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= αi+βiRm+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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