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This should not be that hard, but I can't find any good references.Suppose that I have a lot of data for a fund or an index. In the case of the fund I know the universe of the assets it contains, but not the actual weights for the assets in their portfolio. For an index I know both asset universe and weights.I now want to find the three (or four, or five) assets and corresponding weights to optimally replicate the index, in the sense that my replicating portfolio minimize the variance between itself and the index or fund.How do I do it? Can PCA be of any help?

It is not so clear why you have to do it...however, if I understood what you want I think u can find the efficient forntier of your index and then find the weights of your entities in your fund so that they will match the efficient forntier of the index. At that point you will have a fund that have the same risk/return profile of the index.A PCA method can do it as well...the problem is how to calibrate the volatility structure of your index ...how many autovector you can choose?

It is not so clear why you have to do it...however, if I understood what you want I think u can find the efficient forntier of your index and then find the weights of your entities in your fund so that they will match the efficient forntier of the index. At that point you will have a fund that have the same risk/return profile of the index.A PCA method can do it as well...the problem is how to calibrate the volatility structure of your index ...how many autovector you can choose?

I want to create the minimum variance* tracking portfolio given that I can use N assets.I don't think the efficient frontier ansatz is the way to go. *minimum variance in the sense that if S is a time-series (an M x L matris, where the M rows are daily performances and the L columns are the assets) we have1) I = Sw is the daily index (or fund) performance. The index weights are defined in the vector v2) P = Sv is the daily performance of another portfolio with weight vector v3) tracking error = (w-v)'S'S(w-v)I want to minimize the tracking error, given that I have N non-zero entries in my weight vector v.