sharpe keshav

8/6/2019 Sharpe Keshav

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Members: Deepak SharmaMembers: Deepak SharmaManuManu SadashivSadashiv

VishwambharVishwambhar SinghSinghShrutiShruti ShettyShettyManjunathManjunath PatilPatilKeshavKeshav BhatBhat


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Dividend Discount Model

= Po = current stock price

Di = expected dividend

r = required rate of returng = expected growth rate in perpetuity.

Cons of Growth Model and Discounted Model


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Modern Portfolio Theory or Markowitz Model

Assets in an investment portfolio should not beselected individually on each of their merits.Rather it is important to consider how eachasset in the portfolio changes in price relativeto how every other asset in the portfoliochanges in price.

Tradeoff RISK vs RETURNS.

Best diversification strategy


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Two types of RISK

Systematic or Undiversifiable or Market Risk

Unsystematic or Diversifiable or Portfolio Risk

Markowitz, By actively selecting the

investment instruments an intelligent fundmanager can avoid a certain amount of riskand still reap the rewards over it.


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n number of returns

n number of variances

(n^2 n)/2 number covariance calculations; in total

it requires n(n+3)/2 number of calculation.Institutional Investors with 50 70 stocks, and inputs

more than 5000.

Better off with NAVE or Amateur Diversification.


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In contrast the premises of Markowitzs model, Sharpes

model favors that an individual securities has relationshipwith one common parameter of the market, i.e. index of themarket.

According to Sharpes concept, different securities in the

market do not have any kind of direct relation with eachother; 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 oppositerelationship with the whole market. This association ofindividual securities with the market is through the stockindex of the market


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Stock index (SENSEX) is representative of the market and

every security has a relationship with this Index. Thisrelationship can help in estimating and representing thereturns of these securities. Unlike Markowitz, Sharpe does notbelieve in one to one relationship, of individual securities.

This association of individual securities with the index isrepresented with the help of beta and depending on the Betavalue of the securities, these get classified into following threetypes :

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

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

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


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Defensive stock these are the shares that have beta valueless than 1, which implies that these show a movement in thereturn at a slow pace as compared to the movement of overallmarket. E.g. if a stock has beta of 0.75 than for every 1%change in the overall market this will show a movement of0.75%.

Neutral Stock these shares have a beta value of (1) whichhas an implication that these have the tendency to make amovement as good as that of the overall market.

Aggressive Stocks such shares have the beta value morethan 1 (beta >1) and these move at a faster pace then themovement 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.


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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 securitiescan be represented in the form of characteristic line, which

implies the return and risk of securities can be bifurcated intotwo :

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

Returns and risk on account of company-wide factors -

Non-Systematic Factors


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The model also advocated that an individual security is

desirable only when its returns are in excess of the risk freereturns.

The excess returns of an individual security hold arelationship with the excess return on the market portfolio.

In the absence of the market portfolio a representative indexcan be used to show this relationship.

Returns and risk of individual securities fluctuate, dependingupon the fluctuation in the market portfolio/ market index.This relationship can be used to create portfolio.


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The initial & original work of William Sharpe argued thatthe return of each individual security has two basiccomponents i.e., systematic component and non-systematiccomponent.

Sharpe was of the opinion that each security has anassociation with the market portfolio and the return ofsecurity find an association with the return of such

portfolio. In the absence of market portfolio, arepresentative index of the market (like BSE Sensex orNifty) may be used.


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The changes in the return of a security due to thisassociation are termed as slope of the curve when plottedon a graph. This association is represented with the help ofBeta.

At the same time, each security has returns on account ofthe performance of the company and such returns arecalled non-systematic component of return; in technical

jargon this is called Alpha component of the return. Thisalpha component represents minimum return fromsecurity when return on market portfolio or itsrepresentative index is zero.


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Following concepts are relevant for the model

1) Market Portfolio

2) Systematic Risk3) Non-systematic Risk

4) Residual Error Returns


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Market Portfolio It is a portfolio in which all the securitiesof the market find exactly the same proportion in which thesehave a representation in the overall market capitalization.

Portfolio created like this, represents the movement of wholeof the market and Beta of such market portfolio is always 1.

Such portfolio is the replication of the whole of the marketand 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 beused.


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Systematic Risk By systematic risk, we mean the risk thatarises on account of market-wide factors. This risk can neverbe eliminated because it is an inherent part of the market andinvestment activities. These risk factors affect all investmentavenues. This model assumes that fluctuations in the value of

stock relative to that of another do not depend on thecharacteristic of those two securities alone. The two securitiesare more apt to reflect a broader influence that might bedescribed as general business conditions. Relationshipsbetween securities occur only through their individualrelationship with some index. This relationship with the indexis measured with the help of beta. Beta is a sensitivitymeasurement, representing volatility of the returns from ashare, given particular changes in the overall market or index

of the market.


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Non-systematic Risk This is such component of risk, whichis on account of company-wide factors or factors specific to aparticular investment avenue. This part of the risk can eitherbe eliminated completely with the help of diversification.

Residual Error Returns By residual error returns, we meanthe returns that arise on account of extraordinary eventconcerning the performance of a company. When these eventsare favoring the company, the effect is positive, otherwise it is

negative. Residual error returns are positive when companydeclares bonus, merger, diversification or strategic alliance forthe better. It will be negative when a sudden fall in the profitsis observed, restrictions are applied on company or othernegative aspects take place.


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Risk-return' and Sharpe Model

The return of each security is represented by thefollowing 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 theequation becomes-

im

iiieRR v! FE

i

m

i

i

e

R

R

E

miii RR FE !


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The equation has two coefficients or terms. The alpha value isthe 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 thesecurity even if the market return is zero. This is the non-

market (unsystematic)' component of security's return. Thebeta coefficient is the slope of the regression line and as such,it is a measure of the sensitivity of .the stock's return to themovement 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.mi

RF


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Returns of Portfolio

For portfolio return, we need merely the weightedaverage of the estimated return for each security inthe portfolio. The weights will be the proportionsof the portfolio denoted to each security.

=Expected portfolio return= The proportion of the portfolio devoted to stocki

!

v!n

i

mpiip RWR1

)( FE

i

p

W

R

!

!n

i

iip W1

FF


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

= Value of the Alpha for the portfolio. Portfolioalpha value is the weighted average of theAlpha' values for its component securities,using relative market value as weight.

pF

pE


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Risk of Portfolio

Risk of the security or portfolio is calculated byvariance in return or standard deviation of

return. Total risk of a security is represented bythe following equation.

Total risk = Unsystematic risk + Systematic risk

VarianceVariance = Variance of Security's return

= Unsystematic risk of security i

2

eiW

222

mieiiR

WFW v!iR

2

eiW


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Systematic Risk =

Unsystematic risk = Total variance of security -Systematic risk

22

miWF v


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Variance of Portfolio

Systematic risk of the portfolio =

Non-systematic risk of portfolio =

!

vv!n

i

eiimpp W1

WWFW

22

mp WF v

! v

n

ieiiW1

W


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Efficient Portfolio

An efficient portfolio is the one, which offers maximumreturn for a given level of risk or has minimum risk forthe given level of return. This is identified with thehelp of dominance principle. As investors are risk

averse and are rational decision-makers, they alwaysprefer to accept maximum return by assuming aparticular level of risk. In the long run, only efficientportfolios are feasible. Under Sharpe's single indexmodel, an efficient portfolio can be constructed as

follows:


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Constructing the Efficient Portfolio

Model emphasizes that every individual securitymust generate positive excess return; this implies thatmean return or expected mean return of a securitymust be more than the return from risk-free avenue.

Here, risk-free avenue means an avenue on which anassured and safe (free from default risk) return isgenerated.

According to the model desirability of any securityis directly related to its excess return to beta ratio [(Ri-Rf)/Beta i ],where Rf is the return on risk free assets.


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If securities are ranked by excess return to Beta (fromhighest to lowest) the ranking represents thedesirability of any security's inclusion in a portfolio.

The number of securities selected, depends on aunique cut-off point, such that all securities with

higher ratio of (Ri - Rf)/Beta i, will be includedin the portfolio and all securities with lower ratio willnot be included in the portfolio. To determine whichsecurities are included in the optimum portfolio, thefollowing steps are necessary.


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Steps for Creating Efficient Portfolio:

I. Calculate the excess return. to beta ratio for eachsecurity.

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

3. The optimum portfolio consists of investing in all thesecurities, 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 iscomputed from the characteristics of all the securitiesthat belong to optimum portfolio. To determine c*, it isnecessary to calculate its value as if there weredifferent numbers of securities in the optimum

portfolio.


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Since securities are ranked from highest "Excessreturn 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 betaratio more than the overall cut-off point are included in

the portfolio. Such portfolio is the efficient portfolioand generates the optimum return for the riskcategory.

For a portfolio of i securities, cut-off point (ci) for eachsecurity is calculated as follows:


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Cutoff Rate

!

!

v

v

!n

i ei

im

n

i ei

ifi

m

i

RR

c

1

1

1

]}/){(

[

)(

W

FW

W

FW


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To construct the best portfolio, the proportion offunds invested in each selected security in theoptimum portfolio is to be calculated, using thefollowing formula:

!

i

ii

Z

ZW

]}[{ *cRR

Zi

fi

ei

ii

v!

FW

F


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Conclusion

Sharpe's Model is convenient as compared to themodel of Harry Markowitz. It helps in the creation ofportfolio with less number of calculations as comparedto any other model. In Sharpe's model association of

individual securities/shares with the index of marketis given importance, instead of correlation betweensecurities. Only those securities are desirable in theportfolio, which have positive excess return over riskfree return, All the securities for which excess return tobeta ratio is more than the overall cut-off point-areincluded in the portfolio. Such portfolio is the efficientportfolio and generates the optimum returns.


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What Does Sharpe Ratio Mean?A ratio developed by Nobel laureate William F. Sharpe to measure risk-adjusted performance. The Sharpe ratio is calculated by subtracting therisk-free rate - such as that of the 10-year U.S. Treasury bond - from therate of return for a portfolio and dividing the result by the standard

deviation of the portfolio returns. The Sharpe ratio formula is:

(Rp - Rf)/ Rp = Expected portfolio return

Rf = Risk free return

= Portfolio Standard Deviation

Greater the Sharpe ratio, better its risk adjusted performance has been.Negative ratio indicates that a risk-less asset would perform better thanthe security being analysed.

W

W


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