Specification:

Re-using expression  defining the exceedence level $\chi_t^{\pm_{(k,P)}}$ and expression  defining the set X_t^(k,P) we extend definition  so that  

$$\Phi^{\pm_{(k,P)}}_t(c) = \{t\vert\pm x_t^{_{(k,P)}}\gt\chi^{\pm_{(k,P)}}_{t-1}(c),t\in X_t^{_{(k,P)}}\}.$$ (27)

Note that here we have defined sets $\Phi$ against a universal set of a moving sample of events rather than the entire out-of-sample set of events as expressed in . Next, using $\Phi^{\pm_{(k,P)}}_t(c)$ defined as above, we can construct booleans $R_t^{\pm_{(k,P)}}(c)$ and $G_t^{\pm_{(k,P)}}(c)$ $(\in \{0,1\})$ such that:  

$$\hspace{-0.7cm}(G_t^{\pm_{(k,P)}}(c),R_t^{\pm_{(k,P)}}(c)) =... ... ${\char93 }(\Phi^{\pm_{(k,P)}}_t(c)) \ge 9$}\end{array}\right.$$ (28)

Clearly, through expression , one can identify that $G_t^{\pm_{(k,P)}}(99) = 1$ and $R_t^{\pm_{(k,P)}}(99) = 1$ are the BIS defined colour designation for a risk model with respect to the assets on day t. As usual, the portfolio represented by the superscripts covers the cases of the long (+) or short (-) position in the US Dollar with respect to the series k or portfolio P. Using the booleans defined by expression  we can construct the colour frequency measures

$$(G,R)^{\pm_{(k,P)}}(c) = {1\over751}\sum_{t\in[251,1001]}(G,R)_t^{\pm_{(k,P)}}(c)\\$$ (29)

where the notation (G,R) transparently refers to 2 expressions - one each for $G^{\pm_{(k,P)}}(c)$ and $R^{\pm_{(k,P)}}(c)$ respectively. $(G,R)^{\pm_{(k,P)}}(c)$ is of course the fraction of events in the out-of-sample period when the green and red designation is applicable. Note again that we have extended the BIS definition for this measure and allowed confidence level c to be a variable.

Finally, as specified by expressions  and , we construct the univariate average BIS colour frequency

$$(G,R)_u(c) = {1\over20}\left(\sum_{k=1}^{10}(G,R)^{+_{(k)}}(c) + \sum_{k=1}^{10}(G,R)^{-_{(k)}}(c)\right)$$ (30)

and the multivariate average BIS colour frequency

$$(G,R)_m(c) = {1\over2}\left((G,R)^{+_{(P)}}(c) + (G,R)^{-_{(P)}}(c)\right)$$ (31)

plotted in figures 2u and 2m respectively for all candidate models.