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Covariance and correlation - Main Concepts

Covariance and correlationLet random variablesX,Ywith means X, Yrespectively. The Covariance , denoted withcov(X,Y), is ameasure of the association :cov(X,Y) =E(X X)(Y Y)This can be simplified as follows:cov(X,Y) =E(X X)(Y Y) =E(XY) YE(X) XE(Y) + X YTherefore,cov(X,Y) =E(XY) (EX)(EY)Note: IfX,Yare independent thenE(XY) = (EX)E(Y) Thereforecov(X,Y) = ,X,Y,Zrandom variables, anda,b,c,dconstants: Findcov(a+X,Y)cov(a+X,Y) =E(a+X a+X)(Y Y) =E(a+X X a)(Y Y)Therefore,cov(a+X,Y) =cov(X,Y). Findcov(aX,bY)cov(aX,bY) =E(aX aX)(bY bY) =E(aX a X)(bY b Y)Therefore,cov(aX,bY) =abE(X X)(Y Y) =ab cov(X,Y) Findcov(X,Y+Z)cov(X,Y+Z) =E(X X)(Y Y+Z) =E(X X)(Y+Z Y Z)Orcov(X,Y+Z) =E(X X)(Y Y+Z Z) =E(X X)(Y Y) +E(X X)(Z Z)Therefore,cov(X,Y+Z) =cov(X,Y) +cov(X,Z) Using the results above we can findcov(aW+bX,cY+dZ).cov(aW+bX,cY+dZ) =ab cov(W,Y) +ad cov(W,Z) +bc cov(X,Y) +bd cov(X,Z) correlation :However, the Covariance depends on the scale of measurement and so it is not easy to say whether aparticular Covariance is small or large.

However, the covariance depends on the scale of measurement and so it is not easy to say whether a particular covariance is small or large. The problem is solved by standardize the value of covariance (divide it by ˙ X˙ Y), to get the so called coe cient of correlation ˆ XY. ˆ= cov(X;Y) ˙ X˙ Y; Always, 1 ˆ 1 cov(X;Y) = ˆ˙ X˙ Y

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Transcription of Covariance and correlation - Main Concepts

1 Covariance and correlationLet random variablesX,Ywith means X, Yrespectively. The Covariance , denoted withcov(X,Y), is ameasure of the association :cov(X,Y) =E(X X)(Y Y)This can be simplified as follows:cov(X,Y) =E(X X)(Y Y) =E(XY) YE(X) XE(Y) + X YTherefore,cov(X,Y) =E(XY) (EX)(EY)Note: IfX,Yare independent thenE(XY) = (EX)E(Y) Thereforecov(X,Y) = ,X,Y,Zrandom variables, anda,b,c,dconstants: Findcov(a+X,Y)cov(a+X,Y) =E(a+X a+X)(Y Y) =E(a+X X a)(Y Y)Therefore,cov(a+X,Y) =cov(X,Y). Findcov(aX,bY)cov(aX,bY) =E(aX aX)(bY bY) =E(aX a X)(bY b Y)Therefore,cov(aX,bY) =abE(X X)(Y Y) =ab cov(X,Y) Findcov(X,Y+Z)cov(X,Y+Z) =E(X X)(Y Y+Z) =E(X X)(Y+Z Y Z)Orcov(X,Y+Z) =E(X X)(Y Y+Z Z) =E(X X)(Y Y) +E(X X)(Z Z)Therefore,cov(X,Y+Z) =cov(X,Y) +cov(X,Z) Using the results above we can findcov(aW+bX,cY+dZ).cov(aW+bX,cY+dZ) =ab cov(W,Y) +ad cov(W,Z) +bc cov(X,Y) +bd cov(X,Z) correlation :However, the Covariance depends on the scale of measurement and so it is not easy to say whether aparticular Covariance is small or large.

2 The problem is solved by standardize the value of Covariance (divide it by X Y), to get the so called coefficient of correlation XY. =cov(X,Y) X Y,Always, 1 1cov(X,Y) = X YIfX,Yare independent then XY= 01 Important:var(X+Y) =var(X) +var(Y) + 2cov(X,Y)Proof: Findvar(aX+bY) In general: LetX1,X2, ,Xnbe random variables, anda1,a2, ,anbe constants. Find the varianceof the linear combinationY=a1X1+a2X2+ + risk and returnAn investor has a certain amount of dollars to invest into two stocks (IBMandT EXACO). A portion of the available fundswill be invested into IBM (denote this portion of the funds witha) and the remaining funds into TEXACO (denote it withb)- soa+b= 1. The resulting portfolio will beaX+bY, whereXis the monthly return ofIBMandYis the monthly returnofT EXACO. The goal here is to find the most efficient portfolios given a certain amount of risk. Using market data fromJanuary 1980 until February 2001 we compute thatE(X) = ,E(Y) = ,V ar(X) = ,V ar(Y) = , andCov(X, Y) = first want to minimize the variance of the portfolio.

3 This will be:Minimize Var(aX+bY)subject toa+b= 1 OrMinimizea2V ar(X) +b2V ar(Y) + 2abCov(X, Y)subject toa+b= 1 Therefore our goal is to findaandb, the percentage of the available funds that will be invested in each stock. Substitutingb= 1 ainto the equation of the variance we geta2V ar(X) + (1 a)2V ar(Y) + 2a(1 a)Cov(X, Y)To minimize the above exression we take the derivative with respect toa, set it equal to zero and solve fora. The result is:a=V ar(Y) Cov(X, Y)V ar(X) +V ar(Y) 2 Cov(X, Y)and thereforeb=V ar(X) Cov(X, Y)V ar(X) +V ar(Y) 2 Cov(X, Y)The values ofaandbare:a= + 2( ) a= 1 a= 1 b= Therefore if the investor invests 42% of the available funds intoIBMand the remaining58% intoT EXACOthe variance of the portfolio will be minimum and equal to:V ar( + ) = ( ) + ( ) + 2( )( )( ) = corresponding expected return of this porfolio will be:E( + ) = ( ) + ( ) = can try many other combinations ofaandb(but alwaysa+b= 1) and compute the risk and return for each resultingportfolio.

4 This is shown in the table below and the graph of return against risk on the other possibilities curveRisk (portfolio standard deviation)Expected return4 Efficient frontier with three stocks> summary(returns)ribm rxom rboeingMin. Min. Min. Qu. 1st Qu. 1st Qu. Median : Median : : Mean Mean : Qu.: 3rd Qu.: 3rd Qu.: : Max. : Max. : > cov(returns)ribm rxom rboeingribm possibilities curve with 3 (standard deviation)Expected return5


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