Advances in Risk Management

THADAVILLIL JITHENDRANATHAN 269

The correlation estimator is:

ρ12,t=

q12,t

√

q11,tq22,t

(14.12)

This model will be mean-reverting ifα+β<1. The matrix version of this
model can then be written as:

Qt=S(1−α−β)+α(εt− 1 ε′t− 1 )+βQt− 1 (14.13)

where S is the unconditional correlation matrix of the disturbance terms and
Qt=|q1,2,t|.
The log-likelihood for this estimator can be written as:

L=−

1

2

∑T

t= 1

(

nlog (2π)+2 log|Dt|+log|Rt|+ε′tR−t^1 εt

)

(14.14)

whereDt=diag

{√

hi,t

}

andRtis the time-varying correlation matrix. With

these estimates of variances and correlations, the covariance matrix is
constructed.
The main purpose of this chapter is to study whether the use of time-
varying variances and co-variances in portfolio optimization models will
result in betterex postresults as compared to the traditional rolling estimates.
For this purpose portfolios are created using twenty stocks from Dow Jones
Industrial Average Index. The time period covered is from January 1995 and
December 2004. For consistency the stocks used in this study are the ones
that were part of the Dow Jones Index for the entire period of study, except
for Microsoft, which was included in the list in 1999. The weekly returns for
each of these stocks are obtained from Bloomberg.
Awindow of five years is used in estimating the means, variances and co-
variances using the rolling estimator. This window is moved by one month
for the next five years, creating a total set of 60 separate estimates. These
estimates are the inputs used in the portfolio optimization model. With each
set of monthly inputs, a set of efficient portfolios is estimated. Each of these
efficient portfolios contain one minimum variance portfolio and ten efficient
portfolios with increasing levels of risk compared to the minimum variance
portfolio.
For the DCC estimators I use the same set of five-year rolling windows,
but to capture the time-varying nature of variances and co-variances, the
end of the period values of the same is input into the portfolio optimization
model. For example, using the DCC model one can estimate 260 variances
and correlations for a period of five years. But for estimating the efficient
set of portfolios, only the variances and correlations for the last week of
the sample period is used. For example, for the time period from 3 January
2000 to 27 December 2004, the variances and correlations used are taken