Investment Management Assignment 1

8/3/2019 Investment Management Assignment 1

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Complied By:Priyanka Yadav

B.Com(H) 3rd yr.

Roll no- 438

Group: B-11


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Risk and ReturnRisk and Return


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Expected ReturnExpected Return The future is uncertain.The future is uncertain.

Investors do not know with certainty whether theInvestors do not know with certainty whether the

economy will be growing rapidly or be in recession.economy will be growing rapidly or be in recession. Investors do not know what rate of return theirInvestors do not know what rate of return their

investments will yield.investments will yield.

Therefore, they base their decisions on theirTherefore, they base their decisions on their

expectations concerning the future.expectations concerning the future.

TheThe expected rate of returnexpected rate of return on a stock representson a stock representsthe mean of a probability distribution ofpossiblethe mean of a probability distribution ofpossiblefuture returns on the stock.future returns on the stock.


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Expected ReturnExpected Return The table below provides a probability distribution for the returns onThe table below provides a probability distribution for the returns on

stocksstocks AA andand BB

State Probability Return On Return OnState Probability Return On Return On

Stock A Stock BStock A Stock B

1 20% 5% 50%1 20% 5% 50%

2 30% 10% 30%2 30% 10% 30%

3 30% 15% 10%3 30% 15% 10%

4 20% 20%4 20% 20% --10%10%

The state represents the state of the economy one period in the futureThe state represents the state of the economy one period in the futurei.e. state 1 could represent a recession and state 2 a growth economy.i.e. state 1 could represent a recession and state 2 a growth economy.

The probability reflects how likely it is that the state will occur. The sumThe probability reflects how likely it is that the state will occur. The sumof the probabilities must equal 100%.of the probabilities must equal 100%.

The last two columns present the returns or outcomes for stocksThe last two columns present the returns or outcomes for stocks AA andandBB that will occur in each of the four states.that will occur in each of the four states.


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Expected ReturnExpected Return Given a probability distribution of returns, theGiven a probability distribution of returns, the

expected return can be calculated using the followingexpected return can be calculated using the following

equation:equation: NNE[R] =E[R] = 77(p(piiRRii))

i=1i=1

Where:Where:

E[R] = th

e expected return on th

e stockE[R] = th

e expected return on th

e stock N = the number of statesN = the number of states

ppii = the probability of state i= the probability of state i

RRii = the return on the stock in state i.= the return on the stock in state i.


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Expected ReturnExpected Return In this example, the expected return for stock AIn this example, the expected return for stock A

would be calculated as follows:would be calculated as follows:

E[R]E[R]AA = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%= .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%


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Expected ReturnExpected Return Calculation of expected return of stock B:Calculation of expected return of stock B:

E[R]E[R]BB = .2(50%) + .3(30%) + .3(10%) + .2(= .2(50%) + .3(30%) + .3(10%) + .2(--10%) = 20%10%) = 20%

So we see thatSo we see thatStock BStock B offers a higher expectedoffers a higher expectedreturn thanreturn than Stock A.Stock A.

However, that is only part of the story; we haven'tHowever, that is only part of the story; we haven'tconsidered risk.considered risk.


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Measures of RiskMeasures of Risk Risk reflects the chance that the actual return on anRisk reflects the chance that the actual return on an

investment may be different than the expectedinvestment may be different than the expected

return.return. One way to measure risk is to calculate the varianceOne way to measure risk is to calculate the variance

and standard deviation of the distribution of returns.and standard deviation of the distribution of returns.

We will once again use a probability distribution inWe will once again use a probability distribution inour calculations.our calculations.

The distribution used earlier is provided again forThe distribution used earlier is provided again forease of use.ease of use.


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Measures of RiskMeasures of Risk Probability Distribution:Probability Distribution:

State Probability Return On Return OnState Probability Return On Return OnStock A Stock BStock A Stock B

1 20% 5% 50%1 20% 5% 50%

2 30% 10% 30%2 30% 10% 30%

3 30% 15% 10%3 30% 15% 10%

4 20% 20%4 20% 20% --10%10%

E[R]E[R]AA = 12.5%= 12.5%

E[R]E[R]BB = 20%= 20%


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Measures of RiskMeasures of Risk Given an asset's expected return, its variance can beGiven an asset's expected return, its variance can be

calculated using the following equation:calculated using the following equation:NN

Var(R) =Var(R) = WW22 == 77 ppii(R(Rii E[R])E[R])22

i=1i=1

Where:Where:

N = the number of statesN = the number of states

ppii = the probability of state i= the probability of state i

RRii = the return on the stock in state i= the return on the stock in state i

E[R] = the expected return on the stockE[R] = the expected return on the stock


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Measures of RiskMeasures of Risk The standard deviation is calculated as the positiveThe standard deviation is calculated as the positive

square root of the variance:square root of the variance:

SD(R) =SD(R) = WW == WW22 == ((WW22))1/21/2 = (= (WW22))0.50.5


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Measures of RiskMeasures of Risk The variance and standard deviation for stock A is calculatedThe variance and standard deviation for stock A is calculated

as follows:as follows:

WW22AA = .2(.05= .2(.05 --.125).125)22 + .3(.1+ .3(.1 --.125).125)22 + .3(.15+ .3(.15 --.125).125)22 + .2(.2+ .2(.2 --.125).125)22 ==

.002625.002625

WW%% !! !!!!


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Measures of RiskMeasures of Risk Variance and standard deviation of stock B:Variance and standard deviation of stock B:

WW22BB = .2(.50= .2(.50 --.20).20)22 + .3(.30+ .3(.30 --.20).20)22 + .3(.10+ .3(.10 --.20).20)22 + .2(+ .2(--.10.10 -- .20).20)22 ==

.042.042

WW&& !! !!!!

AlthoughAlthoughStock BStock B offers a higher expected return thanoffers a higher expected return than

Stock AStock A, it also is riskier since its variance and standard, it also is riskier since its variance and standarddeviation are greater thandeviation are greater than Stock AStock A's.'s.

This, however, is still only part of the picture because mostThis, however, is still only part of the picture because mostinvestors choose to hold securities as part of a diversifiedinvestors choose to hold securities as part of a diversifiedportfolio.portfolio.


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ortfolio Risk and ReturnPortfolio Risk and Return Most investors do nothold stocks in isolation.Most investors do nothold stocks in isolation.

Instead, they choose to hold a portfolio of severalInstead, they choose to hold a portfolio of severalstocks.stocks.

When this is the case, a portion of an individual stock'sWhen this is the case, a portion of an individual stock'srisk can be eliminated,risk can be eliminated, i.e.,i.e., diversified away.diversified away.

From our previous calculations, we know that:From our previous calculations, we know that: the expected return on Stock A is 12.5%the expected return on Stock A is 12.5%

the expected return on Stock B is 20%the expected return on Stock B is 20% the variance on Stock A is .00263the variance on Stock A is .00263

the variance on Stock B is .04200the variance on Stock B is .04200

the standard deviation on Stock A is 5.12%the standard deviation on Stock A is 5.12%

the standard deviation on Stock B is 20.49%the standard deviation on Stock B is 20.49%


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Portfolio Risk and ReturnPortfolio Risk and Return TheThe Expected Return on a PortfolioExpected Return on a Portfolio is computed as theis computed as the

weighted average of the expected returns on the stocksweighted average of the expected returns on the stockswhich comprise the portfolio.which comprise the portfolio.

The weights reflect the proportion of the portfolio invested inThe weights reflect the proportion of the portfolio invested inthe stocks.the stocks.

This can be expressed as follows:This can be expressed as follows:NN

E[RE[Rpp] =] = 77 wwiiE[RE[Rii]]i=1i=1

Where:Where: E[RE[Rpp] = the expected return on the portfolio] = the expected return on the portfolio

N = the number of stocks in the portfolioN = the number of stocks in the portfolio

wwii = the proportion of the portfolio invested in stock i= the proportion of the portfolio invested in stock i

E[RE[Rii] = the expected return on stock i] = the expected return on stock i


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Portfolio Risk and ReturnPortfolio Risk and Return For a portfolio consisting of two assets, the aboveFor a portfolio consisting of two assets, the above

equation can be expressed as:equation can be expressed as:

E[RE[Rpp] = w] = w11E[RE[R11] + w] + w22E[RE[R22]]

If we have an equally weighted portfolio of stock AIf we have an equally weighted portfolio of stock Aand stock B (50% in each stock), then the expectedand stock B (50% in each stock), then the expected

return of th

e portfolio is:return of th

e portfolio is:E[RE[Rpp] = .50(.125) + .50(.20) = 16.25%] = .50(.125) + .50(.20) = 16.25%


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Portfolio Risk and ReturnPortfolio Risk and Return The variance/standard deviation of a portfolio reflects notThe variance/standard deviation of a portfolio reflects not

only the variance/standard deviation of the stocks that makeonly the variance/standard deviation of the stocks that makeup the portfolio but also how the returns on the stocks whichup the portfolio but also how the returns on the stocks which

comprise the portfolio vary together.comprise the portfolio vary together. Two measures ofhow the returns on a pair of stocks varyTwo measures ofhow the returns on a pair of stocks vary

together are the covariance and the correlation coefficient.together are the covariance and the correlation coefficient. Covariance is a measure that combines the variance of a stocksCovariance is a measure that combines the variance of a stocks

returns with the tendency of those returns to move up or down at thereturns with the tendency of those returns to move up or down at the

same time other stocks move up or down.same time other stocks move up or down. Since it is difficult to interpret the magnitude of the covariance terms,Since it is difficult to interpret the magnitude of the covariance terms,

a related statistic, the correlation coefficient, is often used to measurea related statistic, the correlation coefficient, is often used to measurethe degree of cothe degree of co--movement between two variables. The correlationmovement between two variables. The correlationcoefficient simply standardizes the covariance.coefficient simply standardizes the covariance.


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Portfolio Risk and ReturnPortfolio Risk and Return TheThe CovarianceCovariance between the returns on two stocks can bebetween the returns on two stocks can be

calculated as follows:calculated as follows:NN

Cov(RCov(RAA,R,RBB) =) = WWA,BA,B == 77 ppii(R(RAiAi -- E[RE[RAA])(R])(RBiBi -- E[RE[RBB])])i=1i=1

Where:Where: WW%&%& = the covariance between the returns on stocks A and B= the covariance between the returns on stocks A and B

N = the number of statesN = the number of states

ppii = the probability of state i= the probability of state i RRAiAi = the return on stock A in state i= the return on stock A in state i

E[RE[RAA] = the expected return on stock A] = the expected return on stock A

RRBiBi = the return on stock B in state i= the return on stock B in state i

E[RE[RBB] = the expected return on stock B] = the expected return on stock B


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Portfolio Risk and ReturnPortfolio Risk and Return TheThe Correlation CoefficientCorrelation Coefficient between the returns on twobetween the returns on two

stocks can be calculated as follows:stocks can be calculated as follows:

WWA,BA,B Cov(RCov(RAA,R,RBB))Corr(RCorr(RAA,R,RBB) =) = VVA,BA,B == WWAAWWBB = SD(R= SD(RAA)SD(R)SD(RBB))

Where:Where:

VVA,BA,B=the correlation coefficient between the returns on stocks A and B=the correlation coefficient between the returns on stocks A and B WWA,BA,B=the covariance between the returns on stocks A and B,=the covariance between the returns on stocks A and B,

WWAA=the standard deviation on stock A, and=the standard deviation on stock A, and

WWBB=the standard deviation on stock B=the standard deviation on stock B


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Portfolio Risk and ReturnPortfolio Risk and Return The covariance between stock A and stock B is as follows:The covariance between stock A and stock B is as follows:

WWA,BA,B = .2(.05= .2(.05--.125)(.5.125)(.5--.2) + .3(.1.2) + .3(.1--.125)(.3.125)(.3--.2) +.2) +.3(.15.3(.15--.125)(.1.125)(.1--.2) +.2(.2.2) +.2(.2--.125)(.125)(--.1.1--.2) =.2) = --.0105.0105

The correlation coefficient between stock A and stock B isThe correlation coefficient between stock A and stock B isas follows:as follows:

--.0105.0105

VVA,BA,B = (.0512)(.2049) == (.0512)(.2049) = --1.001.00


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Portfolio Risk and ReturnPortfolio Risk and Return Using either the correlation coefficient or the covariance,Using either the correlation coefficient or the covariance,

thethe Variance on a TwoVariance on a Two--Asset PortfolioAsset Portfolio can becan becalculated as follows:calculated as follows:

WW22pp = (w= (wAA))22WW22AA + (w+ (wBB))

22WW22BB + 2w+ 2wAAwwBBVVA,BA,B WWAAWWBBOROR

WW22pp = (w= (wAA))

22

WW22AA + (w+ (wBB))

22

WW22BB + 2w+ 2wAAwwBB WWA,BA,B

TheThe Standard Deviation of the PortfolioStandard Deviation of the Portfolio equals theequals thepositive square root of the the variance.positive square root of the the variance.


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Portfolio Risk and ReturnPortfolio Risk and Return Lets calculate the variance and standard deviation of aLets calculate the variance and standard deviation of a

portfolio comprised of 75% stock A and 25% stock B:portfolio comprised of 75% stock A and 25% stock B:

WW22pp =(.75)=(.75)

22

22

+(.25)+(.25)22

(.2049)(.2049)22

+2(.75)(.25)(+2(.75)(.25)(--1)(.0512)(.2049)= .000161)(.0512)(.2049)= .00016

WWpp = .00016 = .0128 = 1.28%= .00016 = .0128 = 1.28%

Notice that the portfolio formed by investing 75% in Stock ANotice that the portfolio formed by investing 75% in Stock A

and 25% in Stock B has a lower variance and standardand 25% in Stock B has a lower variance and standarddeviation than either Stocks A or B and the portfolio has adeviation than either Stocks A or B and the portfolio has ahigher expected return than Stock A.higher expected return than Stock A.

This is the purpose of diversification; by forming portfolios,This is the purpose of diversification; by forming portfolios,some of the risk inherent in the individual stocks can besome of the risk inherent in the individual stocks can be

eliminated.eliminated.


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Investment Management Assignment 1

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