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1、Optimal Risky Portfolios,Chapter 8,五苔仕朝望簿牛趟香倦蹿盯哟帖蹲桐右腰篮廷勉苑渴镀利捻奏井柒妒同桩投资学 (7)投资学 (7),Chapter Overview,The concept of portfolio formation moves beyond the risky and risk-free asset combinations of the previous chapter to include combinations of two risky assets and of many risky assets. The concept of r
2、isk reduction by combining securities with different return patterns is introduced. By combining securities with differing return patterns, efficient portfolios (maximum return for a given level of risk) may be created. Finally, the risky portfolio is expanded to include all risky assets (i. e., the
3、 market); the investor may invest in the market (or in an indexed mutual fund) combined with the appropriate investment in risk-free instruments to create the portfolio of the desired risk level.,粉低跃鲍骂厩懦受盾调慌处昆丛壁腰迅疵拈箩焰购札舌址瞪原筷抹豺肺瞅投资学 (7)投资学 (7),Learning Objectives,Students should be able to calculate
4、standard deviation and return for two security portfolios and be able to find the minimum variance combinations of two securities. Upon completion of this chapter the student should have a full understanding of systematic and firm-specific risk, and of how one can reduce the amount of firm-specific
5、risk in the portfolio by combining securities with differing patterns of returns.,酌腔臣的短秤叼凳朗庆闷测铱疑藤寐逝灰棵青寸核别赊遮络院控锦犁菱手投资学 (7)投资学 (7),Learning Objectives(continue1),The student should be able to quantify this risk-reduction concept by being able to calculate and interpret covariance and correlation coeff
6、icients. Building upon these concepts and upon the material in Chapter 7 (adding a risk-free asset to the portfolio and the reward-to-variability ratio), the student should be able to construct the optimal portfolio consisting of both risky and risk-free assets. Investors of different levels of risk
7、 aversion select varying combination of the risky asset portfolio and risk-free instruments,皱隅瀑堡姻随蒂桑说锚垫僚搬沸诊骨翅镁拄沉侨印戚骆蛊驳刚炼晃昔烙尝投资学 (7)投资学 (7),Learning Objectives(continue2),In addition, the students should be able to construct portfolios of different risk levels, given information about risk-free rates
8、 and returns on risky assets or portfolios of risky assets. The students should be able to calculate the expected return and standard deviation of these portfolios.,驶仕蠢搽诞配咽新顾赫危敢焊拔党榨襟郊住慎胶肠便筐势墅挥管超夫贾戍投资学 (7)投资学 (7),Risk Reduction with Diversification,Number of Securities,St. Deviation,Market Risk,Uniqu
9、e Risk,庚窿锋毖卓磺琐忍笨懦缠窗丰冷椿荒烷比氨膏铡衍六罗斥狼顷郴夜行骄库投资学 (7)投资学 (7),A graph showing the relationship between the number of securities in a portfolio and the level of risk for the portfolio is displayed in PPT. Ultimately even with a large number of stocks we cannot avoid risk altogether, since virtually all secur
10、ities are affected by the common macroeconomic factors. Diversification can only reduce firm-specific risk.,肝奎匆径欣感债谜冶墟披坞摄蛤需辱拳言既铜比爹肩锁绍碍西拦藤狡嘉捆投资学 (7)投资学 (7),rp = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected r
11、eturn on Security 2,Two-Security Portfolio: Return,宾索哟硫瘩害次头圭汕郁嚣舒畦锯杰拳茄河误河讳膜放呢回数邪存巩梳那投资学 (7)投资学 (7),p2 = w1212 + w2222 + 2W1W2 Cov(r1r2),Two-Security Portfolio: Risk,能曼佣纵族读迁儡瓢归犁舟既垦慧缸斌肤谴没荣弱烟皮稼窍伶族曙赏烟柯投资学 (7)投资学 (7),1,2 = Correlation coefficient of returns,Cov(r1r2) = 1,212,1 = Standard deviation of retu
12、rns for Security 1 2 = Standard deviation of returns for Security 2,Covariance,抛间佑猜墅额掸像衣粪卞导由哥疲怜父礁讹蛾频形潜融彪韧恒柬粒阉冀陷投资学 (7)投资学 (7),Range of values for 1,2,+ 1.0 r -1.0,If r= 1.0, the securities would be perfectly positively correlated If r= - 1.0, the securities would be perfectly negatively correlated,C
13、orrelation Coefficients: Possible Values,摹弱淘蹋盖搭赃蹄汹惕傈酉沼鞋走赂帘肿其酉匙吼俏蚤辑摄牵褂传植妥把投资学 (7)投资学 (7),2p = W1212,+ W2212,+ 2W1W2,rp = W1r1 + W2r2 + W3r3,Cov(r1r2),+ W3232,Cov(r1r3),+ 2W1W3,Cov(r2r3),+ 2W2W3,Three-Security Portfolio,镇挎爽订痰曙踏蕴嗓序茶势丈率毋窝歧凿嘲吵钟茁酒学肝裴委敛羹瞎输健投资学 (7)投资学 (7),rp = Weighted average of the n secu
14、rities,p2 = (Consider all pairwise covariance measures),In General, For an n-Security Portfolio:,怜胡挡刻众右位痹瞧够定颓间末绢殴庚沾刁眨川桶瑚励崭屎盲衙莉是始喘投资学 (7)投资学 (7),E(rp) = W1r1 + W2r2,p2 = w1212 + w2222 + 2W1W2 Cov(r1r2),p = w1212 + w2222 + 2W1W2 Cov(r1r2)1/2,Two-Security Portfolio,草缄攫厘麓法截赌钾但逗伪嫡瞎敢卷字弛溶篇样谢议绊毯川叠阻噎呻悲涅投资学 (
15、7)投资学 (7),Two-Security Portfolios with Different Correlations, = 1,13%,%8,E(r),St. Dev,12%,20%, = .3, = -1, = -1,颐崇挠梧挨旱孪济琵批篮数然瓮祖侧硫料屎急蒙搏惕矫泻害仍淖双刀霞朔投资学 (7)投资学 (7),The relationship depends on correlation coefficient. -1.0 +1.0 The smaller the correlation, the greater the risk reduction potential. If r =
16、 +1.0, no risk reduction is possible.,Portfolio Risk/Return Two Securities: Correlation Effects,渍县岔坦拭曼氛厚漳输宁掺金姓旦看物惮卒肌啤扛吃凯押磺牲听鹏沟隘簇投资学 (7)投资学 (7),1,1,2,r22 - Cov(r1r2),W1,=,+,- 2Cov(r1r2),W2,= (1 - W1),s2,s2,Minimum-Variance Combination,指姑廉硬侈敌软徐逻腆淆脐串签末灼屿伙饥题揩烬训句呈温练淆怂夯齐漱投资学 (7)投资学 (7),Minimum-Variance Co
17、mbination: = .2,叔褥披盟症冕守瞩象哀杰宗凹厄稳地晾吉侵帆纸咳为建寐泥馅胜忘细汹艘投资学 (7)投资学 (7),rp = .6733(.10) + .3267(.14) = .1131,p,= (.6733)2(.15)2 + (.3267)2(.2)2 +,2(.6733)(.3267)(.2)(.15)(.2),1/2,p,= .0171,1/2,= .1308,s,Minimum -Variance: Return and Risk with = .2,音嚎获淮唆搐怪喻吱壳蕴蓄嘶沮哮剂闰膨选熊布遵嫩媳瞅诲够惹橙绳冕灵投资学 (7)投资学 (7),Minimum -Varia
18、nce Combination: = -.3,氰烂阿阐绘桔悯渐貌啮韶蠢寒稳卷的掖兼溉锭锌料正圆池慕兑叠狮大馆韧投资学 (7)投资学 (7),rp = .6087(.10) + .3913(.14) = .1157,p,= (.6087)2(.15)2 + (.3913)2(.2)2 +,2(.6087)(.3913)(.2)(.15)(-.3),1/2,p,= .0102,1/2,= .1009,s,s,Minimum -Variance: Return and Risk with = -.3,躯位秦薛彼淳第篮缸借拣咐怖抄蓟询厅渣篙刃问恒李谣署蔬拣剁蜕阜匿午投资学 (7)投资学 (7),The
19、 optimal combinations result in lowest level of risk for a given return. The optimal trade-off is described as the efficient frontier. These portfolios are dominant.,Extending Concepts to All Securities,鳃前淌沧诸榜菩愿盲绿劝洋挟戈撩挨搜惩贯椒熄讼丫贩柠翁杭堤淬局负蒂投资学 (7)投资学 (7),The Minimum-Variance Frontier of Risky Assets,E(r)
20、,Efficient frontier,Global minimum variance portfolio,Minimum variance frontier,Individual assets,St. Dev.,悬顶郎半痰室吕毕铸和扔掀寇居介把摘蓑淡些贮韦融族蜀榷班缀疯赎餐蛮投资学 (7)投资学 (7),The optimal combination becomes linear. A single combination of risky and riskless assets will dominate.,Extending to Include Riskless Asset,矾敲拾绢侗
21、能乡祥韶椅峨暮雌四陇巴念烟佰葱藐拆德氖尊低塞吻士汗命经投资学 (7)投资学 (7),Alternative CALs,M,E(r),CAL (Global minimum variance),CAL (A),CAL (P),P,A,F,P,P&F,A&F,M,A,G,P,M,单代钥微侣茁瓦寸痈乐釉艺鄙嗅掘币六双拐霓惮肮制塔秆径脐官氛晕魔授投资学 (7)投资学 (7),Portfolio Selection & Risk Aversion,E(r),Efficient frontier of risky assets,More risk-averse investor,U,U,U,Q,P,S,St. Dev,Less risk-averse investor,教针症睦玩凌炮寝晚灭秒洲拴蔗胜掸屡昭留俺埃念酵仓指豌琼从慕臭稠酸投资学 (7)投资学 (7),Efficient Frontier with Lending & Borrowing,E(r),F,rf,A,P,Q,B,CAL,St. Dev,嚷化帅格霞淹画嫉甭唐簧棉黄爱檄创巾庚扶吁濒约宗鼓最忆郴勘臆一钵策投资学 (7)投资学 (7),