Multifactor Models

Karl B. Diether
Fisher College of Business

Karl B. Diether (Fisher College of Business)

Multifactor Models

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Factors

A Factor
A variable that explains why a group of stocks have returns that tend to move together. Equivalently a variable that helps explain common components of the variance of security returns.

Example: Oil prices

When oil prices go up, expected cash ows on many rms change. Expected cash ows for oil companies probably increase and the probably 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 security returns which also contributes to the expected return. We usually think of a priced factor in the context of a rational risk-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 ) + E (rit ) rft = iM [E (rMt ) rft ]
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it

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 as the irrationality aected groups of stocks in common ways.

Controversy
Can we identify rational factors that explain the cross-section of expected returns? For example, can we nd a rational factor that explains why value stocks usually do better than growth stocks?

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A Multifactor World

Suppose there are many factors that aect security returns

rit rft = i + i1 F1t + i2 F2t + + iK FKt +
K it

= i +
j=1

ij Fjt +

it

F1t is the rst factor i1 is the beta of security i for the rst factor.

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Factor Betas
Synonyms
Factor loadings Loadings Factor Betas Betas Slopes Coecients 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:
K

rit rft = i +
j=1

ij Fjt +

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

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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 market portfolio and -100% in the riskfree rate (rMt rft ).

Factors can also be non-tradeable variables

Maybe GDP growth. Maybe unexpected ination.

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 positive weights 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 nancing portfolio

In frictionless market it requires no capital outlay.

Return on a zero cost portfolio

Synonyms: excess return or a dierential return.
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Completely Explaining Comovements in Returns

Suppose we have all the factors that aect securities returns. Then we can write the return on a security and the return on a well diversied portfolio as the following:

An Individual Security
K

rit rft = i +
j=1

ij Fjt +

it

A Well Diversied Portfolio

K

rpt rft p +
j=1

pj Fjt

Why are the two equations dierent?

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Pure and Approximate

Approximate factor structure

K

rpt rft p +
j=1

pj Fjt ,

Pure factor structure

K

rpt rft = p +
j=1

pj Fjt ,

Perfectly Diversied
A pure factor structure exists only for perfectly diversied portfolios.

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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 covariances using factor than individual assets. Estimates likely to much more precise because factors should be more stable and persistent over time.

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Multifactor Explanations

Rational asset pricing models

Rational asset pricing models explain expected returns by specifying the sources of risk and exposure to risk. The CAPM is an asset pricing model that implies all dierences in expected returns should be due to dierences in beta. Unfortunately, the CAPM does not seem to explain the average returns we observe.

Empirical failures of the CAPM

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

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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 rm specic risk.

Arbitrage
The creation of riskless prots made possible by relative mispricing among securities. Getting return without risk. An arbitrage opportunity arises if an investor can create a zero-cost portfolio with positive return for certain in the future.

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The APT
In the APT, p 0
Consider a well diversied portfolio (call it P). If there are K factors that completely describe all common movements in securities returns for the economy, then the following is true:
K

rp rf p +
j=1

pj Fj

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 long position and 100% in a short position.
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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 diversied 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 tracks P as closely as possible).

The Weights for the Tracking Portfolio

Convenient: Compact: Weight 120% 50% -70% 100% Riskfree Security Market Portfolio Riskfree Security Oil Portfolio Riskfree Security Total Weight 100% 120% -120% 50% -50% 100%

Market Portfolio Oil Portfolio Riskfree Security Total

Why does this track P as closing as possible?

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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 diversied portfolio as,
K

rp rf
j=1

pj Fj

Expected Returns
Taking the expectation of both sides of the equation we obtain the following:
K

E (rp ) rf
j=1

pj E (Fj )

We now have a model (called the APT) that explains the expected returns of well diversied portfolios.
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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:
K

E (ri ) rf
j=1

ij E (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 is why they aect expected returns. Other factors may account for comovements in returns. However, if they are not priced they do not aect expected returns (they are mean zero). We dont have to include non-priced factors in a model. Thus we really need to nd 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.
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An Example: A Crime Factor

Investors dont like crime
It makes them less happy.

Suppose there is a crime factor in stock returns

If crime goes up, then companies that produce alarms, guns, locks and security guards will see their returns go up. Returns go down for luxury car and jewelry companies. A stock that pays o 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 get little 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?
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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).

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What Are the Factors?

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

The market portfolio is a priced factor

Investors care about the market portfolio because it is something that aects their wealth. It contributes non-diversifable variance to an investors portfolio.

Other priced factors

After the market there is not a lot of agreement (we will get specic about this later).

Non-priced factors
We can certainly nd 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 benet from high crime (G) and portfolio 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 ] + iC E (rg rj )
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Testing a Multifactor Model

For a multifactor model the expected return on a security or portfolio is the following:
K

E (ri ) rf =
j=1

ij E (Fj )

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

K

rit rft = i +
j=1

ij Fjt +

it

the model predicts that i = 0. i is the average abnormal return. It a measure of mispricing with respect to the model.
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Example: Testing a Multifactor Model

In general nding a positive or negative alpha is evidence of mispricing or a bad model.

Maybe we do not have the right factors. Maybe we do not have all the factors. Maybe the market is inecient.

In general nding a positive or negative alpha is not evidence of an arbitrage opportunity.

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

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Mean Variance Analysis and Multifactor models

Math Fact
If some linear combination of the factor portfolios are the tangency portfolio then the alpha must equal zero. A regression of a portfolio or security excess returns on a factor portfolio is a test of whether the some linear combination of the factor is the tangency portfolio.

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Some Final Points

Factors reect common movement in security returns. If a portfolio is well diversied, factors will explain virtually all of the variance in the returns of the portfolio. Individual securities still have idiosyncratic risk. Therefore factors will not explain all of the variance in their returns. Priced factors should explain expected returns for individual securities and portfolios. Figuring out the correct priced factors is dicult, and there is no general consensus. Risk and reward usually go together. If you want high returns on average buy things that do well in good times and poorly in bad times.

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