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1、第四章第四章最优投资组合选择最优投资组合选择1主要内容n资产配置：一种风险资产和一种无风险资产n两种风险资产的最优组合n多种风险资产的最优组合n多种风险资产无风险资产n资产配置：风险资产和无风险资产2n投资者在期初投资，在期末获得回报。n一期模型是对现实的一种近似，如对零息债券、欧式期权的投资。虽然许多问题不是一期模型，但作为一种简化，对一期模型的分析是分析多期模型的基础。一期投资模型3n完美的市场交易是无成本的，市场是可以自由进出的信息是对称的和可以无偿获得地存在很多交易者，没有哪一个交易者的行为对证券的价格产生影响无税收，无买、卖空限制证券无限可分，借贷利率相等假设4资产配置：一种风险资产
2、和一种无风险资产5n在风险资产和无风险资产上配置n无风险资产:国库券、银行存单、n风险资产:股票配置资产6rf=7% rf=0%E(rp)=15% p=22%w=%inp(1-w)=%inrf例子7E(rc)=wE(rp)+(1-w)rfrc=资产组合资产组合Forexample,w=.75E(rc)=.75(.15)+.25(.07)=.13or13%期望收益8pc=Sincerfw=0,then 资产组合的标准差9c=.75(.22)=.165or16.5%Ifw=.75,thenc=1(.22)=.22or22%Ifw=1c=(.22)=.00or0%Ifw=0 组合（无杠杆）10可能的
3、组合E(r)E(rp)=15%rf=7%22%0PF cE(rc)=13%C11TheCALdepictstherisk-returncombinationsforinvestors:Theslopeis:SohereCAL(资本配置线)12假设可以按照同样的利率借入现金，投资与风险资产；投资者拥有资金10000元，借入5000元，则投资组合中风险资产的比例为1.5，无风险资产为-0.5rB=(-.5)(.07)+(1.5)(.15)=.19B=(1.5)(.22)=.33有杠杆13CAL(CapitalAllocationLine)E(r)E(rp)=15%rf=7% p=22%0PFE(r
4、p)-rf=8% B14TotheleftofP:lendingat7%,slopeis0.36.TotherightofP:borrowingat9%,slopeis:TheCAListhereforekinkedatP.借入利率高于贷出利率15CALwithHigherBorrowingRateE(r) 9%7%S=.36S=.27P p=22%16n风险厌恶程度高的投资者会持有更高比例的无风险资产.n风险厌恶程度低的投资者会持有更高比例的风险资产.n寻求高收益并承担高风险将会应用杠杆头寸。风险厌恶与资产配置17效用函数Wehave:So:18最优化First-ordercondition
5、tomaximizeU:Solvingforw:Anexample:if19最优持有n风险资产的最优持有与风险溢价成正比；n风险资产的最优持有与风险资产方差成反比；n风险资产的最优持有与风险厌恶程度成反比；20风险偏好与资本配置E(r) 7%PLenderBorrower p=22%21最优持有LendershavealargerA,steeperindifferencecurvescomparedtotheborrower(moreriskaverse).Theyrequiremoreexpectedreturntocompensatefortakinganadditionalunitofr
6、isk.Investorsseekthehighestpossibleutility,representedbythehighestpossibleindifferencecurvewhichtouchestheCAL.22两种风险资产的最优组合23投资组合收益的均值方差24最优化问题假设：无卖空限制假设：无卖空限制25两种股票:收益26 p2=w12 12+w22 22+2W1W2Cov(r1r2)=w12 12+w22 22+2W1W2 1,2 1 2两种股票:风险27最优化问题MinMinSubjectto28推导29结果30两种股票组合投资机会集14%E(r)St.Dev15%20%1
7、0%31Rangeofvaluesfor 1,2+1.0 -1.0If =1.0,thesecuritieswouldbeperfectlypositivelycorrelatedIf =-1.0,thesecuritieswouldbeperfectlynegativelycorrelated相关系数32If =1.0: p2=w12 12+w22 22+2W1W2 1 2=w1 1+w2 22Withperfectpositivecorrelation,standarddeviationoftheportfolioisjustweightedaverageofeachcomponentss
8、tandarddeviation完全正相关33完全正相关完全正相关34完全正相关完全正相关=114%E(r)St.Dev15%20%10%35If =-1.0: p2=w12 12+w22 222W1W2 1 2=w1 1-w2 22- -完全负相关完全负相关36完全负相关37完全负相关38两种股票组合投资机会集14%E(r)St.Dev15%20%=-1=-110%39不相关40图41n期望收益和标准差相同的股票，组合的效果可能不同n取决于相关系数n-1.0+1.0n相关系数越小，分散化可能达到的效果越好nIf =+1.0,风险无法分散nIf =-1.0,风险可以对冲.分散化42Thefir
9、st-orderconditiontominimize:Therefore:全局最小方差组合(MVP)4312E(r2)=.14=.2012=.2E(r1)=.10=.15 最小方差组合：例子44W1=(.2)2-(.2)(.15)(.2)(.15)2+(.2)2-2(.2)(.15)(.2)W1=.6733W2=(1-.6733)=.3267最小方差组合：例子45rp=.6733(.10)+.3267(.14)=.1131p=(.6733)2(.15)2+(.3267)2(.2)2+2(.6733)(.3267)(.2)(.15)(.2)1/2p=.01711/2=.1308 最小方差组合：
10、例子46有效前沿14%E(r)St.Dev15%20%Minimalvarianceportfolio10%47多种风险资产48投资组合收益的均值方差49最优化问题50可行集可行集n由N种可交易风险证券中的任意K种形成的证券组合构成的集合称为可行集。在均值-标准差平面上来刻画可行集。51三种以上证券形成的可行集可行集的两个重要性质n（1）只要N不小于3，可行集对应于均值-标方差平面上的区域为二维的。n（2）可行集的左边向左凸。可行集52三种证券形成可行集的例子n三点形成地区域53前沿n定义：一个证券组合称为前沿证券组合，如果它在所有具有相同期望回报率的证券组合中具有最小方差。n定义：所有前沿证
11、券组合构成的集合称为证券组合前沿。54性质n证券组合前沿的性质性质1：整个证券组合前沿可以由任何两个前沿证券组合生成。性质2：前沿证券组合的任何凸组合仍然在证券组合前沿上。55n在均值-标准差平面上的证券组合前沿56最小方差前沿/有效前沿E(r)EfficientfrontierGlobalminimumvarianceportfolioMinimumvariancefrontierIndividualassetsSt.Dev.57分散化降低风险nInaportfolioofNassetswithequalweights,58RiskReductionwithDiversificationnT
12、herefore,nAsNbecomeslargeenough,theindividualriskofassetswillbediversifiedaway,whatsleftisthecontributiontothetotalriskcausedbythecovarianceterms.59RiskReductionwithDiversificationNumberofSecuritiesSt.DeviationMarketRiskUniqueRisk60风险资产无风险资产61n可以以相等的利率借贷无风险资产假设62几类不同的资本配置线ME(r)CAL(Globalminimumvaria
13、nce)CAL(A)CAL(P)PAFPP&FA&FMAGPM 63构造过程n风险资产的有效前沿；n无风险资产与切点组合构成新的有效前沿；n新的有效前沿资本配置线64nSincethetangencyportfolioliesonthehighestCALthattouchestheefficientfrontier,wejusthavetomaximizetheslopeoftheCAL:nMaxsubjecttonInthecaseoftwosecurities:n切点组合65PORTFOLIOOFTWORISKYASSETS66OPTIMAL RISKY PORTFOLIOS67Asse
14、t allocation with stocks,bonds and billsTheoptimalriskyportfoliowithtworiskyassetsandariskfreeasset68Asset allocation with stocks,bonds and bills69Asset allocation with stocks,bonds and bills70Asset allocation with stocks,bonds and bills71资产配置72风险资产与无风险资产E(r) RfPLenderBorrower73风险资产E(r)Efficientfron
15、tierofriskyassetsMorerisk-averseinvestorUUUQPSSt.DevLessrisk-averseinvestor74相同无风险利率借贷E(r)FrfAPQBCALSt.Dev75不能借入76借贷利率不等77借贷利率不等风险厌恶高78借贷利率不等风险厌恶低79借贷利率不等风险厌恶中80投资过程n证券分析和市场分析：评估所有可能投资工具的风险和期望回报率特性n在对证券市场进行分析的基础上，投资者确定最优的证券组合：从可行的投资组合中确定最优的风险-收益机会，然后决定最优的证券组合最优投资组合理论选择的目标：使得均值-标准差平面上无差异曲线的效用尽可能的大选择的
16、对象：均值-标准差平面上的可行集81难度优化技术是是投资组合构造过程中最简单的环节；难度最大的是证券分析。GIGO-Garbagein-Garbageout82InternationalCorrelationStructureandRiskDiversificationnSecurityreturnsaremuchlesscorrelatedacrosscountriesthanwithinacountry.Thisissobecauseeconomic,political,institutional,andevenpsychologicalfactorsaffectingsecurityre
17、turnstendtovaryacrosscountries,resultinginlowcorrelationsamonginternationalsecurities.Businesscyclesareoftenhighasynchronousacrosscountries.83InternationalCorrelationStructureStock MarketAUFRGMJPNLSWUKUSAustralia (AU).586France (FR).286.576Germany (GM).183.312.653Japan (JP).152.238.300.416Netherland
18、s (NP).241.344.509.282.624Switzerland (SW).358.368.475.281.517.664United Kingdom (UK).315.378.299.209.393.431.698United States (US).304.225.170.137.271.272.279.439Relativelylowinternationalcorrelationsimplythatinvestorsshouldbeabletoreduceportfolioriskmoreiftheydiversifyinternationallyratherthandome
19、stically.84Domesticvs.InternationalDiversification0.440.270.12PortfolioRisk(%)NumberofStocks11020304050SwissstocksU.S.stocksInternationalstocksWhenfullydiversified,aninternationalportfoliocanbelessthanhalfasriskyasapurelyU.S.portfolio.Afullydiversifiedinternationalportfolioisonly12percentasriskyasho
20、ldingasinglesecurity.85OptimalInternationalPortfolioSelectionnThecorrelationoftheU.S.stockmarketwiththereturnsonthestockmarketsinothernationsvaries.nThecorrelationoftheU.S.stockmarketwiththeCanadianstockmarketis70%.nThecorrelationoftheU.S.stockmarketwiththeJapanesestockmarketis24%.nAU.S.investorwoul
21、dgetmorediversificationfrominvestmentsinJapanthanCanada.86SummaryStatisticsforMonthlyReturns1980-1992($U.S.)Stock MarketCorrelation CoefficientMean (%)SD (%) CNFRGMJPUKCanada (CN).795.830.90France (FR)0.381.427.011.02Germany (GM)0.330.661.236.740.87Japan (JP)0.260.420.361.477.311.22United Kingdom0.5
22、80.540.490.421.525.410.90United States0.700.450.370.240.571.334.560.80.79%monthlyreturn=9.48%peryear87SummaryStatisticsforMonthlyReturns1980-1992($U.S.)Stock MarketCorrelation CoefficientMean (%)SD (%) CNFRGMJPUKCanada (CN).795.830.90France (FR)0.381.427.011.02Germany (GM)0.330.661.236.740.87Japan (
23、JP)0.260.420.361.477.311.22United Kingdom0.580.540.490.421.525.410.90United States0.700.450.370.240.571.334.560.80bmeasuresthesensitivityofthemarkettotheworldmarket.ClearlytheJapanesemarketismoresensitivetotheworldmarketthanistheU.S.88TheOptimalInternationalPortfolioUSCNGMUKJPFR4.2%1.53OIPEfficients
24、etRf89CompositionoftheOIPforaU.S.InvestorBelgianmarket14.66%Italianmarket0.37%Japanesemarket9.25%Dutchmarket14.15%Swedishmarket20.26%U.S.market41.31%Total100.00%901.53%1.33%GainsfromInternationalDiversificationnForaU.S.investor,therisk-returntradeofffortheoptimalinternationalportfolioandoptimaldomes
25、ticportfolioareshownbelowandatright.OIPODPMean Return1.53%1.33%Standard Deviation4.27%4.56%riskreturn4.27%4.56%OIPODP91example1.Youmanageariskyportfoliowithanexpectedrateofreturnof18%andastandarddeviationof28%.TheT-billrateis8%.nYourclientchoosestoinvest70%ofaportfolioinyourfundand30%inaT-billmoneym
26、arketfund.Whatistheexpectedvalueandstandarddeviationoftherateofreturnonhisportfolio?nSupposethatyourriskyportfolioincludesthefollowinginvestmentsinthegivenproportionstockA25%;stockB32%;stockC43%.Whataretheinvestmentproportionsofyourclientsoverallportfolio,includingthepositioninT-bill?nWhatistherewar
27、dtovariabilityratio(S)ofyourriskyportfolio?YourclientsnDrawtheCALofyourportfolioonanexpectedreturn-standarddeviationdiagram,whatistheslopeoftheCAL?ShowthepositionofyourclientonyourfundsCal92nSupposethatyourclientdecidestoinvestinyourportfolioaproportionyofthetotalinvestmentbudgetsothattheoverallport
28、foliowillhaveanexpectedrateofreturnof16%.Whatistheproportiony?WhatareyourclientsinvestmentproportionsinyourthreestocksandtheT-billfund?Whatisthestandarddeviationoftherateofreturnonyourclientsportfolio?nSupposethatyourclientpreferstoinvestinyourfundaproportionythatmaximizestheexpectedreturnonthecompl
29、eteportfoliosubjecttotheconstraintthatthecompleteportfoliosstandarddeviationwillnotexceed18%.Whatistheinvestmentproportiony?Whatistheexpectedrateofreturnonthecompleteportfolio?nYourclientsdegreeofriskaversionisA=3.5.Whatproportionyofthetotalinvestmentshouldbeinvestedinyourfund?Whatistheexpectedvalueandstandarddeviationoftherateofreturnonyourclientsoptimizedportfolio?example9394
