a vanishing total volatility, as given in Eqs.(\[4840\])-(\[485\]).*]{}

4. Concluding Remarks {#concluding-remarks .unnumbered}
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The main objective of this work, namely recasting the efficient portfolio problem in terms of principal portfolios whereby the selection is made from an uncorrelated set of portfolios instead of the original asset set has been implemented in detail. As we have emphasized throughout, principal portfolios are the natural instruments for analyzing the efficient frontier when short sales are allowed. More generally, they are the natural instruments for any stock selection process based on the mean-variance formulation. In effect, the analysis is transformed from the original, correlated environment of individual assets to one of uncorrelated portfolios, thus simplifying both the conceptual framework as well as the practical procedures involved. This is of course another example of the golden rule in applied analysis which teaches us that when the basic object in the problem involves a quadratic form, it is often advantageous to treat it in the principal axes basis.

In order to illustrate the concepts and methods of this paper, we have analyzed the single-index model as well as the constant residual variance version of it in considerable detail. Indeed our perturbative treatment of the general single-index model for large asset sets has revealed interesting new spectral features for that model. In particular, the bifurcation into market-aligned and market-orthogonal portfolios found in §2 is an important observation on the volatility structure of the model. This is particularly so in view of the fact that for sufficiently large asset sets the market-aligned portfolio is all but excluded from the efficient frontier thereby eliminating the component of volatility commonly referred to as market risk. The constant residual variance version of the model, while admittedly oversimplified, brings out the above-mentioned bifurcation as well as the elimination of the market ri