 hedged and leveraged, and nearly free of systematic market fluctuations. Equations (\[439\])-(\[442\]) provide approximate expressions valid to first order in $1/N$ for the properties of these portfolios.*]{}

3. Single-Index Model with Constant Residual Variance {#single-index-model-with-constant-residual-variance .unnumbered}
=====================================================

To provide an explicit illustration of the principal portfolio structure within the single-index model described in the preceding section, we now turn to an exactly solvable, albeit oversimplified, version of that model. This model is defined by the assumption that the residual variance of the $i$th asset in the original set, ${\bar{{\alpha}_{i}^{2}}}$, is equal to ${\bar{{\alpha}^{2}}}$ for all assets. Observe that this assumption does not affect the expected rate of return for the $i$th asset, which is given by ${r}_{i}={\bar{{\alpha}_{i}}}+{\beta}_{i} {\bar{\rho}}_{mkt}$ as before. This simplification will allow us to derive an exact solution for the model and illustrate the concepts and methods of the previous section in more explicit terms. The price for this simplification is of course the unrealistic assumption of constant residual variance which defines the model.

The covariance matrix with the above simplification appears as $${\sf \sigma}_{ij}^{crv}={\bar{{\alpha}^{2}}} {\delta}_{ij}+
{\beta}_{i}{\beta}_{j} {\bar{{\rho}^{2}}}_{mkt},  \label{443}$$ whose rescaled version is $${\tilde{\sf \sigma}}_{ij}^{crv}={\gamma}^{2} {\delta}_{ij}+
{\hat{\beta}}_{i}{\hat{\beta}}_{j}  \label{444}$$ These equations are of course specialized versions of Eqs. (\[432\]) and (\[433\]).

Referring to the results of the previous section, one can readily see that the spectrum of ${\tilde{\sf \sigma}}^{crv}$ consists of a major eigenvalue (exactly) equal to $1+{\gamma}^{2}$ \[cf. Eq. (\[437\])\], and $N-1$ minor eigenvalues, all equal to ${\gamma}^{2}$. Recall that these eigenvalues respectively correspond to the market-aligned and market-orthogo