Index Models and Portfolio Computation: Overview

Mean-variance optimisation tells you what to do: maximise expected return for a given variance. It does not tell you how to do it without breaking under the weight of its own input requirements. A portfolio of

N = 200

stocks needs

N(N-1)/2 = 19{,}900

pairwise correlation estimates. Most of those estimates are noisy, organisationally unnatural to produce, and numerically fragile when assembled into a large covariance matrix. This topic addresses that problem — not by weakening the objective, but by imposing structure on how returns co-move.

The key insight is due to William Sharpe (1963): if the only reason stocks move together is a shared sensitivity to a single market index, then all

N(N-1)/2

correlations collapse into

N

betas. A portfolio analyst with

N

stock-specific parameters plus one market model replaces a statistical task that scales quadratically with one that scales linearly. Chapter 7 of MPTA derives this reduction precisely and shows how beta can be estimated from historical time-series data. Chapter 8 extends the idea: when one common factor is not enough — when industries, macro variables, or style tilts produce residual co-movement — multi-index models restore fidelity while keeping the input count manageable. Chapter 9 closes the loop by showing that the single-index structure produces a ranking device (excess return to beta) that lets a portfolio manager select and weight an optimal portfolio by hand, without solving a quadratic programme.

By the end of this topic you should be able to:

state the single-index model, identify its key assumption (residuals are uncorrelated across stocks), and use it to reduce a full covariance matrix to $N$ betas plus $N$ idiosyncratic variances;
estimate $\alpha_i$ , $\beta_i$ , and $\sigma^2_{e_i}$ by time-series regression and verify that the three-part variance decomposition $\sigma^2_i = \beta_i^2 \sigma^2_m + \sigma^2_{e_i}$ holds;
explain where the single-index model fails and how multi-index models — grouped/industry indexes, fundamental multi-index models, average-correlation models — patch the gaps;
apply the excess-return-to-beta ranking rule to select the optimal portfolio, compute the cutoff rate $C^*$ , and find the portfolio weights under both the short-sales-allowed and long-only constraints;
interpret the constant-correlation model as a special case of a multi-index model and apply its simple selection algorithm.

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