3. In index tracking, one wishes to create a portfolio weighting a number n of assets...

  

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3. In index tracking, one wishes to create a portfolio weighting a number n of assets so that at any given time point, the portfolio return matches closely that of a given index. In this exercise, we examine how to create a constant portfolio weight vector wER", so that the weighted average (with the weights given by w) of the returns of then assets matches well a given index over a time period t= 1,....T. Denote by XR the matrix with elements the return at time t of asset i, denoted r(i). 1st≤T, 1≤ i ≤n; and by y ER the "index" vector with elements . 1S1ST. (a) Explain how to express the tracking error vector in terms of X, w, and y. (b) How would you minimize the tracking error using regularized least-squares? You can ignore the fact that in practice shorting (negative elements in w) may not be allowed. (c) Often it is desirable to find a sparse portfolio weight vector w, in order to reduce the number of assets to be traded, and the associated transaction costs. Explain how you would modify the regularized least-squares problem found previously. (d) The models you have introduced above may contain one or two regularization parameters. Explain how you would choose those parameters via cross-validation in this time-series context. 3. In index tracking, one wishes to create a portfolio weighting a number n of assets so that at any given time point, the portfolio return matches closely that of a given index. In this exercise, we examine how to create a constant portfolio weight vector wER", so that the weighted average (with the weights given by w) of the returns of then assets matches well a given index over a time period t= 1,....T. Denote by XR the matrix with elements the return at time t of asset i, denoted r(i). 1st≤T, 1≤ i ≤n; and by y ER the "index" vector with elements . 1S1ST. (a) Explain how to express the tracking error vector in terms of X, w, and y. (b) How would you minimize the tracking error using regularized least-squares? You can ignore the fact that in practice shorting (negative elements in w) may not be allowed. (c) Often it is desirable to find a sparse portfolio weight vector w, in order to reduce the number of assets to be traded, and the associated transaction costs. Explain how you would modify the regularized least-squares problem found previously. (d) The models you have introduced above may contain one or two regularization parameters. Explain how you would choose those parameters via cross-validation in this time-series context.

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a The tracking error vector can be expressed as TE X w y b The tracking error can be minimiz... View the full answer