1 marks ii Show that the variance of the portfolio var n y can be written as w

1 marks ii show that the variance of the portfolio

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(1 marks) ii) Show that the variance of the portfolio )) ( var( n y can be written as w C w xx T n y )) ( var( , where JJ J J J J T xx n n E ... : : : : ... ... ] ) ( ) ( [ 2 1 2 22 12 1 21 11 x x C is the covariance of ) ( n x . x n n μ x x ) ( ) ( is the centered return. (0.5 marks) iii) The mean-variance portfolio optimization problem can be written as: 1 t subject to min 2 1 J T x T xx T 1 w μ w w C w w ,
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where t is a specified/desired expected return of the portfolio. J 1 is a 1 J vector of ones. By differentiating the Lagrangian ) ( ) 1 ( ) , , ( 2 1 2 1 2 1 t L x T J T xx T μ w 1 w w C w w w.r.t. w , the Lagrange multipliers 1 , 2 and set the gradient to zero. Show that the optimum weight vector * w that minimizes the variance and reaching the target return is ) ( * 2 1 1 x J xx μ 1 C w . . (1 marks) iv) (*) By substituting ) ( * 2 1 1 x J xx μ 1 C w into the constraints t x T μ w and 1 J T 1 w , show that the optimal Lagrange multipliers are 2 1 * b ac bt c and 2 2 * b ac b at , where J xx T J a 1 C 1 1 , J xx T J b μ C 1 1 and J xx T J c μ C μ 1 . (0.5 marks) Part 2. MATLAB Implementation of Markowitz mean-variance portfolio b) Run the MATLAB® code “Ass2.m”. It contains three parts: 1. 200 time-instants of the return of 10 assets are randomly generated. They are assumed to be independent of each other and are normally distributed. 2. The autoregressive moving average and general autoregressive conditional heteroscedasticity (ARMA(P,Q)-GARCH(R,S)) is a time series model in which the mean and variance of the assets are modelled by the ARMA and the GARCH processes respectively. The mean and variance of the assets in future
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