外文翻譯---markowitz投資組合選擇模型

1、<p><b>　　英文原文：</b></p><p>　　10　The Markowitz Investment Portfolio Selection Model</p><p>　　The first nine chapters of this book presented the basic probability theory with which

2、 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 Mark

3、owitz. 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.</p><p>　　The Markowitz portfolio selection mod

4、el 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 trac

5、ed 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 t

6、hat it is possible to beat </p><p>　　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 th

7、is 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

8、 bonds. We then consider portfolios of two risky securities and a risk-free asset, the prototype being a portf</p><p>　　We conclude this chapter by briefly discussing an important consequence of the Markowit

9、z 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 equilibriu

10、m. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice.</p><p>　　10.1Portfolios of Two Securities</p><p>　　In this section, we consider portfolios c

11、onsisting of only two securities, and . 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 els

12、e. Our objective is to determine the “best mix” of and in the portfolio.</p><p>　　Portfolio Opportunity Set</p><p>　　Let's begin by describing the set of possible portfolios that can be co

13、nstructed 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 th

14、at begins now and ends at a fixed future point in time and let be the corresponding simple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideratio</p>

15、<p><b>　　.</b></p><p>　　Hence, the return on the portfolio over the given time period is</p><p><b>　　,</b></p><p>　　where is the fraction of the portf

16、olio currently invested in . Consequently, by varying , we can change the return characteristics of the portfolio.</p><p>　　Now if and are risky securities, as we will assume throughout this section, then

17、, , 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 assumption. However, data on stock pric

18、e 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 o</p><p>　　

19、To see this more clearly, note that from the identity and the properties of means and variances, we have</p><p><b>　　,</b></p><p><b>　　,</b></p><p>　　where

20、 is the correlation between and , Eliminating from these two equations by substituting , which we obtain from the equation for , into the equation for , we obtain</p><p><b>　　,</b></p>&

21、lt;p>　　which describes a curve in the plane as claimed.</p><p>　　Notice that and change with , while , , , , remain fixed. To emphasize the fact that and are variables, let’s drop the subscript fro

22、m now on. Then, the preceding equation for can be written as</p><p><b>　　,</b></p><p>　　where , , are parameters depending only on and with and . Indeed,</p><p>　　(

23、the inequality holding since ), and</p><p>　　(again since ). Further,</p><p><b>　　.</b></p><p>　　Consequently, the possible portfolios lie on the curve</p><p&

24、gt;<b>　　,，</b></p><p>　　which we recognize as being the right half of a hyperbola with vertex at . (Figure 10.1). </p><p>　　Notice that the hyperbola describes a trade-off between r

25、isk (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,

26、 no gain.” The portfolios on the lower branch of</p><p>　　the hyperbola, while theoretically possible, will never be selected risk level , the portfolio on the upper branch with standard deviation will alwa

27、ys 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 portf

28、olios that need be considered further are the ones on the upper branch. These portfolios are referred to as efficient portfolios. I</p><p>　　Determining the Optimal Portfolio</p><p>　　Now let’s

29、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 investo

30、r to have a different optimal portfolio. We will soon see that this is indeed the case.</p><p>　　Let’s consider one particular investor and let’s suppose that this investor is able to assign a number to eac

31、h possible investment return distribution with the following properties:</p><p>　　1. if and only if the investor prefers the investment with return to the investment with return .</p><p>　　2.

32、 if and only if the investor is indifferent to choosing between the investment with return and the investment with return .</p><p>　　The functional , which maps distribution functions to the real numbers, i

33、s called a utility functional. Note that different investors in general have different utility functionals.</p><p>　　There are many different forms of utility functionals. For simplicity, we assume that ever

34、y investor has a utility functional of the form</p><p><b>　　,</b></p><p>　　where is a number that measures the investor’s level of risk aversion and is unique to each investor. (Her

35、e, 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

36、. 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.</p><p>　　The portfolio optimization problem

37、 for an investor with risk tolerance level can then be stated as follows:</p><p>　　Maximize:</p><p>　　Subject to:.</p><p>　　This is a simple constrained optimization problem th

38、at 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 L

39、agrange multiplier method from multivariable calculus.</p><p>　　Graphically, the maximum value of is the number such that the parabola is tangent to the hyperbola . (See Figure 10.2. The optimal portfolio

40、 in this figure is denoted by .) Clearly, the optimal portfolio depends on the value of , which specifies the investor’s level of risk aversion.</p><p>　　Carrying out the details of the optimization, we find

41、 that when and are both risky securities (i.e. and ), the risk-reward coordinates of the optimal portfolio are</p><p><b>　　,</b></p><p><b>　　.</b></p><p>

42、　　Since , it follows that the portion of the portfolio that should be invested in is</p><p><b>　　.</b></p><p>　　CommentWe have assumed that short selling without margin posting is

43、 possible (i.e., 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

44、just determined.</p><p>　　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 correl

45、ation between the returns is 0.60. 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 poss

46、ible.</p><p>　　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, the

47、n</p><p><b>　　,</b></p><p><b>　　.</b></p><p>　　Note that such a calibration can always be achieved by proper selection of .</p><p>　　From the fo

48、rmulas that have been developed, the expected return on the optimal portfolio is</p><p><b>　　,</b></p><p><b>　　where ,</b></p><p><b>　　and</b><

49、;/p><p><b>　　.</b></p><p>　　Hence, the portion of the portfolio that should be invested in bonds is</p><p><b>　　.</b></p><p>　　Thus, for a

50、portfolio of $1000, it is optimal to sell short $33351.56 worth of bonds and invest $4351.56 in stocks. ■</p><p>　　Special Cases of the Portfolio

51、Opportunity Set</p><p>　　We 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</p><p&

52、gt;<b>　　and .</b></p><p>　　(The situation where and is not interesting since then is always preferable to .) We also assume that no short positions are allowed.</p><p>　　Assets Ar

53、e Perfectly Positively Correlated　Suppose that (i.e. and are perfectly positively correlated). Then the set of possible portfolios is a straight line, as illustrated in Figure 10.3a.</p><p>　　Assets Are P

54、erfectly Negatively Correlated　Suppose 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

55、 perfectly hedged portfolio (i.e., portfolio with ).</p><p>　　Assets Are Uncorrelated　Suppose that . Then the portfolio opportunity set has the form illustrated in Figure 10.3c. From this picture, it is clea

56、r that starting from a portfolio consisting only of the low-risk security , it is possible to decrease risk and increase expected return simultaneously by adding a portion of the high-risk security to the portfolio. Hen

57、ce, even investors with a low level of risk tolerance should have a portion of their portfolios invested in the high-risk security . (S</p><p>　　One of the Assets Is Risk Free　Suppose that is a risk-free as

58、set (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</p><p><b>　　, .</b></p><p>　　This is a line

59、 in risk-reward space with slope and -intercept (see Figure 10.3d).</p><p>　　10.2Portfolios of Two Risky Securities and a Risk-Free Asset</p><p>　　Suppose now that we are to construct a portf

60、olio 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 , denote the returns on the risky securities an

61、d suppose that and . Further, let denote the risk-free rate.</p><p>　　The Efficient Set</p><p>　　From our discussion in §10.1, we know that the portfolios consisting only of the two risky

62、 securities , must lie on a hyperbola of the type illustrated in Figure 10.4.</p><p>　　We claim that when a risk-free asset is also available, the efficient set consists of the portfolios on the tangent lin

63、e through (0,) (Figure 10.5). Note that in this figure is the -intercept of the tangent line through .</p><p>　　To see why this is so, consider a portfolio consisting only of and and let be the tangency

64、 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 and the risk-free asset lies on the

65、 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, from Figur</p><p>　　Conseq

66、uently, the efficient portfolios lie on the line through (0,) and as claimed. Note, in particular, that the efficient portfolios all have the same risky part ; the only difference among them is the portion allocated to

67、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 theo

68、rem, the portfolio selection problem is reduced to det</p><p>　　Determining the Tangency Portfolio</p><p>　　The tangency portfolio has the property of being the portfolio on the hyperbola for w

69、hich the ratio</p><p>　　is maximal. (Convince yourself that this is so.) Hence, one method of determining the coordinates of is to solve the following optimization problem:</p><p>　　Maximize:

70、</p><p>　　Subject to:.</p><p>　　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.</p&g

71、t;<p>　　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,) and ,

72、 can be determined by solving the following collection of optimization problems (one for each ):</p><p>　　Minimize:</p><p>　　Subject to:.</p><p>　　Let , , be the amounts alloca

73、ted to , , and the risk-free asset, respectively. Then the return on such a portfolio is</p><p><b>　　,</b></p><p><b>　　and so</b></p><p><b>　　and</b

74、></p><p><b>　　,</b></p><p>　　where , , and . Note that does not contain any terms in ! Consequently, the optimization problem can be written as</p><p>　　Minimize:&

75、lt;/p><p>　　Subject to:,</p><p><b>　　.</b></p><p>　　Note that the conditions in the optimization are equivalent to the conditions</p><p><b>　　,</b&g

76、t;</p><p><b>　　.</b></p><p>　　(Substitute into the first condition.) Since the only place that now occurs is in the condition , this means that we can solve the general optimizatio

77、n problem by first solving the simpler problem</p><p>　　Minimize:</p><p>　　Subject to:</p><p>　　and then determining by . Indeed, will still be minimized because the required

78、, will be the same in both optimization problems.</p><p>　　The simpler optimization problem can be solved using the Lagrange multiplier method. In general, we will have</p><p><b>　　and ,&

79、lt;/b></p><p>　　where and is the Lagrange multiplier. The letter is generally reserved in investment theory for the reward-to-variability ratio and, hence, will not be used to represent a Lagrange mul

80、tiplier here. Performing the required differentiation, we obtain</p><p><b>　　,</b></p><p>　　or equivalently,</p><p><b>　　,</b></p><p><b>　

81、　where</b></p><p><b>　　, .</b></p><p>　　Note that the Lagrange multiplier will depend in general on .</p><p>　　Now the tangency portfolio lies on the efficient s

82、et and has the property that (i.e., no portion of the tangency portfolio is invested in the risk-free asset). Hence, the values of and for the tangency portfolio are given by</p><p><b>　　, ,</b&g

83、t;</p><p>　　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 coordinates (,) for t

84、he tangency portfolio are then determined using the equations</p><p><b>　　,</b></p><p><b>　　,</b></p><p>　　where , are the fractions just calculated.</p&

85、gt;<p><b>　　中文譯文：</b></p><p>　　第十章：Markowitz投資組合選擇模型</p><p>　　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。在最后一章里，我們討論基本理論的一種重要應(yīng)用：歸功于Harry?Markowitz的贏取諾貝爾獎(jiǎng)的投資組合選擇模型。這材料不在其它這個(gè)水平的概率材料討論；但

86、是，它是基本理論的一種很好的應(yīng)用并且它是非常容易理解的。</p><p>　　Markowitz組合選擇模型已經(jīng)對投資產(chǎn)業(yè)有一個(gè)深刻作用。確實(shí)，共同基金的普及（跟蹤一個(gè)指數(shù)的表現(xiàn)譬如S&P 500和不試圖“擊打市場”的共同基金）可以被跟蹤成Markowitz模型的一個(gè)驚奇后果： 每個(gè)投資者，不考慮風(fēng)險(xiǎn)容忍，應(yīng)該拿著同樣風(fēng)險(xiǎn)保障的組合。這個(gè)結(jié)果表示了對于傳統(tǒng)經(jīng)驗(yàn)的置疑，用“正確的”投資管理人擊打市場是理性的

87、，并且這樣做改革了投資產(chǎn)業(yè)。</p><p>　　我們對Markowitz模型的介紹用下面的方法組織。我們從考慮兩個(gè)保障的組合開始。這種形式組合的一個(gè)重要例子是由一個(gè)股票共同基金和一個(gè)債券共同基金組成的組合。從這個(gè)觀點(diǎn)看到，有二個(gè)保障的組合選擇問題與股票和證券之間的資產(chǎn)組合問題是等價(jià)的。我們?nèi)缓罂紤]二個(gè)風(fēng)險(xiǎn)保障和無風(fēng)險(xiǎn)資產(chǎn)的組合，原型是股票共同基金、債券共同基金和貨幣市場基金的組合。最后，對于包含在組合中無窮多個(gè)

88、保障可利用時(shí)，我們考慮組合選擇。</p><p>　　我們由簡要地談?wù)揗arkowitz模型的一個(gè)重要結(jié)果，即歸功于William Sharpe的諾貝爾獎(jiǎng)贏取資本資產(chǎn)定價(jià)模型結(jié)束本章。資本定價(jià)模型（CAPM），如所提到的，當(dāng)整體市場是處于平衡時(shí)給對于風(fēng)險(xiǎn)保障公平的回報(bào)一個(gè)公式。像Markowitz模型一樣，資本定價(jià)模型（CAPM）對組合管理實(shí)踐有著深刻的影響。</p><p>　　10.1

89、兩個(gè)保障的組合</p><p>　　本節(jié), 我們考慮只包括兩個(gè)保證金和的組合。這兩保證金可以是一個(gè)股票共同基金和一個(gè)債券共同基金, 在這種情形組合選擇問題相當(dāng)于資產(chǎn)分配, 或可能是其它別的。我們的目標(biāo)是要在組合中確定和的“最佳匹配”。</p><p><b>　　組合機(jī)會(huì)集合</b></p><p>　　讓我們由描述可以被由和建立的可能的投資

90、組合集合開始。假設(shè)我們組合的當(dāng)前值是美元并且讓和分別表示投資在和的美元數(shù)。讓和表示在和上在經(jīng)歷一個(gè)現(xiàn)在開始且在未來某固定點(diǎn)及時(shí)結(jié)束的這一未來時(shí)間段上簡單的回報(bào)，并讓表示投資組合對應(yīng)的簡單回報(bào)。然后，如果在考慮的這段時(shí)期內(nèi)對投資組合匹配不做變動(dòng)，那么</p><p><b>　　。</b></p><p>　　這時(shí)，在指定時(shí)期內(nèi)投資組合的回報(bào)是</p>&

91、lt;p><b>　　，</b></p><p>　　其中是當(dāng)前被投資在中的組合比例。所以，由變化，我們能改變組合的回報(bào)特征。</p><p>　　現(xiàn)在如果和是風(fēng)險(xiǎn)保障，本節(jié)我們都要這樣假設(shè)，那么、，和都是隨機(jī)變量。假設(shè)和都是正態(tài)分布的并且它們的聯(lián)合分布是二元正態(tài)分布。這也許是一個(gè)條件強(qiáng)的假定。但是，關(guān)于股票價(jià)格回報(bào)的數(shù)據(jù)表明，作為最初的近似，它不是不合情理的。

92、然后，從正態(tài)分布的性質(zhì)（見§6.3.1），可以得出是正態(tài)分布的并且、和的分布完全由它們各自的均值和標(biāo)準(zhǔn)差所刻畫。因此，由于是和的一個(gè)線性組合，由和組成的可能的投資組合集合可以由平面中的曲線所描述。</p><p>　　為了看起來更加明顯，從等式和均值和方差的性質(zhì)，我們有</p><p><b>　　，</b></p><p><

93、b>　　，</b></p><p>　　這里是和的相關(guān)系數(shù)，從的方程中得到，將它代入的方程中，就從這兩方程中消去了，我們得到</p><p><b>　　，</b></p><p>　　它描述的是如前提到的平面內(nèi)的一條曲線。</p><p>　　注意到當(dāng)，，，，保持不變時(shí)，和隨著改變。為了強(qiáng)調(diào)和是變量，

94、我們從現(xiàn)在開始舍去下標(biāo)。則前面關(guān)于的等式可以被寫為</p><p><b>　　，</b></p><p>　　其中，，是只依賴和的參數(shù)，且，。的確，</p><p>　　(不等式成立是因?yàn)?</p><p><b>　　和</b></p><p><b>　　(同

95、樣因?yàn)?</b></p><p><b>　　進(jìn)一步，</b></p><p><b>　　。</b></p><p>　　因此，可能的投資組合位于曲線</p><p><b>　　，，</b></p><p>　　這是以為頂點(diǎn)的一個(gè)雙曲線的

96、右邊一半（見圖10.1）。</p><p>　　注意，雙曲線描述的是在風(fēng)險(xiǎn)（用測量）和收益（用測量）之間的交易。的確，沿雙曲線的上半支，明顯獲得一個(gè)更加巨大的收益，我們必須以更大的風(fēng)險(xiǎn)去投資一份組合投資；換句話說，“沒有痛苦，沒有取得?！痹陔p曲線的下半分支的組合，雖理論上是可能，但不會(huì)在實(shí)際中被選擇。原因是對任一個(gè)選擇的風(fēng)險(xiǎn)水平， 具有標(biāo)準(zhǔn)差為的上半支的投資組合總比具有標(biāo)準(zhǔn)差為的下半支的投資組合有更高的期望回報(bào)（

97、即有更高的收益）。所以，將總是更喜歡在下半支的組合。結(jié)果，只需要被進(jìn)一步考慮的組合就是在上半分支的那些。這些組合被認(rèn)為是有效資產(chǎn)組合。一般地，一份有效資產(chǎn)組合是對一個(gè)給定的風(fēng)險(xiǎn)水平能提供最高收益的組合。</p><p><b>　　確定最優(yōu)投資組合</b></p><p>　　現(xiàn)在我們考慮在有效集中哪份組合是最佳的。要做到這點(diǎn)，我們需要考慮投資者對風(fēng)險(xiǎn)的承受力。因?yàn)橥?/p>

98、常不同的投資者有不同的風(fēng)險(xiǎn)承受力，我們應(yīng)該期望每個(gè)投資者有一份不同的最優(yōu)投資組合。我們很快看見這的確是實(shí)際情形。</p><p>　　考慮一個(gè)特殊投資者，假設(shè)這個(gè)投資者有能力對每個(gè)可能的投資回報(bào)分布去指定一個(gè)數(shù)，它具有以下性質(zhì)：</p><p>　　1. 當(dāng)且僅當(dāng)這個(gè)投資者更喜歡以回報(bào)投資，而不是以回報(bào)投資。</p><p>　　2. 當(dāng)且僅當(dāng)這個(gè)投資者對選擇以回報(bào)

99、投資和以回報(bào)投資漠不關(guān)心。</p><p>　　將分布函數(shù)映射到實(shí)數(shù)的函數(shù)稱為效用函數(shù)。注意通常不同的投資者有不同的效用函數(shù)。</p><p>　　效用函數(shù)有許多不同的形式。簡單起見，我們假設(shè)每個(gè)投資者有以下形式的效用函數(shù)</p><p><b>　　，</b></p><p>　　其中是一個(gè)測量投資者風(fēng)險(xiǎn)厭惡水平的值并

100、且它對各個(gè)投資者是唯一的。（這里，和代表回報(bào)概率分布的平均值和標(biāo)準(zhǔn)差) 。對于假設(shè)一個(gè)這種形式的效用函數(shù)有真正的理論原因。但是，為簡單起見，我們略去細(xì)節(jié)。注意到在假設(shè)一個(gè)效用函數(shù)具有這種形式時(shí)，我們暗含著假設(shè)在具有同樣風(fēng)險(xiǎn)水平的投資組合中，您是選擇期望收益更大的，而在具有同樣期望收益的投資組合中，選擇風(fēng)險(xiǎn)較少的。</p><p>　　對于一個(gè)有風(fēng)險(xiǎn)承受力水平的投資者，投資組合最優(yōu)化問題可以被如下陳述：</p

101、><p><b>　　最大化:</b></p><p><b>　　滿足:.</b></p><p>　　這是一個(gè)可求解的簡單約束最優(yōu)化問題，只須將條件代入目標(biāo)函數(shù)，然后由單變量微積分，使用標(biāo)準(zhǔn)最優(yōu)化方法求解?；蛘哌@個(gè)優(yōu)化問題也可用多元微積分中的拉格朗日乘子法方法解決。</p><p>　　從

102、圖來看，的最大值是滿足拋物線是雙曲線的切線的數(shù)（參見圖10.2。最優(yōu)的投資組合在這個(gè)圖用表示）。清楚地，最優(yōu)的投資組合依賴于的值，而表示這個(gè)投資者的風(fēng)險(xiǎn)厭惡水平。</p><p>　　詳細(xì)進(jìn)行最優(yōu)化，我們發(fā)現(xiàn)當(dāng)和都是兩個(gè)風(fēng)險(xiǎn)保障時(shí)（即和)，最優(yōu)投資組合的風(fēng)險(xiǎn)-收益坐標(biāo)是</p><p><b>　　，</b></p><p><b>

103、　　。</b></p><p>　　從，可得出應(yīng)該在上投資的投資組合比例是</p><p><b>　　。</b></p><p>　　評(píng)述 我們已經(jīng)假設(shè)，沒有保證金的短期銷售是可能的（即我們已經(jīng)假設(shè)可以是任一個(gè)實(shí)數(shù)值，包括在區(qū)間[0,1]之外的值）。在短期銷售被限制的更加現(xiàn)實(shí)的條件下，最優(yōu)的投資組合也許與剛決定的不同。<

104、/p><p>　　例1:設(shè)債券基金的收益期望值為5%和標(biāo)準(zhǔn)差為12%，股票基金的收益期望值為10%和標(biāo)準(zhǔn)差為20%。兩種收益的相關(guān)系數(shù)是0.60。假設(shè)一個(gè)投資者的效用函數(shù)具有形式。假設(shè)沒有保證金的短期銷售是可能的，確定投資者在股票和債券之間的最優(yōu)份額。</p><p>　　對這種類型的問題，習(xí)慣上假設(shè)，效用函數(shù)是用百分?jǐn)?shù)度量。因此，如果，分別代表在證券和股票基金上的收益，則</p>

105、;<p><b>　　，</b></p><p><b>　　。</b></p><p>　　注意，這樣的度量也總可由的適當(dāng)選擇而達(dá)到。</p><p>　　由已推出的公式，最優(yōu)組合的期望收益是</p><p><b>　　，</b></p><

106、;p><b>　　這里，</b></p><p><b>　　和</b></p><p><b>　　。</b></p><p>　　因此，應(yīng)該投資在債券中的投資組合的比例是</p><p>　　。</p><p&g

107、t;　　這樣，對一個(gè)$1000的投資組合，對賣出空頭$3351.56的債券和投資$4351.56到股票是最優(yōu)的?！?lt;/p><p>　　投資組合機(jī)會(huì)集合的特殊情況</p><p>　　我們用在一些特殊情況下強(qiáng)調(diào)投資組合機(jī)會(huì)集合的形式結(jié)束本節(jié)。從頭到尾，我們假設(shè)和是滿足和的保證金（且的情況不使人感興趣，因?yàn)橹罂偸潜群茫?。我們也假設(shè)不允許有空頭位置。</p><p>