that the hyperbola describes a trade-off between risk (as measured by ) and reward (as measured by ). Indeed, along the upper branch of the hyperbola, it is clear that to obtain a greater reward, we must invest in a portfolio with greater

18、risk; in other words, “no pain, no gain.” The portfolios on the lower branch ofFIGURE10.1Set 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 exp

19、ected return (i.e., 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 referr

20、ed 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 lets consider which portfolio in the efficient set is best. To do this, we need to consider the investors tolerance for risk. Since

21、 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.Lets consider one particular investor and lets suppose that this investor is able to assign a number to each possible in

22、vestment 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 , wh

23、ich 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,w

24、here is a number that measures the investors 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 brevit

25、y, 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

26、: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 multipl

27、ier method from multivariable calculus.Graphically, the maximum value of is the number such that the parabola is tangent to the hyperbola . (See Figure 10.2. The optimal portfolio in this figure is denoted by .) Clearly, the optimal portfolio depends on the value of , which specifies the investors l

28、evel of risk aversion.FIGURE10.2Portfolio with Greatest UtilityCarrying out the details of the optimization, we find that when and are both risky securities (i.e. and ), the risk-reward coordinates of the optimal portfolio are,.Since , it follows that the portion of the portfolio that should be inve

29、sted in is.CommentWe have assumed that short selling without margin posting is possible (i.e., we have assumed that can assume any real value, including values outside the interval0,1). In the more realistic case, where short selling is restricted, the optimal portfolio may differ from the one just

30、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 0.60. Suppose that an investors utility functional is of the form . Determine th

31、e investors 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, respectivel

32、y, 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

33、optimal to sell short $33351.56 worth of bonds and invest $4351.56 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 sit

34、uation where and is not interesting since then is always preferable to .) We also assume that no short positions are allowed.Assets Are Perfectly Positively CorrelatedSuppose that (i.e. and are perfectly positively correlated). Then the set of possible portfolios is a straight line, as illustrated i

35、n Figure 10.3a.Assets Are Perfectly Negatively CorrelatedSuppose that (i.e, and are perfectly negatively correlated). Then the set of possible portfolios is as illustrated in Figure 10.3b. Note that, in this case, it is possible to construct a perfectly hedged portfolio (i.e., portfolio with ).a. b.

36、 c. d. One of the Assets Is Risk FreeFIGURE 10.3 Special Cases of the Portfolio Opportunity SetAssets Are UncorrelatedSuppose that . Then the portfolio opportunity set has the form illustrated in Figure 10.3c. From this picture, it is clear that starting from a portfolio consisting only of the low-r

37、isk security , it is possible to decrease risk and increase expected return simultaneously by adding a portion of the high-risk security to the portfolio. Hence, even investors with a low level of risk tolerance should have a portion of their portfolios invested in the high-risk security . (See also

38、 the discussion on the standard deviation of a sum in §8.3.3.)One of the Assets Is Risk FreeSuppose that is a risk-free asset (i.e., ) and put , the risk-free rate of return. Further, let denote and write , in place of , . Then the efficient set is given by, .This is a line in risk-reward space

39、 with slope and -intercept (see Figure 10.3d).10.2Portfolios of Two Risky Securities and a Risk-Free AssetSuppose now that we are to construct a portfolio from two risky securities and a risk-free asset. This corresponds to the problem of allocating assets among stocks, bonds, and short-term money-m

40、arket securities. Let , denote the returns on the risky securities and suppose that and . Further, let denote the risk-free rate.The Efficient SetFrom our discussion in §10.1, we know that the portfolios consisting only of the two risky securities , must lie on a hyperbola of the type illustrat

41、ed in Figure 10.4.We claim that when a risk-free asset is also available, the efficient set consists of the portfolios on the tangent line through (0,) (Figure 10.5). Note that in this figure is the -intercept of the tangent line through .FIGURE 10.4Portfolio Opportunity Set for Two SecuritiesFIGURE

42、 10.5Efficient Set as a Tangent LineFIGURE 10.6Portfolios 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 (i.e., the portfolio which is on both the hyperbola and the tangent line). From our disc

43、ussion in §10.1, we know that every portfolio consisting of the risky portfolio and the risk-free asset lies on the straight line through and (0,), and every portfolio consisting of the tangency portfolio and the risk-free asset lies on the tangent line through and (0,) (Figure 10.6). Hence, fr

44、om Figure 10.6, it is clear that every portfolio consisting of and the risk-free asset is dominated by a portfolio consisting of and the risk-free asset. Indeed, for any given risk level , there is a portfolio on the line through and (0,) with greater than the corresponding portfolio on the line thr

45、ough and (0,). Hence, given a choice between holding as the risky part of our portfolio and holding as the risky part, we should always choose .Consequently, the efficient portfolios lie on the line through (0,) and as claimed. Note, in particular, that the efficient portfolios all have the same ris

46、ky part ; the only difference among them is the portion allocated to the risk-free asset. This surprising result, which provides a theoretical justification for the use of index mutual funds by every investor, is known as the mutual fund separation theorem. In view of this separation theorem, the po

47、rtfolio selection problem is reduced to determining the fraction of an investors portfolio that should be invested in the risk-free asset. This is a straightforward problem when the utility functional has the form (Figure 10.7). The investors optimal portfolio in this figure is denoted by . Details

48、are left to the reader.FIGURE 10.7Optimal Portfolio for a Given Utility FunctionalDetermining the Tangency PortfolioThe tangency portfolio has the property of being the portfolio on the hyperbola for which the ratiois maximal. (Convince yourself that this is so.) Hence, one method of determining the

49、 coordinates of is to solve the following optimization problem:Maximize:Subject to:.We will determine the coordinates of in a slightly different way, which is more easily adapted when the number of risky securities is greater than two.Recall that the efficient portfolios are the ones with the least

50、risk (i.e., smallest ) for a given level of expected return . Hence, the efficient set, which we already know is the line through (0,) and , can be determined by solving the following collection of optimization problems (one for each ):Minimize:Subject to:.Let , , be the amounts allocated to , , and

51、 the risk-free asset, respectively. Then the return on such a portfolio is,and soand,where , , and . Note that does not contain any terms in ! Consequently, the optimization problem can be written asMinimize:Subject to:,.Note that the conditions in the optimization are equivalent to the conditions,.

52、(Substitute into the first condition.) Since the only place that now occurs is in the condition , this means that we can solve the general optimization problem by first solving the simpler problemMinimize:Subject to:and then determining by . Indeed, will still be minimized because the required , wil

53、l 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 reward-to-variability ratio and, hence, w

54、ill 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 (i.e., no portion of the tan

55、gency portfolio is invested in the risk-free 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 risk-reward coordin

56、ates (,) for the tangency portfolio are then determined using the equations,where , are the fractions just calculated.中文译文：第十章：Markowitz投资组合选择模型这本书前面九个章节提出了保险和投资任一名学生应该熟悉的基本的概率理论。在最后一章里，我们讨论基本理论的一种重要应用：归功于HarryMarkowitz的赢取诺贝尔奖的投资组合选择模型。这材料不在其它这个水平的概率材料讨论；但是，它是基本理论的一种很好的应用并且它是非常容易理解的。Markowitz组合选择模型已

57、经对投资产业有一个深刻作用。确实，共同基金的普及（跟踪一个指数的表现譬如S&P 500和不试图“击打市场”的共同基金）可以被跟踪成Markowitz模型的一个惊奇后果： 每个投资者，不考虑风险容忍，应该拿着同样风险保障的组合。这个结果表示了对于传统经验的置疑，用“正确的”投资管理人击打市场是理性的，并且这样做改革了投资产业。我们对Markowitz模型的介绍用下面的方法组织。我们从考虑两个保障的组合开始。这种形式组合的一个重要例子是由一个股票共同基金和一个债券共同基金组成的组合。从这个观点看到，有二个保障的组合选择问题与股票和证券之间的资产组合问题是等价的。我们然后考虑二个风险保障和无风险资产的组合，原型是股票共同基金、债券共同基金和货币市场基金的组合。最后，对于包含在组合中无穷多个保障可利用时，我们考虑组合选择。我们由简要地谈论Markowitz模型的一个重要结果，即归功于William Sharpe的诺贝尔奖赢取资本资产定价模型结束本章。资本定价模型（CAP