外文翻译---Markowitz投资组合选择模型

外文翻译---Markowitz合选择模型

投资组

英文原文：

10 The Markowitz Investment Portfolio Selection Model

The 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 Prize 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 investment 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 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. 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 allocation 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 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 when the overall market is in equilibrium. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice.

10.1 Portfolios of Two Securities In this section, we consider portfolios consisting of only two securities, S1 and S2. These two securities could be a stock mutual fund 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

S1 and S2 in the portfolio.

Portfolio Opportunity Set

Let's begin by describing the set of possible portfolios that can be constructed from S1 and S2. Suppose that the current value of our portfolio is d dollars and let d1 and d2 be the dollar amounts invested in S1 and S2, respectively. Let R1 and R2 be the simple returns on S1 and S2 over a future time period that begins now and ends at a fixed future point in time and let R be the corresponding simple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideration,

d1Rd11R1d21R2.

Hence, the return on the portfolio over the given time period is

RxR11xR2,

where xd1d is the fraction of the portfolio currently invested in S1. Consequently, by varying x, we can change the return characteristics of the portfolio.

Now if S1 and S2 are risky securities, as we will assume throughout this section, then R1, R2, and R are all random variables. Suppose that R1 and

R2 are both normally distributed and their joint distribution has a bivariate normal distribution. This may appear to be a strong assumption. 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 R is normally distributed and that the distributions of R1, R2, and R are completely characterized by their respective means and standard deviations. Hence, since R is a linear combination of R1 and R2, the set of possible investment portfolios consisting of S1 and S2 can be described by a curve in the  plane.

To see this more clearly, note that from the identity RxR11xR2 and the properties of means and variances, we have

RxR1xR,

122222, Rx2R2x1xRR1xR1122where  is the correlation between R1 and R2, Eliminating x from these two equations by substituting xRR2R1R2, which we obtain from

2the equation for R, into the equation for R, we obtain

2RRR1R2R2222R21RR2R1R2R1R22RR12R1RR2R122, R22which describes a curve in the RR plane as claimed.

Notice that R and R change with x, while R1, R1, R2, R2,  remain fixed. To emphasize the fact that R and R are variables, let’s drop

2the subscript R from now on. Then, the preceding equation for R can be

written as

2, 2A020where A, 0, 02 are parameters depending only on S1 and S2 with A0 and 020. Indeed,

A1R1R222R12 2R1R2R2

1R1R22R1R2221R1R2

0

(the inequality holding since 11), and

022212RR1R1R22221R1R2

(again since 11). Further,

022RRRRRRRR121R1R221221221R1R2.

Consequently, the possible portfolios lie on the curve

2,0， 2A020which we recognize as being the right half of a hyperbola with vertex at 0,0. (Figure 10.1).

2Notice that the hyperbola 2A00 describes a trade-off

2between 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 risk; in other words, “no pain, no gain.” The portfolios on the lower branch of

FIGURE10.1 Set of Possible Portfolios consisting of S1 and S2

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 (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 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 Portfolio

Now 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 UFR to each possible investment return distribution

FR with the following properties:

1.UFRaUFRb if and only if the investor prefers the investment with return

Ra to the investment with return Rb.

2. UFRaUFRb if and only if the investor is indifferent to choosing between the investment with return Ra and the investment with return Rb.

The functional U, 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

UFRk2,

where k0 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 FR.) 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

k can then be stated as follows:

Maximize: Subject to:

Uk2

2. 2A020This 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 U is the number u such that the

2parabola k2u is tangent to the hyperbola 2A00. (See

2Figure 10.2. The optimal portfolio in this figure is denoted by O.) Clearly, the optimal portfolio depends on the value of k, which specifies the investor’s level of risk aversion.

FIGURE10.2 Portfolio with Greatest Carrying out the details of the optimization, we find that when S1 and S2

are both risky securities (i.e.R10 and R20), the risk-reward coordinates of the optimal portfolio O are

*120, 24Ak1. 2kA*0Since xR11xR2, it follows that the portion of the portfolio that should be invested in S1 is

*R. xRR212Comment We have assumed that short selling without margin posting is

possible (i.e., we have assumed that x 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 0.60. Suppose

12that an investor’s utility functional is of the form U. Determine the

100investor’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 R1,R2 represent the returns on the bond and stock funds, respectively, then

11223.56, 1001UFR2101226.

100Note that such a calibration can always be achieved by proper selection of k.

UFR15From the formulas that have been developed, the expected return on the optimal portfolio is

1, 2kA*0where k1100,

22RRRRRRRR1021R1R221221221R1R2

52025100.60122010122 2122020.401220 21.875

and

A1R1R2

2R1R2221R1R2

12122020.401220 251010.24.

Hence, the portion of the portfolio that should be invested in bonds is

*R xRR212 26.757812510

510 3.3515625.

Thus, for a portfolio of $1000, it is optimal to sell short $33351.56 worth of bonds and ■

invest

$4351.56

in

stocks.

Special Cases of the Portfolio Opportunity Set

We conclude this section by high lighting the form of the portfolio opportunity set in some special cases. Throughout, we assume that S1 and S2 are securities such that

RR and RR.

1212(The situation where R1R2 and R1R2 is not interesting since then S2 is always preferable to S1.) We also assume that no short positions are allowed. Assets Are Perfectly Positively Correlated Suppose that 1 (i.e. R1 and

R2 are perfectly positively correlated). Then the set of possible portfolios is a straight line, as illustrated in Figure 10.3a.

Assets Are Perfectly Negatively Correlated Suppose that 1 (i.e, R1 and

R2 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 0).

 a.

b.

 0

c.

d. One of the

FIGURE 10.3 Special Cases of the Portfolio Opportunity Set

Assets Are Uncorrelated Suppose that 0. 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-risk security S1, it is possible to decrease risk and increase expected return simultaneously by adding a portion of the high-risk security S2 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 S2. (See also the discussion on the standard deviation of a sum in §8.3.3.)

One of the Assets Is Risk Free Suppose that S1 is a risk-free asset (i.e.,

R0) and put Rrf, the risk-free rate of return. Further, let S denote

11S2 and write S, S in place of R2, R2. Then the efficient set is given by

Srfrf, 0.

SThis is a line in risk-reward space with slope SrfS and -intercept rf (see Figure 10.3d).

10.2 Portfolios of Two Risky Securities and a Risk-Free Asset Suppose 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-market securities. Let R1, R2 denote the returns on the risky securities and suppose that R1R2 and R1R2. Further, let

rf denote the risk-free rate.

The Efficient Set

From our discussion in §10.1, we know that the portfolios consisting only of the two risky securities S1, S2 must lie on a hyperbola of the type illustrated 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,rf) (Figure 10.5). Note that rf in this figure is the -intercept of the tangent line through T.

FIGURE 10.4 Portfolio Opportunity Set for Two Securities FIGURE 10.5 Efficient Set as a Tangent Line

FIGURE 10.6 Portfolios Containing the Tangency Portfolio Dominate All Others To see why this is so, consider a portfolio P consisting only of S1 and S2 and let T be the tangency portfolio (i.e., the portfolio which is on both the hyperbola and the tangent line). From our discussion in §10.1, we know that every portfolio consisting of the risky portfolio P and the risk-free asset lies on the straight line through P and (0,rf), and every portfolio consisting of the tangency portfolio T and the risk-free asset lies on the tangent line through T and (0,rf) (Figure 10.6). Hence, from Figure 10.6, it is clear that every portfolio consisting of P and the risk-free asset is dominated by a portfolio consisting of

T and the risk-free asset. Indeed, for any given risk level , there is a portfolio on the line through T and (0,rf) with greater  than the corresponding portfolio on the line through P and (0,rf). Hence, given a choice between holding P as the risky part of our portfolio and holding T as the risky part, we should always choose T.

Consequently, the efficient portfolios lie on the line through (0,rf) and T as claimed. Note, in particular, that the efficient portfolios all have the same risky part T; 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 portfolio selection problem is

reduced to determining the fraction of an investor’s portfolio that should be invested in the risk-free asset. This is a straightforward problem when the utility functional has the form Uk2(Figure 10.7). The investor’s optimal portfolio in this figure is denoted by O. Details are left to the reader.

FIGURE 10.7 Optimal Portfolio for a Given Utility Functional Determining the Tangency Portfolio

The tangency portfolio T has the property of being the portfolio on the hyperbola for which the ratio

rf is maximal. (Convince yourself that this is so.) Hence, one method of determining the coordinates of T is to solve the following optimization problem:

Maximize: Subject to:

rf 2. 2A020We will determine the coordinates of T 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 risk (i.e., smallest ) for a given level of expected return . Hence, the efficient set,

which we already know is the line through (0,rf) and T, can be determined by solving the following collection of optimization problems (one for each ):

Minimize: Subject to:

VarR ER.

Let y1, y2, y3 be the amounts allocated to S1, S2, and the risk-free asset, respectively. Then the return on such a portfolio is

Ry1R1y2R2y3rf,

and so

22VarR12y12212y1y22y2

and

ER1y12y2rfy3,

where jERj, 2jVarRj, and ijCovRi,Rj. Note that VarR does not contain any terms in y3! Consequently, the optimization problem can be written as

Minimize: Subject to:

2212y12212y1y22y2

1y12y2rfy3,

y1y2y31.

Note that the conditions in the optimization are equivalent to the conditions

1rfy12rfy2rf,

y1y2y31.

(Substitute y1y2y31 into the first condition.) Since the only place that

y3 now occurs is in the condition y1y2y31, this means that we can solve

the general optimization problem by first solving the simpler problem

Minimize:



Subject to:

1rfy12rfy2rf

and then determining y3 by y31y1y2. Indeed,  will still be minimized because the required y1, y2 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

g and g0,

where g1rfy12rfy2rf and  is the Lagrange multiplier. The letter  is generally reserved in investment theory for the reward-to-variability ratio rf2 and, hence, will not be used to represent a Lagrange multiplier here. Performing the required differentiation, we obtain

1212y11rf2122y222rfor equivalently,

, 1212z11rf2122z22rfwhere

z122, y1, z2y2.

Note that the Lagrange multiplier  will depend in general on .

Now the tangency portfolio T lies on the efficient set and has the property that y30 (i.e., no portion of the tangency portfolio is invested in the risk-free asset). Hence, the values of y1 and y2 for the tangency portfolio are given by

y1z1z2y, 2,

z1z2z1z2where (z1,z2) is the unique solution of the preceding matrix equation. Indeed,

since T lies on the efficient set, we must have (y1,y2)=(z1,z2), and since

2y1y21y31, we must have 21z1z2. The risk-reward coordinates

(R,R) for the tangency portfolio are then determined using the equations

TTRTy11y22,

2222RTy2112y1y212y22,

where y1, y2 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Harry・Markowitz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（CAPM），如所提到的，当整体市场是处于平衡时给对于风险保障公平的回报一个公式。像Markowitz模型一样，资本定价模型（CAPM）对组合管理实践有着深刻的影响。

10.1 两个保障的组合

本节, 我们考虑只包括两个保证金S1和S2的组合。这两保证金可以是一个股票共同基金和一个债券共同基金, 在这种情形组合选择问题相当于资产分配, 或可能是其它别的。我们的目标是要在组合中确定S1和S2的“最佳匹配”。

组合机会集合

让我们由描述可以被由S1和S2建立的可能的投资组合集合开始。假设我们组合的当前值是d美元并且让d1和d2分别表示投资在S1和S2的美元数。让R1和

R2表示在S1和S2上在经历一个现在开始且在未来某固定点及时结束的这一未来时间段上简单的回报，并让R表示投资组合对应的简单回报。然后，如果在考虑的这段时期内对投资组合匹配不做变动，那么

d1Rd11R1d21R2。

这时，在指定时期内投资组合的回报是

RxR11xR2，

其中xd1d是当前被投资在S1中的组合比例。所以，由变化x，我们能改变组合的回报特征。

现在如果S1和S2是风险保障，本节我们都要这样假设，那么R1、R2，和R都是随机变量。假设R1和R2都是正态分布的并且它们的联合分布是二元正态分布。这也许是一个条件强的假定。但是，关于股票价格回报的数据表明，作为最初的近似，它不是不合情理的。然后，从正态分布的性质（见§6.3.1），可以得出R是正态分布的并且R1、R2和R的分布完全由它们各自的均值和标准差所刻画。因此，由于R是R1和R2的一个线性组合，由S1和S2组成的可能的投资组合集合可以由平面中的曲线所描述。

为了看起来更加明显，从等式RxR11xR2和均值和方差的性质，我们有

RxR1xR，

122222， Rx2R2x1xRR1xR1122这里是R1和R2的相关系数，从R的方程中得到xRR2R1R2，将

2它代入R的方程中，就从这两方程中消去了x，我们得到

2RRR1R2R2222R21RR2R1R2R1R22RR12R1RR2R1222R，

2

它描述的是如前提到的RR平面内的一条曲线。

注意到当R1，R1，R2，R2，保持不变时，R和R随着x改变。为

2了强调R和R是变量，我们从现在开始舍去下标R。则前面关于R的等式可

以被写为

2， 2A020其中A，0，02是只依赖S1和S2的参数，且A0，020。的确，

A1R1R2122R12 2R1R2R2

R1R22R1R2221R1R2

0 (不等式成立是因为11)

和

2212RR102进一步，

R1R22221R1R20 (同样因为11)

022RRRRRRRR121R1R221221221R1R2。

因此，可能的投资组合位于曲线

2，0， 2A020这是以0,0为顶点的一个双曲线的右边一半（见图10.1）。

2 注意，双曲线2A00描述的是在风险（用测量）和收益（用

2测量）之间的交易。的确，沿双曲线的上半支，明显获得一个更加巨大的收

益，我们必须以更大的风险去投资一份组合投资；换句话说，“没有痛苦，没有取得。”在双曲线的下半分支的组合，虽理论上是可能，但不会在实际中被选择。原因是对任一个选择的风险水平， 具有标准差为的上半支的投资组合总比具有标准差为的下半支的投资组合有更高的期望回报（即有更高的收益）。所以，将总是更喜欢在下半支的组合。结果，只需要被进一步考虑的组

合就是在上半分支的那些。这些组合被认为是有效资产组合。一般地，一份有效资产组合是对一个给定的风险水平能提供最高收益的组合。

确定最优投资组合

现在我们考虑在有效集中哪份组合是最佳的。要做到这点，我们需要考虑投资者对风险的承受力。因为通常不同的投资者有不同的风险承受力，我们应该期望每个投资者有一份不同的最优投资组合。我们很快看见这的确是实际情形。

考虑一个特殊投资者，假设这个投资者有能力对每个可能的投资回报分布

FR去指定一个数UFR，它具有以下性质：

1. UFRaUFRb当且仅当这个投资者更喜欢以回报Ra投资，而不是以回报

Rb投资。

2. UFRaUFRb当且仅当这个投资者对选择以回报Ra投资和以回报Rb投资漠不关心。

将分布函数映射到实数的函数U称为效用函数。注意通常不同的投资者有不同的效用函数。

效用函数有许多不同的形式。简单起见，我们假设每个投资者有以下形式的效用函数

UFRk2，

其中k0是一个测量投资者风险厌恶水平的值并且它对各个投资者是唯一的。（这里，和代表回报概率分布FR的平均值和标准差) 。对于假设一个这种形式的效用函数有真正的理论原因。但是，为简单起见，我们略去细节。注意到在假设一个效用函数具有这种形式时，我们暗含着假设在具有同样风险水平的投资组合中，您是选择期望收益更大的，而在具有同样期望收益的投资组合中，选择风险较少的。

对于一个有风险承受力水平k的投资者，投资组合最优化问题可以被如下陈述： 最大化: 满足:

Uk2

2. 2A020

这是一个可求解的简单约束最优化问题，只须将条件代入目标函数，然后由单变量微积分，使用标准最优化方法求解。或者这个优化问题也可用多元微积分中的拉格朗日乘子法方法解决。

从图来看，U的最大值是满足抛物线k2u是双曲线

2的切线的数u（参见图10.2。最优的投资组合在这个图用O2A020表示）。清楚地，最优的投资组合依赖于k的值，而k表示这个投资者的风险厌恶水平。

详细进行最优化，我们发现当S1和S2都是两个风险保障时（即R10和

R0)，最优投资组合O的风险-收益坐标是

2*12， 024Ak1。 2kA*0从xR11xR2，可得出应该在S1上投资的投资组合比例是

x

*R。

RR212评述 我们已经假设，没有保证金的短期销售是可能的（即我们已经假设x可以是任一个实数值，包括在区间[0,1]之外的值）。在短期销售被限制的更加现实的条件下，最优的投资组合也许与刚决定的不同。

例1: 设债券基金的收益期望值为5%和标准差为12%，股票基金的收益期望值为10%和标准差为20%。两种收益的相关系数是0.60。假设一个投资者的效用函

12数具有形式U。假设没有保证金的短期销售是可能的，确定投资者

100在股票和债券之间的最优份额。

对这种类型的问题，习惯上假设，效用函数是用百分数度量。因此，如果R1，

R2分别代表在证券和股票基金上的收益，则

UFR15UFR211223.56， 1001101226。

100

注意，这样的度量也总可由k的适当选择而达到。 由已推出的公式，最优组合的期望收益是

1， *02kA这里k1100，

022RRRRRRRR121R1R221221221R1R2

52025100.60122010122  2122020.401220 21.875

和

A1R1R2

2R1R2221R1R2

12122020.401220 2510 10.24。

因此，应该投资在债券中的投资组合的比例是

*R x

RR212

26.757812510

510 3.3515625。



这样，对一个$1000的投资组合，对卖出空头$3351.56的债券和投资$4351.56到股票是最优的。

投资组合机会集合的特殊情况

我们用在一些特殊情况下强调投资组合机会集合的形式结束本节。从头到尾，我们假设S1和S2是满足R1R2和R1R2的保证金（R1R2且R1R2的情况不使人感兴趣，因为之后S2总是比S1好）。我们也假设不允许有空头位置。

资产是完全正相关的 假设1（即R1和R2完全正相关的）。则可能的投资组

合集合是一条直线，如图10.3.a所示。

资产是完全负相关的 假设1（即R1和R2完全负相关的）。则可能的投资组合集合如图10.3.b所示。注意，在这种情形，构造一个完全套期保值资产组合是可能的（即0的投资组合) 。

资产是不相关的 假设0。则投资组合机会集合具有如图10.3.c所示的形式。从这张图，很明显从只包括低风险保证金S1的投资组合开始，利用将高风险保证金S2的一部分加到投资组合来减少风险和同时增加期望回报是可能的。因此，即使具有低风险承受力水平的投资者也应该让他们投资组合的一部分投资在高风险保证金S2里（也可参见在§8.3.3关于一个和的标准差的讨论）。 资产之一是无风险的 假设S1是无风险资产（即R10），令R1rf，无风险回报率。进一步，让S表示S2，并且用S，S代替R2，R2。则有效集如下给出

Srfrf，0。

S这是在斜率为SrfS和-截断为rf的风险-收益空间里的一条直线（参见图10.3.d）。

10.2 两个风险保证金和一个无风险资产的投资组合

现在假设，我们要从两个风险保证金和一个无风险资产构造一个投资组合。这与在股票、债券、和短期货币市场保证金中配置资产问题相一致。让R1，R2表示在风险保证金的收益并且假设R1R2和R1R2。进一步，让rf表示无风险率。

有效集

从我们在§10.1的讨论，我们知道只包括两个风险保证金S1，S2的投资组合一定位于如图10.4所示的双曲线上。

我们主张当无风险资产也可利用时，有效集由过(0,rf)点的切线上的投资组合组成（图10.5）。注意在这个图中rf是通过T的切线的-截距。

为看清为什么是这样，考虑一份只包括S1和S2的投资组合P，并且让T是切触投资组合（既在双曲线上又在切线上的投资组合）。从我们在§10.1的讨论，我们知道，由风险投资组合P和无风险资产组成的每一投资组合位于过P和(0,rf)点的直线上，由切触投资组合T和无风险资产组成的每一投资组合位于过T和(0,rf)的切线上（图10.6）。因此，从图10.6，很明显由P和无风险资产组成的每一投资组合被由T和无风险资产的投资组合控制。的确，对任意给定的风险水平，在过T和(0,rf)的直线上都有一投资组合比在过P和(0,rf)的切线上的相应投资组合具有更大的。因此，在将P作为我们投资组合的风险部分还是将T作为风险部分的选择中，我们应该总是选择T。

因此，如前所述有效资产投资组合位于过(0,rf)和T的直线上。注意，特别地，有效资产投资组合全部都有同样的风险部分T；在他们之中唯一的区别是被配置到无风险资产中的比例。这个惊奇的结果，它为每个投资者对指数共同基金的使用提供了一个理论的辩解，这是著名的共同基金分离定理。按照这个分离定理的观点，投资组合选择问题被退化为确定一个投资者应该投资在无风险资产上的投资组合的比例。当效用函数具有形式Uk2时，这是一个直接的问题（图10.7)。投资者的优选投资组合在这个图上用O表示。细节留给读者。

确定切触投资组合

rf达最大的投资组合（说切触投资组合T具有性质：是双曲线上使比率

服自己它就是这样的）。因此，确定T的坐标的一个方法是解以下最优化问题： 最大化： 满足：

rf 2。 2A020我们将用一个稍微不同的方式确定T的坐标，当风险保证金的数量大于２的时候

它更容易被采用。

回顾一下有效的投资组合是对于给定期望收益的一个水平，具有最少风险（即最小的）的组合。因此，有效集（我们已经知道它是过(0,rf)和T的直线）可以由解下列最优化问题集族确定（对每一个）： 最小化： 满足：

VarR ER。

让y1，y2，y3分别是被分配到S1，S2，和无风险资产的数额。则在这样一个投资组合上的收益是

Ry1R1y2R2y3rf，

因此

22VarR12y12212y1y22y2

和

ER1y12y2rfy3，

这里jERj，2jVarRj，且ijCovRi,Rj。注意VarR不含y3的任何项！因此，最优化问题可以被写成 最小化： 满足：

2212y12212y1y22y2

1y12y2rfy3

y1y2y31。

注意在这个优化中的条件与以下条件是等价的

1rfy12rfy2rf，

y1y2y31。

（将y1y2y31代入第一个条件）。因为y3唯一出现的位置是在条件

y1y2y31中，这意味着我们能解决一般优化问题，途径是首先解决下面更

加简单的问题 最小化：



满足：

1rfy12rfy2rf

然后由y31y1y2确定y3。事实上，因为在两个优化问题中要求y1，y2是相同的，所以仍然能被最小化。

更加简单的优化问题可以运用拉格朗日乘子法解决。一般地，我们将有

g和g0，

这里g1rfy12rfy2rf，是拉格朗日乘子。字母为收益变化比率rf2，一般被用在投资理论中，所以在这里将不被用于代表拉格朗日乘子。进行必要的微分，我们获得

1212y11rf2122y222rf或者等价地，

， 1212z11rf2122z22rf其中

z122， y1，z2y2。

注意拉格朗日乘子将一般依赖于。

现在切触投资组合T位于有效集上并且有y30的性质（即切触投资组合的任何部份都不被投资在无风险资产上）。因此，对于切触投资组合y1和y2的值被

y1z1z2，y2 z1z2z1z2给出，这里(z1,z2)是前面矩阵方程的唯一解。的确，因为T位于有效集上，我们必须有(y1,y2)=

(z1,z2)，并且因为y1y21y31，我们必须有2TT21z1z2。则对于切触投资组合，风险—收益坐标(R,R)被等式

Ry11y22，

T

222Ry12122y1y212y22，

T决定，这里y1，y2是刚被计算出来的比例。