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1、BAFI1045InvestmentsMarkowitz Portfolio TheoryMarkowitz Portfolio TheoryRMIT University貌想辐彩输磕柿赚剔查管蚤挽止抛头绥镊衷菠腆遵雇声葛秒幅篷弧才猪渊Markowitz Portfolio TheoryMarkowitz Portfolio Theory1RMIT UniversityRMIT UniversitySlide 2ReferencesReilly, Frank K. and Keith C. Brown, Analysis of Investment and Management Portfol
2、ios (9th Edition), Thomson South-Western, 2009.Chapter 7桩琐薛栋洽裁漱殴发贬躇茬皋生冕蝉瞩冤箱朝丝赴影芳秒跟味抓厨顺赖属Markowitz Portfolio TheoryMarkowitz Portfolio Theory2RMIT UniversityTopic OverviewAfter discussing this topic, you should be able to answer the following questions:What do we mean by risk aversion?What do we mean
3、 by risk?How do we compute the expected returns for an individual asset and a portfolio of assets?How do we compute the standard deviation for an individual asset and a portfolio of assets?What is covariance? How is it computed? What is the relationship between covariance and correlation?Given the p
4、ortfolio standard deviation, how do we diversify a portfolio?What happens to the standard deviation when we change the correlations between assets in the portfolio?What is the risk-return efficient frontier of risky assets?Why would you expect different investors would select different portfolios on
5、 the efficient frontier?RMIT UniversitySlide 3缅婉眶僳席旭矿绅毯逐蠕阀赚移双疟军劈铬望弧涅拣建帖毒窘宠培液盎锗Markowitz Portfolio TheoryMarkowitz Portfolio Theory3RMIT UniversityIntroductionTHE PORTFOLIO SELECTION BASIC PROBLEM:Given uncertain outcomes, what risky securities should an investor own?How can we build the optimal port
6、folio given the infinite number of portfolios that can be composed from the existing risky securities?夫狼豢钢塘忙暴多毁匆邦帝训平睦嘲儡氧戳回赶府哎京的淤钥裁挽迁熙材Markowitz Portfolio TheoryMarkowitz Portfolio Theory4RMIT UniversityConsider the problem Consider a situation where you have three stocks to choose from: Stock A, Sto
7、ck B and Stock C. You can invest your entire wealth in one of these three securities. Or you could purchase two securities, investing 10% in A and 90% in B, or 20% in A and 80% in B, or 70% in A and 30% in B, or 50% in each, or 60% in A and 40% in B, orthere is a huge number of possible combinations
8、 and this in a simple case when considering two securities. Imagine the different combinations you have to consider when you have thousands of stocks 估是焊迄家夹三汝姨翻孪酣仆督未芯溜舆岿洋轩慌罢午冯妇贴甄持央彻冬Markowitz Portfolio TheoryMarkowitz Portfolio Theory5RMIT Universityand the SolutionThe investor does not need to eval
9、uate all the possible portfolios. The answer is provided by theefficientsettheorem.Any investor will choose the optimal portfolio from the set of portfolios that1.Maximizeexpectedreturnforagivenlevelofrisk;and2.Minimizerisksforagivenlevelofexpectedreturns.听酬锄亏冉叫蛤并倘待孜俏援琼偿嫌委耍曼史拷饲热糯娘侧恫挡控御局兄Markowitz Po
10、rtfolio TheoryMarkowitz Portfolio Theory6RMIT UniversitySomeBackgroundAssumptionsAs an investor you want to maximize the returns for a given level of risk Your portfolio includes all of your assets and liabilities.The relationship between the returns for assets in the portfolio is important.A good p
11、ortfolio is not simply a collection of individually good investments.RMIT University瞻黎彩绿蒸拌条姚铝裸芦垫则榜磅吞士镐问袱肩濒付讲草艇旺李鞭殉邻沥Markowitz Portfolio TheoryMarkowitz Portfolio Theory7RMIT UniversitySomeBackgroundAssumptionsRisk Aversion Given a choice between two assets with equal rates of expected return, risk-a
12、verse investors will select the asset with the lower level of riskIf you give such an investor a choice between $A for sure, or a risky gamble in which the expected payoff is $A, a riskaverseinvestor will go for the sure payoff.Risk AverseReject investment portfolios that are fair gambles or worsePr
13、ospect with zero risk premiumRisk-averse investors will go only for risk-free or speculative prospects with positive risk premiumsRisk involved should be compensated with something more than risk-free rateRMIT University对撑睡虎词夸掣裴彭夏哆蹄云霜牵氢件瓦爹啮藉战沏完蛋受效酉信没虏廓Markowitz Portfolio TheoryMarkowitz Portfolio Th
14、eory8RMIT UniversitySomeBackgroundAssumptionsEvidenceMany investors purchase insurance for: Life, Automobile, Health, and Disability Income. Yield on bonds increases with risk classifications from AAA to AA to A, etc.Not all Investors are risk averseIt may depend on the amount of money involved:Risk
15、ing small amounts, but insuring large lossesIndividuals are generally risk averse when large fractions of their wealth is at risk.RMIT University容萌竭深拧瀑父彤想虽值效娱烈碧政诧歼绝庭嚼蚊拖沤珍枚鹊唐讶胖公烯Markowitz Portfolio TheoryMarkowitz Portfolio Theory9RMIT UniversityMarkowitzPortfolioTheoryAssumptions for Investors Consi
16、der investments as probability distributions of expected returns over some holding periodMaximize one-period expected utility, which demonstrate diminishing marginal utility of wealthEstimate the risk of the portfolio on the basis of the variability of expected returnsBase decisions solely on expect
17、ed return and riskPrefer higher returns for a given risk level. Similarly, for a given level of expected returns, investors prefer less risk to more riskRMIT University砚廓鸟刨剪鄙躺党术浊钧叶炊宴炼怜拣买卉汉圈靳彼掀辜戎羹腕胁缀办锭Markowitz Portfolio TheoryMarkowitz Portfolio Theory10RMIT UniversityMarkowitzPortfolioTheoryUsing t
18、hese five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.RMIT University馏打只裕摄芯渗忿了寞润站粗豹升喉怨茂谨历抠招度刽卓颜箕轮枕推鬼巨Mark
19、owitz Portfolio TheoryMarkowitz Portfolio Theory11RMIT UniversityDefinition of RiskUncertainty:Risk means the uncertainty of future outcomes. For instance, the future value of an investment in Googles stock is uncertain; so the investment is risky. On the other hand, the purchase of a six-month CD h
20、as a certain future value; the investment is not risky.Probability:Risk is measured by the probability of an adverse outcome. For instance, there is 40% chance you will receive a return less than 8%.RMIT University聪桶讨顾悄卓旱造傍识穴乳桩塞练复浩抵朔桅隅滋敝矫磊桔俺模狱吩栈傲Markowitz Portfolio TheoryMarkowitz Portfolio Theory12
21、RMIT UniversityMeasuring RiskThe Advantages of Using Standard Deviation of ReturnsThis measure is somewhat intuitiveIt is a correct and widely recognized risk measure It has been used in most of the theoretical asset pricing modelsRMIT University窿琅沥家痹息帐碘蒋淳碘啮削秆盯陶负脖榜龙熬溺甘上圣畅抬溺孪且徐劲Markowitz Portfolio Th
22、eoryMarkowitz Portfolio Theory13RMIT UniversityRule 1The return for an asset is the probability weighted average return in all scenarios碗直焰典除器狡瞧吾髓蚂假髓权雍阐骨敌验渡厌盒遭瘸铬哀艳兵寐莽单杨Markowitz Portfolio TheoryMarkowitz Portfolio Theory14RMIT UniversityExpectedRatesofReturnIndividual Investments where Pi = Probabil
23、ity for possible return iRi = Possible return i势战沾嫌栽鸿集性调沁柿茁源登憋糖测响砍蜘殊香释沉粹萝你跃勒挎衬茅Markowitz Portfolio TheoryMarkowitz Portfolio Theory15RMIT UniversityExample1RMIT University训相葵颂贪活逢寺诲熏折爷尉萄瑚档中浴墅淬沮吭吾柒卸供斩驳拒鸳喊逢Markowitz Portfolio TheoryMarkowitz Portfolio Theory16RMIT UniversityRule 2Portfolio of Investmen
24、tsThe rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weightsW1+ W2 = 1肢斋古滋洛筛猎勋新嘘分取臆洋呆哺辑熔交梁纳禾妇明皇广壶蛊明官篇局Markowitz Portfolio TheoryMarkowitz Portfolio Theory17RMIT UniversityExpectedRatesofReturn-Portf
25、olioIf you want to construct a portfolio of n risky assets, what will be the expected rate of return on the portfolio if you know the expected rates of return on each individual assets?The formula RMIT University抱疫裔汤犬卫菇咖汀胎衰哄玲散宝链蕾拟絮熏捧渺面爪吼脊廉除挟共贬徽Markowitz Portfolio TheoryMarkowitz Portfolio Theory18RM
26、IT UniversityExampleRMIT University淘攒相桂坟碾铺衡套纬筒魏挠余瘦诧洒带崖褐俏绸杉懒佑猛诈醒陷蹈现队Markowitz Portfolio TheoryMarkowitz Portfolio Theory19RMIT UniversityRule 3The variance of an assets return is the expected value of the squared deviations from the expected returna measure that estimates the extent to which the actu
27、al outcome is likely to diverge from the expected outcome召锁憨讫墓躁彭她镀引湖抵敬蔽华混疑墙班窄舰谱透窟市藻佛迄卢匹波倘Markowitz Portfolio TheoryMarkowitz Portfolio Theory20RMIT UniversityIndividualInvestmentRiskMeasureVarianceIt is a measure of the variation of possible rates of return Ri, from the expected rate of return E(Ri)
28、where Pi is the probability of the possible rate of return, RiStandard Deviation () It is simply the square root of the varianceRMIT University尼份揪伊疾雀初纱嫡袭顿匝醋恫福希荚靳撮自虱舵弥荔昨炊希亏梧砰夺祷Markowitz Portfolio TheoryMarkowitz Portfolio Theory21RMIT UniversityIndividualInvestmentRiskMeasureExampleVariance ( 2) =Sta
29、ndard Deviation ( ) =RMIT University恐照柒绰频茫帘庇逗方哉够度渡勾例膀谨沈技需手屏觅伞若妙匡睫慨胚德Markowitz Portfolio TheoryMarkowitz Portfolio Theory22RMIT UniversityCovarianceofReturnsA measure of the degree to which two variables “move together” relative to their individual mean values over timeFor two assets, i and j, the co
30、variance of rates of return is defined as: Covij=ERi-E(Ri)Rj-E(Rj)possible values:positive: variables move togetherzero: no relationshipnegative: variables move in opposite directions Example The Wilshire 5000 Stock Index and Lehman Brothers Treasury Bond Index during 2007RMIT University邢爸勋约瞩卯脏晚罩隙妊貌
31、眯躯帽委山桅锦斡尘召椒蛰罩耕吁位纂晨涅嘘Markowitz Portfolio TheoryMarkowitz Portfolio Theory23RMIT UniversityExampleRMIT University道馁脏盲细祷酋迅迄摈朱绘盼参惺隘陌隆屠凸嘲钝瓮酞纂巫怎锌茶比取嘎Markowitz Portfolio TheoryMarkowitz Portfolio Theory24RMIT UniversityExampleRMIT University过先溜亩戈鸯赊专销谬耸冠井正痪骚枕筷嘶睡接粳病饲宵赊亩孰捅不右蒸Markowitz Portfolio TheoryMarkowit
32、z Portfolio Theory25RMIT UniversityCovarianceandCorrelationThe correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviationsComputing correlation from covariance RMIT University甸翌捍啃焕十铂屏略房塔其地毖概红俏缀厄底黔勘埔疼蠕蚜桔府腑诉厄霍Markowitz Portfolio T
33、heoryMarkowitz Portfolio Theory26RMIT UniversityCorrelationCoefficientThe coefficient can vary in the range +1 to -1. -1 ij 1 A value of +1 would indicate perfect positive correlation. This means that returns for the two assets move together in a positively and completely linear manner. A value of 1
34、 would indicate perfect negative correlation. This means that the returns for two assets move together in a completely linear manner, but in opposite directions. RMIT University豺曝岛坟累驮饶囤胰孕器曼厚尝锻楞椅淬之昧姆浙舆念拽乡株籍孔己繁励Markowitz Portfolio TheoryMarkowitz Portfolio Theory27RMIT UniversityAn ExampleLet us say y
35、ou consider two securities: Goa-Sze Inc and Xi Xi Inc.You do some basic calculations and find the following characteristics:SecurityERSDRABC15%18.6%XYZ21%28.0%溪窃咯梦淆丈爪泌馒淮蝉七蝉速郸吸巫狂像憎瓤诡榆锦钮命习初砒梦荔芥Markowitz Portfolio TheoryMarkowitz Portfolio Theory28RMIT UniversityExpected Return of the PortfolioLet us s
36、ay that you decide to invest 60% of your portfolio in ABC Inc and 40% in XYZ Inc.What is the Portfolios Expected Return?Remember: take the weighted average of the individual securitys returns 更巳受掩蹦擎舒稳齐博旗坟梦拣救逆滋萨主各蛤丈橇轨锭在迅肚烯圆箭嘿Markowitz Portfolio TheoryMarkowitz Portfolio Theory29RMIT UniversityPortfol
37、ios RiskIt depends on the covariance or correlation between the two securities? Scenario 1: Assume correlation = 1Scenario 2: Assume correlation = -1Scenario 3: Assume correlation = +0.2茵伟答螺骗逾溢隘貉思啮藻即凋挨榷攘奇铅司然扁就匹宠侦畸代眨仰藤梢Markowitz Portfolio TheoryMarkowitz Portfolio Theory30RMIT UniversityConsider the
38、Two Asset CaseE(R)Asset1Asset2泡承逗瞥衙说业抉狮戏措者观姓搁取裳跺骏异锋左霍充漱很衅梨倘涸谩抓Markowitz Portfolio TheoryMarkowitz Portfolio Theory31RMIT UniversityMinimum Variance PortfolioOne way to understand the benefits of diversification is to look at the portfolio that provides the investor with the smallest possible varianc
39、e: the global minimum-variance portfolio.In the case of a two stock portfolio (security 1 and 2), the weights in such a portfolio are given by the following:梧评曹沦庚泵孝柯农佣狞犯橙包峡讲硒顿肩剩呸里茎葬宾芭寸抨每馆认恤Markowitz Portfolio TheoryMarkowitz Portfolio Theory32RMIT UniversityPerfect Negative CorrelationWith perfect n
40、egative correlation, , we face a situation where we can reduce portfolio risk to zero. In this case, the global minimum variance portfolio will have a variance of zero.The weights for security 1 and 2 in the global minimum variance portfolio are given by:蜀若谊彪硫蓑坡鹰美镇践场磕待摹坤延废粒迅业址策宫敛岛批其腺梦圣蝗Markowitz Por
41、tfolio TheoryMarkowitz Portfolio Theory33RMIT UniversityPerfectly Negative CorrelationE(R)Asset1Asset2Zero-VariancePortfolio罗雹镀韧拟砖嘴斟叠瓶耶皿滞田帕穷沼掀纱眨青务滞网疏竭名凭动便岁瞻Markowitz Portfolio TheoryMarkowitz Portfolio Theory34RMIT UniversityPerfectly Positive CorrelationWhen there are perfectly positive correlation
42、, , the investor can ONLY reduce portfolio risk to zero if short-selling is available.When one of the assets has a low risk and low return, portfolio returns will be less than the return offered by the low return asset. In the case when short-selling is not available: The investor has to put his/her
43、 wealth in the lower risk asset.幢端苍带币筒枯障碗堕蚌烬痔雇斗倦唾烷技耀势塞改找冈拄劝冻驯觅价响Markowitz Portfolio TheoryMarkowitz Portfolio Theory35RMIT UniversityPerfectly Positive CorrelationE(R)Asset1Asset2Minimum-VariancePortfolio氟麻辕肥驭蹿幻呜陪黑禾氛掠债泻肖软瞎园颧疽终灿晕吏操景配熙济袱赊Markowitz Portfolio TheoryMarkowitz Portfolio Theory36RMIT Unive
44、rsityThe General Case: Imperfect CorrelationThe case that we face in real life is that of imperfect correlation, that is whenIn this case, diversification will contribute to the reduction of risk, but cannot totally eliminate risk. In such a case and when two securities are used, the minimum varianc
45、e portfolio will have positive weights in both securities. 龚滋菩嗣巢规密啮墒网反仕花芒迅牙泳箭迫怎搞牟针府壳完汇陇酪赶斑娘Markowitz Portfolio TheoryMarkowitz Portfolio Theory37RMIT UniversityNon-Perfect CorrelationE(R)Asset1Asset2Minimum-VariancePortfolioPortfoliosmadeupofmostlySecurity2PortfoliosmadeupofmostlySecurity1仅也家敬棚滔涛畔衣墒
46、魔辈液疲盖釉粪癣眨序惨固做进奏奢抱己舆溅挚赚Markowitz Portfolio TheoryMarkowitz Portfolio Theory38RMIT UniversityPerfectlyPositivelyCorrelatedSecuritiesReturnonAReturnonB仓排炸像槽寇率侥腑抢咒狂戴轩堪写凄槐职攫哲洱穴涨朝将乃绰多孩猴卖Markowitz Portfolio TheoryMarkowitz Portfolio Theory39RMIT UniversityPerfectlyNegativelyCorrelatedSecuritiesReturnonARet
47、urnonB斜酬蛋捆午懒徘哎智叶锈双迄缀勇签痘掇装瑚突介遍需实别指搏命赋冬肚Markowitz Portfolio TheoryMarkowitz Portfolio Theory40RMIT UniversityUncorrelatedSecuritiesReturnonAReturnonB督饭姑膘爹糜决搽捣舌肩傣甫叮支污殆孺钦募犯蹋斡沃陨缠孟辞泛辈发学Markowitz Portfolio TheoryMarkowitz Portfolio Theory41RMIT UniversityRule 4When two risky assets with variances 12 and 22
48、, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by: p2 = w1212 + w2222 + 2W1W2 Cov(r1r2) Cov(r1,r2) = Covariance of returns for Security 1 and Security 2佯藤峻儿障胞苹淌搓涣咯恿衔李戏盎搞宇蜗赚难刷铁维漾杉钞嚏置畜橇点Markowitz Portfolio TheoryMarkowitz P
49、ortfolio Theory42RMIT UniversityStandardDeviationofaPortfolioThe Formula RMIT University福爽核慑栽拙珠弛焦豹蔬艇饵寂训读孝民制埔璃星晓钝勉征悼涸慧纸蹈糟Markowitz Portfolio TheoryMarkowitz Portfolio Theory43RMIT UniversityStandardDeviationofaPortfolioComputations with A Two-Stock Portfolio Any asset or a portfolio may be described
50、by two characteristics:The expected rate of returnThe expected standard deviations of returnsThe correlation, measured by covariance, affects the portfolio standard deviationLow correlation reduces portfolio risk while not affecting the expected returnRMIT University寇您悍渗绦授芽罕厢袁悍赶晰坡粮砍冬卑玫靖并杠瘦铁疥氖疥廉立房嘶盟M
51、arkowitz Portfolio TheoryMarkowitz Portfolio Theory44RMIT UniversityStandardDeviationofaPortfolioTwo Stocks with Different Returns and Risk1 0.10 0.50 0.0049 0.07 2 0.20 0.50 0.0100 0.10 W)E(R Asset ii2ii Case Correlation Coefficient Covariance a +1.00 b +0.50 c 0.00 d -0.50 e -1.00 RMIT University刺
52、龄极俗睫补白勋冠稗晰涯唆吞挨齐拈拇俘识橱瘁捅瘩法钵猴税颧鼻径平Markowitz Portfolio TheoryMarkowitz Portfolio Theory45RMIT UniversityStandardDeviationofaPortfolioAssets may differ in expected rates of return and individual standard deviationsNegative correlation reduces portfolio riskCombining two assets with +1.0 correlation will
53、not reduces the portfolio standard deviationCombining two assets with -1.0 correlation may reduce the portfolio standard deviation to zeroProposition: A portfolio of less than perfectly correlated assets always offer better risk-return opportunities than the individual component assets on their own.
54、See Exhibits 7.10 and 7.12RMIT University彻透槽怨株骑寻裳误基中尼擦黔址荫眷忍硫险俄次候妓佬上遮阶挺蜂标笺Markowitz Portfolio TheoryMarkowitz Portfolio Theory46RMIT UniversityExhibit7.10RMIT University糟楼撇初充席郡来碍趟敲早垄膛栓哪钠浦炕臃羚蓝殉枫牲缩仇凑箱怯骡浮Markowitz Portfolio TheoryMarkowitz Portfolio Theory47RMIT UniversityExhibit7.12RMIT University非绳嘛戌固
55、梆轴怨癣弯扣晦虎梧寺襄牌鼻瞥轨儿授押谦峡趟窒戎待趟伯掺Markowitz Portfolio TheoryMarkowitz Portfolio Theory48RMIT UniversityStandardDeviationofaPortfolioConstant Correlation with Changing WeightsAssume the correlation is 0 in the earlier example and let the weight vary as shown in the next slide. Portfolio return and risk are
56、(See Exhibit 7.13):RMIT University思颊市誊硒扑粥选肋衣衫违咸恢秒廷娱沼祝录颧耍毡人点逼衡沽母铣腑饵Markowitz Portfolio TheoryMarkowitz Portfolio Theory49RMIT UniversityRMIT UniversitySlide 50Various weights and constant correlation yield the following risk-return combinationsCasew1w2E(Ri)f0.001.000.20g0.200.800.18h0.400.600.16i0.50
57、0.500.15j0.600.400.14k0.800.200.12l1.000.000.10w1w2E(Ri)E(port)f0.001.000.200.1000g0.200.800.180.0812h0.400.600.160.0662i0.500.500.150.0610j0.600.400.140.0580k0.800.200.120.0595l1.000.000.100.0700酚瞥图迸缸椭淑促茵灵瘟伪抛板购参糟哑绕恬溯粪诌碎挡侥棠葛吓石航爹Markowitz Portfolio TheoryMarkowitz Portfolio Theory50RMIT UniversityExh
58、ibit7.13RMIT University母竖凿咨抵超掸格痈铅蔗嫌摸鹊驮悄霉朵扮仪敷弓华蹄额孩台邦卢脉撮妥Markowitz Portfolio TheoryMarkowitz Portfolio Theory51RMIT UniversityStandard Deviation of a Portfolio Standard Deviation of a Portfolio A Three-Asset Portfolio The results presented earlier for the two-asset portfolio can extended to a portfoli
59、o of n assetsAs more assets are added to the portfolio, more risk will be reduced everything else being the sameThe general computing procedure is still the same, but the amount of computation has increase rapidlyFor the three-asset portfolio, the computation has doubled in comparison with the two-a
60、sset portfolioRMIT University奏若孕循勘谣凛古货馋徽是豪刽畸记哆底庙羞赚梢跟款缨却添忌贝勺胡讶Markowitz Portfolio TheoryMarkowitz Portfolio Theory52RMIT UniversityRMIT UniversitySlide 533AssetPortfolioStandardDeviation3 classes of assets: Stocks (S), Bonds (B) and Cash Equivalents (C)We can see the dynamism of the portfolio process
61、 when assets are addedCan also see the rapid growth in computations required犹雨眠晕撑债原附杖暗僵死惧藕范溃坏淑糙酱董纯盈啮袄娠哲畴握篱递惑Markowitz Portfolio TheoryMarkowitz Portfolio Theory53RMIT UniversityPortfolio RiskThe total risk of a portfolio (indeed of a security) consists of two parts: - Market (or systematic) Risk - U
62、nique (or firm-specific) RiskDiversification reduces the Unique Riskhence, diversification reduces total risk毡迹裳祁盆魁阎肤蛰毙胃默局岛髓藻奴捻示晃俄椿世辆滤淳晃踞时失炕甸Markowitz Portfolio TheoryMarkowitz Portfolio Theory54RMIT UniversityEstimationIssuesResults of portfolio allocation depend on accurate statistical inputsEstim
63、ates ofExpected returns Standard deviationCorrelation coefficient Among entire set of assetsWith 100 assets, 4,950 correlation estimatesEstimation risk refers to potential errorsRMIT University遵毅盟估踌驳琢链港厚侵侈雕滋氛惭逊烬浑涝臣铜邱衅创叛渴呵谐欺络狄Markowitz Portfolio TheoryMarkowitz Portfolio Theory55RMIT UniversityEstima
64、tionIssuesWith the assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assetsSingle index market model:bi = the slope coefficient that relates the returns for security i to the returns for the aggregate marketRm = the
65、returns for the aggregate stock marketRMIT University陀挎疤陨枫谓宣郴弊傀罢灯柱珐胳忱档昆猫哭猾监墅丘醇囚警谋魁唯活钙Markowitz Portfolio TheoryMarkowitz Portfolio Theory56RMIT UniversityEstimationIssuesIf all the securities are similarly related to the market and a bi derived for each one, it can be shown that the correlation coef
66、ficient between two securities i and j is given as:RMIT University壬囤皮宵峰条固威纠主赋铸潭尚冀舅喻卜旋花觅溯纶狰洪旺医飞卤钥考术Markowitz Portfolio TheoryMarkowitz Portfolio Theory57RMIT UniversityTheEfficientFrontierThe efficient frontier represents that set of portfolios with the maximum rate of return for every given level of
67、 risk, or the minimum risk for every level of returnEfficient frontier are portfolios of investments rather than individual securities except the assets with the highest return and the asset with the lowest riskThe efficient frontier curvesExhibit 7.14 shows the process of deriving the efficient fro
68、ntier curveExhibit 7.15 shows the final efficient frontier curveRMIT University靖熬牢访濒磊目农讳褂迫琴哇羡雏顺针怀溶蔫坡喳诱原身编杯践是齐讨花Markowitz Portfolio TheoryMarkowitz Portfolio Theory58RMIT UniversityExhibit7.14RMIT University傅虽怖棉肺值能滩溯赘浴鹅吕懦插病诧无哟辰脚弥东粒四厂牵冻豺广匣翘Markowitz Portfolio TheoryMarkowitz Portfolio Theory59RMIT Uni
69、versityMinimum-Variance FrontierE(R)Minimum-VariancePortfolioIndividualAssetsIndividualAssetsMinimum-VarianceFrontierXYZ年艇筏喷悟轻胁游饶脊撰孔毕审充宇毛宇颂巴痪曰人敬唐箭赫乔刊扁窟圣Markowitz Portfolio TheoryMarkowitz Portfolio Theory60RMIT UniversityMinimum-Variance Frontier The outcome of risk-return combinations generated by
70、portfolios of risky assets that gives you the minimum variance for a given rate of return. Intuitively, any set of combinations formed by two risky assets with less than perfect correlation will lie inside the triangle XYZ and will be convex.卒搞互邪攫诌釜自词垛旷气奈烈揽曹黍火惯脏钳热区讣诽汽柠嫂泛讽然停Markowitz Portfolio Theory
71、Markowitz Portfolio Theory61RMIT UniversityThe Efficient Frontier Investors will never want to hold a portfolio below the minimum variance point. They will always get higher returns along the positively sloped part of the minimum-variance frontier. The Efficient Frontier is the set of mean-variance
72、combinations from the minimum-variance frontier where for a given risk no other portfolio offers a higher expected return.慰袄协剿盼捶划截涵禁为碘复牲茅簿镭晒镇氓但立仪笑肇盼贞贿方促寐质Markowitz Portfolio TheoryMarkowitz Portfolio Theory62RMIT UniversityEfficient FrontierE(R)EfficientFrontier(portfolioslyingbetweenpointsEandF)ABC
73、DEF岭嗅猫是山衍尸鸯控濒胖枯键槽汪砰谍铅乾岂援衍蛰拽戮孵绩择啦疽哎疏Markowitz Portfolio TheoryMarkowitz Portfolio Theory63RMIT UniversityEfficient setEfficient set the set of mean-variance choices from the investment opportunity set where for a given variance (or standard deviation) no other investment opportunity offers a higher m
74、ean return.E(rp)pRMIT University龙掂姚幌柒找呢陕砸尼绳坤手洽置磨校荡昼钝迟聂耻送刽泻熙咽奴凹禄挪Markowitz Portfolio TheoryMarkowitz Portfolio Theory64RMIT UniversityRMIT UniversitySlide 65Efficient Frontier CurveDecreases steadily as we move upward the curvesimplication: adding increments of risk as we move up the frontier gives d
75、iminishing increments of expected return Slope of efficient frontier肤拖律刊爵龋糜栅荧惜品货封笋万鹅簿檀员办妮泽茧政饵缆猛腮阮荤陀老Markowitz Portfolio TheoryMarkowitz Portfolio Theory65RMIT UniversitySelection of the Optimal Portfolio How will the investor go about selecting the optimal portfolio? Investors will have to consider
76、their indifference curves. Put the investors indifference curves and the efficient frontier and go for the portfolio on the farthest northwest indifference curve, where the indifference curve is tangent to the efficient frontier. 桅冕抨负辈智顽肃座渡溯娄冲汕危洋猫郁到培厨噎聪催免泞佣勃巧褥淀摹Markowitz Portfolio TheoryMarkowitz Po
77、rtfolio Theory66UtilityAnIntroductionWhatwillinfluencethetrade-offdecisions?Such factors as different degrees of unwillingness to bear risk which will be reflected in how much additional expected return the investor requires for taking an additional unit of risk.Another factor will be how much the i
78、nvestment could affect the investors total wealth. A potential loss of $1,000 would probably worry a millionaire less than someone with earnings of $100 per week. 轨损蠢副扬仙柠昌图匿奸榔岛桨独卖缕乳崩朽眨肄屈谚奏溃趟锰考田挨把Markowitz Portfolio TheoryMarkowitz Portfolio Theory67RMIT UniversityWhyisThisImportant?Knowledge of the
79、investors utility function enables him/her to choose between securities. In a single measure we have the investors attitudes to risk and return at each level of wealth.Difficult without knowledge of the utility function to make decisions between different securities with different expected returns a
80、nd different risks. 爵肢交制灭匣早携磷短允贰园挎塞术傲汇颁楞微膨说戚谜可矣萨帖副缘我Markowitz Portfolio TheoryMarkowitz Portfolio Theory68RMIT UniversityThe Markowitz ApproachINVESTOR UTILITYDEFINITION: is the relative satisfaction derived by the investor from the economic activity.It depends upon individual tastes and preferences
81、It assumes rationalityi.e. people will seek to maximize their utility敷赫陋朵短条跃结儒恭品蔽土驻姆杏耿穗筑镀苦呵秀鼻梢票走柱议协船各Markowitz Portfolio TheoryMarkowitz Portfolio Theory69RMIT UniversityThe Markowitz ApproachMARGINAL UTILITYeach investor has a unique utility-of-wealth functionincremental or marginal utility differs
82、 by individual investorAssumesdiminishing characteristicNon-satiationConcave utility-of-wealth function峨瞅忱磋敦僵超腿懈棚宰乏荧隋泊促娃几锰花闹寥亚甄硅外桶霖娶蛀誓帖Markowitz Portfolio TheoryMarkowitz Portfolio Theory70RMIT UniversityThe Markowitz ApproachUTILITY OF WEALTH FUNCTIONWealthUtilityUtility of Wealth苗缴形章痔嚼甸肝幅眉们易稿商美镰椅摘
83、惭砧衙凸血嘛形箱尤莽予辅烟役Markowitz Portfolio TheoryMarkowitz Portfolio Theory71RMIT UniversityRisk Aversion and UtilityHow do you choose among risky portfolios?RFR = 5%PortfolioRisk Premium (%)Expected Return (%)Risk (SD) (%)A: Low risk275B: Medium Risk4910C: High Risk81320时婶扔挖起属众胜甲傈受圃叉詹筏换铝屋伯烯宣拳抱址搓幸谭海瑟洋怯烈Marko
84、witz Portfolio TheoryMarkowitz Portfolio Theory72RMIT UniversityQuadraticUtilityFunctions When we use quadratic utility functions, we can nicely express our utility functions;Where, U = utility; E ( r ) = expected return on the asset or portfolio; A = coefficient of risk aversion; 2 = variance of re
85、turnsEmployed by both Financial Theorist and the CFA Institute Hence, the investor just considers risk and return. In other words, you require a higher expected return, the higher the risk of the investment. If A 0, then the investor is risk averse. If A 0, then the investor is risk-loving.婆累尧职暇躲叭凝吴
86、称歌频冲胞荐霉浓栖闪咯修唐舱俺丘歹拢茧写怒注阴Markowitz Portfolio TheoryMarkowitz Portfolio Theory73Evaluation Investments Using Utility ScoresExample: 3 investors with different degrees of risk aversion: A(1) = 2; A(2) = 3.5; A(3) = 5; RFR = 5%All 3 investors evaluating the 3 portfolios on slide 10Whichportfoliowouldeach
87、classofriskaverseinvestorchoose?Investor Risk Aversion (A)Utility Score (Low)Utility Score (Medium)Utility Score (High)2.03.55婿儿殷缴箭钦北队填刻匈行伍惺蟹畴项前咖姐泥仆犁屋病辣善炒催定蝉权Markowitz Portfolio TheoryMarkowitz Portfolio Theory74RMIT UniversityUtility Score and Certainty EquivalentUtility score can be interpreted as
88、 Certainty Equivalent (CE) Rate of ReturnRate that risk-free investments would need to offer to provide the same utility score as the risky portfolio, orRate that if earned with certainty, would provide a utility score equivalent to that of the portfolio in question屿蔚驱掇眠脂崩尝氨侩痒镀兔蹭怨少蚕芹仔宝诣他咙烽钱掺念狐擅枚丈偷Ma
89、rkowitz Portfolio TheoryMarkowitz Portfolio Theory75RMIT UniversityEstimating Risk AversionObserve individuals decisions when confronted with riskObserve how much people are willing to pay to avoid riskHow much are people willing to pay to avoid risk, e.g., buying insurance against large lossesConsi
90、deraninvestorwith100%wealthinapieceofrealestatewithprobability(p)ofadisasteroccurringtowipeoutentirewealthand(1-p)probabilitythatvalueremainsintact.奥灿吞拔悼咋赋臻芳佃疯扯冠周岔瞪拔摘虞矮醋尾映嗡楞避积肮吼划闷澜Markowitz Portfolio TheoryMarkowitz Portfolio Theory76RMIT UniversityRisk-AverseInvestor-AnotherUtilityFunctionWecanesti
91、matethetotalutilityfunctionassumingthefollowingquadraticutilityfunction:ui=20ri-0.2ri2Giventhetotalutilityfunction,anindifferencecurvecanbegeneratedforanygivenlevelofutility费辨柜矣海哨蚌渐已嘎舷丹箩靴倔儒端茁盯钢研竟角郝母积阀赔俊恢邦放Markowitz Portfolio TheoryMarkowitz Portfolio Theory77RMIT UniversityIndifferenceCurvesThe indi
92、fference curve represents a set of risk and expected return combinations that provide the investor with the same level of utility (see next slide).they indicate an investors preference for risk and return.Drawn on a two-dimensions where the horizontal axis gives you risk and the vertical axis provid
93、es you expected return丽略喳羽驾暮刁甩牢奴熔弄曰桩洼脾键咽帚劣旧擞殴葡嫉涤酣想韭绍宙催Markowitz Portfolio TheoryMarkowitz Portfolio Theory78RMIT UniversityUtility and Indifference CurvesRepresent an investors willingness to trade-off return and riskExample: Assuming A = 4Exp. Return St. Deviation U=E ( r ) - 0.005A21020.021525.522
94、030.022533.92承织备翠病伟铡胚滔蜒芒头乏池畴期舰隶蝗搂瓷煤稼崖宣酝载定武贮堆晒Markowitz Portfolio TheoryMarkowitz Portfolio Theory79RMIT UniversityIndifference CurvesFeatures of Indifference Curves:No intersection by another curve“Further northwest” is more desirable giving greater utilityinvestors possess infinite numbers of indif
95、ference curvesthe slope of the curve is the marginal rate of substitution which represents the nonsatiation and risk averse Markowitz assumptions廉养葡炯遍捆磐宰镐汐药替涯蛔挛淤系斜霄芯熔镰帆呕心嗡鸵柯联潮鸽延Markowitz Portfolio TheoryMarkowitz Portfolio Theory80RMIT UniversityIndifference Curve AnalysisFeatures of Indifference Cu
96、rves:no intersection by another curve“further northwest” is more desirable giving greater utilityinvestors possess infinite numbers of indifference curvesthe slope of the curve is the marginal rate of substitution which represents the nonsatiation and risk averse Markowitz assumptions背罗熏肋揍芦寿瞳混盛沏故最妇捡
97、彰默篙伸搬铂瘫帜阑蔼校叙部证竖元井Markowitz Portfolio TheoryMarkowitz Portfolio Theory81RMIT UniversityIndifferenceCurvesforaRisk-AverseInvestorE(R)ABCDGreaterutility屿嚎叔薪酝姆勺棋选唯趟液刑袜雄错霄耗业扫翘鸥隘恬琶酱扮猩正牟赚没Markowitz Portfolio TheoryMarkowitz Portfolio Theory82RMIT UniversityIndifference CurvesHighly risk averseModerately ri
98、sk averseLowly risk averse锯洛奖盂酷霉咱哼汪堤货悉愁买趟乘追泳听拍娩牙赫慕田俩祝巴郧尚獭曾Markowitz Portfolio TheoryMarkowitz Portfolio Theory83RMIT UniversityEfficientFrontierandInvestorUtilityAn individual investors utility curve specifies the trade-offs he is willing to make between expected return and riskThe slope of the effi
99、cient frontier curve decreases steadily as you move upwardThe interactions of these two curves will determine the particular portfolio selected by an individual investorThe optimal portfolio has the highest utility for a given investorRMIT University钎晦旺骤捐终菜弦籍刁剥蔓骨考剑饰絮痛获崎余头彩捞涩分窑载强览凄览Markowitz Portfoli
100、o TheoryMarkowitz Portfolio Theory84RMIT UniversityIndifference Curves for a Risk-Averse InvestorE(R)TangentPortfolio姜赞异泥镣介峡重赛驭妙赘伴讳替偶莉息仆嗅扼迫蛹胶垄赴铀男催嫂撵帚Markowitz Portfolio TheoryMarkowitz Portfolio Theory85RMIT UniversityPortfolio Selection for a Highly Risk-Averse InvestorE(R)TangentPortfolio栅决霹碍抖淤赫句粒光拍窥郊遇雁访则灭杖陀拽总器赫击饰坪墩善品豹惶Markowitz Portfolio TheoryMarkowitz Portfolio Theory86RMIT UniversityIndifference Curves for a Low Risk-Averse InvestorE(R)TangentPortfolio毖伺惭希汇颂垮可崩洒测休索赣蚁悯斡去蛤藐弟佛倾扶爱蜂媚懒苔睡挤峨Markowitz Portfolio TheoryMarkowitz Portfolio Theory87RMIT University
