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Consider two random variables X1 and X2 having finite variances, representingdaily return values of stock prices

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Consider two random variables X1 and X2 having finite variances, representingdaily return values of stock prices. Let µj = E(Xj) and j = sd(Xj) denote themean and standard deviation of Xj for j = 1, 2 and let = Cor(X1,X2) denote(Pearson) correlation.Consider the weighted random variable X(w) = wX1 + (1 - w)X2 for w ? [0, 1]representing a portfolio consisting of two stocks. Let µ(w) = E[X(w)] and (w) =sd(X(w)) denote mean and standard deviation of portfolio return X(w). Withoutany loss of generality assume that 2 = 1 > 0.(a) Show that for any w ? [0, 1],µ(w) = c0 + c1w where c0 = µ2, and c1 = µ1 - µ2;and 2(w) = aw2 - 2bw + cwhere a = 21 + 22 - 212, b = 22 - 12, and c = 22.Justify that a = 0 for any ? (-1, 1).(b) Suppose our goal is to find w ? [0, 1] that maximizes average return valueµ(w) of the portfolio subject to the constraint 2(w) = 21.i. If µ1 = µ2, show that the w = 1 solves the above optimization problem.(Recall that we have already assumed 2 = 1)ii. If µ1 < µ2, show that the optimal w is given by the smallest w ? [0, 1]satisfying 2(w) = 21.