# What are Sharpe, Single, and Multi-Index Model?

Sharpe Index Model

One Simplification of the CAPM formula was done by Sharpe (1963), who developed the Single-Index Model. The single-index model imposes restrictions on how security returns can covary. In particular, it is assumed that all covariance arises through an “index.” As we will see, this leads to a dramatic reduction in complexity. Sharpe’s model has since been extended to multi-index models and leads to a more general theory called the Arbitrage Pricing Theory, developed by Ross (1976). Besides simplifying the covariance matrix, this approach is easily extended to take account of non-financial factors. In the multi-index model, for example, one of the indexes could easily be the rate of inflation.

Single-Index Model

The major assumption of Sharpe’s single-index model is that all the covariation of security returns can be explained by a single factor. This factor is called the index, hence the name “single-index model.”

Beta Coefficient is the slope of the regression line and is a measure of the changes in the value of the security relative to changes in values of the index.

A beta of +1.0 means that a 10% change in index value would result in a 10% change in the same direction in the security value. A beta of 0.5 means that a 10% change in index value would result in a 5% change in the security value. A beta of – 1.0 means that the returns on the security are inversely related.

Multi-Index Model

The single index model is in fact an oversimplification. It assumes that stocks move together only because of a common co-movement with the market. Many researchers have found that there are influences other than the market that cause stocks to move together. Multi-index models attempt to identify and incorporate these nonmarket or extra-market factors that cause securities to move together also into the model. These extra-market factors are a set of economic factors that account for common movement in stock prices beyond that accounted for by the market index itself. Fundamental economic variables such as inflation, real economic growth, interest rates, exchange rates, etc. would have a significant impact in determining security returns and hence, their co-movement.

A multi-index model augments the single index model by incorporating these extra market factors as additional independent variables.

The model says that the return of an individual security is a function of four factors – the general market factor Rm and three extra-market factors R1, R2, R3. The beta coefficients attached to the four factors have the same meaning as in the single-index model. They measure the sensitivity of the stock return to these factors. The alpha parameter ai and the residual term ei also have the same meaning as in the single-index model.

Calculation of return and risk of individual securities as well as portfolio return and variance follows the same pattern as in the single-index model. These values can then be used as inputs for portfolio analysis and selection.

A multi-index model is an alternative to the single-index model. However, it is more complex and requires more data estimates for its application. Both the single-index model and the multi-index model have helped to make portfolio analysis more practical. 