Treatment of pathological VaR estimates

 All the problems listed above deal with the existence of unreliable regions of portfolio space in which risk is underestimated due to estimation errors in the covariance matrix. Our proposal to handle this problem is to determine $\kappa(\lambda,\Sigma_t,c)$ as the minimum acceptable risk for a portfolio at confidence level c. The reported risk (at level c) is then the larger of $\hbox{VaR}(\lambda,\Sigma_t,c)$ and $\kappa(\lambda,\Sigma_t,c)$. As a corollary, we assert that $\hbox{VaR}(\lambda,\Sigma_t,c)$ is defined as pathologically underestimated if $\hbox{VaR}(\lambda,\Sigma_t,c) < \kappa(\lambda,\Sigma_t,c)$.

To build this machinery we first introduce the function:  

$$\nu(\lambda) = \sum_{i=1}^d\vert\lambda_i\vert$$ (46)

and refer to this as the collateral function - since this function approximates the collateral needed to support the investment. (The picture here is consistent with the view that trading is conducted without leverage and via a broker who allows one to take long or short positions on paper - as long as the value of the collateral function is deposited with the broker. The true collateral may be different due to several reasons - but modeling the true collateral is tangential to our objective.) Our purpose is simply to define a meaningful norm, $\Vert\lambda\Vert = \nu(\lambda)$, against which the size of $\hbox{VaR}(\lambda,\Sigma_t,c)$ may be compared, and which satisfies the condition that the ratio  

$${\hbox{VaR}(\lambda,\Sigma_t,c)\over\nu(\lambda)}$$ (47)

is invariant under scalar multiplication of $\lambda$ (true because both the numerator and the denominator satisfy the property, $f(\alpha\lambda) = \alpha f(\lambda)$).

The scale invariance of the ratio  has the important repercussion that distinct values must correspond to distinct directions in the portfolio space of dimension d (from the origin). Since only direction matters, for further analysis it is convenient to define the surface S of normalized portfolios, as the locus of all portfolios for which $\nu = 1$.In this notation, the components of $\lambda^{_{(S)}} = {\lambda\over\Vert\lambda\Vert}\in{\cal S}$may be denoted:

$$\lambda_i^{_{\cal S}} = {\lambda_i\over \sum_{i=1}^d\vert\lambda_i\vert}.$$ (48)

(For example, if d=3, then $\cal S$ is a regular octahedron joining the points on each axis at positive and negative unity.)

We still need two more mathematical objects to define the methodology for determining $\kappa(\lambda,\Sigma_t,c)$.One is the function $\cal P$ defined by:

$${\cal P}(\lambda^{(1)},\lambda^{(2)}) = {1\over d}\sum_{i=1}^d \log\left({\lambda_i^{(1)}\over\lambda_i^{(2)}}\right)^2$$ (49)

which measures the strength of a perturbation that transforms $\lambda^{(1)}$to $\lambda^{(2)}$.The other is the more complicated object, $\lambda_o^{_{\cal S}}$, determined as a function of $\lambda^{_{\cal S}}$.This object has different definition depending on whether $\hbox{VaR}(\lambda^{_{\cal S}},\Sigma_t,c)$ is negative or positive.

• If $\hbox{VaR}(\lambda^{_{\cal S}},\Sigma_t,c) \gt 0$ then $\lambda_o^{_{\cal S}}$is the local minimum for the VaR on surface $\cal S$ which can be reached from $\lambda^{_{\cal S}}$ via a continuous path of decreasing VaR. If the local minima is negative - $\lambda_o^{_{\cal S}}$ is the first point on the steepest descent where VaR becomes zero.
• If $\hbox{VaR}(\lambda^{_{\cal S}},\Sigma_t,c) < 0$ then $\lambda_o^{_{\cal S}}$is the point on the surface $\cal S$ such that $\hbox{VaR}(\lambda_o^{_{\cal S}},\Sigma_t,c) = 0$ which requires the smallest perturbation ${\cal P}(\lambda^{_{\cal S}},\lambda_o^{_{\cal S}})$.
• If $\hbox{VaR}(\lambda^{_{\cal S}},\Sigma_t,c) = 0$ then $\lambda_o^{_{\cal S}} = \lambda^{_{\cal S}}$.

We are now in a position to describe a procedure for reporting VaR which largely handles the underestimation problems when reporting risk:

1.

From $\lambda$, determine $\nu(\lambda)$ and $\lambda^{_{(S)}} = {\lambda\over\Vert\lambda\Vert}\in{\cal S}$.

2.

From $\lambda^{_{(S)}}$, determine $\lambda_o^{_{\cal S}}$.

3.

From $\lambda_o^{_{\cal S}}$ follow the path of steepest ascent (of VaR) - constrained to the surface $\cal S$ - until $\tilde\lambda^{_{\cal S}}$ such that ${\cal P}(\lambda_o^{_{\cal S}},\tilde\lambda^{_{\cal S}}) = p$,where p is a pre-established (but ad-hoc) perturbation strength.

4.

From $\tilde\lambda^{_{\cal S}}$, define:

$$\kappa(\lambda,\Sigma_t,c) = \nu(\lambda)\hbox{VaR}(\tilde{\lambda}^{_{\cal S}},\Sigma_t,c)$$ (50)

5.

The reported VaR at confidence level c is then the larger of $\hbox{Var}(\lambda,c)$ and $\kappa(\lambda,\Sigma_t,c)$.

Admittedly, our skeletal presentation above, is not an exhaustive consideration of all possible topological complexities that can arise - particularly when $\Sigma_t$ admits negative eigen-values. Our motivation, in this section, is only to introduce the concept of minimum acceptable risk for a portfolio $\lambda$ with respect to perturbation level p - as a useful concept which can go a long way in handling the various problems discussed in this section.