Barra-Type Factor Structurefaculty.baruch.cuny.edu/lwu/890/LuluWang_Barra.pdf · In the Barra USE4...

Barra-Type Factor Structure

Lulu Wang

September 18, 2019

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Overview

1 The Model Set UpAnalysis Portfolio with Factor ModelStyle Exposure

2 Barra USE4 ModelFactor ReturnsFactor Covariance Matrix

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The Model Set Up—Returns of a Stock

The intuition behind the model is that there exists a set of commonfactors that drive stocks return.

rn =∑k

Xnk fk + un (1)

where rn is the return of stock n.fk is the return of the factor k.Xnk is the exposure of stock n to factor k.un is the stock specific return, which cannot explain by the factors.

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The Model Set Up—Return of a Portfolio

Consider a portfolio made by N stocks, a weight of stock n is wn thenreturn of this portfolio is the weighted average of individual stockreturns:

Rp =∑n

wnrn (2)

The portfolio’s exposure to factor K is given by the weighted averageof the stock exposure, i.e.,

X pk =

∑n

wnXnk (3)

Therefore, the return of portfolio can be further expressed as theweighted form of single factor’s return, plus the weighted average ofspecific return.

Rp =∑k

X pk fk +

∑n

wnun (4)

Here, return is computed daily

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The Model Set Up—Variance of a Portfolio

Since (1) factor returns are uncorrelated with specific returns. (2)specific returns are uncorrelated among themselves. This allow thevariance of the portfolio to be expressed as:

var(Rp) =∑kl

X pk FklX

pk +

∑n

w2n var(un) (5)

where Fkl is the covariance between factors k and l.

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The Model Set Up—Standardize the Exposure to theFactor

Example: Exposure to earning factor.

EPIBS, ETOP and CETOP are descriptors of Earnings. We need tostandardize descriptors to have a mean of 0 and standard deviation of 1.

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The Model Set Up—Standardize the Exposure to theFactor

We need to standardizes descriptors to have mean of 0 and standarddeviation of 1.

dnl =dRawnl − µlσl

(6)

Then, combine descriptors into factors.

Xnk =∑l∈k

wldnl (7)

Additional to standardization, we also need to address multicollinearity andheteroscedasticity as well. Common methods are VIF( Variance InflationFactor) and WLS.

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Factor Returns USE4 —Return of a stock

In the Barra USE4 model, stock returns are explained by returns ofcountry factor, industry factors and style factors.

rn = fc +∑i

Xni fi +∑s

Xns fs + un (8)

where fc is the return of the Country factor.fi is the return of industry factor i .fs is the return of style factor s.Xni are the exposures of stock n to industry i , if the stock n in theindustry i , Xni = 1, otherwise Xni = 0.Xns are the exposures of stock n style s .Un is the specific return.Every stock has an exposure of 1 to the Country factor.

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Factor Returns USE4 —Market Return

Notice for any stock, sum of industry factor exposures is 1:∑i

Xni = 1. (9)

which is exactly the USE4 Country factor exposure. A constraint,therefore, must be applied to obtain a unique regression solution.

In USE4, the model adopts a constraint that the cap-weightedindustry factor returns sum to zero, which is:∑

i

wi fi = 0. (10)

where wi is the capitalization weight in industry i .

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Factor Returns USE4 —Market Return

Consider a cap-weighted estimation universe (e.g.: all US stocks), if aweight of stock n is hEn , then return of this stock market is

RE = fc +∑i

wi fi +∑s

XEs fs +

∑n

hEn un (11)

Because the constrain we put before, the second sum is equal to 0,the standardization makes the third sum equal to 0, and the finalterm is the sum of specific returns of a broadly diversified portfolio,and is therefore approximately zero, we have

RE ≈ fc (12)

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Factor Returns USE4 —Solve the model

We need solve a constrained Least Squares.

min∑n

wn · (rn − fc −∑i

Xni fi −∑s

Xns fs)2 (13)

such that∑i

wi fi = 0. (14)

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Factor Returns USE4 —Example 1

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Factor Covariance Matrix —Bias Statistics

Recall the variance of the portfolio is expressed as

var(Rp) =∑kl

X pk FklX

pk +

∑n

w2n var(un) (15)

Where F is the factor covariance matrix (FCM) of returns of factors, andu is the variance matrix of specific returns. The (FCM) predicts thevolatilities and correlations of the factors, thus represents a second keypillar for constructing a high-quality risk model.

How do we know if our risk model is accurate?

A commonly used measure to assess a risk models accuracy is biasstatistic. Conceptually, the bias statistic represents the ratio ofrealized risk to forecast risk.

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Bias statistics is constructed as follows:

First, assume perfect forecasts. Rt+h be the return to a portfolio overperiod h , and let σt be the beginning-of-period volatility forecast.Then the standardized return,

bt,h =Rt+h

σt(16)

has an expected standard deviation of 1.

The bias statistic for portfolio n is the realized standard deviation ofstandardized returns,

B =

√√√√ 1

T − 1

T∑t=1

(bt − bt)2 (17)

where T is the number of periods in the observation window.

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Therefore1 If the forecast standard deviation perfectly matches the realized one,

we have Bn = 1.2 If we underestimate risk, Bn > 1.3 If we overestimate risk, Bn < 1.

Assuming normally distributed returns and perfect risk forecasts, forsufficiently large T, Bn is approximately normally distributed withmean approximately to 1. Roughly 95 percent of the observations fallwithin the confidence interval,

Bn ∈ [1−√

2/T , 1 +√

2/T ]. (18)

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Factor Covariance Matrix —EWMA (dealing with datanon-stationarity)

The first step is to compute FCM from daily factor returns. The modelemploys exponentially weighted moving averages(EWMA).

The idea of EWMA is that recent events have bigger impact on risk.

This approach gives more weight to recent observations and is aneffective method for dealing with non-stationary data.

FEWMAkl = cov(fk , fl)t =

t∑s=t−h

λt−s(f sk −fk)(f sl −fl)/t∑

s=t−hλt−s (19)

where, λ = 0.51/τ ,and τ is factor correlation half-life.

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Factor Covariance Matrix —Newey-West (deal withauto-correlation)

We calculate the factor covariance matrix from daily factor returns, so weneed to consider auto-correlated between factor returns.

Use the Newey-West approach, we have:

FNW = 22[FEWNA +D∑

∆=1

(1− ∆

D + 1)(C

(d)+∆ + C

(d)−∆)] (20)

C(d)+∆ = cov(f t−δk , f tl ) =

t∑s=t−h+∆

λt−s(f s−∆k −fk)(f sl −fl)/

t∑s=t−h+∆

λt−s

(21)

C(d)−∆ = cov(f tk , f

t−δl ) =

t∑s=t−h+∆

λt−s(f sk −fk)(f s−∆l −fl)/

t∑s=t−h+∆

λt−s

(22)where D is Newey-West correlation lags.

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Factor Covariance Matrix —Newey-West (deal withauto-correlation)

Factor covariance matrix parameters for the USE4 model.

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(what is the problem?)

In 1952, Markowitz (1952) established the mean-variance frameworkfor constructing efficient portfolios. This paradigm provided thefoundation upon which the modern theory of finance was built.

But Muller (1993) found that risk models have a systematic tendencyto under predict the risk of optimized portfolios.

Recently, Shepard (2009) derived an analytic result for the magnitudeof the bias.

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(Shepard 2009)

Under assumptions of normality, stationarity, and many assets(i.e.,the large N limit), he found:

σtrue ≈σpred

1− (K/T )(23)

where, σtrue is true volatility of the optimized portfolio.σpred is predicted volatility from the risk model.K is the number of factors, and T is the effective number ofobservations used to compute the covariance matrix.

For example: if we estimate the covariance matrix by using 60 tradingdays and 30 stocks, then the predicted volatility from the risk modelis half of the true risk.

An important innovation in the Barra USE4 model is to identify thesebiases, and to correct the biases by adjusting the factor covariancematrix.

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(concept of eigenfactors)

The underestimation of risk for optimized portfolios is closely linked withthe concept of eigenfactors.

For FCM, eigenfactors are the eigenvectors of the matrix;

They represent uncorrelated portfolios of pure factors.

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(Identify the bias)

Let F0 = FNW denote the KxK sample factor covariance matrix(FCM), which can be expressed in diagonal form as:

D0 = U′0F0U0 (24)

where U0 is the KxK rotation matrix whose columns are given by theeigenvectors of F0.

U0 = [v1, v2, ..., vk ]. (25)

It’s worthy to point out that an element in this matrix, uij , representa weight of factor i in the portfolio j .

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The predicted variances of the eigenfactors(portfolios) are given bythe diagonal elements of D0 .

D0=

var(v1). . .

var(vk)

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Bias statistics of eigenfactors using the unadjusted covariance matrix(95% CI is indicated by the two dashed horizontal lines) Figure 28

The lower volatility eigenfactors have realized volatilities higher thantheir predicted volatilities and fall well outside the 95% CI. The largereigenfactors, by contrast, fall mostly within the confidence interval.

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(volatility biases Monte Carlo simulation)

Since we identified biases, next step is to compute how much we need toadjust according them. We estimate the magnitude of the empiricaleigenfactor biases via Monte Carlo simulation.

We consider the sample FCM F0 as the true FCM.Step 1: generate a KxT matrix bm that simulateeigenfactors(portfolios) return.

bm =

b11 b12 b13 . . . b1T

b21 b22 b23 . . . b2T...

......

. . ....

bk1 xK2 xK3 . . . xKT

(26)

i.e. b1 ∼N(0, var(v1))That is, the elements of row K of bm are drawn from a randomnormal distribution with mean zero and variance given by thediagonal element D0(k) of matrix D0.

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Step 2: we can generate a set of factor returns for simulation m as:

fm = U0bm (27)

where U0 is the KxK rotation matrix whose columns are given by theeigenvectors of F0.

We can think of eigenvectors vs in U0 as different portfolios. RecallEach element in a vector v is the weight of a corresponding asset. Forbm, we can think of it as the return of the asset.

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Step 3: Compute estimated FCM :

Fm = cov(fm, fm) (28)

where Fm is unbiased.

Step 4: diagonalize the simulated FCM

Dm = U′mFmUm (29)

where Um denotes the simulated eigenfactors with estimatedvariances given by the diagonal elements of Dm , i.e., Dm(k).

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Step 5: Since we know the true distribution that governs thesimulated factor returns, we can compute the true FCM of thesimulated eigenfactors,

D̃m = U′mF0Um (30)

Note that since Um is not composed of the true eigenfactors, thematrix D̃m is not diagonal. Nevertheless, current focus is on thediagonal elements of the matrix.

Repeat step 1 to step 5 M times.

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Compute the simulated volatility biases according to

v(k) =

√√√√ 1

M

∑m

D̃m(k)

Dm(k)(31)

Mean simulated volatility bias computed per Equation (33)Figure 33

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Factor Covariance Matrix —Eigenfactor RiskAdjustment(Correct the bias)

Qualitatively, Figure 33 is in good agreement with Figure 28.However, our simulation assume both normality and stationarity. Realfinancial data, of course, violate both of these assumptions.

In practice, therefore, additional scaling is required to fully remove thebiases of the eigenfactors.

vs(k) = a[v(k)− 1] + 1 (32)

where vs(k) is the scaled value. a = 1.4 is an empirically determinedconstant.

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We now assume that the sample FCM, F0 , which uses the samecovariance estimator as the simulated FCM, Fm , also suffers from thesame biases. Let D̃0 denote the diagonal FCM whose eigenvarianceshave been adjusted

D̃0 = v2s D0 (33)

where v2s is a diagonal matrix whose elements are given by v2

s (k) .

The FCM in Equation (33) is now rotated from the diagonal basis tothe pure factor basis using the sample eigenfactors. That is,

F̃0 = U0D̃0U′0 (34)

where F̃0 denotes the eigen-adjusted factor covariance matrix.

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Bias statistics of eigenfactors using the eigen-adjusted covariance matrix ofEquation (34)Figure 36

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Barra-Type Factor Structurefaculty.baruch.cuny.edu/lwu/890/LuluWang_Barra.pdf · In the Barra USE4...

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