portfolio-theory-sharpe-index-model1-110921030315-phpapp02.ppt

7/27/2019 portfolio-theory-sharpe-index-model1-110921030315-phpapp02.ppt

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SECURITY ANALYSIS &PORTFOLIO MGT.

Sharpe Index Model


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Group Members1. Esha Khosla 61

2. Amit Goyal 623. Samarpit Nagpal 63

4. Sakshi Sarin 64


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THE SHARPE INDEX MODEL

Generally, when the Sensex rises, stock prices

also tend to increase and vice versa.

Thus, stock prices are related to the market

index and this relationship can be used to

estimate the return on stock.


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In that regard, the following equation can be used;

Ri= i + iRm + ei

where,

Ri= expected return of security ii= alpha coefficient(intercept of the straight line)

i= beta coefficient (slope of the straight line)

Rm = the rate of return of market index

ej =error term


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According to the above equation, the

return of a stock can be divided into

two components;1)The return due to the market,

2)The return independent of the

market

i of 1 indicates that the market

return and security return aremoving in tandem.


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The Sharpes Single Index Model is

based on the assumption that stocks

vary together because of the

common movement in the stock

market and there are no effects

beyond the market(i.e. any

fundamental factor effects) thataccount the stocks co-movement.


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e expec e re urn, s an ar ev a on an co-var ance othe single index model represent the joint movement of

securities.

The mean return is:Ri= i + iRm + ei

The variance of securitys return is:=im + ei

The covariance of returns between securities i and j is:

ij = ijm

The variance of the security has two components:


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1)Systematic risk is the variance explained by the

market index.

Systematic Risk =i x variance of market index

=im

2)Unsystematic Risk=total variancesystematic risk

= ei = i systematic risk

Thus,

Total Risk=Systematic Risk-Unsystematic Risk

=i + ei


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From this, the portfolio variance can be derived;

p = [ (Xii)m] + [ Xi ei]Where,

p = variance of portfolio

m = expected variance of market index

ei= variation in security return not related to the

market index

xi = the portion of stock i in the portfolio


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Likewise, expected return on the portfolio also can be

estimated.

For each security ai and i should be estimated.

Rp=Xi(ai + iRm)

Portfolio return is the weighted average of the

estimated return for each security in the portfolio. The

weights are the respective stocks proportions in the

portfolio.


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A portfolios Alpha is a weighted average of the alpha

values for its component securities using the proportion

of the investment in a security as weight.

ap= Xiai

Where,

ap= value of the alpha for the portfolio

Xi= proportion of the investment on security i

ai= value of alpha for security i


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Similarly, a portfolios beta value is the weightedaverage of the beta values of its component stocks

using relative share of them in the portfolio as

weights.

p= xii

where,

p= portfolio beta


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Problem 1:

The following information is given for stocks of

Company A and Company B and the BSE Sensex for a

period of one year.Calculate the systematic and unsystematic risks for the

stocks. If equal amount of money is allocated for the

stocks what would be the portfolio risk?

Particulars Stock A Stock B Sensex

Average

return

0.15 0.25 0.06

Variance of

return

6.30 5.86 2.25

0.71 0.27

Correlation

coefficient

0.424

Coefficient of

determination

0.18


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Ans.Company A:

Systematic Risk=i x Variance of Market Index

=(0.71) x 2.25=1.134

Unsystematic Risk=Total Variance Systematic Riskei = i systematic risk

=6.30-1.134=5.166

Total Risk =Systematic Risk + Unsystematic Risk

= i + ei=1.134 + 5.166=6.30


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Company B:

Systematic Risk =(0.27) x 2.25=0.1640

Unsystematic Risk=(5.86-0.1640)=5.696

Total Risk =(0.1640+5.696)=5.86

p = [ (Xii)m] + [ Xi ei]=[(.5 x .71 + .5 x .27) 2.25] +

[(.5)(5.166) + (.5)(5.696)]

=[(.355 +1.35) 2.25] + [(1.292 +

1.424)]=(.540 + 2.716)

=3.256


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SHARPES OPTIMAL PORTFOLIO

Sharpe had provided a model for the selection ofappropriate securities in a portfolio.

The selection of any stock is directly related to its

excess return-beta ratio.

Ri-Rf/iwhere,

Ri = the expected return on stock i

Rf= the return on a risk less security

i = the expected change in the rate of return on stock iassociated with one unit change in the market retrun.


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The excess return to beta ratio measures the additional

return on a security(excess of the risk less security

return) p.u. of systematic risk or non-diversifiable risk.

This ratio provides a relationship between potential risk

and reward.

Ranking of the stocks are done on the basis of theirexcess return to beta.


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The selection of the stocks depends on a unique cut-off

rate such that all stocks with higher ratios of Ri-Rf/iare included and the stocks with lower ratios are left

out

C*


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The cut-off rate is denoted by C*The steps for finding out the stocks to be included in

the optimal portfolio are given below:

Step 1:Find out the excess return to beta ratio for each stock

under consideration.

Step 2:

Rank them from the highest to the lowest.Step 3:

Calculate C, for all the stocks according to the ranked

order using the following formula:

Step 4:

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.


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N

m2 (RiRf)i

ei2

i=1

Ci = N

1 + m2 i

2

ei2

i =1


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Where,

m2 = Variance of the Market Index

ei

2= Variance of a stocks movement that is not

associated with the movement of Market Index i.e.

stocks unsystematic risk.


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EXAMPLE -1:


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SOLUTION OF EXAMPLE- 1:


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Thank u.

portfolio-theory-sharpe-index-model1-110921030315-phpapp02.ppt

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