Single index model

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The single index model ( SIM for short , also single index model ) is a theory of optimal portfolio selection . The aim of the single index model is to simplify to just one influencing factor. The aim is to create a structure that explains the returns in purely statistical terms . It's a kind of regression relationship .

Fewer parameters have to be estimated in the SIM than in the full Markowitz model .

Data requirements

In the single index model, the data requirement is significantly reduced compared to the full Markowitz model. The time series analysis only requires n estimates for , and and one estimate for and . In the Markowitz model, expected values are required for returns and correlations for risk variables. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \sigma _{\epsilon _{i}}}$ ${\displaystyle \sigma _{I}}$ ${\displaystyle \mu _{I}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (n^{2}-n)/2}$

Structural assumptions

• The return on the index drives the return of all stocks.
• The share-specific ( idiosyncratic ) risk has no influence on the individual risks of the other shares: ${\displaystyle \operatorname {Cov} (\epsilon _{i},\epsilon _{j})=0}$
• The company-specific risks have no influence on the macro risk and, conversely, the macro risk has no influence on the company-specific risks. ${\displaystyle \operatorname {Cov} (r_{I},\epsilon _{j})=0}$
• The individual risks are not systematically distorted

Consequences

• The expected return on a stock is a constant plus the index return ${\displaystyle i}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle \beta _{i}}$

• The variance of the i-th stock is made up of the index variance and the individual standard error ${\displaystyle \beta _{i}^{2}}$
• The covariance between two stocks and is the index variance weighted with and ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle \beta _{i}}$ ${\displaystyle \beta _{j}}$
• Correlation is correspondingly the covariance divided by the standard deviations of and . ${\displaystyle i}$ ${\displaystyle j}$
• The data requirement is significantly reduced compared to the Markowitz model.

Intermediate result

• The risk of a share consists of the market risk and the individual company risk.
• The returns on two stocks are correlated when the product of their betas is positive.
• The risk of a portfolio consists of a market component and an individual component.

Regression line

${\displaystyle E(r_{i})=\alpha +{\frac {\operatorname {Cov} (I,M)}{\sigma _{\text{Markt}}^{2}}}\cdot E(r_{\text{Markt}})}$

in which

${\displaystyle E(r_{i})}$: expected return of the individual component

${\displaystyle \operatorname {Cov} (I,M)}$: Covariance of the individual component and the market

${\displaystyle \beta ={\frac {\operatorname {Cov} (I,M)}{\sigma _{\text{Markt}}^{2}}}}$

So it follows

${\displaystyle E(r_{i})=\alpha +\beta \cdot E(r_{\text{Markt}})}$

Stationary estimation of the beta (technical procedure)

The beta can be estimated technically over a historical period, for example one year. With an observation frequency of, for example, one week, one receives 52 observations from which the beta can be estimated. This is based on the fact that the beta will be about the same in the future. It is assumed to be stationary.

Mean Reverting as an example procedure

The mean reverting method assumes that the beta observed fluctuate around a long-term value . The deviations in the direction of are corrected by means of a reversion factor . ${\displaystyle \beta _{0}}$ ${\displaystyle \beta _{0}}$

Fundamental factors influencing beta

The idea here is that the beta represents the sensitivity of the stock return versus the return on the index. If we consider the return on equity as the return on equity , the level of indebtedness , the size of the company , the capital intensity or the production program have an influence on this value.

application

The findings can be used in portfolio management for asset allocation .

The full market model is used at the top portfolio level. The portfolio is divided into two segments for stocks and bonds. The efficient portfolio is selected using the first efficiency line at a higher level. In the equity segment, stock picking then takes place using a single index model. This results in a second, conditional efficiency line.

Although this approach is theoretically not optimal in its entirety, it requires significantly less input data than the Markowitz model and thus also reduces the computational effort.