SINGLE INDEX MODEL

The basic notion underlying the single index model is that all stocks are affected by

movements in the stock market. Casual observation of share prices reveals that when the

market moves up (as measured by any of the widely used stock market indices), prices of

most shares tend to increase. When the market goes down, the prices of most shares tend

to decline. This suggests that one reason why security returns might be correlated and there

is co-movement between securities, is because of a common response to market changes.

This co-movement of stocks with a market index may be studied with the help of a simple

linear regression analysis, taking the returns on an individual security as the dependant

variable (Ri) and the returns on the market index (Rm) as the independent variable.

The return of an individual security is assumed to depend on the return on the market index.

The return of an individual security may be expressed as:

Where,

= Component of security’s return that is independent of the market’s performance.

= Rate of return on the market index.

= Constant that measures the expected change in Ri given a change in Rm.

= Error term representing the random or residual return.

This equation breaks the return on a stock into two components, one part due to the market

and the other part independent of the market. The beta parameter in the equation, ,

measures how sensitive a stock’s return is to the return on the market index. It indicates how

extensively the return of a security is expected to increase by 20 percent when the market

return increases by 10 percent. In this case, if the market returns decrease by 10 percent,

the security return is expected to decrease by 20 percent. For a security with beta of 0.5,

when the market return increases or decreases by 10 percent, the security return is

expected to increase or decrease by 5 percent (that is 10*0.5). A beta coefficient greater

than one would suggest greater responsiveness on the part of the stock in relation to the

market and vice versa.

The alpha parameter , indicates what the return of the security would be when the market

return is zero. For example, a security with an alpha of +3 percent would earn 3 percent

return even when the market return is zero and it would earn an additional 3 percent at all

levels of market return. Conversely, a security with an alpha of -4.5 percent would lose 4.5

percent when the market return is zero, and would earn 4.5 percent less at all levels of

market return. The positive alpha thus represents a sort of bonus return and would be highly

desirable aspect of a security, whereas a negative alpha represents a penalty to the investor

and is an undesirable aspect of a security.

The final term is the equation, , is the unexpected return resulting from influences not

identified by the model. It is referred to as the random or residual return. It may take on any

value, but over a large number of observations it will average out to zero.

William Sharpe, who tried to simplify the data inputs and data tabulation required for the

Markowitz model of portfolio analysis, suggested that a satisfactory simplification would be

achieved by abandoning the covariance of each security with each other security and

substituting in its place the relationship of each security with a market index as measured by

the single index model suggested above. This is known as Sharpe Index Model.

In the place of [N (N-1)/2] co variances required for the Markowitz model. Sharpe model

would require only N measures of beta coefficients.

SHARPE’S OPTIMAL PORTFOLIO

Sharpe had provided a model for the selection of appropriate securities in a portfolio. The

selection of any stock is directly related to its excess return – beta ratio.

(Ri –Rf) / β

Where,

– The expected return of stock

– The return on risk free asset

– The expected change in the rate of return on stock i associated with one unit

change in the market return.

The excess return is the difference between the expected return on the stock and the

riskless rate of interest such as the rate offered on the government security or Treasury bill.

The excess return to beta ratio measures the additional return on a security (excess of the

riskless asset return) per unit of systematic risk or non-diversified risk. This ratio provides a

relationship between potential risk and reward.

Ranking of the stocks are done on the basis of their excess return to beta. Portfolio

managers would like to include stocks with higher ratios. The selection of the stocks

depends on a unique cut-off rate such that all stocks with higher ratios of Ri-Rf / Beta are

included and the stocks with lower ratios are left off. The cut-off point is denoted by C*.

The steps for finding out the stocks to be included in the optimal portfolio are given below:

1. Find out the ―excess return to beta‖ ratio for each stock under consideration.

2. Rank them from the highest to the lowest.

3. Proceed to calculate Ci for all the stocks according to the ranked order using the

following formula.

Where,

= Variance of the market index

= Variance of a stock’s movement that is not associated with the movement of the

market index i.e. stock’s unsystematic risk.

The cumulated values of Ci start declining after a particular Ci and that point is taken as the

cut-off point and that stock ratio is the cut-off ratio C.

CONSTRUCTION OF THE OPTIMAL PORTFOLIO

After determining the securities to be selected, the portfolio manager should find out how

much should be invested in each security. The percentage of funds to be invested in each

security can be estimated as follows:

Where,

The 1st expression indicates the weights on each security and they sum up to one.

The 2nd shows the relative investment in each security. The residual variance has a role in

determining the amount to be invested in each security.

Assumptions:

Sharpe’s model is based on the following assumptions:

1. The securities’ returns are related to each other.

2. The expected return and variances of indices are the same.

3. The return on individual securities is determined by unpredictable factors.