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Multifactor Models

Karl B. Diether

Fisher College of Business

Karl B. Diether (Fisher College of Business) Multifactor Models 1 / 29

Factors

A Factor

A variable that explains why a group of stocks have returns that tendto move together.

Equivalently a variable that helps explain common components of thevariance of security returns.

Example: Oil prices

When oil prices go up, expected cash flows on many firms change.

Expected cash flows for oil companies probably increase and theprobably decreases for chemical companies (they use oil as an input).


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Priced Factors

A priced or risk factor

A variable which helps explain the expected returns on a cross-section

of assets.Put another way: a common component of the variance of securityreturns which also contributes to the expected return.

We usually think of a priced factor in the context of a rationalrisk-based asset pricing model.

Example: The CAPM

The CAPM is a single risk (or priced) factor model.

The excess return on the market is the risk factor:

rit rft = iM(rMt rft) + it

E(rit) rft = iM[E(rMt) rft]


There Are Factors

There are factors in security returns?

Everyone agrees that there are factors in security returns.

Even if markets are totally irrational there will be factors as long asthe irrationality affected groups of stocks in common ways.

Controversy

Can we identify rational factors that explain the cross-section ofexpected returns?

For example, can we find a rational factor that explains why valuestocks usually do better than growth stocks?


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A Multifactor World

Suppose there are many factors that affect security returns

rit rft = i + i1F1t + i2F2t + + iKFKt + it

= i +K

j=1

ijFjt + it

F1t is the first factor

i1 is the beta of securityi

for the first factor.


Factor Betas

Synonyms

Factor loadings

Loadings

Factor Betas

Betas

Slopes

Coefficients

Sensitivities

Mathematics of Factor Betas

Just like CAPM betas, the factor beta of a portfolio is the weighted sum(based on the portfolio weights) of the individual security factor betas.


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Estimating Betas

Run a multiple (i.e., multiple dependent variables) regression

You can estimate the factor loadings (betas) using a regression:

rit rft = i +K

j=1

ijFjt + it

The regression estimates i, i1, i2, . . ., and iK.

We did this for the CAPM

We ran the following univariate regression to estimate alpha and beta:

rit rft = i + iM(rMt rft) + it


Types of Factors

Factors can be tradeable portfolios

They must be zero cost portfolios (weights add up to zero).

For example, in the CAPM the factor portfolio is 100% in the marketportfolio and -100% in the riskfree rate (rMt rft).

Factors can also be non-tradeable variables

Maybe GDP growth.

Maybe unexpected inflation.

In this course . . .

factors will always be zero cost portfolios.


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Zero Cost Portfolios

A zero cost portfolio

Any portfolio where the weights add up to zero.

Usually we put zero cost portfolios in units such that the positiveweights add up to 100% and the negative weights add up to -100%.

Examples

You go long IBM 100% and short T-bills 100%: ribm rf

You go 100% long in IBM and short 100% in GE: ribm rge

Also called a self financing portfolio

In frictionless market it requires no capital outlay.

Return on a zero cost portfolio

Synonyms: excess return or a differential return.


Completely Explaining Comovements in Returns

Suppose we have all the factors that affect securities returns. Then we canwrite the return on a security and the return on a well diversified portfolioas the following:

An Individual Security

rit rft = i +K

j=1

ijFjt + it

A Well Diversified Portfolio

rpt rft p +K

j=1

pjFjt

Why are the two equations different?


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Pure and Approximate

Approximate factor structure

rpt rft p +K

j=1

pjFjt,

Pure factor structure

rpt rft = p +K

j=1

pjFjt,

Perfectly DiversifiedA pure factor structure exists only for perfectly diversified portfolios.


Practical Uses of Factors

Controlling portfolio risk

Factor models can help you take the bets you want to take.

Factor models can help avoid the bets you dont want to take.

Maximizing Sharpe ratio

Can decompose returns into factors.

Build frontier from factors.

Much easier to estimate expected returns, variance, and covariancesusing factor than individual assets.

Estimates likely to much more precise because factors should be morestable and persistent over time.


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Multifactor Explanations

Rational asset pricing models

Rational asset pricing models explain expected returns by specifyingthe sources of risk and exposure to risk.

The CAPM is an asset pricing model that implies all differences inexpected returns should be due to differences in beta.

Unfortunately, the CAPM does not seem to explain the averagereturns we observe.

Empirical failures of the CAPM

Because of the failures of the CAPM we turn to multifactor asset pricingmodel called Arbitrage Pricing Theory (APT).


The APT: Assumptions

The APT needs very few assumptions

Returns follow a factor model.

There are no arbitrage opportunities.

There are lots of securities so that we can eliminate almost all firmspecific risk.

Arbitrage

The creation of riskless profits made possible by relative mispricingamong securities.

Getting return without risk.

An arbitrage opportunity arises if an investor can create a zero-costportfolio with positive return for certain in the future.


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The APT

In the APT, p 0

Consider a well diversified portfolio (call it P). If there are K factors

that completely describe all common movements in securities returnsfor the economy, then the following is true:

rp rf p +K

j=1

pjFj

The no arbitrage opportunities assumption implies that p 0.

A minor technical condition

We also need to assume that all K factors are tradeable portfolios.

Thus, the factors are all zero cost portfolio with 100% in a longposition and 100% in a short position.


Example: Why the APT predicts 0

Suppose there are two factors in security returns

The market factor: rM rf

A small stock factor: rs rf

The factor model and a well diversified portfolio (P)

rp rf p + pM(rM rf) + ps(rs rf)

rp rf + p + pM(rM rf) + ps(rs rf)

For concreteness, let pM = 1.2 and ps = 0.5

rp rf + p + 1.2(rM rf) + 0.5(rs rf)


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Example: A tracking portfolio

What we need

To derive the APT we need to form a tracking portfolio (a portfolio that

tracksP

as closely as possible).

The Weights for the Tracking Portfolio

Compact:

WeightMarket Portfolio 120%Oil Portfolio 50%

Riskfree Security -70%Total 100%

Convenient:

WeightRiskfree Security 100%Market Portfolio 120%Riskfree Security -120%

Oil Portfolio 50%Riskfree Security -50%

Total 100%

Why does this track P as closing as possible?


Example: Long P and Short the Tracking Portfolio

the return on the tracking portfolio is the following

rtracking = rf + 1.2M(rM rf) + 0.5(rs rf)

Go long 100% in P and go short 100% in the tracking portfolio

rp rtracking rf + p + 1.2M(rM rf) + 0.5(rs rf)

rf + 1.2M(rM rf) + 0.5(rs rf)

p

The no arbitrage condition implies p 0.

If p = 0, then you have found an arbitrage opportunity. Why?


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The APT

Returns

Since p 0, we can write the excess return on the well diversified

portfolio as,

rp rf

K

j=1

pjFj

Expected Returns

Taking the expectation of both sides of the equation we obtain thefollowing:

E(rp) rfK

j=1

pjE(Fj)

We now have a model (called the APT) that explains the expectedreturns of well diversified portfolios.


The APT: Individual Securities

No compensation for idiosyncratic risk

If an investor can use portfolios to diversify away idiosyncratic risk,then they should not be compensated for holding idiosyncratic risk:

Thus for an individual security i:

E(ri) rf

K

j=1

ijE(Fj),


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Thinking about Risk Factors

A priced or risk factor

A variable which helps explain the expected returns on a cross-section of

assets.

Priced factor should link with utility

Risk factors are linked to investor happiness (marginal utility). That iswhy they affect expected returns.

Other factors may account for comovements in returns. However, ifthey are not priced they do not affect expected returns (they aremean zero). We dont have to include non-priced factors in a model.

Thus we really need to find the priced factors.

Finding factors

We need variables that explain expect returns, that are linked to utility,and are not already captured by the market factor.


An Example: A Crime Factor

Investors dont like crimeIt makes them less happy.

Suppose there is a crime factor in stock returns

If crime goes up, then companies that produce alarms, guns, locksand security guards will see their returns go up.

Returns go down for luxury car and jewelry companies.

A stock that pays off more money when crime is high is desirable

since it gives you money when you are less happy.A stock that has low returns during high crime is undesirable; you getlittle money when you are already unhappy.

The crime factor will have low expected returns.

Really a crime factor?

Is there a crime factor in stock returns in the real world?Karl B. Diether (Fisher College of Business) Multifactor Models 22 / 29

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Good Times and Bad

Factors that are positively correlated with good things

Have high risk premia: for example, the market factor.

Factors that are negatively correlated with bad things

Have high low premia (or even negative).


What Are the Factors?

What are the priced factors that actually affect security returns?We dont really know.

The market portfolio is a priced factor

Investors care about the market portfolio because it is something thataffects their wealth.

It contributes non-diversifable variance to an investors portfolio.

Other priced factorsAfter the market there is not a lot of agreement (we will get specific aboutthis later).

Non-priced factors

We can certainly find non-priced factors: for example, industry returns.


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Some Practical Issues

Multifactor model: Market + Crime Factor

We cant use the following as the model:

E(ri) rf iM[E(rM) rf] + iC[E(crime rate)]

Need all factors to be tradeable portfolios

Solution: make the crime factor a zero cost portfolio

Form a portfolio of securities that benefit from high crime (G) andportfolio of securities hurt by high crime (J).

The crime factor as a zero cost portfolio is, rg rj.

The multifactor model is,

E(ri) rf iM[E(rM) rf] + iCE(rg rj)


Testing a Multifactor Model

For a multifactor model the expected return on a security or portfoliois the following:

E(ri) rf =K

j=1

ijE(Fj)

We can test a multifactor model by running the following regression:

rit rft = i +K

j=1

ijFjt + it

the model predicts that i = 0.

i is the average abnormal return. It a measure of mispricing withrespect to the model.


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Example: Testing a Multifactor Model

In general finding a positive or negative alpha is evidence ofmispricing or a bad model.

Maybe we do not have the right factors.

Maybe we do not have all the factors.

Maybe the market is inefficient.

In general finding a positive or negative alpha is not evidence of anarbitrage opportunity.

It is only an arbitrage opportunity if the security or portfolio on theleft hand side has no idiosyncratic risk.


Mean Variance Analysis and Multifactor models

Math Fact

If some linear combination of the factor portfolios are the tangencyportfolio then the alpha must equal zero.

A regression of a portfolio or security excess returns on a factorportfolio is a test of whether the some linear combination of the

factor is the tangency portfolio.


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Some Final Points

Factors reflect common movement in security returns.

If a portfolio is well diversified, factors will explain virtually all of thevariance in the returns of the portfolio.

Individual securities still have idiosyncratic risk. Therefore factors willnot explain all of the variance in their returns.

Priced factors should explain expected returns for individual securitiesand portfolios.

Figuring out the correct priced factors is difficult, and there is nogeneral consensus.

Risk and reward usually go together. If you want high returns onaverage buy things that do well in good times and poorly in bad times.


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