SHARPE SINGLE INDEX MODEL

SHARPE SINGLE INDEX MODEL

The Markowitz model is adequate and conceptually sound in analyzing the risk and return of the portfolio. The problem with Markowitz model is that a number of co-variances have to be estimated. If a financial institution buys 150 stocks, it has to estimate 11,175 (N2-N)/2 correlation co-efficient. Sharpe had developed a simplified model to analyze the portfolio.
Concept:
This model was developed by William Sharpe. He simplified the method of diversification of portfolios. Sharpe published a model simplifying the mathematical calculations done by the Markowitz model. According to Sharpe’s model, the theory estimate, the expected return and variance of indices which may be one or more and are related to economic activity. This theory has come to be known as market model. He assumed that the return of a security is linearly related to a single index like the market index. The market index should consist of all the securities trading on the exchange. In the absence of it, a popular index can be treated as a surrogate for the market index.
Assumptions:
Sharpe’s portfolio theory is based on the following assumptions:
1.      The securities returns are related to each other.
2.      The expected return and variance of indices are the same.
3.      The return on individual securities is determined by unpredictable factors.
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. For this following equation can be used:
Ri= αiiRm+ei
Where Ri= expected return on security i
αi=intercept of the straight line or alpha co-efficient
βi = slope of straight line or beta co-efficient
Rm=the rate of return on market index
ei=error term
Beta is a measure of volatility faced by a financial asset between actual earned or a project returns.
Alpha is the measurement of difference between actual earned return and expected return at a level of systematic risk.
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 effect beyond the market. The variance of the security has two components systematic risk and unsystematic risk.
Total risk= systematic risk + unsystematic risk
Total risk= βi2 2m+e2i
Systematic risk= β2i * variance of market index
Unsystematic risk = total variance – systematic risk.

e2i

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