Define Single Index Model

Single Index Model

Casual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index. When the Sensex increases, stock prices also tend to increase and vice-versa. This indicates that some underlying factors affect the market index as well as the stock prices. Stock prices are related to the market index and this relationship could be used to estimate the return on stock. Towards this purpose, the following equation can be used

R_i = α_i + β_iR_m + e_i

Where

According to the equation, the return of a stock can be divided into t components, the return due to the market and the return independent of the market. 13. indicates the sensitiveness of the stock return to the changes in the market return. For example 13 of 1.5 means that the stock returns is expected to increase by 1.5% when the market index return increases by 1% and vice-versa. Likewise, 13.of 0.5 expresses that the individual stock return would change by 0.5 per cent when there is a change of 1 per cent in the market return. 13 of 1 indicate that the market return and the security return are moving in tandem. The estimates of 13.and a are obtained from regression analysis.

The single index model is based on the assumption that stocks vary together because of the common movement in the stock market and there are no effects beyond the market (i.e. any fundamental factor effects) that account the stocks co-movement. The expected return, standard deviation and co-variance of the single index model represent the joint movement of securities. The mean return is

The variance of the security has t components namely, systematic risk or market risk and unsystematic risk or unique risk. The variance explained by the index is referred to systematic risk. The unexplained variance is called residual variance or unsystematic risk. Systematic risk = β_i2 x variance of market index.

Unsystematic risk = Total variance — Systematic risk.

Thus, the total risk = Systematic risk + Unsystematic risk.

From this, the portfolio variance can be derived

Likewise expected return on the portfolio also can be estimated. For each security α₁and β₁ should be estimated. N

Portfolio return is the weighted average of the estimated return for each security in the portfolio. The f weights are the respective stocks’ proportionsin the portfolio.

A portfolio’s alpha value is a weighted average of the alpha values for its component securities using the F proportion of the investment in a security as weight.

Similarly, a portfolio’s beta value is the weighted average of the beta values of its component stocks using relative share of them in the portfolio as weights.