Question

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1. explain the components of the single index model, linking diversifiable and systematic risk to your answer.
2. Is the full covariance efficient frontier or the index model efficient frontier better?

Answer 1

A Single Index Model is a model which believes that the returns of a secuirty can be explained by an economic factor through a linear relationship. Most common economic factor is a market index like S&P 500 Index or Interest rates or cost of labor.

The equation that Single Index Model uses is below:

Return of security - Risk free rate = Abnormal Returns (Alpha) + Systematic risk (Beta) * (Market return - Risk free rate) + Company specific risk or Unsystematic risk (Epsilon or error term)

It believes that each stock varies differently to economic factors, hence, their covariances can be found by multiplying their Betas with Market's variance.

Covariance (R_k, R_L) = Beta_k* Beta_L * Market Variance

In terms of Diversification in an equally weighted portfolio, when number of stocks gets large enough, the company-specific or Unsystematic risk component of Single index model gets diversified away.

The Full Covariance Efficient Frontier seems better than Index Model efficient frontier.

However, the limitations of Full Covariance model is that it involves estimation of risks of 1000s of terms which is prone to errors. These errors would output a worse portolio than what Single index model can provide. The Single index model reduces the estimation and makes portfolion construction more practical by reducing macro and security analysis.