SHARPE’S MODEL – SINGLE INDEX MODEL

Why Sharpe’s ModelMarkowitz’s approach of model portfolio requires a considerable amount of calculations; for n securities portfolio, one need to have (a) n number of returns (b) n number of variances (c) (n^2 – n)/2 number covariance calculations; in total it requires n(n+3)/2 number of calculations, which is a cumbersome task. Markowitz also emphasized that each security has correlation with another. In contrast to this, William Sharp was of the opinion that each security has a perfect link with the market portfolio or index of the market. Using this relationship of the security with the single index of the market, one can estimate characteristic line of the security as well as can construct efficient portfolio. In contrast the premises of Markowitz’s model, Sharpe’s model favors that an individual securities has relationship with one common parameter of the market, i.e. index of the market. According to Sharpe’s concept, different securities in the market do not have any kind of direct relation with each other; instead, these have a link with the index of the market, which is representative of the entire market. There are stocks(shares) in the market, which show an upward movement as soon as market moves up and vice – versa. Certain shares in the market have an opposite relationship with the whole market. This association of individual securities with the market is through the stock index of the market.

Stock index (SENSEX) is representative of the market and every security has a relationship with this Index. This relationship can help in estimating and representing the returns of these securities. Unlike Markowitz, Sharpe does not believe in one to one relationship of individual securities.

This association of individual securities with the index is represented with the help of beta and depending on the Beta value of the securities, these get classified into following three types :

Defensive stock (shares) i.e. beta < 1

Neutral stock (shares) i.e. beta = 1

Aggressive stock (shares) i.e. beta >1

Defensive stock – These are the shares that have beta value less than 1, which implies that these show a movement in the return at a slow pace as compared to the movement of overall market. E.g. if a stock has beta of 0.75 than for every 1% change in the overall market this will show a movement of 0.75%.

Neutral Stock – These shares have a beta value of (1) which has an implication that these have the tendency to make a movement as good as

that of the overall market.

Aggressive Stocks – Such shares have the beta value more than 1 (beta >1) and these move at a faster pace then the movement of the overall market. E.g. if beta of a share is 1.45, then this will show a movement of 1.45% for every 1% movement in the overall index of the market. It is the simplification over the modern portfolio theory given my Harry Markowitz. In this model, it is favored that returns and risk of a securities can be represented in the form of characteristic line, which implies the return and risk of securities can be bifurcated into two :

Returns and risk on account of market-wide factors – Systematic Factors

Returns and risk on account of company-wide factors Non-Systematic Factors

The model also advocated that an individual security is desirable only when its returns are in excess of the risk free returns. The excess returns of an individual security hold a relationship with the excess return on the market portfolio. In the absence of the market portfolio a representative index can be used to show this relationship. Returns and risk of individual securities fluctuate, depending upon the fluctuation in the market portfolio/ market index. This relationship can be used to create portfolio.

ORIGINAL VERSION OF WILLIAM SHARPE’S SINGLE INDEX MODEL

The initial and original work of William Sharpe argued that the

return of each individual security has two basic components i.e., systematic component and non-systematic component. Sharpe was of the opinion that each security has an association with the market portfolio and the return of security find an association with the return of such portfolio. In the absence of market portfolio, a representative index of the market (like BSE Sensex or Nifty) may be used. The changes in the return of a security due to this association are termed as slope of the curve when plotted on a graph. This association is represented with the help of Beta. At the same time, each security has returns on account of the performance of the company and such returns are called non-systematic component of return; in technical jargon this is called Alpha component of the return. This alpha component represents minimum return from security when return on market portfolio or its representative index is zero.

Following concepts are relevant for the model

1) Market Portfolio

2) Systematic Risk

3) Non-systematic Risk

4) Residual Error Returns

1) Market Portfolio – It is a portfolio in which all the securities of the market find exactly the same proportion in which these have a representation in the overall market capitalization. Portfolio created like this, represents the movement of whole of the market and Beta of such market portfolio is always ‘1’. Such portfolio is the replication of the whole of the market and moves in alignment with the market. In the absence of such portfolio general index of the market, which is true representative of whole of the market, can be used.

2) Systematic Risk – By systematic risk, we mean the risk that arises on account of market-wide factors. This risk can never be eliminated because it is an inherent part of the market and investment activities. These risk factors affect all investment avenues. This model assumes

that fluctuations in the value of stock relative to that of another do not depend on the characteristic of those two securities alone. The two securities are more apt to reflect a broader influence that might be described as general business conditions. Relationships between securities occur only through their individual relationship with some index. This relationship with the index is measured with the help of beta. Beta is a sensitivity measurement, representing volatility of the returns from a share, given particular changes in the overall market or index of the market.

3) Non-systematic Risk – This is such component of risk, which is on account of company-wide factors or factors specific to a particular investment avenue. This part of the risk can either be eliminated completely with the help of diversification.

4) Residual Error Returns – By residual error returns, we mean the returns that arise on account of extraordinary event concerning the performance of a company. When these events are favouring the company, the effect is positive, otherwise it is negative. Residual error returns are positive when company declares bonus, merger, diversification or strategic alliance for the better. It will be negative when a sudden fall in the profits is observed, restrictions are applied on company or other negative aspects take place.

Characteristic line:It represents decomposition of risk and return into its components; it is

believed that both of these parameters of an individual security and portfolio are on account of two broad factors. Characteristic line can be used to calculate estimated return of a portfolio or security.

Characteristic line shows the bifurcation of security’s return (Ri) into the following :

Market-wide component of return and risk- Systematic Component

Company-wide component of return and risk – Non-systematic – Alpha Component

Random returns

Graph showing characteristic line figure 1.1

This graph shows characteristic line and alpha of a security is positive and return on market portfolio is zero.

Sometimes Alpha of security is negative and return on market portfolio is zero; this relationship has been shown in the next graph:

Graph showing characteristic line figure 1.2

Systematic component of returns is such a component, which is on account of association of the security with the general index of the market and is represented with the help of beta. Mathematically, it is shown as follows:

stematic component Alpha is the residual return resulting from the performance of the company. This changes on account of changes in the performance of the company.

Random returns are the returns generated on account of random factors like mergers, acquisitions, extraordinary performance etc. Generally, it is considered zero, due to this, alpha of a security can be calculated as residual returns or residual component of risk.

Calculation of Return and Risk of Portfolio Under Sharpe'sModel:

The equation has two coefficients or terms. The alpha value is the value of (Ri), in the equation when the value of (Rm) is zero; in other words, it is part of return which is realized from the security even if the market return is zero. This is the non-market (unsystematic)' component of security's return. The beta coefficient is the slope of the regression line and as such, it is a measure of the sensitivity of .the stock's return to the movement in the market's return. The

combined term denote that part of return, which is due to market movement. This is the systematic component of the Security's return.

Returns of Portfolio

For portfolio return, we need merely the weighted average of the estimated return for each security in the portfolio. The weights will be the proportions of the portfolio denoted to each security.

= Beta of the portfolio is the weighted beta of the individual securities comprised in the portfolio. = Value of the Alpha for the portfolio. Portfolio alpha value is the weighted average of the Alpha' values for its component securities, using relative market value as weight

Systematic Risk =

Unsystematic risk = Total variance of security - Systematic risk

Variance of Portfolio

Systematic risk of the portfolio =

Non-systematic risk of portfolio =

Construction of Efficient Portfolio:Efficient Portfolio:An efficient portfolio is the one, which offers maximum return for a given level of risk or has minimum risk for the given level of return. This is identified with the help of dominance principle. As investors are risk averse and are rational decision-makers, they always prefer to accept maximum return by assuming a particular level of risk. In the long run, only efficient portfolios are feasible. Under Sharpe's single index model, an efficient portfolio can be constructed as follows:

Constructing the Efficient Portfolio

Model emphasizes that every individual security must generate positive excess return; this implies that mean return or expected mean return of a security must be more than the return from risk-free avenue. Here, risk-free avenue means an avenue on which an assured and safe (free from default risk) return is generated. According to the model desirability of any security is directly related to its excess return to beta ratio [(Ri- Rf)/Beta i ],where Rf is the return on risk free assets.

If securities are ranked by excess return to Beta (from highest to lowest) the ranking represents the desirability of any security's inclusion in a portfolio. The number of securities selected, depends on a unique cut-off point, such that all securities with higher ratio of (Ri - Rf)/Beta i, will be included in the portfolio and all securities with lower ratio will not be included in the portfolio. To determine which securities are included in the optimum portfolio, the following steps are necessary.

Steps for Creating Efficient Portfolio:

I. Calculate the excess return. to beta ratio for each security.

2. Review and rank from highest to lowest excess return to beta ratio

3.The optimum portfolio consists of investing in all the securities, for which excess return to beta ratio [(Ri- Rf)/Beta i] is greater than the overall cut-off point c*.

The value of c* is the overall cut-off point It is the cut-off point of the last security included in the portfolio. It is computed from the characteristics of all the securities that belong to optimum portfolio. To determine c*, it is necessary to calculate its value as if there were different numbers of securities in the optimum portfolio.

Since securities are ranked from highest "Excess return to beta" to lowest, securities with individual cut-off point more than c* are eligible to be included in the portfolio. All the securities, which have excess return to beta ratio more than the overall cut-off point are included in the portfolio. Such portfolio is the efficient portfolio and generates the optimum return for the risk category. For a portfolio of i securities, cut-off point (ci) for each security is

calculated as follows:

To construct the best portfolio, the proportion of funds invested in each selected security in the optimum portfolio is to be calculated, using the following formula:

Conclusion

Sharpe's Model is convenient as compared to the model of Harry Markowitz. It helps in the creation of portfolio with less number of calculations as compared to any other model. In Sharpe's model association of individual securities/shares with the index of market is given importance, instead of correlation between securities. Only those securities are desirable in the portfolio, which have positive excess return over risk free return, All the securities for which excess return to beta ratio is more than the overall cut-off point-are included in the portfolio. Such portfolio is the efficient portfolio and generates the optimum returns.

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