portfolio sharpe index model_2

7/29/2019 Portfolio Sharpe Index Model_2

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The investor always likes to purchase a combination of stocks thatThe investor always likes to purchase a combination of stocks that

provides the highest return and has lowest risk. He want to maintainprovides the highest return and has lowest risk. He want to maintain

a satisfactory reward to risk ratio. Now a days risk has received increaseda satisfactory reward to risk ratio. Now a days risk has received increased

attention and analysts are providing estimates of risk as well as return.attention and analysts are providing estimates of risk as well as return.

the Markowitz model is adequate and conceptually sound in analyzingthe Markowitz model is adequate and conceptually sound in analyzingthe risk and return of the portfolio. The problem with Markowitz modelthe risk and return of the portfolio. The problem with Markowitz model

is that a number of co-variances have to be estimated. If the financialis that a number of co-variances have to be estimated. If the financial

institution buys 150 stocks, it has to estimate 11175 (N2-N)/2 correlationinstitution buys 150 stocks, it has to estimate 11175 (N2-N)/2 correlation

co-efficient. Sharpe has developed a simplified model to analysis theco-efficient. Sharpe has developed a simplified model to analysis the

portfolio. He assumed that the return of a security is linearly relatedportfolio. He assumed that the return of a security is linearly related

to a single index like the market index.to a single index like the market index.

SHARPE INDEX MODELSHARPE INDEX MODEL


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SHARPE INDEX MODELSHARPE INDEX MODELSINGLE INDEX MODELSINGLE INDEX MODEL

casual observation of the stock prices over a period of time reveals thatcasual observation of the stock prices over a period of time reveals thatmost of the stock prices move with the market index. When the sensexmost of the stock prices move with the market index. When the sensex

stock prices also tend to increase and vice-versa. This indications thatstock prices also tend to increase and vice-versa. This indications that

some underlying factors affect the market index as well as the stocksome underlying factors affect the market index as well as the stock

prices are related to the market index and this relationship could be usedprices are related to the market index and this relationship could be used

to estimate the return on stock.to estimate the return on stock.

Ri = a i + B i Rm + e iRi = a i + B i Rm + e i

RiRi = Expected return in security i= Expected return in security i

aiai = Intercept of the straight line or alpha co-efficient= Intercept of the straight line or alpha co-efficient

BiBi = Slop of straight line or beta co-efficient= Slop of straight line or beta co-efficient

eiei = Error term= Error term

RmRm = The rate of return on market index= The rate of return on market index


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According to the equation, the return of a stock can be dividend into twoAccording to the equation, the return of a stock can be dividend into two

components, the return due to the market and the return independent ofcomponents, the return due to the market and the return independent ofthe market. Bi indicates the sensitiveness of the stock return to thethe market. Bi indicates the sensitiveness of the stock return to the

changes in the market return.changes in the market return.

The single index model is based on the assumption that stocks varyThe single index model is based on the assumption that stocks vary

together because of the common movement in the stock market andtogether because of the common movement in the stock market and

there are no effects beyond the market ( fundamental factor effects )there are no effects beyond the market ( fundamental factor effects )

that account the stocks co-movement. The expected return, standardthat account the stocks co-movement. The expected return, standard

deviation and co-variance of the single index model represent the jointdeviation and co-variance of the single index model represent the joint

movement of securities.movement of securities.

The variance of securitys return has two components namely systematicThe variance of securitys return has two components namely systematic

risk or market risk and unsystematic risk or unique risk, The variancerisk or market risk and unsystematic risk or unique risk, The variance

explained by the index is referred to systematic risk. The unexplainedexplained by the index is referred to systematic risk. The unexplained

variance is called residual variance or unsystematic risk.variance is called residual variance or unsystematic risk.



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Systematic Risk = B2i * Variance of market indexSystematic Risk = B2i * Variance of market index

= B2i 62 m= B2i 62 mUnsystematic Risk = Total Variance Systematic RisUnsystematic Risk = Total Variance Systematic Ris

e2ie2i = 62i systematic risk= 62i systematic risk

SHARPE OPTIMAL PORTFOLIOSHARPE OPTIMAL PORTFOLIO

Sharpe had provided a model for the selection of appropriate securitiesSharpe had provided a model for the selection of appropriate securities

in a portfolio. The selection of any stock is directly related to its excessin a portfolio. The selection of any stock is directly related to its excess

return-beta ratioreturn-beta ratio

RRi i RRff

BiBi

The excess return is the difference between the expected return on theThe excess return is the difference between the expected return on the

stock and the risk less rate of interest such as the rate offered on thestock and the risk less rate of interest such as the rate offered on the

Government security or treasury bill. The excess return to beta ratioGovernment security or treasury bill. The excess return to beta ratio



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Measures the additional return on a security (excess of the risk less assetMeasures the additional return on a security (excess of the risk less asset

Return) per unit of systematic risk or non diversifiable risk. This ratioReturn) per unit of systematic risk or non diversifiable risk. This ratioProvides a relationship between potential risk and rewards.Provides a relationship between potential risk and rewards.

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

Portfolio managers would like to include stocks with higher ratios. ThePortfolio managers would like to include stocks with higher ratios. The

Selection of the stocks depends on a unique cut-off rate such that allSelection of the stocks depends on a unique cut-off rate such that allStocks with higher ratios of Ri - Rf / Bi are included and the stocks withStocks with higher ratios of Ri - Rf / Bi are included and the stocks with

Lower ratios are left off. The cut-off point is denoted by c*Lower ratios are left off. The cut-off point is denoted by c*

STEPS FOR FINDING OUT THE STOCKS TO BE INCULDED IN THESTEPS FOR FINDING OUT THE STOCKS TO BE INCULDED IN THE

OPTIMAL PORTFOLIO :OPTIMAL PORTFOLIO :

1. Find out the excess return to beta ratio for each stock under1. Find out the excess return to beta ratio for each stock under

consideration.consideration.

2. Rank them from the highest to the lowest2. Rank them from the highest to the lowest

3. Proceed to calculate ci for all the stocks according to the ranked3. Proceed to calculate ci for all the stocks according to the ranked

order.order.



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62 N (Ri Rf ) Bi62 N (Ri Rf ) Bi

mm EE 6262

ci i=1ci i=1 eiei

N B2iN B2i

1+ 62 m1+ 62 m EE 6 26 2i=1 eii=1 ei

6 2m = variance of the market index6 2m = variance of the market index

62ei = variance of a stocks movement that is not associated with62ei = variance of a stocks movement that is not associated with

the movement of market index.the movement of market index.4. The cumulated valued of C start declining after a particular Ci and4. The cumulated valued of C start declining after a particular Ci and

that point is taken as the cut off point and that stocks ratio is thethat point is taken as the cut off point and that stocks ratio is the

cut off ratio Ccut off ratio C



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Ci can be mathematically equivalent wayCi can be mathematically equivalent way

Ci = Bip ( Rp Rf )Ci = Bip ( Rp Rf )

BiBi

Bip = the expected change in the rate of return on stock I associatedBip = the expected change in the rate of return on stock I associated

with 1 per cant change in the return on the optimal portfolio.with 1 per cant change in the return on the optimal portfolio.

Rp = the expected return on the optimal portfolioRp = the expected return on the optimal portfolioBip and Rp can not be determined until the optimal portfolio is found.Bip and Rp can not be determined until the optimal portfolio is found.

To find out the optimal portfolio the formula given previously should beTo find out the optimal portfolio the formula given previously should be

Used. Securities are added to the portfolio as long asUsed. Securities are added to the portfolio as long as

Ri RfRi RfBiBi

Less than CiLess than Ci



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DATA FOR FINDING OUT THEDATA FOR FINDING OUT THE

OPTIMAL PORTFOLIOOPTIMAL PORTFOLIO

Security

Number

Mean

Return

Excess

Return

Beta Unsystematic

Risk

Excess

Return to

Beta

1

2

3

4

56

7

19

23

11

25

139

14

14

18

6

20

84

9

1.0

1.5

0.5

2.0

1.00.5

1.5

20

30

10

40

2050

30

14

12

12

10

88

6


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DATA FOR FINDING OUT THEDATA FOR FINDING OUT THE

OPTIMAL PORTFOLIOOPTIMAL PORTFOLIO

Security

Number

1

Ri-Rf / Bi

2

Ri-Rf * Bi /62ei

3

En (Ri-Rf)* Bi/

62ei

4

Bi2/62ei

5

En Bi2/

62ei

6

Ci

7

1

2

3

45

6

7

14

12

12

108

8

6

0.7

0.9

0.3

1.00.4

0.04

0.45

0.7

1.6

1.9

2.93.3

3.34

3.79

0.05

0.075

0.025

0.10.05

0.005

0.075

0.05

0.125

0.15

0.250.30

0.305

0.38

4.67

7.11

7.60

8.298.25

8.25

7.90

portfolio sharpe index model_2

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