Estimating High Dimensional Covariance Matrices Using a Factor Model_Sun_2013_Slides

7/27/2019 Estimating High Dimensional Covariance Matrices Using a Factor Model_Sun_2013_Slides

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Estimating High Dimensional

Covariance Matrix and

Volatility Index by makingUse of Factor Models

Celine Sun

R/Finance 2013


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Outline

Introduction

Proposed estimation of covariance matrix:

Estimator 1: Factor-Model Based Estimator 2: SVD based

Empirical testing results

Proposed volatility estimation: Cross-section volatility (CSV)

Empirical Results

Conclusion


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Two new estimators are

proposed in this work: We propose two new covariance matrix

estimators :

1. Allow non-parametrically time-varying:Estimate the monthly realized covariance matrix using daily data

2. Allow full rank for N>T:

Using the factor model and SVD to estimate such that the

covariance estimator is full rank

The new estimators are different from the commonly used

estimators and approaches


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Covariance matrix estimation

based on FM (factor models) We propose an estimation of

covariance matrix, based on a statistical

factor model with k factors (k< N).

Here, { } are the loadings,

{ } are the regression errors.

Note: The estimator matrix is full rank.

T

t

Nt

T

t

it

Nkk

N

NkN

k

FMRCOV

1

2

1

2

1

111

1

111

0

0

ij

ij

FMRCOV

FMRCOV


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Covariance matrix estimation

based on SVD method I propose the 2nd estimation of

covariance matrix, based on SVD:

Here, { } and { } are from the usual eigen

decomposition of the NxN realized variance matrix, andhaving , with k< N.

{ } = the remaining terms from reconstructing

the return matrix by { } and { }

SVDRCOV

T

t

Nt

T

t

it

kNk

N

kkNN

k

SVD

d

d

ee

ee

ee

ee

RCOV

1

2

1

2

1

111

2

2

1

1

111

0

0

2

i ije

01 k

itd

i ije


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

192612

192901

193102

193303

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193705

193906

194107

194308

194509

194710

194911

195112

195401

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195803

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196808

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197411

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198102

198303

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198705

198906

199107

199308

199509

199710

199911

200112

200401

200602

200803

201004

Volatility

Global minimum portfolio

Shrinkage

FM

SVD

Empirical testing:

1 Year Rolling Volatility for S&P 500


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Empirical testing:


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

192612

192901

193102

193303

193504

193705

193906

194107

194308

194509

194710

194911

195112

195401

195602

195803

196004

196205

196406

196607

196808

197009

197210

197411

197612

197901

198102

198303

198504

198705

198906

199107

199308

199509

199710

199911

200112

200401

200602

200803

201004

Volatility

Mean variance efficient portfolio with mean=8%

Shrinkage

FM

SVD


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Volatility Index

A number of drawbacks of current volatility index

Not based on actual stock returns

The index only available to liquid options

Only available at broad market level

Advantage of CSV

Observable at any frequency

Model-free Available for every region, sector, and style of the

equity markets

Don't need to resort option market

8


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Cross-sectional volatility

Cross-sectional volatility (CSV) is defined

as the standard deviation of a set of asset

returns over a period.

The relationship between cross-sectionalvolatility, time-series volatility and averagecorrelation is given by:

1x


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0

0.1

0.2

0.3

0.4

0.5

0.6

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199909

200111

200401

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201007

Monthly cross-sectional volatility vs.

average volatility & average correlation

cross-vol

vol*sqrt(1-corr)

Correlation: 0.85

Empirical testing:



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Decomposing Cross-Sectional

Volatility Apply the factor model on return

The change of beta is more persistent

Cross-sectional volatility of the specific

return is a proxy for the future volatility

The correlation between VIX and CSV of

specific return is 0.62.

)()()( itii CSVfCSVRCSV


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Conclusion

Constructed covariance matrix estimators

which are full rank

The portfolios constructed based on myestimators have lower volatility

Applying factor model structure to CSV

gives us a good estimation of the volatility. It could be used at any frequency and at

any set of stocks

Documents

Estimating High Dimensional Covariance Matrices Using a Factor Model_Sun_2013_Slides