Estimating High Dimensional
Covariance Matrix and
Volatility Index by making
Use of Factor Models
Celine Sun
R/Finance 2013
1
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
2
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
3
Covariance matrix estimation
based on FM (factor models)
– We propose an estimation RCOV FM of
covariance matrix, based on a statistical
factor model with k factors (k < N).
RCOV FM
 ˆ11
ˆ1k   ˆ11





ˆ   ˆ
 ˆ

N
1
Nk   1k

T 2
ˆ N 1   ˆit
  t 1



ˆ Nk   0



0 


T
2 
ˆ


Nt 
t 1

– Here, { ̂ ij } are the loadings,
– { ˆij} are the regression errors.
– Note: The estimator matrix RCOV FM is full rank.
4
Covariance matrix estimation
based on SVD method
– I propose the 2nd estimation RCOVSVDof
covariance matrix, based on SVD:
ek1  
 e11

RCOV SVD  


e1N
ekN  
2
1
–
 e11



2
k  ek1
T 2
e1N   d it
 t 1




ekN   0



0 


T
2 
d

Nt 
t 1

Here, {i2 } and { eij } are from the usual eigen
decomposition of the NxN realized variance matrix, and
having 1      k  0 , with k < N.
– { d it} = the remaining terms from reconstructing
the return matrix by { i } and {eij }
5
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
Empirical testing:
1 Year Rolling Volatility for S&P 500
Global minimum portfolio
0.1
0.09
0.08
0.07
0.06
0.05
Shrinkage
0.04
FM
SVD
0.03
0.02
0.01
0
6
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
Empirical testing:
1 Year Rolling Volatility for S&P 500
Mean variance efficient portfolio with mean=8%
0.7
0.6
0.5
0.4
Shrinkage
0.3
FM
SVD
0.2
0.1
0
7
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
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-sectional
volatility, time-series volatility and average
correlation is given by:
 x   1   
9
0.6
0.4
192601
192803
193005
193207
193409
193611
193901
194103
194305
194507
194709
194911
195201
195403
195605
195807
196009
196211
196501
196703
196905
197107
197309
197511
197801
198003
198205
198407
198609
198811
199101
199303
199505
199707
199909
200111
200401
200603
200805
201007
Empirical testing:
1 Year Rolling Volatility for S&P 500
Monthly cross-sectional volatility vs.
average volatility & average correlation
0.5
Correlation: 0.85
0.3
cross-vol
vol*sqrt(1-corr)
0.2
0.1
0
10
Decomposing Cross-Sectional
Volatility
• Apply the factor model on return
CSV ( Ri )  CSV ( i ) f t  CSV ( i )
• 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.
11
Conclusion
• Constructed covariance matrix estimators
which are full rank
• The portfolios constructed based on my
estimators 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
12