44315933 portfolio markowitz model

8/3/2019 44315933 Portfolio Markowitz Model

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PORTFOLIOPORTFOLIO

MARKOWITZMARKOWITZMODELMODEL


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3

Risk and ReturnRisk and Return -- MPTMPT

Prior to the establishment of Modern PortfolioPrior to the establishment of Modern PortfolioTheory, most people only focused uponTheory, most people only focused uponinvestment returnsthey ignored risk.investment returnsthey ignored risk.

With MPT, investors had a tool that they couldWith MPT, investors had a tool that they coulduse to dramatically reduce the risk of theuse to dramatically reduce the risk of theportfolio without a significant reduction in theportfolio without a significant reduction in theexpected return of the portfolio.expected return of the portfolio.


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DIVERSIFICATIONDIVERSIFICATION

DiversificationDiversification is a riskis a risk--managementmanagementtechnique that mixes a wide variety oftechnique that mixes a wide variety ofinvestments within a portfolio in order toinvestments within a portfolio in order to

minimize the impact that any one securityminimize the impact that any one securitywill have on the overall performance ofwill have on the overall performance ofthe portfolio. Diversification lowers thethe portfolio. Diversification lowers therisk of your portfolio. Academics haverisk of your portfolio. Academics have

complex formulas to demonstrate how thiscomplex formulas to demonstrate how thisworks, but we can explain it clearly withworks, but we can explain it clearly withan example:an example:


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DIVERSIFICATIONDIVERSIFICATION

Suppose thatSuppose that you live on an island where the entireyou live on an island where the entireeconomy consists of only two companies: one sellseconomy consists of only two companies: one sellsumbrellas while the other sells sunscreen. If you investumbrellas while the other sells sunscreen. If you investyour entire portfolio in the company that sells umbrellas,your entire portfolio in the company that sells umbrellas,

you'll have strong performance during the rainy season, butyou'll have strong performance during the rainy season, butpoor performance when it's sunny outside. The reversepoor performance when it's sunny outside. The reverseoccurs with the sunscreen company, the alternativeoccurs with the sunscreen company, the alternativeinvestment; your portfolio will be high performance wheninvestment; your portfolio will be high performance whenthe sun is out, but it will tank when the clouds roll in.the sun is out, but it will tank when the clouds roll in.Chances are you'd rather have constant, steady returns.Chances are you'd rather have constant, steady returns.

The solution is to invest 50% in one company and 50% inThe solution is to invest 50% in one company and 50% inthe other.the other. Because you have diversified your portfolio, youBecause you have diversified your portfolio, youwill get decent performance year round instead of havingwill get decent performance year round instead of havingeither excellent or terrible performance depending on theeither excellent or terrible performance depending on theseason.season.


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DiversificationDiversification

Diversification of a portfolio isDiversification of a portfolio islogically a good idealogically a good idea

Virtually all stock portfolios seek toVirtually all stock portfolios seek todiversify in one respect or anotherdiversify in one respect or another

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TotalRisk

UniqueRisk

MarketRisk

10 15 205

Diversification and portfolioRisk

Number of stocks

Total Risk = Unique risk + Market risk

Relationship Between diversification andRelationship Between diversification and

RiskRisk

WWpp


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MARKET RISKMARKET RISK

TheThe market risk of a stock representsmarket risk of a stock representsthat portion of its risk which isthat portion of its risk which is

attributable to economyattributable to economy--wide factorswide factorslike the growth rate of GNP, the levellike the growth rate of GNP, the levelof government spending, moneyof government spending, moneysupply, interest rate structure, andsupply, interest rate structure, and

inflation rate. Market risk is alsoinflation rate. Market risk is alsoreferred to as systematic risk or nonreferred to as systematic risk or non--

diversifiable riskdiversifiable risk..


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UNIQUE RISKUNIQUE RISK

UniqueUnique risk of a security representsrisk of a security representsthat portion of its total risk whichthat portion of its total risk which

stems from firmstems from firm--specific factors likespecific factors likethe development of a new product, athe development of a new product, alabourlabour strike, or the emergence of astrike, or the emergence of anew competitor. Unique risk is alsonew competitor. Unique risk is also

referred to as diversifiable risk orreferred to as diversifiable risk orunsystematic risk.unsystematic risk.


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Carrying Your Eggs in MoreCarrying Your Eggs in More

Than One BasketThan One Basket Investments in your own egoInvestments in your own ego

The concept of risk aversion revisitedThe concept of risk aversion revisited

Multiple investment objectivesMultiple investment objectives

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Investments in Your Own EgoInvestments in Your Own Ego

Never put a large percentage ofNever put a large percentage ofinvestment funds into a single securityinvestment funds into a single security

If the security appreciates, the ego is strokedIf the security appreciates, the ego is strokedand this may plant a speculative seedand this may plant a speculative seed

If the security never moves, the ego views thisIf the security never moves, the ego views thisas neutral rather than an opportunity costas neutral rather than an opportunity cost

If the security declines, your ego has a veryIf the security declines, your ego has a verydifficult time letting godifficult time letting go

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The Concept ofThe Concept of

Risk Aversion RevisitedRisk Aversion Revisited Diversification is logicalDiversification is logical

If you drop the basket, all eggs breakIf you drop the basket, all eggs break

Diversification is mathematicallyDiversification is mathematicallysoundsound

Most people are risk averseMost people are risk averse

People take risks only if they believePeople take risks only if they believethey will be rewarded for taking themthey will be rewarded for taking them

1111


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The Concept of RiskThe Concept of Risk

Aversion Revisited (contd)Aversion Revisited (contd) Diversification is more important nowDiversification is more important now

Journal of FinanceJournal of Finance article shows thatarticle shows thatvolatility of individual firms hasvolatility of individual firms hasincreasedincreased

Investors need more stocks to adequatelyInvestors need more stocks to adequatelydiversifydiversify

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Multiple Investment ObjectivesMultiple Investment Objectives

Multiple objectives justify carryingMultiple objectives justify carryingyour eggs in more than one basketyour eggs in more than one basket

Some people find mutual fundsSome people find mutual fundsunexcitingunexciting

Many investors hold their investmentMany investors hold their investmentfunds in more than one account so thatfunds in more than one account so that

they can play with part of the totalthey can play with part of the total E.g., a retirement account and a separateE.g., a retirement account and a separate

brokerage account for trading individualbrokerage account for trading individualsecuritiessecurities

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Portfolio return and riskPortfolio return and risk

Portfolio expected returnPortfolio expected return

The expected return on a portfolio isThe expected return on a portfolio is

simply the weighted average of thesimply the weighted average of theexpected returns on the individualexpected returns on the individualsecurities in the portfolio:securities in the portfolio:


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Portfolio Expected ReturnPortfolio Expected Return

iRE

iw

REwRE

i

i

n

i

iip

assetonreturnexpectedtheis)(

securityofweighttheis

portfolioon thereturnexpectedtheis)E(R:Where

)()(

p

1

!

!

!

!!


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exampleexample

A portfolio consists of four securities A,B,CA portfolio consists of four securities A,B,C

and D with expected returns of12%,15%,and D with expected returns of12%,15%,

18% and 20% respectively. The proportions of18% and 20% respectively. The proportions of

portfolio value invested in these securities areportfolio value invested in these securities are

0.20,0.30,0.30 and 0.20 respectively. The0.20,0.30,0.30 and 0.20 respectively. The

expected return on the portfolio is :expected return on the portfolio is :

E(E(RpRp)= 0.20(12%)+0.30(15%)+0.30(18%)+0.20(20%))= 0.20(12%)+0.30(15%)+0.30(18%)+0.20(20%)

== 16.3%16.3%


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Measurement of coMeasurement of co--movements inmovements in

security returnssecurity returns

CoCo--movements between the returnsmovements between the returns

of securities are measured byof securities are measured bycovariance and coefficient ofcovariance and coefficient ofcorrelation.correlation.


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covariancecovariance

Covariance reflects the degree to whichCovariance reflects the degree to whichthe returns of the two securities vary orthe returns of the two securities vary orchange together. A positive covariancechange together. A positive covariance

means that the returns of the twomeans that the returns of the twosecurities move in the same directionsecurities move in the same directionwhereas a negative covariance implieswhereas a negative covariance impliesthat the returns of the two securitiesthat the returns of the two securities

move in opposite direction. The covariancemove in opposite direction. The covariancebetween any two securitiesbetween any two securities ii and j isand j iscalculated as follows:calculated as follows:


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covariancecovariance

COV(COV(RRii,, RRjj)=)= PP11[R[Ri1i1 E(E(RRii)] [R)] [Rj1j1 E(E(RRjj)])]

+ P+ P22[R[Ri2i2 E(E(RiRi)] [R)] [Rj2j2 E(E(RRjj)])]


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exampleexample

The returns on securities 1 and 2The returns on securities 1 and 2under five possible states of natureunder five possible states of natureare given below:are given below:

State ofnature

probability Return onsecurity 1(%)

Return onsecurity 2(%)

1 0.10 -10 52 0.30 15 12

3 0.30 18 19

4 0.20 22 15

5 0.10 27 12


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solutionsolution

The expected return on security 1 is:The expected return on security 1 is:

E(RE(R11) =) =

0.10(0.10(--10%)+0.30(15%)+0.30(18%)+10%)+0.30(15%)+0.30(18%)+0.20(22%)+0.10(27%) = 16%0.20(22%)+0.10(27%) = 16%

The expected return on securityThe expected return on security 22 is:is:

E(RE(R22)) ==

0.10(5%)+0.30(12%)+0.30(19%)+0.10(5%)+0.30(12%)+0.30(19%)+

0.20(15%)+0.10(12%)0.20(15%)+0.10(12%) == 14%14%


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The covariance between the returns on securitiesThe covariance between the returns on securities

1 And 2 is calculated below :1 And 2 is calculated below :

State ofnature

probability Return onsecurity 1

(%)

Deviationof thereturn onsecurity 1

Return onsecurity 2

(%)

Deviationof thereturn onsecurity 2

Product ofthedeviationstimesprobability

(1) (2) (3) (4) (5) (6) (2)*(4)*(6)

1 0.10 -10 -26 5 -9 23.4

2 0.30 15 -1 12 2 0.6

3 0.30 18 2 19 5 3.0

4 0.20 22 6 15 1 1.2

5 0.10 27 11 12 -2 -2.2Sum=26.0

Thus the covariance between the returns on the two securities is 26.0


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Coefficient of correlationCoefficient of correlation

jandiuritiesonreturnstheofdeviationsdardsthearejiand

jandiuritiesonreturnsthebetweenariancetheisRjRi

jandiuritiesonreturnsthe

betweentcoefficienncorrelatiotheisRjRiCorwhere

ji

RjRiRjRiCor

sectan.

seccov),cov(

sec

),(

.

),cov(),(

!

!

!

!

WW

WW


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Example of Portfolio CombinationsExample of Portfolio Combinations

and Correlationand Correlation

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 1

B 14.0% 40.0%

Weight of A Weight of B

Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 17.5%

80.00% 20.00% 6.80% 20.0%

70.00% 30.00% 7.70% 22.5%

60.00% 40.00% 8.60% 25.0%50.00% 50.00% 9.50% 27.5%

40.00% 60.00% 10.40% 30.0%

30.00% 70.00% 11.30% 32.5%

20.00% 80.00% 12.20% 35.0%

10.00% 90.00% 13.10% 37.5%

0.00% 100.00% 14.00% 40.0%

Portfolio Components Portfolio Characteristics

Perfect PositiveCorrelation nodiversification


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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 15.9%

80.00% 20.00% 6.80% 17.4%

70.00% 30.00% 7.70% 19.5%

60.00% 40.00% 8.60% 21.9%

50.00% 50.00% 9.50% 24.6%

40.00% 60.00% 10.40% 27.5%

30.00% 70.00% 11.30% 30.5%

20.00% 80.00% 12.20% 33.6%

10.00% 90.00% 13.10% 36.8%

0.00% 100.00% 14.00% 40.0%


PositiveCorrelation

weakdiversification

potential


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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 14.1%

80.00% 20.00% 6.80% 14.4%

70.00% 30.00% 7.70% 15.9%

60.00% 40.00% 8.60% 18.4%

50.00% 50.00% 9.50% 21.4%

40.00% 60.00% 10.40% 24.7%

30.00% 70.00% 11.30% 28.4%

20.00% 80.00% 12.20% 32.1%

10.00% 90.00% 13.10% 36.0%

0.00% 100.00% 14.00% 40.0%


No Correlation some

diversificationpotential

Lower

risk than

asset A


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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 12.0%

80.00% 20.00% 6.80% 10.6%

70.00% 30.00% 7.70% 11.3%

60.00% 40.00% 8.60% 13.9%

50.00% 50.00% 9.50% 17.5%

40.00% 60.00% 10.40% 21.6%

30.00% 70.00% 11.30% 26.0%

20.00% 80.00% 12.20% 30.6%

10.00% 90.00% 13.10% 35.3%

0.00% 100.00% 14.00% 40.0%


NegativeCorrelation

greaterdiversification potential


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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -1

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 9.5%

80.00% 20.00% 6.80% 4.0%

70.00% 30.00% 7.70% 1.5%

60.00% 40.00% 8.60% 7.0%

50.00% 50.00% 9.50% 12.5%

40.00% 60.00% 10.40% 18.0%

30.00% 70.00% 11.30% 23.5%

20.00% 80.00% 12.20% 29.0%

10.00% 90.00% 13.10% 34.5%

0.00% 100.00% 14.00% 40.0%


PerfectNegative

Correlation greatestdiversification potential

Risk of the

portfolio is

almost

eliminated at70% asset A


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The Effect of Correlation on Portfolio Risk:The Two-Asset Case

Expected Return

Standard Deviation

0%

0% 10%

4%

8%

20% 30% 40%

12%

B

VAB= +1

A

VAB = 0

VAB = -0.5

VAB = -1

Diversification of a Two Asset Portfolio Demonstrated Graphically


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Grouping Individual Assets intoGrouping Individual Assets into

PortfoliosPortfolios

The riskiness of a portfolio that is made ofThe riskiness of a portfolio that is made ofdifferent risky assets is a function of threedifferent risky assets is a function of threedifferent factors:different factors:

the riskiness of the individual assets that makethe riskiness of the individual assets that makeup the portfolioup the portfolio

the relative weights of the assets in thethe relative weights of the assets in theportfolioportfolio

the degree of cothe degree of co--movement of returns of themovement of returns of theassets making up the portfolioassets making up the portfolio


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Risk of aRisk of a TwoTwo--assetasset PortfolioPortfolio

The standard deviation of a twoThe standard deviation of a two--assetassetportfolio may be measured using theportfolio may be measured using theMarkowitz model:Markowitz model:

BABABABBAAp wwwwWW

VWWW

,

22222!


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exampleexample

A portfolio consists of two securities,1 and 2 in theA portfolio consists of two securities,1 and 2 in the

proportions 0.6 and 0.4. the standard deviations of the returnsproportions 0.6 and 0.4. the standard deviations of the returns

on securities 1 and 2 areon securities 1 and 2 are 1 = 10 and1 = 10 and 2 = 16. The coefficient2 = 16. The coefficient

of correlation between the returns on securities 1 and 2 is 0.5.of correlation between the returns on securities 1 and 2 is 0.5.

What is the standard deviation of the portfolio returns?What is the standard deviation of the portfolio returns?

p = [0.6*10 + 0.4*16+2*0.6*0.4*0.5*10*16]p = [0.6*10 + 0.4*16+2*0.6*0.4*0.5*10*16]

= 10.7%= 10.7%


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Risk of a ThreeRisk of a Three--asset Portfolioasset Portfolio

CACACACBCBCBBABABACCBBAApwwwwwwwww WWVWWVWWVWWWW

,,,

222222222 !

The data requirements for a three-asset portfoliogrows dramatically if we are using MarkowitzPortfolio selection formulae.

We need 3 (three) correlation coefficients between Aand B; A and C; and B and C.

A

B C

a,b

b,c

a,c


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Risk of a FourRisk of a Four--asset Portfolioasset Portfolio

The data requirements for a four-asset portfolio growsdramatically if we are using Markowitz Portfolio selectionformulae.

We need 6 correlation coefficients between A and B; A andC; A and D; B and C; C and D; and B and D.

A

C

B D

a,b a,d

b,c c,d

a,cb,d


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EFFICIENT FRONTIEREFFICIENT FRONTIER

The efficient frontier describes the relationshipThe efficient frontier describes the relationshipbetween the return that can be expected from abetween the return that can be expected from aportfolio and the riskiness (portfolio and the riskiness (volatilityvolatility) of the) of theportfolio. It can be drawn as a curve on a graphportfolio. It can be drawn as a curve on a graphof risk againstof risk against expected returnexpected return of a portfolio. Theof a portfolio. Theefficient frontier gives the best return that can beefficient frontier gives the best return that can beexpected for a given level of risk or the lowestexpected for a given level of risk or the lowest

level of risk needed to achieve a given expectedlevel of risk needed to achieve a given expectedrate of return.rate of return.


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The concept of an efficient frontier can beThe concept of an efficient frontier can beused to illustrate the benefits ofused to illustrate the benefits ofdiversificationdiversification. An undiversified portfolio. An undiversified portfolio

can be moved closer to the efficientcan be moved closer to the efficientfrontier by diversifying it. Diversificationfrontier by diversifying it. Diversificationcan, therefore, increase returns withoutcan, therefore, increase returns withoutincreasing risk, or reduce risk withoutincreasing risk, or reduce risk without

reducing expected returns.reducing expected returns.


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Feasible portfolioFeasible portfolio

AA feasible portfolio is a possible set offeasible portfolio is a possible set ofinvestments, which are chosen frominvestments, which are chosen fromavailable alternatives from within a setavailable alternatives from within a set

limited of anlimited of an investorsinvestors investmentinvestmentobjectives, risk tolerances, and capitalobjectives, risk tolerances, and capitalresources. Every feasible portfolio comesresources. Every feasible portfolio comeswith its own set of rewards and riskswith its own set of rewards and risks

profile, and is not really an efficient orprofile, and is not really an efficient orbalance portfolio.balance portfolio.


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TwoTwo--AssetAsset Markowitz Feasible SetMarkowitz Feasible Set


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OPTIMAL PORTFOLIOOPTIMAL PORTFOLIO

AnAn efficient portfolioefficient portfolio most preferredmost preferredby anby an investorinvestor because itsbecause itsriskrisk/reward characteristics/reward characteristicsapproximate theapproximate the investorinvestor's's utilityutilityfunctionfunction. A. A portfolioportfolio that maximizesthat maximizesanan investorinvestor's's preferencepreferences withs with

respect torespect to returnreturn andand riskrisk..


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OPTIMAL PORTFOLIOOPTIMAL PORTFOLIO


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CONCLUSIONCONCLUSION

The Modern Portfolio Theory still holds its relevance today,The Modern Portfolio Theory still holds its relevance today,and its precepts are routinely used by modern managers forand its precepts are routinely used by modern managers forportfolio management and investment analysis. The Theoryportfolio management and investment analysis. The Theoryis considered an important advancement in the field ofis considered an important advancement in the field ofmathematicalmathematical modelingmodeling of Finance. In its time itof Finance. In its time itrevolutionizedrevolutionized the investment world. Till then, investorsthe investment world. Till then, investorsmademade investmentinvestment only becauseonly because of the expected return. Theof the expected return. TheTheory, for the first time revealed that the risk of anTheory, for the first time revealed that the risk of aninvestment is as important as its return and that investinginvestment is as important as its return and that investingis a trade off between risk and return. Hence, if risk &is a trade off between risk and return. Hence, if risk &

return could be calculated, then investment can bereturn could be calculated, then investment can bediversified into different asset classes to maximize return &diversified into different asset classes to maximize return &minimize risk for an investor.minimize risk for an investor.


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44315933 portfolio markowitz model

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