ssrn-id2411493 the role of covariance matrix forecasting method in the performance of...

8/12/2019 SSRN-id2411493 The Role of Covariance Matrix Forecasting Method in the Performance of Minimum-Variance Port

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The Role of Covariance Matrix Forecasting Methodin the Performance of Minimum-Variance Portfolios

Valeriy Zakamulin

This revision: March 19, 2014

Abstract

Providing a more accurate covariance matrix forecast can improve substantially theperformance of optimized portfolios. In this paper we evaluate different covariance matrixforecasting methods by looking at their ability, in out-of-sample tests, to optimize risk andimprove risk-adjusted performance of minimum-variance portfolios. We nd large differ-ences between the methods. Our results suggest that shrinkage of sample covariance matriximproves neither the forecast accuracy nor the performance of minimum-variance portfo-lios. We demonstrate that the simple exponentially weighted covariance matrix forecastproduces superior performance compared to the commonly used sample covariance matrix.Our ndings also reveal that the exponentially weighted covariance matrix forecast usuallyproduces the best risk-adjusted performance of minimum-variance portfolios. Yet, whenit comes to the portfolio risk optimization, multivariate GARCH models generally demon-strate the best results. Specically, switching from the sample covariance matrix forecastto a multivariate GARCH forecast reduces the forecasting error and portfolio tracking errorby half at least.

Key words : covariance matrix forecasting, minimum-variance portfolio optimization,sample covariance, covariance shrinkage, exponentially weighted covariance, multivariateGARCH, model comparison

JEL classication : C30, C52, G11, G17

The author is grateful to Steen Koekebakker for his insightful comments. All other comments are welcomed. a.k.a. Valeri Zakamouline, School of Business and Law, University of Agder, Service Box 422, 4604 Kris-

tiansand, Norway, Tel.: (+47) 38 14 10 39, E-mail: [email protected]

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1 Introduction

There is an ongoing discussion on whether the optimized portfolios can outperform equally

weighted portfolios. Whereas DeMiguel, Garlappi, and Uppal (2009) and Duchin and Levy

(2009) show that optimized portfolios cannot consistently beat equally weighted portfolios,

Kritzman, Page, and Turkington (2010) demonstrate the superiority of optimized portfolios.

Specically, the latter researchers nd that portfolios that optimize the mean-variance tradeoff

show better performance than equally weighted portfolios. What is especially noteworthy is

that their ndings reveal that portfolios that minimize the variance of returns outperform the

portfolios that optimize the mean-variance tradeoff.

Clarke, de Silva, and Thorley (2006) were the rst to document the superiority of minimum-

variance portfolios. The advantage of minimum-variance portfolios over mean-variance opti-

mized portfolios is two-fold. First, minimum-variance portfolios deliver superior risk-adjusted

performance compared to that of mean-variance optimized portfolios. Second, while the imple-

mentation of the mean-variance model requires forecasting both the mean returns and covari-

ance matrix of returns, the implementation of the minimum-variance model requires forecasting

the covariance matrix of returns only.

There is a common consensus that mean returns are notoriously difficult to forecast, whereas

forecasting the covariance matrix can be easily done using a rolling sample covariance matrix.

A standard approach is to use the monthly rolling n-year covariance matrix, where n varies from

5 to 20 years, as a forward-looking estimate of future covariance matrix (see Chan, Karceski,

and Lakonishok (1999), DeMiguel et al. (2009), Duchin and Levy (2009), and Kritzman et al.

(2010) among others). Yet, this is a valid approach if one assumes that the covariance matrix

is either constant over time or varying very slowly over time. This assumption does not seem

to hold for nancial asset returns. On the contrary, there is a large strand of literature that

demonstrate that nancial asset returns exhibit heteroskedasticity with volatility clustering.

The assumption of constant correlation between nancial asset returns also seems to be vi-

olated. As a motivation, Figure 1 plots the monthly realized standard deviations of returns

on the US stock market and bond market indices as well as the monthly realized correlation

coefficient between the returns on these indices. The graphs in this gure suggest that both the

standard deviation and correlation coefficient can change dramatically over the course of a few

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years. For example, there was a ten-fold increase in the standard deviation of the stock market

returns over the period 2006-08. During the same period the standard deviation of the bond

market returns increased by a factor of four, whereas the correlation coefficient experienced

a change from virtually zero to a signicantly negative value. Therefore it is only logical to

assume that in forecasting the covariance matrix one has to take into account the time-varying

nature of variances and covariances.

The objective of this paper is to compare the performances of different covariance matrix

forecasting methods. The different forecasting methods are evaluated on a statistical basis and

on a practical basis. For this purpose we perform three studies. In the rst study we directly

evaluate the covariance matrix forecasting methods by comparing their out-of-sample forecast

accuracy. In the second study we evaluate different methods by looking at their ability, in out-of-sample tests, to optimize risk and improve risk-adjusted performance of minimum-variance

portfolios. In the nal study we evaluate different methods by comparing out-of-sample per-

formance of minimum-variance portfolios with a volatility target mechanism. Specically, we

evaluate the ability of different covariance forecasting methods to keep the portfolio volatility

at a target level and improve the portfolio risk-adjusted performance.

In all our studies, employing a xed size data window of 10 years that is rolled over time,

we generate out-of-sample forecasts of covariance matrix using 5 distinct methods. The rstmethod, which is most common in practice, uses the (rolling) sample covariance matrix as

a predictor for the future covariance matrix. In other words, in this approach the equally

weighted covariance matrix is estimated using a 10-year lookback period. In order to decrease

the estimation error of the sample covariance matrix, in the second method we use the shrink-

age estimation of the covariance matrix proposed by Ledoit and Wolf (2004). This is the

application of the idea of shrinking estimation pioneered by Stein (1956). The third method

of estimation of covariance matrix is popularized by RiskMetrics TM group and uses the expo-

nentially weighted covariance matrix. This method of forecasting is usually denoted as the

Exponentially Weighted Moving Average (EWMA) model.

The other two methods of covariance matrix forecasting employ multivariate Generalized

AutoRegressive Conditional Heteroskedasticity (GARCH) models. The univariate GARCH

modeling started with the seminal paper by Bollerslev (1986) and proved to be indispensable

in modeling and forecasting time-varying nancial asset volatility. Unfortunately, a direct

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0.00

0.05

0.10

0.15

1995 2000 2005 2010

Panel A: Standard deviation of USA Govt. Bond Index

0.00

0.05

0.10

0.15

0.20

0.25

1995 2000 2005 2010

Panel B: Standard deviation of MSCI USA Stock Index

0.5

0.0

0.5

1995 2000 2005 2010

Panel C: Correlation between MSCI USA Stock Index and USA Govt. Bond Index

Figure 1: Panels A and B plot the monthly realized standard deviations of returns on theUS stock market and bond market indices. Panel C plots the monthly realized correlationcoefficient between the returns on these indices. The standard deviations and correlationcoefficient are computed using daily returns.

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2 Data and Methodology

2.1 Data

In our study we use nine empirical datasets that are listed in Table 1. These datasets are

similar to the datasets used in the previous studies by DeMiguel et al. (2009) and Kritzman

et al. (2010). The rst eight datasets come from the data library of Kenneth French. 1 All

of them represent value-weighted portfolios formed using different criteria. For example, the

dataset of portfolios formed on size consists of returns on 10 stock portfolios sorted by market

equity. The portfolios are constructed at the end of each June using the June market equity and

NYSE breakpoints. As another example, the dataset of industry portfolios consists of returns

on 10 industry portfolios in the US. The 10 industries considered are Consumer Non-Durables,

Consumer Durables, Manufacturing, Energy, High-Tech, Telecommunication, Wholesale and

Retail, Health, Utilities, and Others.

# Dataset N1 Portfolios formed on size 102 Portfolios formed on book-to-market 103 Portfolios formed on momentum 104 Portfolios formed on size and book-to-market 65 Industry portfolios 106 Portfolios formed on size and momentum 107 Portfolios formed on size and short-term reversal 108 Portfolios formed on size and long-term reversal 109 Core asset classes 7

Table 1: List of datasets considered. N denotes the number of portfolios in each dataset.

The ninth dataset covers the seven major asset classes and is obtained from the Thomson

Reuters Datastream database. The asset classes in this dataset are listed in Table 2. The

data for all nine datasets come at the daily frequency and cover the period from January 1st,1986 to December 31st, 2013. In addition we use the 90-day nominal US TBill rate as a proxy

for the risk-free rate of return. The daily TBill rate is also obtained from the data library of

Kenneth French.1 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html

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return on the complete portfolio at time t is given by

r Ct = a t r P t + (1 a t )r f t .

We consider two distinct cases. In the rst case a t = 1, that is, all money is allocated to the

risky portfolio only. In the second case the value of a t is chosen to keep the volatility of the

complete portfolio at a target level denoted by target

a t = min target

P t, a max ,

where a max is the maximum allowable exposure to the risky portfolio.

2.3 Covariance Matrix Forecasting Methods

We assume that the vector of daily asset returns is given by

r t = + t ,

where is the n 1 vector of asset mean returns, and t is the n 1 vector of random

disturbances at time t such that E [ t ] = 0. The covariance matrix of asset returns at time t isthe n n matrix t = Cov [ t ]. In order to compute the composition of the global minimum

variance portfolio for time t, the covariance matrix of asset returns for time t needs to be

forecasted using the information available at time t 1. Below we review the covariance

matrix forecasting methods employed in this study.

2.3.1 Rolling Historical Covariance

In this method, the covariance matrix of asset returns for time t is forecasted using an equally-

weighted moving average calculated on a xed size data window that is rolled over time.

Denoting the length of this window by L (also called the length of the lookback period), the

covariance matrix for time t is forecasted using

Histt =

1L

t 1

i= t L

ii .

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Since our data come at the daily frequency, but the composition of the global minimum variance

portfolio is revised at the month-end, we actually need to forecast the covariance matrix for

the subsequent month. This is done by multiplying each element of the daily covariance matrix

by the number of days in the subsequent month.

2.3.2 Rolling Historical Covariance with Shrinkage

The shrinkage estimation of the covariance matrix is designed to decrease the error in estimation

by using the estimator of the form

Shrinkt = (1 t )

Histt + t

Targett ,

where 0 < t < 1, Histt is the historical covariance over rolling window of length L , and

Targett

is the shrinkage target which is also computed over rolling window of length L . In essence, this

estimator shrinks the covariance matrix toward the shrinkage target. In our study, we use

the shrinkage estimator proposed by Ledoit and Wolf (2004), who take the shrinkage target to

be the covariance matrix in the constant correlation model.

2.3.3 Exponentially Weighted Moving Average

This approach to forecast the covariance matrix is popularized by RiskMetricsTM

group. In

this approach the exponentially weighted covariance matrix is estimated using the following

recursive form

EWMAt = (1 ) t 1 t 1 +

EWMAt 1 ,

where 0 < < 1 is the decay constant. We follow the recommendations of RiskMetricsTM

group and estimate the daily EWMA covariance matrix with = 0 .97. The monthly EWMA

covariance matrix is obtained by multiplying each element of the daily EWMA covariance

matrix by the number of days in the subsequent month.

2.3.4 Dynamic Conditional Correlation GARCH

The Dynamic Conditional Correlation (DCC) GARCH model belongs to the family of mul-

tivariate GARCH models. In the DCC-GARCH model the covariance matrix is decomposed

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into the matrix of standard deviations and the correlation matrix as

DCC-GARCHt = D t R t D t ,

where D t = diag( 1t , . . . , nt ) is the diagonal matrix containing the standard deviations of

asset returns, and R t = [ijt ] is the correlation matrix.

The elements of D t are modelled as univariate GARCH processes. In our study we use

the DCC-GARCH(1,1) where the variances of each asset return are modelled as GARCH(1,1)

which is given by

2it = 0i + 1i 2it 1 + 1i

2it 1 . (1)

The standardized residuals from the estimation of the GARCH(1,1) models are used to estimatethe DCC-GARCH(1,1) correlation specication

R t = Q 1t Q t Q1t ,

where

Q t = (1 a b)Q + ae t 1e t 1 + bQ t 1 ,

e t = D 1t t are standardized errors, Q = E [e t e t ] is the unconditional covariance matrix of

standardized errors, and Q is a diagonal matrix with the square root of the diagonal elements

of Q t at the diagonal, Q= (diag( Q t ))12 .

We estimate the DCC-GARCH(1,1) model using the daily data and rolling window of L

observations. Then the estimated parameters are used to perform the N -day ahead forecast

of the (daily) covariance matrix where N is the number of days in the subsequent month. The

monthly covariance matrix is obtained by summing N daily covariance matrixes.

2.3.5 Generalized Orthogonal GARCH

The Generalized Orthogonal (GO) GARCH model also represents a multivariate generalization

of a univariate GARCH model. In the GO-GARCH model the vector of random disturbances

t is modelled as a linear combination of n unobserved independent factors f t

t = Zf t ,

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where Z is the n n invertible matrix that links the unobserved components with the observed

variables. The unobservable components are normalized to have unit variance such that the

unconditional covariance matrix of t

= E [ t t ] = ZZ .

The independent factors f t are modelled as univariate GARCH. In our study we use the GO-

GARCH(1,1) where the factors are modelled by a GARCH(1,1) process H t = diag( h 1t , . . . , h nt )

h2it = 0i + 1i h2it 1 + 1i f

2it 1 . (2)

The conditional covariance matrix of t is given by

GO-GARCHt = ZH t Z .

Denote by P the n n matrix that contains the orthogonal elements of and by the n n

diagonal matrix that contains the corresponding eigenvalues, = diag( 1 , . . . , n ). These

matrixes can be estimated since the matrix can be estimated consistently by 1L t 1i= t L i

i .

The singular value decomposition of Z yields

Z = P12 U ,

where U is the n n orthogonal matrix that can be identied from the structure of the

conditional covariance matrix t .

Similarly to the DCC-GARCH model, we estimate the GO-GARCH(1,1) model using the

daily data and rolling window of L observations. Then the estimated parameters are used to

perform the N -day ahead forecast of the (daily) covariance matrix where N is the number of

days in the subsequent month. The monthly covariance matrix is obtained by summing N

daily covariance matrixes.

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3 Empirical Results

In this section we evaluate different covariance matrix forecasting methods by performing three

studies. In the rst study we directly evaluate the covariance matrix forecasting methods

by comparing their forecast accuracy. In the second and the third study we evaluate the

covariance matrix forecasting methods in practical settings. In particular, we compare the

performances of different methods by looking at their ability to optimize risk and improve

risk-adjusted performance of minimum variance portfolios and the minimum variance portfolios

with a volatility target. In all our studies we perform out-of-sample forecast of the monthly

covariance matrix. That is, the covariance matrix for month t is forecasted on the basis of

information available at the end of month t 1. More specically, our forecasts are based on

the rolling-window estimation scheme by using the lookback period of L = 120 months. With

this choice, our initial in-sample period is from January 1st, 1986 to December, 31st 1994 and,

consequently, the performances of different forecasting methods are evaluated over the period

from January 1st, 1995 to December 31st, 2013.

3.1 Covariance Matrix Forecast Accuracy

In this study we evaluate the covariance matrix forecast accuracy provided by each distinct

method of forecasting. Let t = [ ijt ] denote the monthly realized covariance matrix for month

t and t = [ ijt ] denote a forecast of t made at the end of month t 1. Both the matrixes

are obtained using the daily returns. The month t squared forecast error for single covariance

is given by ( ijt ijt )2 . To avoid double accounting of covariance forecast errors, we compute

the month t total Squared Forecast Error (SFE) for covariance matrix t by summing up

the squared forecast errors for single covariances in the upper-triangular part of matrix t

including the main diagonal

SF E t =n

i=1

i

j =1

( ijt ijt )2 .

The Mean Squared Forecast Error (MSFE) is computed as

MSFE = 1M

M

t =1

SF E t ,

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0.0000

0.0025

0.0050

0.0075

0.0100

HIST SHRINK EWMA DCCGARCH GOGARCH

M e a n S q u a r e d F o r e c a s t i n g E r r o r

Figure 2: Average forecast accuracy over the period 1995-2013 versus the method of covariancematrix forecasting. For each distinct method of forecasting, this gure plots the average of

Mean Squared Forecast Errors over all datasets. HIST denotes the rolling historical covari-ance; SHRINK denotes the rolling historical covariance with shrinkage; EWMA denotes theexponentially weighted moving average covariance; DCC-GARCH denotes the Dynamic Con-ditional Correlation GARCH; GO-GARCH denotes the Generalized Orthogonal GARCH. Inall methods the length of the rolling estimation window amounts to 120 months.

where M is the number of months in the out-of-sample period.

Table 3 reports the MSFE produced by different covariance matrix forecasting methods

for each dataset separately and the average MSFEs over all datasets. Figure 2 plots the

average of MSFEs over all datasets for different forecasting methods. The results reported inTable 3 advocate that both the rolling historical covariance method and the rolling historical

covariance with shrinkage method provide identical forecast accuracy. That is, shrinkage does

not produce better forecast of the covariance matrix for the subsequent month than the rolling

historical covariance. In contrast, as compared with the two rst methods of forecasting, the

remaining three methods allow to reduce the MSFE by approximately 50%. The differences

in forecast accuracy between these three methods are marginal. In particular, the methods

based on the multivariate GARCH models produce slightly better forecasts than the method

that uses EWMA, and the forecast produced by the DCC-GARCH model is marginally better

than that of the GO-GARCH model.

3.2 Performance of Minimum Variance Portfolios

In this study we evaluate the out-of-sample performance of minimum variance portfolios where

the composition of the (global minimum variance) portfolio for month t is determined on

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C o v a r

i a n c e m a t r i x

f o r e c a s t

i n g m e t

h o d

D a t a s e t

H I S T

S H R I N K

E W M

A

D C C - G

A R C H

G O - G A R C H

P o r t f o l i o s

f o r m e d o n s i z e

0 . 0 0 6 9

0 . 0 0 6 9

0 . 0 0 4 0

0 . 0 0 3 3

0 . 0 0 2 8

P o r t f o l i o s

f o r m e d o n

b o o

k - t o - m a r

k e t

0 . 0 0 8 0

0 . 0 0 8 0

0 . 0 0 4 2

0 . 0 0 3 5

0 . 0 0 4 3

P o r t f o l i o s

f o r m e d o n m

o m e n t u m

0 . 0 1 4 9

0 . 0 1 4 9

0 . 0 0 6 9

0 . 0 0 6 1

0 . 0 0 7 0

P o r t f o l i o s

f o r m e d o n s i z e a n

d b o o

k - t o - m a r

k e t

0 . 0 0 8 1

0 . 0 0 8 1

0 . 0 0 4 2

0 . 0 0 3 6

0 . 0 0 4 0

I n d u s t r y p o r t f o l i o s

0 . 0 0 8 1

0 . 0 0 8 1

0 . 0 0 5 2

0 . 0 0 4 5

0 . 0 0 4 3

P o r t f o l i o s


d m o m e n t u m

0 . 0 1 4 9

0 . 0 1 4 9

0 . 0 0 6 9

0 . 0 0 6 1

0 . 0 0 7 0

P o r t f o l i o s


d s h o r t - t e r m r e v e r s a l

0 . 0 0 7 9

0 . 0 0 7 9

0 . 0 0 4 8

0 . 0 0 4 0

0 . 0 0 3 9

P o r t f o l i o s


d l o n g - t e r m r e v e r s a l

0 . 0 1 5 3

0 . 0 1 5 3

0 . 0 0 9 3

0 . 0 0 8 1

0 . 0 0 7 4

C o r e a s s e t s

0 . 0 0 6 5

0 . 0 0 6 5

0 . 0 0 3 2

0 . 0 0 2 3

0 . 0 0 3 2

A v e r a g e s o v e r a l

l d a t a s e t s

0 . 0 1 0 1

0 . 0 1 0 1

0 . 0 0 5 4

0 . 0 0 4 6

0 . 0 0 4 9

T a b

l e 3 : F o r e c a s t a c c u r a c y o v e r t h e p e r i o d 1 9 9 5 - 2

0 1 3 v e r s u s t h e m e t

h o d o f c o v a r i a n c e m a t r i x

f o r e c a s t i n g . T h i s t a b l e r e p o r t s t h e M e a n

S q u a r e

d F o r e c a s t E r r o r

( M S F E

) p r o d u c e d

b y d i ff e r e n t c o v a r i a n c e m a t r i x

f o r e c a s t i n g m e t

h o d s

. H I S T d e n o t e s t h e r o

l l i n g

h i s t o r i c a l c o v a r i a n c e ;

S H R I N K d e n o t e s t h e r o

l l i n g

h i s t o r i c a l c o v a r i a n c e w i t h s h r i n k a g e ; E W M A d e n o t e s t h e e x p o n e n t i a l

l y w e i g h t e d m o v i n g a v e r a g e c o v a r i a n c e ;

D C C - G

A R C H d e n o t e s t h e D

y n a m i c C o n

d i t i o n a l

C o r r e

l a t i o n

G A R C H ;

G O - G

A R C H d e n o t e s t h e

G e n e r a l i z e d

O r t

h o g o n a

l G A R C H

. I n a l

l

m e t

h o d s t h e

l e n g t h o f t h e r o l l i n g e s t i m a t i o n w i n d o w a m o u n t s t o 1 2 0 m o n t h s .

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the basis of the covariance matrix forecast provided at the end of month t 1. Given the

daily returns generated by the minimum variance portfolio for each dataset and each distinct

covariance forecasting method, we compute two measures of performance. The rst measure is

the Mean Tracking Error (MTE) which is computed as follows. Denote by w t the ex-ante vector

of weights of the global minimum variance portfolio for month t and by w t the ex-post vector

of weights of the global minimum variance portfolio for month t . The former one is determined

on the basis of t provided by each distinct method of forecasting at the end of month t 1,

and the latter one is determined on the basis of t which is the realized covariance matrix for

month t . Further denote by oosP t the out-of-sample (i.e., realized) monthly standard deviation

of the minimum variance portfolio and by isP t the in-sample monthly standard deviation of

the minimum variance portfolio. These standard deviations are given by

oosP t = w t t w t , isP t = w t t w t .The month t Tracking Error (TE) is dened as the squared difference between the in-sample

and out-of-sample standard deviations of the minimum variance portfolio

T E t = isP t oosP t

2.

The MTE is computed as

MT E = 1M

M

t =1

T E t ,

where M is the number of months in the out-of-sample period. The second measure of perfor-

mance is the (out-of-sample) Sharpe ratio of the minimum variance portfolio.

Table 4 reports the out-of-sample performance of minimum variance portfolios over the

period 1995-2013 versus the method of covariance matrix forecasting. Specically, this tablereports the portfolio MTE and the Sharpe ratio produced by different covariance matrix fore-

casting methods. Panel A in Figure 3 plots the average of MTEs over all datasets whereas

Panel B in the same gure plots the average of Sharpe ratios over all datasets. The results

reported in Table 4 suggest that both the rolling historical covariance method and the rolling

historical covariance with shrinkage method provide virtually identical MTEs and Sharpe ra-

tios. That is, shrinkage does not produce better performance of minimum variance portfolios

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Covariance matrix forecasting methodStatisticsHIST SHRINK EWMA DCC-GARCH GO-GARCH

Portfolios formed on sizeMean tracking error 5.37e-05 7.23e-05 1.22e-05 2.16e-05 1.90e-05Sharpe ratio 0.52 0.51 0.61 0.61 0.61Portfolios formed on book-to-marketMean tracking error 1.18e-04 1.15e-04 2.35e-05 2.43e-05 2.80e-05Sharpe ratio 0.66 0.66 0.77 0.72 0.71Portfolios formed on momentumMean tracking error 1.24e-04 1.23e-04 4.98e-05 4.23e-05 8.50e-05Sharpe ratio 0.51 0.54 0.58 0.53 0.52Portfolios formed on size and book-to-marketMean tracking error 1.16e-04 1.16e-04 2.46e-05 1.86e-05 1.69e-05Sharpe ratio 0.71 0.70 0.78 0.71 0.76Industry portfoliosMean tracking error 6.27e-05 6.21e-05 2.68e-05 3.24e-05 3.25e-05Sharpe ratio 0.71 0.72 0.64 0.69 0.73Portfolios formed on size and momentumMean tracking error 1.24e-04 1.23e-04 4.98e-05 4.23e-05 8.50e-05Sharpe ratio 0.51 0.54 0.58 0.53 0.52Portfolios formed on size and short-term reversalMean tracking error 5.34e-05 5.45e-05 3.08e-05 3.66e-05 5.00e-05Sharpe ratio 0.76 0.75 0.63 0.66 0.69Portfolios formed on size and long-term reversal

Mean tracking error 1.21e-04 1.18e-04 1.28e-04 7.27e-05 8.13e-05Sharpe ratio 0.51 0.53 0.49 0.56 0.51Core assetsMean tracking error 4.07e-05 4.09e-05 2.14e-05 2.01e-05 2.87e-05Sharpe ratio 0.74 0.74 1.19 1.15 0.99Averages over all datasetsMean tracking error 9.03e-05 9.17e-05 4.08e-05 3.45e-05 4.74e-05Sharpe ratio 0.63 0.63 0.70 0.68 0.67

Table 4: Out-of-sample performance of minimum variance portfolios over the period 1995-2013 versus the method of covariance matrix forecasting. This table reports the portfolioMean Tracking Error and the Sharpe ratio produced by different covariance matrix forecast-

ing methods. HIST denotes the rolling historical covariance; SHRINK denotes the rollinghistorical covariance with shrinkage; EWMA denotes the exponentially weighted moving av-erage covariance; DCC-GARCH denotes the Dynamic Conditional Correlation GARCH;GO-GARCH denotes the Generalized Orthogonal GARCH. In all methods the length of therolling estimation window amounts to 120 months.

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0.0e+00

2.5e05

5.0e05

7.5e05


M e a n

t r a c k

i n g e r r o r

0.0

0.2

0.4

0.6


S h a r p e

r a t i o

Panel A Panel B

Figure 3: Average out-of-sample performance of minimum variance portfolios over the period1995-2013 versus the method of covariance matrix forecasting. For each distinct method of forecasting, Panel A plots the average of the Mean Tracking Errors over all datasets whereasPanel B plots the average of the Sharpe ratios over all datasets. HIST denotes the rolling his-torical covariance; SHRINK denotes the rolling historical covariance with shrinkage; EWMAdenotes the exponentially weighted moving average covariance; DCC-GARCH denotes theDynamic Conditional Correlation GARCH; GO-GARCH denotes the Generalized Orthog-onal GARCH. In all methods the length of the rolling estimation window amounts to 120months.

than the rolling historical covariance. As opposed to the two rst methods of forecasting, the

remaining three methods allow to reduce the average Mean Tracking Error by 50-65%. The

forecast obtained using the DCC-GARCH model produces the least MTE. Somewhat surpris-

ingly, the use of the EWMA model produces a smaller MTE than the use of the much more

advanced GO-GARCH model; yet the differences between their MTEs are not substantial.

When it comes to the Sharpe ratios of the minimum variance portfolios, even though the per-

formance of the portfolios that use either EWMA, DCC-GARCH, or GO-GARCH method of

forecasting is higher than the performance of the portfolios that use the other two methods of

forecasting, the differences are marginal. In particular, the maximum average improvement inthe Sharpe ratio of the minimum variance portfolio amounts to 11% increase in the value of

this performance measure.

3.3 Performance of Minimum Variance Portfolios with a Volatility Target

In the last study we evaluate the out-of-sample performance of minimum variance portfolios

with a volatility target mechanism. The performance of these portfolios is also measured by the

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Mean Tracking Error and the Sharpe ratio. To implement this portfolio construction method,

we rst dene the composition of the global minimum variance portfolio for month t on the

basis of the covariance matrix forecast provided at the end of month t 1. The forecasted

standard deviation of the minimum variance portfolio for month t is given by

P t = w t t w t ,where w t the ex-ante vector of weights of the global minimum variance portfolio for month t

and t is the covariance matrix forecast for month t . Then, given the volatility target target ,

the global minimum variance portfolio is mixed with the risk-free asset in order to keep the

volatility of the complete portfolio at the target level. Specically, the weight of the minimum

variance portfolio in the complete portfolio is determined as

a t = min target

P t, a max .

Given the daily returns generated by the minimum variance portfolio with a volatility target,

we compute the portfolio out-of-sample monthly standard deviation which is denoted by Ct .

The month t Tracking Error and the Mean Tracking Error are computed as

T E t = Ct target2 , MT E =

1M

M

t =1

T E t ,

where M is the number of months in the out-of-sample period. In our study the annual

volatility target, target , is set to be 10% and the maximum risk exposure, amax , is set to be

150%.

Table 5 reports the out-of-sample performance of minimum variance portfolios with a

volatility target over the period 1995-2013 versus the method of covariance matrix forecasting.

Specically, this table reports the portfolio MTE and the Sharpe ratio produced by different

covariance matrix forecasting methods. Panel A in Figure 4 plots the average of MTEs over

all datasets, whereas Panel B in the same gure plots the average of Sharpe ratios over all

datasets. The results on the portfolio MTE reported Table 5 agree with those presented in

the previous subsection. In brief, there is no differences in the MTEs produced by the rolling

historical covariance method and the rolling historical covariance with shrinkage method. The

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Covariance matrix forecasting methodStatisticsHIST SHRINK EWMA DCC-GARCH GO-GARCH

Portfolios formed on sizeMean tracking error 3.61e-04 3.66e-04 1.83e-04 1.59e-04 1.62e-04Sharpe ratio 0.46 0.45 0.72 0.75 0.73Portfolios formed on book-to-marketMean tracking error 3.38e-04 3.41e-04 1.50e-04 1.28e-04 1.22e-04Sharpe ratio 0.59 0.60 0.89 0.84 0.82Portfolios formed on momentumMean tracking error 2.99e-04 3.01e-04 1.44e-04 1.26e-04 1.20e-04Sharpe ratio 0.44 0.47 0.71 0.66 0.65Portfolios formed on size and book-to-market

Mean tracking error 3.70e-04 3.75e-04 1.59e-04 1.48e-04 1.50e-04Sharpe ratio 0.65 0.65 0.93 0.88 0.89Industry portfoliosMean tracking error 2.85e-04 2.83e-04 1.39e-04 1.19e-04 1.18e-04Sharpe ratio 0.63 0.64 0.80 0.81 0.90Portfolios formed on size and momentumMean tracking error 2.99e-04 3.01e-04 1.44e-04 1.26e-04 1.20e-04Sharpe ratio 0.44 0.47 0.71 0.66 0.65Portfolios formed on size and short-term reversalMean tracking error 2.80e-04 2.85e-04 1.48e-04 1.28e-04 1.20e-04Sharpe ratio 0.71 0.71 0.82 0.85 0.84Portfolios formed on size and long-term reversalMean tracking error 2.90e-04 2.95e-04 1.43e-04 1.34e-04 1.25e-04Sharpe ratio 0.45 0.48 0.66 0.70 0.65Core assetsMean tracking error 1.44e-04 1.43e-04 8.15e-05 6.13e-05 7.20e-05Sharpe ratio 0.75 0.76 1.45 1.37 1.17Averages over all datasetsMean tracking error 2.96e-04 2.99e-04 1.43e-04 1.25e-04 1.23e-04Sharpe ratio 0.57 0.58 0.85 0.84 0.81

Table 5: Out-of-sample performance of minimum variance portfolios with a volatility targetover the period 1995-2013 versus the method of covariance matrix forecasting. This table re-ports the portfolio Mean Tracking Error and the Sharpe ratio produced by different covariancematrix forecasting methods. The annualized target volatility of minimum variance portfolios is10%. HIST denotes the rolling historical covariance; SHRINK denotes the rolling historicalcovariance with shrinkage; EWMA denotes the exponentially weighted moving average covari-ance; DCC-GARCH denotes the Dynamic Conditional Correlation GARCH; GO-GARCHdenotes the Generalized Orthogonal GARCH. In all methods the length of the rolling estima-tion window amounts to 120 months.

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other three methods of forecasting allow to reduce the MTE by 50-60%. In contrast to the

results presented in the previous subsection, the EWMA, DCC-GARCH, and GO-GARCH

methods of covariance matrix forecasting allow not only to reduce substantially the portfolio

tracking error, but at the same time these methods allow to increase considerably the portfo-

lio risk-adjusted performance. In particular, better methods of covariance matrix forecasting

allow to increase the portfolio Sharpe ratio by 40-50%.

The explanation for why better forecast accuracy allows one to achieve better risk-adjusted

portfolio performance lies in the fact that the contemporaneous relation between the market

return and volatility is found to be negative, see French, Schwert, and Stambaugh (1987). That

is, usually over the periods with above average volatility the market return is below average.

Since the volatility is predictable, keeping the volatility at a target level reduces exposure tostocks when volatility is high, and increases that exposure when volatility is low. The advantage

of the minimum-variance portfolio with a volatility target over the pure minimum-variance

portfolio comes from two effects: a reduction in volatility and an increase in returns; both

these effects contribute to a better risk-return tradeoff. Yet in order to realize this advantage

one needs to forecast the future volatility with high accuracy. Poor forecast accuracy results

in deterioration of the performance of minimum-variance portfolios. This effect is clearly seen

by comparing the Sharpe ratios of the minimum-variance portfolios and minimum-varianceportfolios with a volatility target (see Tables 4 and 5) when one uses the sample covariance

matrix (with and without shrinkage) as a forecast for future covariance matrix.

4 Conclusions

Our research showed that there are large differences between the covariance matrix forecast-

ing methods. For those researchers and practitioner who routinely use the sample covariancematrix as a forward-looking estimate for the future covariance matrix our empirical results

imply that the out-of-sample performance of minimum-variance portfolios can be improved

substantially by adopting multivariate GARCH models. We found that minimum-variance

portfolios, constructed by using multivariate GARCH forecasts, deliver the best performance

in terms of risk optimization and portfolio tracking error. In particular, switching from the

sample covariance matrix forecast to a multivariate GARCH forecast allows one to reduce the

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0e+00

1e04

2e04

3e04


M e a n

t r a c k

i n g e r r o r

0.0

0.2

0.4

0.6

0.8


S h a r p e

r a t i o

Panel A Panel B

Figure 4: Average out-of-sample performances of minimum variance portfolios with a volatil-ity target over the period 1995-2013 versus the method of covariance matrix forecasting. Theannualized target volatility of minimum variance portfolios is 10%. For each distinct methodof forecasting, Panel A plots the average of Mean Tracking Errors over all datasets whereasPanel B plots the average of Sharpe ratios over all datasets. HIST denotes the rolling histor-ical covariance; SHRINK denotes the rolling historical covariance with shrinkage; EWMAdenotes the exponentially weighted moving average covariance; DCC-GARCH denotes theDynamic Conditional Correlation GARCH; GO-GARCH denotes the Generalized Orthog-onal GARCH. In all methods the length of the rolling estimation window amounts to 120months.

forecasting error and portfolio tracking error by more than 50%. The risk-adjusted perfor-

mance of the portfolios based on multivariate GARCH forecast is also superior compared to

that of the portfolios based on sample covariance matrix forecast. The differences in perfor-

mances is especially large when minimum-variance portfolios are implemented with a volatility

target mechanism; in this case the risk-adjusted performance can be improved up to 50%. We

tested two multivariate GARCH models, DCC-GARCH and GO-GARCH, and in our tests

the performance of the DCC-GARCH model is found to be marginally better than that of the

GO-GARCH model.

There is a common impression that shrinkage of sample covariance matrix improves the

accuracy of covariance matrix forecast. Our tests revealed that this impression is wrong. We

found that shrinkage is a computationally intensive method that improves neither the forecast

accuracy nor the performance of minimum-variance portfolios. We also found that the simple

EWMA covariance matrix forecast performs only slightly worse than the multivariate GARCH

forecast. Therefore those researchers and practitioner who nd that multivariate GARCH

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models are cumbersome in implementation are advised to adopt the EWMA method. This

method is not only computationally simple and fast, but also generates the best risk-adjusted

performance of minimum-variance portfolios in the majority of our empirical tests.

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