optimal portfolio selection

8/6/2019 Optimal Portfolio Selection

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OptimalOptimal PortfolioPortfolioSelectionSelection


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ReturnReturnReturn :Return : It is the primary motivating forces that drivesIt is the primary motivating forces that drives

investment.investment. It represents the reward for undertaking the investment.It represents the reward for undertaking the investment.

In security analysis we are primarily and particularlyIn security analysis we are primarily and particularlyconcerned with returns from investors perspective.concerned with returns from investors perspective.

The return of an investment consists two components : (i)The return of an investment consists two components : (i)Current return (ii) Capital return.Current return (ii) Capital return.

(i)(i) Current returnCurrent return: Periodic cash flow (income) such as: Periodic cash flow (income) such as

dividend and interest. This can be zero or positive.dividend and interest. This can be zero or positive.(ii)(ii) Capital returnCapital return: The price appreciation or price changes.: The price appreciation or price changes.This can be zero, positive and negative also.This can be zero, positive and negative also.

Thus Total Return = Current return + Capital returnThus Total Return = Current return + Capital return


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Historical (exHistorical (ex--post) Returnpost) Return

Historical capital return excluding Dividend.Historical capital return excluding Dividend.Period : 0Period : 0 11 22 33 44

Price : 100Price : 100 110110 108108 130130 115115Solution: RSolution: R11 = (P= (P11--PP00)/P)/P00 =(110=(110 --100)/100 =0.10=10%100)/100 =0.10=10%


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Historical capital return including Dividend.Historical capital return including Dividend.Period : 0Period : 0 11 22 33 44

Price : 100Price : 100 110110 108108 130130 115115Dividend:Dividend: 55 77 88 33Solution:RSolution:R11 = [(P= [(P11--PP00)+D)+D11]/P]/P00 =[(110=[(110 --100)+5]/100100)+5]/100

=0.15=15%=0.15=15%This return will give the investors an insight or theThis return will give the investors an insight or the

prediction about the future return.prediction about the future return.


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Historical ( exHistorical ( ex--post) Riskpost) Risk

The majority of investors tend to emphasize theThe majority of investors tend to emphasize thereturn. They also tend to view the risk inreturn. They also tend to view the risk insubjective as well as comparative term.subjective as well as comparative term.

Suppose you are evaluating two shares A & B forSuppose you are evaluating two shares A & B for

investment. You have collected data of returninvestment. You have collected data of returnearned for the last 5 years.earned for the last 5 years.Stock A:Stock A: 30%30% 28%28% 34%34% 32%32% 31%31%Stock B:Stock B: 26%26% 13%13% 48%48% 11%11% 57%57%

You have to choose one stock among these two.You have to choose one stock among these two.


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In this context , we interpret risk essentially in the terms ofIn this context , we interpret risk essentially in the terms ofthe variability of the security return. The most commonthe variability of the security return. The most common

measures of risk ness of security is SD and Variance ofmeasures of risk ness of security is SD and Variance ofreturn.return.Given below returns of two stocks X and Y.Given below returns of two stocks X and Y.PeriodPeriod Return of stock X(%)Return of stock X(%) Return ofReturn of stockYstockY(%)(%)

11 --66 4422 33 6633 1010 1111

44 1313 151555 1616 1919

Calculate which stock is more risky ?Calculate which stock is more risky ?


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Mean of X = 36/5 = 7.2 =XMean of X = 36/5 = 7.2 =XMean of Y = 55/5 =11=YMean of Y = 55/5 =11=Y

Variance of X = (XVariance of X = (X-- X)X)22/(n/(n --1)= 310.80/(51)= 310.80/(5--1)= 77.7 and the S.D1)= 77.7 and the S.D=77.7=8.815=77.7=8.815

PeriodPeriod X X YY (X (X X)X) (Y (Y Y)Y) (X (X X)X)22 (Y(Y Y)Y)22

11 --66 44 --13.213.2 -7 174.24 494922 33 66 --4.24.2 --55 17.6417.64 252533 1010 1111 2.82.8 00 7.847.84 0044 1313 1515 5.85.8 44 33.6433.64 161655 1616 1919 8.88.8 88 77.4477.44 6464

SumSum 3636 5555 310.84310.84 154154


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Measuring Expected (exMeasuring Expected (ex--ante) Returnante) ReturnWhen we invest in a stock we try to anticipate the futureWhen we invest in a stock we try to anticipate the future

streams return based on the past performance of thestreams return based on the past performance of thestock. It may bestock. It may be --5%, 15% or 35%.5%, 15% or 35%.Further the likely hood of these possible returns canFurther the likely hood of these possible returns can

vary. Hence we should think in terms of probabilityvary. Hence we should think in terms of probability

distribution.distribution.The probability of an events represents the likelihood ofThe probability of an events represents the likelihood of

its occurrence. For example there is a 70% chance thatits occurrence. For example there is a 70% chance that

the price of the stock will increase and 30% chancethe price of the stock will increase and 30% chancethat the price of the stock will not increase during thethat the price of the stock will not increase during thenext quarternext quarter

..


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EconomicEconomicscenarioscenario

Probability ofProbability ofoccurrenceoccurrence

Return fromReturn fromstock A(%)stock A(%)

BoomBoom 0.250.25 3636

StagnationStagnation 0.500.50 2626

RecessionRecession 0.250.25 1212


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The expected return would be E(R) =The expected return would be E(R) =(0.25*36)+(0.50*26)+(0.25*12) = 25%(0.25*36)+(0.50*26)+(0.25*12) = 25%

The risk of the stock :The risk of the stock :22 =P=Pii *[*[RRiiE(R)]E(R)]22

=(36=(36--25)25)22 0.250.25 + (26+ (26--25)25)22 0.50 + (120.50 + (12--25)25)22

0.250.25= 73%= 73%SD = 8.54%SD = 8.54%


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Reduction of risk throughReduction of risk through

DiversificationDiversificationInvestment Analysis:Investment Analysis: Measuring risk & return.Measuring risk & return.PortfolioPortfolio MgtMgt:: minimize risk or maximize return.minimize risk or maximize return.

Example:Example:Concept of Covariance:Concept of Covariance: the degree to which thethe degree to which the

return of two securities vary or change together.return of two securities vary or change together.

Concept of Correlation:Concept of Correlation: the degree of relationshipthe degree of relationshipbetween two variables.between two variables.


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Portfolio Risk & ReturnPortfolio Risk & Return

The performance of the three stocks A, B and CThe performance of the three stocks A, B and Cfor the four years are given below. Compute thefor the four years are given below. Compute theaverage return, variance and S.D.average return, variance and S.D.

PeriodPeriod A(%)A(%) B(%)B(%) C(%)C(%)

11 1010 1111 88

22 1212 99 121233 1414 1313 99

44 1616 1717 1515


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CovCovabab =22/n=22/n--1 =7.31 =7.3CovCovacac = 6= 6

CovCovbcbc = 6= 6

__

(A(A A)A)__

(B(B B)B)__

(C(C C)C)11 22 11 33 22 33

-- 33 -- 1.51.5 -- 33 4.54.5 99 4.54.5

-- 11 -- 3.53.5 11 3.53.5 -- 11 -- 3.53.5

11 0.50.5 -- 22 0.50.5 -- 22 -- 11

33 4.54.5 44 13.513.5 1212 1818


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WWAA=50%, W=50%, WBB=20%, W=20%, WCC =30%=30%_ _ __ _ _

Return of the portfolio = AReturn of the portfolio = A WWAA + B+ B WWBB +C+C WcWc=12.3%=12.3%

Variance of portfolio(Variance of portfolio(22) =) = 22AA WW22AA ++ 22BB WW22 BB ++

22CC WW22CC + 2[+ 2[ CovCovABAB WWAA WWBB ++ CovCovBCBC WWBB WWCC ++CovCovACAC WWAA WWCC]]

==7.02%7.02%

S.D. = 2.65%S.D. = 2.65%


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The expected return would beThe expected return would beE(A) = (0.40*15)+(0.35*12)+(0.25*8) = 12.2%E(A) = (0.40*15)+(0.35*12)+(0.25*8) = 12.2%SD(A) = 2.75%SD(A) = 2.75%E(B) = 12.45% SD (B) = 1.24%E(B) = 12.45% SD (B) = 1.24%E(C) = 9.85% SD(C) = 2.82%E(C) = 9.85% SD(C) = 2.82%

EconomicEconomicscenarioscenario

Probability ofProbability ofoccurrenceoccurrence

A(%)A(%) B(%)B(%) C(%)C(%)

BoomBoom 0.400.40 1515 1111 1313StagnationStagnation 0.350.35 1212 1313 99

RecessionRecession 0.250.25 88 1414 66


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Portfolio Return = ?Portfolio Return = ?

Portfolio Risk = ?Portfolio Risk = ?

AA--E(A)E(A) BB--E(B)E(B) CC--E(C)E(C) 1122PP 2233PP 1122PP

2.82.8 --1.451.45 3.153.15 -- 1.621.62 -- 1.831.83 3.533.53

-- 0.20.2 0.550.55 -- 0.850.85 -- 0.040.04 --0.160.16 0.060.06

-- 4.24.2 1.551.55 -- 3.853.85 --1.631.63 --1.491.49 4.044.04

COVCOV --3.293.29 --3.483.48 7.637.63


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Decomposition ofRiskDecomposition ofRisk

Systematic riskSystematic risk: This risk refers to that portion of: This risk refers to that portion oftotal variability of return caused by the factors affectingtotal variability of return caused by the factors affectingthe price of all securities.the price of all securities.

This risk affects the market as a whole. The economicThis risk affects the market as a whole. The economicconditions, political situations and the sociologicalconditions, political situations and the sociologicalchanges affect the security market.changes affect the security market.Example : A steep increase in the international oil pricesExample : A steep increase in the international oil pricesis almost certain to affect the entire market adversely.is almost certain to affect the entire market adversely. Unsystematic risk :Unsystematic risk :This risk is the portion of totalThis risk is the portion of totalrisk that is unique to a firm or industry. This risk is alsorisk that is unique to a firm or industry. This risk is also

called diversifiable risk. Contcalled diversifiable risk. Cont


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Example : managerial inefficiency, technologicalExample : managerial inefficiency, technological

changes, availability of raw materials, change inchanges, availability of raw materials, change incustomer preferences, labor strikes, unexpectedcustomer preferences, labor strikes, unexpectedentry of new competitor.entry of new competitor.

The nature and magnitude differ from industry toThe nature and magnitude differ from industry toindustry and company to company.industry and company to company.


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Unsystematic riskUnsystematic risk

Business riskBusiness risk::This risk caused by the operatingThis risk caused by the operating

environments of the business. This risk can be divided intoenvironments of the business. This risk can be divided intotwo broad categoriestwo broad categories -- external and internal business risk.external and internal business risk.

(i) Internal business risk is largely associated with the efficiency(i) Internal business risk is largely associated with the efficiencywith which a firm conducts its operations within the broadwith which a firm conducts its operations within the broadoperating environment.operating environment.

(ii) External business risk is result of operating conditions(ii) External business risk is result of operating conditionsimposed upon the firm by circumstances beyond its control.imposed upon the firm by circumstances beyond its control.

Financial riskFinancial risk: This risk associated with the way in which a: This risk associated with the way in which acompany finance its activities.We usually gauge financial riskcompany finance its activities.We usually gauge financial riskby looking at the capital structure of the firm.by looking at the capital structure of the firm.


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BetaBeta


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Beta EstimationBeta Estimation

Month Return of X Ltd. Market return

January 23% 21%

February -14% -12%

March 18% 13%

April -9% -11%

May 16% -19%

June 7% 5%

The following Table gives the rate of return of X Ltd. and the market

over a period of time. Calculate beta of X Ltd.

The characteristic line is the regression line of best fit through a

scatter plot of rate of return for the individual risky asset and for

the market portfolio of risky assets over some designated past

period.


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Three Key DecisionsThree Key Decisions

(i)(i) The length of estimation periodThe length of estimation period(ii)(ii) The return intervalThe return interval(iii)(iii)The choice of market index.The choice of market index.


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Fundamental DeterminantsFundamental Determinants

(i)(i) The type of businessThe type of business(ii)(ii) Degree of OperatingDegree of Operating LLeverageeverage

(iii)(iii) Degree of Financial LeverageDegree of Financial Leverage

Beta Smoothing:Beta Smoothing:


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Portfolio Beta:Portfolio Beta:

CompanyCompany BetaBeta ProportionProportion Weighted BetaWeighted Beta

InfosysInfosys 1.371.37 35%35% 0.47950.4795ICICIICICI 0.990.99 20%20% 0.19800.1980

RanbaxyRanbaxy 0.910.91 20%20% 0.18200.1820

TISCOTISCO 1.191.19 10%10% 0.11900.1190GACLGACL 0.950.95 15%15% 0.14250.1425

Portfolio BetaPortfolio Beta 100%100% 1.21101.2110


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Capital Asset Pricing ModelCapital Asset Pricing Model

CAPM: It is an equation that express the equilibriumCAPM: It is an equation that express the equilibrium

relationship between securities required return & itsrelationship between securities required return & itssystematic risk or beta.systematic risk or beta.

RRAA == RRff+ Beta (+ Beta (RRmm RRff))

Expected rate of return:Expected rate of return: It is the rate of return from anIt is the rate of return from anasset that investors anticipate or expect to earn overasset that investors anticipate or expect to earn oversome future time.some future time.

Required rate of return:Required rate of return: It is the minimum rate ofIt is the minimum rate ofreturn needed to induce to purchase a security.return needed to induce to purchase a security.


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AlphaAlpha

Mathematically this is the difference betweenMathematically this is the difference betweenthe expected return & the required rate returnthe expected return & the required rate returnfrom a security or a portfolio.from a security or a portfolio.

A high positive alpha indicates that the securityA high positive alpha indicates that the securityis undervalued & vice versa.is undervalued & vice versa. Investors invest in a stock having higher alphaInvestors invest in a stock having higher alpha

given the beta level of the stock suits their riskgiven the beta level of the stock suits their riskappetite.appetite.


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The Risk free rate of return is 4% . Which stock shouldThe Risk free rate of return is 4% . Which stock shouldwe select?we select?

Condition Prob HPCL BPCL BSE 200

Recession &

High Interest

0.20 - 13% - 4% - 9%

Recession &Low Interest

0.15 16% - 2% 8%

Boom & HighInterest

0.40 32% 21% 16%

Boom & LowInterest

0.25 12% 20% 20%


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Security Market LineSecurity Market Line The SML represents the average or normal trade off between risk andThe SML represents the average or normal trade off between risk and

return for a group of securities, where the risk is measured typically inreturn for a group of securities, where the risk is measured typically interms of the security beta.terms of the security beta.So, we can say it is the graphical representation of the CAPM model.So, we can say it is the graphical representation of the CAPM model.

Application of SML :Application of SML :

oo Large positive alpha indicates above normal performance andLarge positive alpha indicates above normal performance and

negative alpha indicate below normal performance.negative alpha indicate below normal performance.oo If the alpha of a stock is positive we can conclude that theIf the alpha of a stock is positive we can conclude that thestock is underpriced and indication of buying signal and vicestock is underpriced and indication of buying signal and vice

versa.versa.oo This measure we can also apply in measuring the performanceThis measure we can also apply in measuring the performance

of a portfolio.of a portfolio.oo Drawbacks :Drawbacks : (i) beta depends upon the index(i) beta depends upon the index(ii) Predicting performance of the portfolio managers(ii) Predicting performance of the portfolio managers..


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Estimation ofRisk PremiumEstimation ofRisk Premium(i)(i) Large investors can be surveyed about their futureLarge investors can be surveyed about their future

expectations.expectations.(ii)(ii) The average actual premium earned over the pastThe average actual premium earned over the past

period.period.

(iii)(iii) Implied premium extracted from current marketImplied premium extracted from current marketdata.data.

The most common approach is the second one.The most common approach is the second one.

The premium is computed to be the differenceThe premium is computed to be the differencebetween average return on stocks and average returnbetween average return on stocks and average returnon risk free securities over the extended period ofon risk free securities over the extended period of

history.history.


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Major issuesMajor issues

(i)(i) Time period usedTime period used(ii)(ii) Arithmetic Average Vs. Geometric AverageArithmetic Average Vs. Geometric Average

(iii)(iii) Risk free rate (TRisk free rate (T--bill Vs. Tbill Vs. T--Bond /GBond /G--Sec).Sec).


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Equity Risk Premium in USEquity Risk Premium in US

T-Bill T-Bond

AM GM AM GM

1928-2004 7.92% 6.53% 6.02% 4.84%

1964-2004 5.82% 4.34% 4.59% 3.47%

1994-2004 8.60% 5.82% 6.85% 4.51%


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Equity Risk Premium inEquity Risk Premium in IndiaIndiaAM GM

1979-1991 15.34% 11.87%1991-2008 12.83% 6.52%

1979-2008 12.79% 9.75%

Reference:J.R. Varma & S Barua; IIMA Working Paper.They use Sensex as a proxy of market return and shortterm G-sec rate as a proxy of risk free rate of return.


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Risk Aversion & Utility ValueRisk Aversion & Utility Value

One would rank each portfolio as more attractiveOne would rank each portfolio as more attractivewhen the expected return is higher and lower whenwhen the expected return is higher and lower whenrisk is high.risk is high.

How can investor quantify the rate at which theyHow can investor quantify the rate at which theyare willing to tradeare willing to trade--off risk & return?off risk & return?

We willWe will assume that eachassume that each investor can assign ainvestor can assign autility score to competing investment portfoliosutility score to competing investment portfolios

based on expected risk & return.based on expected risk & return. Portfolio receives high utility scores for highPortfolio receives high utility scores for high

expected return and lower score for high volatility.expected return and lower score for high volatility.


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U = E(r)U = E(r) A A22

Where,Where,U = Utility ValueU = Utility ValueE(r) = Expected return of the portfolioE(r) = Expected return of the portfolio22= Variance of portfolio returns= Variance of portfolio returns

A = An index of investors risk aversion.A = An index of investors risk aversion.

A = 0 for risk neutral investorsA = 0 for risk neutral investorsA < 0 for risk loversA < 0 for risk loversA > 0 for risk averseA > 0 for risk averse


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Utility scores of alternative portfolios forUtility scores of alternative portfolios for

investors with varying degrees of risk aversioninvestors with varying degrees of risk aversion

Investors Risk

Aversion (A)

Portfolio L

E(r) = 0.07, = 0.05

Portfolio M

E(r) = 0.09, = 0.10

Portfolio H

E(r) = 0.13, = 0.20

2.0 0.07-1/2 20.052=0.0675 = 0.080 =0.09

3.5 =0.0656 =0.0725 =0.06

5.0 =0.0638 =0.065 =0.03

The portfolio lies with the highest utility scores that would be assignedby each investors appears in bold.

High risk portfolio would be chosen by the investors with the lowestdegree of risk aversion A = 2.


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Indifference Curve [A=4]Indifference Curve [A=4]

Expected Return S.D. Utility

0.10 0.200 0.10 0.5 40.2002=0.02

0.15 0.255 0.02

0.20 0.300 0.02

0.25 0.339 0.02

These equally preferred portfolios lie in the mean-S.D.

plane on a curve called the indifference curve that connectsall the portfolio points with same utility.


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Capital Market TheoryCapital Market Theory

Markowitz Efficient Frontier.Markowitz Efficient Frontier. Covariance & Correlation of risk free asset withCovariance & Correlation of risk free asset with

risky asset.risky asset. Combination of risky assets with risk free asset.Combination of risky assets with risk free asset. Risk return possibilities with leverage.Risk return possibilities with leverage. Lending & borrowing at risk free rate.Lending & borrowing at risk free rate. Dominant portfolioDominant portfolio The Separation Theorem.The Separation Theorem.


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Markowitz EfficientMarkowitz Efficient FrontierFrontier

Using this model an investor can identify a set ofUsing this model an investor can identify a set of

portfolios that maximize expected returns at eachportfolios that maximize expected returns at eachlevel of risk.level of risk.

A set of efficient portfolios thus obtained.A set of efficient portfolios thus obtained.

A portfolio is said to be an efficient portfolio, if itA portfolio is said to be an efficient portfolio, if itoffers the maximum expected return for a givenoffers the maximum expected return for a givenlevel of risk or the minimum risk for a given levellevel of risk or the minimum risk for a given level

of expected return.of expected return.


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Portfolio of one risky asset & a risk free assetPortfolio of one risky asset & a risk free asset

Suppose the investment budget, Y, to be allocatedSuppose the investment budget, Y, to be allocatedto risky portfolio P, the remaining (1to risky portfolio P, the remaining (1--Y) is to beY) is to beinvested in the risk free asset F.invested in the risk free asset F.

The base rate of return for any portfolio is the riskThe base rate of return for any portfolio is the riskfree rate plus the risk premium multiplied withfree rate plus the risk premium multiplied withproportion of risky assets.proportion of risky assets.

When we combined a risky asset & a risk free assetWhen we combined a risky asset & a risk free asset

in a portfolio, the S.D. of the portfolio would be thein a portfolio, the S.D. of the portfolio would be theS.D. of the risky asset multiplied by the weightage ofS.D. of the risky asset multiplied by the weightage ofthe risky asset in the portfolio.the risky asset in the portfolio.


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Risk tolerance & asset allocationRisk tolerance & asset allocation Individual investors have different risk aversion.Individual investors have different risk aversion. Given an identical opportunity set (risky & risk freeGiven an identical opportunity set (risky & risk free

securities) different investor will choose differentsecurities) different investor will choose differentposition in the risky asset.position in the risky asset.

In particular the more risk averse investors willIn particular the more risk averse investors willchoose to hold less of the risky asset & more riskchoose to hold less of the risky asset & more riskfree asset.free asset.

Investors attempts to maximizeInvestors attempts to maximize utility byutility by choosingchoosingthe best allocation to the risky asset.the best allocation to the risky asset.

Utilit l l f r ri p iti n f ri k tUtilit l l f r ri p iti n f ri k t


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Utility level for various position of risky assetUtility level for various position of risky asset

for an investor with risk aversion A=4for an investor with risk aversion A=4

Y E(Rc

) c

Utility0 0.07 0.00 0.07

0.1 0.078 0.022 0.077

0.2 0.086 0.044 0.0821

0.3 0.094 0.066 0.0853

0.4 0.102 0.088 0.0865

0.5 0.110 0.110 0.0858

0.6 0.118 0.132 0.0832

0.7 0.126 0.154 0.0736

0.8 0.134 0.176 0.07200.9 0.142 0.198 0.0636

1.00 0.150 0.220 0.0532

optimal portfolio selection

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