Consider time series of returns and make the usual assumption that returns are serially uncorrelated. Then, we can define a vector of zero-mean white noises , where is the vector of returns and is the vector of expected returns.
Despite being serially uncorrelated, the returns may present contemporaneous correlation. That is:
may not be a diagonal matrix. Moreover, this contemporaneous variance may be time-varying, depending on past information. The Exponentially Weighted Moving Average () covariance model assumes a specific parametric form for this conditional covariance. More specifically, we say that if:
V-Lab uses , the parameter suggested by RiskMetrics for daily returns, and is the sample average of the returns.
Notice that the elements from the main diagonal of give us conditional variances of the returns, i.e. is the conditional variance of the return . Analogously, the elements outside of the main diagonal give us conditional covariances, i.e. is the conditional covariance between the returns and . Hence, we can easily back out the conditional correlations,
This is what is plotted by V-Lab.
More concisely, we can define the whole correlation matrix by:
where is a matrix such that, :
where is the Kronecker delta, i.e. if and otherwise. That is, is a matrix with with all elements outside of the main diagonal set to zero, and the main diagonal set to the conditional volatilities, i.e. the elements in the main diagonal are equal to the square root of the elements in the main diagonal of .
Then, is again the correlation between and . Note that , .
Notice that the is actually a multivariate version of an model, which is a particular case of the model.
Notice also that after iterating the conditional variance expression, we obtain, if :
which is a weighted average, with weights decaying exponentially at rate , hence the name of the model, Exponentially Weighted Moving Average.
Engle, R. F., 2009. Anticipating Correlations: A New Paradigm for Risk Management. Princeton University Press.
Tsay, R. S., 2005. Analysis of Financial Time Series — 2nd Ed. Wiley-Interscience.