V-Lab

### EWMA Covariance Model

#### Definition

Consider $n$ 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 ${\epsilon }_{t}={r}_{t}-\mu$, where ${r}_{t}$ is the $n⨯1$ vector of returns and $\mu$ is the vector of expected returns.

Despite being serially uncorrelated, the returns may present contemporaneous correlation. That is:

$∑t ≔ 𝔼t-1 rt-μ rt-μ '$

may not be a diagonal matrix. Moreover, this contemporaneous variance may be time-varying, depending on past information. The Exponentially Weighted Moving Average ($\mathrm{EWMA}$) covariance model assumes a specific parametric form for this conditional covariance. More specifically, we say that ${r}_{t}-\mu ~\mathrm{EWMA}\left(\lambda \right)$ if:

$∑t+1 = 1-λ rt-μ rt-μ ' + λ∑t$

V-Lab uses $\lambda =0.94$, the parameter suggested by RiskMetrics for daily returns, and $\mu$ is the sample average of the returns.

#### Correlations

Notice that the elements from the main diagonal of ${\sum }_{t}$ give us conditional variances of the returns, i.e. ${\sum }_{t}^{i,i}$ is the conditional variance of the return ${r}_{t}^{i}$. Analogously, the elements outside of the main diagonal give us conditional covariances, i.e. ${\sum }_{t}^{i,j}$ is the conditional covariance between the returns ${r}_{t}^{i}$ and ${r}_{t}^{j}$. Hence, we can easily back out the conditional correlations,

$Γ t i,j ≔ ∑ti,j ∑ti,i ∑tj,j$

This is what is plotted by V-Lab.

More concisely, we can define the whole correlation matrix by:

$Γ t ≔ Dt-1 ∑t Dt-1$

where ${D}_{t}$ is a matrix such that, $\forall i,j\in \left\{1,...,n\right\}$:

$D t i,j ≔ δ i,j ∑ti,j$

where ${\delta }_{i,j}$ is the Kronecker delta, i.e. ${\delta }_{i,j}=1$ if $i=j$ and ${\delta }_{i,j}=0$ otherwise. That is, ${D}_{t}$ 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 ${\sum }_{t}$.

Then, ${\text{Γ}}_{t}^{i,j}$ is again the correlation between ${r}_{t}^{i}$ and ${r}_{t}^{j}$. Note that ${\text{Γ}}_{t}^{i,j}=1$, $\forall i\in \left\{1,...,n\right\}$.

#### Relation to The GARCH(1,1) Model

Notice that the $\mathrm{EWMA}$ is actually a multivariate version of an $\mathrm{IGARCH}\left(1,1\right)$ model, which is a particular case of the $\mathrm{GARCH}\left(1,1\right)$ model.

Notice also that after iterating the conditional variance expression, we obtain, if $\lambda \in \left(0,1\right)$:

$∑t+1 = 1-λεtεt' + λ1-λεt-1εt-1' + λ21-λεt-2εt-2' + ... = 1-λ εtεt' + λεt-1εt-1' + λ2εt-2εt-2' + ... = εtεt' + λεt-1εt-1' + λ2εt-2εt-2' + ... 1 1-λ = εtεt' + λεt-1εt-1' + λ2εt-2εt-2' + ... 1+λ+λ2+...$

which is a weighted average, with weights decaying exponentially at rate $\lambda$, hence the name of the model, Exponentially Weighted Moving Average.

#### Bibliography

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.