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FACTOR MODELS(Chapter 6)

Markowitz Model

Employment of Factor Models

Essence of the Single-Factor Model

The Characteristic Line

Expected Return in the Single-Factor Model

Single-Factor Models Simplified Formula for

Portfolio Variance

Explained Versus Unexplained Variance

Multi-Factor Models

Models for Estimating Expected Return

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Markowitz Model

Problem: Tremendous data requirement.

Number of security variances needed = M.

Number of covariances needed = (M2 - M)/2

Total = M + (M2 - M)/2

Example: (100 securities)

100 + (10,000 - 100)/2 = 5,050

Therefore, in order for modern portfolio theory to be usable forlarge numbers of securities, the process had to be simplified.

(Years ago, computing capabilities were minimal)

-

!

! !

)(r......)r,Cov(r

....

....

)r,Cov(r...)(r)r,Cov(r

)r,Cov(r...)r,Cov(r)(r

)r,Cov(rxx)(r

M2

1M

M222

12

M12112

M

1j

M

1kkjkjp

2

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Employment of Factor Models

To generate the efficient set, we need estimates ofexpected return and the covariances between the

securities in the available population. Factor models

may be used in this regard.

Risk Factors (rate of inflation, growth in industrial

production, and other variables that induce stock prices

to go up and down.)

May be used toevaluate covariances ofreturn between

securities.

Expected Return Factors (firm size, liquidity, etc.)May be used toevaluateexpectedreturns ofthe securities.

In the discussion that follows, we first focus on risk

factor models. Then the discussion shifts to factors

affecting expected security returns.

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Essence of the Single-Factor Model

Fluctuations in the return of a security relative to that ofanother (i.e., the correlation between the two) do not depend

upon the individual characteristics of the two securities.

Instead, relationships (covariances) between securities occur

because of their individual relationships with the overall

market (i.e., covariance with the market).

IfStock (A) is positively correlated with the market, and if

Stock (B) is positively correlated with the market, then Stocks

(A) and (B) will be positively correlated with each other.

Given the assumption that covariances between securitiescan be accounted for by the pull of a single common factor

(the market), the covariance between any two stocks can be

written as:)(r)r,Cov(r M

2kjkj !

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The Characteristic Line(See Chapter 3 for a Review of the Statistics)

Relationship between the returns on an individualsecurity and the returns on the market portfolio:

Aj = intercept of the characteristic line (the expectedrate of return on stock (j) should the market happen

to produce a zero rate of return in any given period).

Fj

= beta of stock (j); the slope of the characteristic

line.

Ij,t = residual of stock (j) during period (t); the vertical

distance from the characteristic line.

(t)periodduring(j)stockonreturnofratetheisr

rAr

tj,

tj,tM,jjtj, !

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Graphical Display of the Characteristic

Line

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6

rj,t

rM,t

= Fj

Aj

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The Characteristic Line (Continued)

Note: A stocks return can be broken down into twoparts:

Movement along the characteristic line (changes

in the stocks returns caused by changes in the

markets returns).

Deviations from the characteristic line (changes in

the stocks returns caused by events unique to the

individual stock).

Movement along the line: Aj + FjrM,t

Deviation from the line: Ij,t

tj,tM,jjtj, rAr !

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Major Assumption of the Single-Factor Model

There is no relationship between the residuals of one

stock and the residuals of another stock (i.e., thecovariance between the residuals of every pair of

stocks is zero).0),Cov( kj !

-10

0

10

-10 0 10 20

Stock js Residuals (%)

Stock ks Residuals (%)

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Expected Return in the Single-Factor Model

Actual Returns:

Expected Residual:Given the characteristic line is truly the line of best

fit, the sum of the residuals would be equal to zero

Therefore, the expected value of the residual for

any given period would also be equal to zero:

Expected Returns:

Given the characteristic line, and an expected

residual of zero, the expected return of a security

according to the single-factor model would be:

tj,tM,jjtj, rAr !

0

n

1t

tj, !!

0)E(j !

)E(rA)E(rMjjj

!

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Single-Factor Models Simplified Formula for

Portfolio VarianceVarianceofan IndividualSecurity:

Given:

It Follows That:

!

!

n

1i

2jij,ij

2 )]E(r[rh)(r

)E(rA)E(r

rAr

Mjjj

ij,iM,jjij,

!

!

!!

!

!

n

1i

2ij,iij,MiM,

n

1i

ij

n

1=i

2MiM,i

2j

n

1=i

2ij,ij,MiM,j

2MiM,

2ji

n

1=i

2ij,MiM,ji

2Mjj

n

1i

ij,iM,jjij2

h)]E(r[rh2)]E(r[rh=

))]E(r[r2)]E(r[r(h=

))]E(r[r(h=

)])E(rA[]r([Ah)(r

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Note:

Therefore:

3)Chapter(SeeVarianceResidual)(h

.regressionsimpleinlinefitbest

theforzerotoequalis),Cov(rSince

0=

),Cov(r2=

0)E(Since

)]E()][E(r[rh2)]E(r[rh2

)(r)]E(r[rh

j2

n

1i

2ij,i

jM

jMj

j

jij,MiM,

n

1i

ijij,MiM,

n

1i

ij

n

1=i

M22

j2

MiM,i2j

!!

!

!

!

!

!!

)()(r)(r j2

M22

jj2

!

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Explained Vs. Unexplained Variance

(Systematic Vs. Unsystematic Risk)

Total Risk = Systematic Risk + UnsystematicRisk

Systematic: That part of total variance which

is explained by the variance in the markets

returns.

Unsystematic: The unexplained variance, orthat part of total variance which is due to the

stocks unique characteristics.

)()(r)(r j2

M22

jj2

!

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Note:

[i.e., Fj2W2(rM) is equal to the coefficient of

determination (the % of the variance in the securitys

returns explained by the variance in the marketsreturns) times the securitys total variance]

Total Variance = Explained + Unexplained

As the number of stocks in a portfolio increases, the

residual variance becomes smaller, and the

coefficient of determination becomes larger.

)(r)(r

)(r)(r:Therefore

)(r

)(r

)(r

)(r)(r

)(r

)r,Cov(r

j22

Mj,M22

j

jMj,Mj

M

jMj,

M2

MjMj,

M2

Mjj

!

!

!!!

)()(r=

)()(r)(r

p2p22 Mp,

p2

M22

pp2

!

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Explained Vs. Unexplained Variance

(A Graphical Display)

0

2

4

6

8

10

1 5 9 13 17

0

0.2

0.4

0.6

0.8

1

1.2

1 5 9 13 17

ResidualVariance

NumberofStocks

Coefficient ofDetermination

NumberofStocks

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Explained Vs. Unexplained Variance

(A Two Stock Portfolio Example)

-

-

)(),Cov(),Cov()(

)(r)(r)(r)(r

B2

AB

BAA

2

M22

BM2

AB

M

2

BAM

22

A

Covariance Matrix for

Explained Variance

Covariance Matrix for

Unexplained Variance

0),Cov( mingAssu

VariancedUnexplaine)(x)(x+

VarianceExplained

)(rxx2)(rx)(rx)(r

BA

B

22

BA

22

A

M2

BABAM22

B2BM

22A

2Ap

2

!

!

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Explained Vs. Unexplained Variance (A Two

Stock Portfolio Example) Continued

.overstatedbewillvarianceresidual0,),Cov(If

.dunderstatebewillvarianceresidual0,),Cov(If

)()(r=

)(x)(rx=

dUnexplaineExplained

)(x)(r)x(x)(r

BA

BA

p2

M22

p

j2

m

1=j

2jM

2

2m

1=j

jj

j2m

1j

2jM

22BBAAp

2

"

-

!

!

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A Note on Residual Variance

The Single-Factor Model assumes zero correlationbetween residuals:

In this case, portfolio residual variance is expressed

as:

In reality, firms residuals may be correlated with

each other. That is, extra-market events may impact

on many firms, and:

In this case, portfolio residual variance would be:

0),Cov( kj !

!!

m

1j

j22

jp2 )(x)(

0),Cov( kj {

! !!

!

m

1j

m

1jk

kjkj

m

1j

j22

jp2 ),Cov( xx2)(x)(

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Markowitz Model Versus the Single-Factor Model

(A Summaryof the Data Requirements)

Markowitz Model

Number of security variances = m

Number of covariances = (m2 - m)/2

Total = m + (m2 - m)/2

Example - 100 securities:

100 + (10,000 - 100)/2 = 5,050

Single-Factor Model

Number of betas = m

Number of residual variances = mPlus one estimate ofW2(rM)

Total = 2m + 1

Example - 100 securities:

2(100) + 1 = 201

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Multi-Factor Models

Recall the Single-Factor Models formula for portfolio

variance:

If there is positive covariance between the residuals

of stocks, residual variance would be high and thecoefficient of determination would be low. In this

case, a multi-factor model may be necessary in order

to reduce residual variance.

A Two Factor ModelExample

where: rg = growth rate in industrial production

rI = % change in an inflation index

)()(r=

)()(r)(r

p2

p22

Mp,

p2

M22

pp2

!

tj,tI,jI,tg,jg,jtj, rrAr !

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Two Factor Model Example - Continued

Once again, it is assumed that the covariance

between the residuals of the the individual stocks areequal to zero:

Furthermore, the following covariances are also

presumed:

Portfolio Variance in a Two Factor Model:

0),Cov( kj !

0)r,Cov(r

0),Cov(r

0),Cov(r

Ig

jI

jg

!

!

!

)()(r)(r)(r p2

I22

pI,g22

pg,p2 !

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where:

Note that if the covariances between the residuals ofthe individual securities are still significantly different

from zero, you may need to develop a different model

(perhaps a three, four, or five factor model).

0),Cov( Assuming

)(x)(

x

x

kj

j2

m

1j

2jp

2

jI,

m

1j

jpI,

jg,

m

1j

jpg,

!

!

!

!

!

!

!

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Note on the Assumption Cov(rg,rI ) = 0

Ifthe Cov(rg,rI) is not equal to zero, the twofactormodelbecomes a bit more complex. In general,fora twofactormodel, the systematic risk ofaportfolio can be computed using thefollowingcovariancematrix:

To simplify matters, we will assume that thefactors in a multi-factormodel are uncorrelated

with each other.

-

W

W

)Ir(2

)Ir,gCov(r)gr(2Fg,p

FI,p

Fg,p

FI,p

)Ir,gr(Covp,Ip,g2)Ir(22

p,I)gr(

22p,g)pr(

2 FFWFWF!W

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Models for Estimating Expected Return

One Simplistic Approach

Use past returns to predict expected futurereturns. Perhaps useful as a starting point.

Evidence indicates, however, that the future

frequently differs from the past. Therefore,

subjective adjustments to past patterns of returns

are required.

Systematic Risk Models

One FactorSystematic Risk Model:

Given a firms estimated characteristic line and an

estimate of the future return on the market, the

securitys expected return can be calculated.

)E(rA)E(r Mjjj !

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Models forEstimatingExpected Return

(Continued)

Two FactorS

ystematic Risk Model:

N FactorSystematic Risk Model:

Other Factors That MayBe Used in Predicting

Expected Return

Note that the author discusses numerous factors

in the text that may affect expected return. Areview of the literature, however, will reveal that

this subject is indeed controversial. In essence,

you can spend the rest of your lives trying to

determine the best factors to use. The following

summarizes some of the evidence.

)E(r)E(rA)E(r 2j2,1j1,jj !

)E(r+...)E(r)E(rA)E(r NjN,2j2,1j1,jj !

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Other Factors That May Be Used in Predicting

Expected Return

Liquidity (e.g.,bid-asked spread)

Negatively related to return [e.g., Low liquiditystocks (high bid-asked spreads) should provide

higher returns to compensate investors for the

additional risk involved.]

ValueStock Versus Growth StockP/E Ratios

Low P/E stocks (value stocks) tend to

outperform high P/E stocks (growth stocks).

Price/(Book Value) Low Price/(Book Value) stocks (value stocks)

tend to outperform high Price/(Book Value)

stocks (growth stocks).

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Other Factors That May Be Used in Predicting Expected

Return(continued)

TechnicalAnalysisAnalyze past patterns of market data (e.g., pricechanges) in order to predict future patterns ofmarket data. Volumes have been written on thissubject.

Size EffectReturns on small stocks (small market value) tendto be superior to returns on large stocks. Note:Small NYSE stocks tend to outperform smallNASDAQ stocks.

January Effect

Abnormally high returns tend to be earned(especially on small stocks) during the month ofJanuary.

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Other Factors That May Be Used in Predicting Expected

Return(continued)

And the List Goes On

If you are truly interested in factors that

affect expected return, spend time in the

library reading articles in Financial Analysts

Journal, Journal of Portfolio Management,

and numerous other academic journals.

This could be an ongoing venture the rest

of your life.

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Building a Multi-FactorExpected Return Model:

One Possible Approach

Estimate the historical relationship between returnand chosen variables. Then use this relationship to

predict future returns.

Historical Relationship:

Future Estimate:

...(Beta)a+Size)(FirmaRatio)(P/Eaar

1t3

1t21t10t

!

...(Beta)a+

Size)(FirmaRatio)(P/Eaar

t3

t2t101t

!

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Using the Markowitz and Factor Models

to Make Asset Allocation Decisions

Asset Allocation Decisions

Portfoliooptimization is widely

employed to allocatemoney between

themajorclasses of investments: Large capitalization domestic stocks

Small capitalization domestic stocks

Domestic bonds

International stocks

Internationalbonds

Realestate

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Using the Markowitz and Factor Models

to Make Asset Allocation Decisions Continued:

Strategic Versus Tactical Asset Allocation

Strategic Asset Allocation

Decisions relate torelative amounts invested

in different asset classes over thelong-term.

Rebalancingoccurs periodically toreflect

changes in assumptions regardinglong-term

risk andreturn, changes in therisk tolerance

ofthe investors, and changes in the weights of

the asset classes due to past realizedreturns.

TacticalAsset AllocationShort-term asset allocation decisions basedon

changes in economic andfinancial conditions,

and assessments as to whethermarkets are

currently underpricedoroverpriced.

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Using the Markowitz and Factor Models

to Make Asset Allocation Decisions Continued

Markowitz Full Covariance ModelUse to allocate investments in the portfolio among thevarious classes ofinvestments (e.g., stocks,bonds,cash). Note that the numberofclasses is usually rathersmall.

FactorModels

Use todetermine which individual securities to includein thevarious asset classes. The numberofsecuritiesavailablemay bequitelarge. Expectedreturn factormodels could alsobeemployed to provide inputsregardingexpectedreturn into the Markowitz model.

FurtherInformationInterestedreaders may referto Chapter7,AssetAllocation,fora more indepth discussion ofthissubject. In addition, the authorhas provided handson examples ofmanipulatingdata using thePManagersoftware in the process ofmaking asset

allocation decisions.