Rank defects

In typical implementation of VaR methodologies it is not uncommon that the number of risk factors involved, d, is far greater than the number of data, n, in the history used for forecasting the covariance matrix $\Sigma_t$ (of dimension d). In particular, if n<d and $\Sigma_t$is constructed with the rectangular moving average model (model 2) of section  or the RiskMetrics model (model 3) of section  it is guaranteed that the covariance matrix will have a rank defect. The existence of a rank defect implies that $\Sigma_t$ admits the existence of a portfolio (or a set of portfolios) $\lambda = \lambda(\Sigma_t)$ such that $\lambda^T\Sigma_t\lambda = 0$ (or $\hbox{VaR}(\lambda,\Sigma_t,c) = 0$ for all c, where c is the confidence level used - usually $99\%$).

In the rectangular moving average model every x_^(j)t-i entering the computation  of s^_(jk) for any k does so with the same weight - unity. In the J. P. Morgan RiskMetrics case the computation  expresses s^_(jk) in recursive form - but when expanded in terms of the data history it is stll true that every participation of x_^(j)t-i in the construction of the covariance matrix occurs with the same weight or coefficient (although not unity).

In general - any covariance matrix construction which can be expressed in the form:

$$\begin{array} {l} \Sigma_t = W^T_tW_t\quad\hbox{where}\\ W^T... ...}}_{t-n+1}x^{_{(d)}}_{t-n+1}\\ \end{array}\right)\\ \end{array}$$ (45)

(W^T is a matrix of the weighted data history) will suffer from a rank defect if n<d. (Theoretically, the rank of $\Sigma_t$ is the the rank of W_t - which is the minimum of d and n. If n<d, then the rank n of $\Sigma_t$ is smaller than its dimension d and hence $\Sigma_t$ must have a rank defect.)