Sharpe's Model

Welcome to Unitedworld

SHARPE’S MODEL – SINGLE INDEX MODEL

SHARPE’S MODEL – SINGLE INDEX MODEL

Markowitz’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

SHARPE’S MODEL – SINGLE INDEX MODEL….contd….

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.

Original version of william sharpe’s single index model…. Contd…..

Following concepts are relevant for the model

1)Market Portfolio

2)Systematic Risk

3)Non-systematic Risk

4)Residual Error Returns


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.


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.


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.

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

Alpha

Return on Market Portfolio

Ret

urn

on in

divi

dual

sha

re Undervalued securities or portfolio

Overvalued securities or portfolio

Characteristic line

Graph showing characteristic line figure 17.1

Graph showing characteristic line figure 17.1 contd……

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 17.2

Return on Market Portfolio

Ret

urn

on in

divi

dual

sha

reUndervalued shares

Overvalued securities or portfolio

Characteristic line

Alpha

Characteristic line contd…….

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:Systematic Return = ‾

Systematic Risk = ‾


Non-systematic component ( ) is the residual return resulting from the performance of the company. This changes on account of changes in the performance of the company. Residual Return ( ) : ‾ ‾

Residual Risk =


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.


Characteristic Line Showing Return

= Mean return of the security =Alpha component, i.e., non-systematic

return (residual return) = Beta of the security = Mean of the market or index representing

the market = Residual error return, which is considered

as 'zero'Using characteristic line, a portfolio manager can

estimate expected return.

imiii eRR )(iRi

iei

mR

ie


Characteristic Line Showing risk :-

= Variance of security's return

= Beta of the security

= Variance of market or index

= Non-systematic risk i.e., residual risk

2

2

2

2222 )(

ei

m

i

i

eimii

Calculation of Return and Risk of Portfolio Under Sharpe's Model

Risk-return' and Sharpe ModelThe return of each security is represented by

the following equation:

= Expected return on security= Intercept of straight line or Alpha coefficient= Expected mean return on market= Random error or error term with mean and S.D. equal to zero which is a constant.

The mean value of (ei) is zero and hence the equation becomes-

imiii eRR

i

m

i

i

e

R

R

miii RR

Calculation of Return and Risk of Portfolio Under Sharpe's Model contd...

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.

miR


Returns of PortfolioFor 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.

=Expected portfolio return= The proportion of the portfolio devoted to stock i

n

i

mpiip RWR1

)(

i

p

W

R

n

iiip W

1


= 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.

p

p


Risk of Portfolio

Risk of the security or portfolio is calculated by variance in return or standard deviation of return. Total risk of a security is represented by the following equation.

Total risk = Unsystematic risk + Systematic risk

Variance

Variance = Variance of Security's return

= Unsystematic risk of security i

2ei

222mieiiR

iR2ei


Systematic Risk =

Unsystematic risk = Total variance of security - Systematic risk

22mi


Variance of Portfolio

Systematic risk of the portfolio =

Non-systematic risk of portfolio =

n

ieiimpp W

1

22222

22mp

n

ieiiW

1

22

Construction of Efficient Portfolio

Efficient PortfolioAn 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:

Construction of Efficient Portfolio contd…

Constructing the Efficient PortfolioModel 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 ratio3. 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:


Cutoff Rate

n

i ei

im

n

i ei

ifim

i

RR

c

12

22

12

2

1

]}/){(

[

)(


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:

i

ii Z

ZW

]}[{ *2

cRR

Zi

fi

ei

ii


ConclusionSharpe'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.

Subsequent Modification in the Model by Considering Risk-Free Return

The original work of William Sharpe insisted on only two components of return and risk - systematic and nonsystematic components. At a later stage, few thinkers and portfolio planners have amended the components of return and risk at the time of representing these in the form of characteristic line. These thinkers are of the view that total return of an individual security as well as that of a portfolio can first be represented as excess return over risk-free rate of return and then this excess return can further be divided into two - systematic component of excess return and nonsystematic component of excess return. Accordingly, following concepts are relevant in the modified model:

1. Concept of Risk-free Return and Excess Return2. Market Portfolio3. Systematic Risk4. Non-systematic Risk5. Residual Error Returns

Subsequent Modification in the Model by Considering Risk-Free Return contd…

I. Concept of Risk-free Return and Excess Return

Sharpe's model advocates that an individual share is desirable for investment only when it has average/mean returns more than the risk-free return; this is preferred by investors because investment in individual shares entails certain degree of risk and investors need extra returns to compensate for the risk. Accordingly, the model assumes that prima-facie selection of securities in a portfolio completely depends on the amount of excess return over the risk free returns. This concept of excess return is also relevant in representing the characteristic line representing returns of an individual share.


Characteristic line is a way to represent excess return of the security as originating from non-systematic factors and systematic factor. The excess return of an individual security are generated on account of the company's performance and the association of the share with the market portfolio or representative index of the market, respectively called non-systematic component part of the excess return and systematic component of the excess return. Accordingly, mean return of an individual share are represented as follows:


= Mean return of the security= Risk-free return= Alpha component, i.e. non-systematic return (residual return), it is such component of the excess return when excess return on the market portfolio is zero= Beta of the security= Mean of the market or index representing the market= Residual error return, which is considered as 'zero'

ifmiifi eRRRR )(

ie

mRi

ifRiR


2. Market PortfolioIt is portfolio in which all the securities of the market find exactly the same proportion in which these have a representation in the overall market. Portfolio created like this represents the movements of the whole of the market and beta of such market portfolio is always one “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.


3. Systematic RiskBy systematic risk, we mean the risk, which arises on account of market wide factors. This risk can never be eliminated because it is inherit part of the market and investment activities. These risk factors affect all investment avenues. This model assumes that fluctuation in the value of stock relative to that of another do not depend primarily on the characteristics 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.


4. Non-systematic Risk

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


5. Residual Error ReturnsBy residual error returns, we mean those returns which 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, restriction 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 these parameters of an individual security and portfolio are on account of two broad factor, i.e. systematic factor affecting the whole of the economy and market and another set of factors related to the company itself called as non-systematic factors. Returns on account of both of these factors is called excess return. This excess return when represented in these two segments, is called characteristic line of the security. The characteristic line developed on the basis of historical data is helpful in predicting the returns of an individual share, given the excess return on the market portfolio/index of the market. The characteristic line so developed, can be used to calculate estimated return of a portfolio.

Characteristic line shows the bifurcation of security’s excess return into the following:

(a)Market wide component of return and risk – Systematic Component

(b) Company wide component of return and risk i.e. , Non-Systematic- Alpha Component

(c) Random returns always considered to be zeroSystematic component of excess return is such component

which is on account of association of the security with the general index of the market and it is represented with the help of beta. Mathematically it is shown as follows

Systematic Return = Systematic Risk =

Characteristic line contd….

fi RR

)( fmi RR

)(2fmi RRVar

Non-systematic component (alpha i) is the residual part of the excess return resulting from the performance of the company. This changes on account of changes in the performance of the company as well as on account of the factors solely affecting the performance of the company.

Residual Return here ei is the random error of the returns, always considered to

be zeroResidual Risk = Random returns are returns which are generated on account of

random factors like merger, acquisition, extraordinary performance, etc. generally it is considered zero, due to this alpha(alpha i) of the security can be calculated as residual return or residual components of risk.


ifmifii eRRRR )()(

)(22fmii RRVar


Characteristic line Showing Return

= Mean return of the security

= Risk free return

= Alpha components of excess returns, i.e. non-systematic return( residual return)


= Mean of the market or index representing the market

= Residual error return which is considered as ‘zero’

Using characteristic line a portfolio manager can estimate expected return

ifmiifi eRRRR )(

iemRi

ifR

iR


Characteristic Line Showing Risk

= Variance of security’s excess return


= variance of the market or index = Var

= Non-systematic risk, i.e. residual risk

fm RR

222 )( eifmii RRVar

2ei

i

2i

2m

Calculation of return and Risk of the Portfolio Under Sharpe’s Model

Risk-Return and Sharpe Model The return of each security is represented by the following

equation:

= Mean excess return of the security

= Risk free return

= Intercept of straight line or Alpha coefficient it such amount of excess return when excess return on the market portfolio is zero.

= Expected mean return on market

= Random error or error term with mean of zero and S.D. which is a constant.

The mean value of ei is zero and hence the equation becomes

ifmiifi eRRRR )(

iemR

ifR

fi RR

Calculation of return and Risk of the Portfolio Under Sharpe’s Model contd..

The equation has the two coefficients or terms. The alpha value is the value of excess returns in the equation when the value of excess returns on the market portfolio is zero, in other words it is part of excess returns which is realized from the security even if the market’s excess 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 measure of the sensitivity of the stock’s excess return to the movement in the market’s excess return/market return. The combined term denotes that part of excess return, which is due to market movement. This is the systematic component of security’s excess return.

)( fmiifi RRRR

)( fmi RR


Returns of Portfolio For portfolio return we need merely the weighted

average of the estimated return of each portfolio. The weights will be the proportions of the portfolio devoted to each security.

= expected portfolio return= Risk free return= The proportion of the portfolio devoted to stock i

= Beta of the portfolio is weighted beta of the individual securities comprised in the portfolio.

n

ifmpiifp RRWRR

1

)()(

pR

fR

iW

p

n

iiip W

1

n

iiip W

1


n

iiip W

1

= 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.

p


Risk of PortfolioRisk of the security or portfolio is calculated by variance in return or standard deviation of return. Total risk of a security is represented by the following equation.

Total risk = Unsystematic risk+ Systematic risk

Variance Variance = Variance of Security’s excess

return

)()( 22fmieifi RRVarRR

iR


= Unsystematic risk of security I

Systematic Risk =

Unsystematic Risk = Total variance of security – Systematic risk

2ei

)(2fmi RRVar


Variance of Portfolio

Systematic Risk of the portfolio

Non-systematic Risk of the portfolio

)(2fmp RRVar

n

ieiiW

1

22

n

ieiimpp W

1

22222

Thank You

Documents

Sharpe's Model