Single Index Model

Overview


The single index model is a model for calculating asset covariances that relies on each asset being driven by 1 systemic risk factor, usually taken to be the market return, and a set of independent idiosyncratic factors. It is closely related to the Capital Asset Pricing Model. An extension of the single index model that uses multiple indices are the multi index models

Single Index Model


The single index model formulates an assets returns as the sum of exposure to the market risk factor and an idiosyncratic risk factor.
{% r_i = \alpha_i + \beta_i \times r_m + e_i %}
{% r_i %} is the rate of return of a given asset and {% r_m %} is the return on the market. Typically, these returns are measured as excess returns over the risk free rate of return. This is due to considerations that come from the capital asset pricing model
{% \mathbb{E} [e_i] = 0 %}

Assumptions


The key assumption that makes the single index model easy to work with is the assumptions that the idiosyncratic asset returns are uncorrelated with the market, and with each other.
{% cov(e_i, r_m) = 0 %}
{% cov(e_i, e_j) = 0 %}
Then we have the following:
{% \mathbb{E} [r_i] = \alpha_i + \beta_i \times \mathbb{E} [r_m] %}
{% \sigma_i^2 =\beta_i^2 \sigma_m^2 + \sigma _{ei}^2 %}

Porfolio Beta


Portfolio beta is the weighted sum of the individual asset betas.
{% \beta_{port} =\sum w_i \beta_i %}
Total portfolio risk is then
{% \sigma_{port}^2 =\beta_{port}^2 \sigma_m^2 + \sum w_i^2\sigma_{ei}^2 %}

Estimating Beta


The single index equation
{% r_i = \alpha_i + \beta_i \times r_m + e_i %}
is just a simple linear equation. It can be estimated by a standard OLS regression.

Managing to a Benchmark


Using a single index factor model is an effective tool when managing against a benchmark. When trying to perform against a benchmark, the typical strategy is to measure your portfolio's beta (sensitivity) to that benchmark, and then to target a beta of 1.

Regression of Beta to 1


Using historical data to obtain an estimate of beta, as above, does have some limitations. In particular, it is known that there is a certain variability in measured betas that creates a jumping around effect. In general, betas that are measured as greater than 1 tend to jump back towards 1, and betas less than 1 tend to jump back up, in effect regressing toward the mean. Many authors have suggested corrections to measured beta to compensate for this effect.

Blume Technique

Blume suggests the following correction to measured beta:
{% \beta = 0.343 + 0.677 \times \beta_{measured} %}

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