Refresher on Random Variables Let X be a financial asset described by the discrete random variable: X1,P1 X=X2, P2 , X <.. .< XN, N . . XN , PN i=1 ,?pi=1 Interpretation: the financial asset X gives payoff x; with probability pi- Note: we can also write X as X={ X1, P1 ; ... ; XN , PN } Recall the following operators: Expectation of X: N i=1 E[x]= > pi Xi Variance of X: Var [X]=E (X-E[X]2] Let X and Y denote discrete random variables. Let a and b be real numbers. Show the following results and provide an interpretation in terms of financial assets: 1. E[aX+b] = aE[X]+b 2. E[X+Y] = E[X] + E[Y] 3. if a?Xsb then a?E[X]?b 4. Var[aX+b] = a2 Var[X] Now define the covariance between (jointly distributed) random variables X and Y: Cov[X, Y]=E(X-E[X] (Y-E[Y ]] Covariance measures how X and Y move together relative to their means. * Cov[X,Y]>0 means that X and Y tend to move in the same direction (both higher or both lower) relative to their means. Example, stocks of BMW and Mercedes. * Cov[X,Y]<0 means that X and Y tend to move in opposite directions (when one is lower the other is higher) relative to their means. Example, exchange rates from GBP to USD and its reverse.
* Cov[X,Y]=0 means that X and Y are linearly independent. Show: 5. Cov[X,X]=Var[X] 6. Cov[X,a]=0 7. Cov[aX,bY]=abCov[X,Y] (in particular, Cov[X,-X] =- Var[X]) Show and provide an interpretation in terms of financial assets: 8. Var[X+Y]= Var[X]+Var[Y]+2Cov[XY] Numerical example: Consider the following two assets: X={10, 0.3; 15, 0.4; 20, 0.3} Y={5, 0.4; 15, 0.2; 25, 0.4} Compute E[X], Var[X], E[Y], Var[Y]. Suppose the joint distribution of X and Y is given by: X,Y={(10,5),0.3; (15,5), 0.1; (15,15),0.2; (15,25), 0.1; (20,25),0.3} Compute Cov[X.Y].