Specification:

Using expressions  and we can directly construct

$$\ell^{\pm_{(k,P)}}(c) = {\sum_{t\in\Phi^{\pm_{(k,P)}}(c)}{\cal L}_{t-1}^{_{(k,P)}}. \over{\char93 }(\Phi^{\pm_{(k,P)}}(c))}$$ (37)

which is the mean log-likelihood of all loss exceedences over confidence level c for a long (+) or short (-) position in the US Dollar with respect to the series k or portfolio P.

Finally, as specified by expressions  and , we construct the univariate average mean log-likelihood

$$\ell_u(c) = {1\over20}\left(\sum_{k=1}^{10}\ell^{+_{(k)}}(c) + \sum_{k=1}^{10}\ell^{-_{(k)}}(c)\right)$$ (38)

and the multivariate average mean log-likelihood

$$\ell_m(c) = {1\over2}\left(\ell^{+_{(P)}}(c) + \ell^{-_{(P)}}(c)\right)$$ (39)

plotted in figures 4u and 4m respectively for all candidate models. Plots are always for the confidence range of $50\%$ and higher, since this defines the division between loss making and profitable events. Due to symmetrization (consideration of both long and short, positions) the value of $\ell_u(c = 50)$ is based on all 10000 out-of-sample events x_t^(k) $(t\in[2,1001])$ and the value $\ell_m(c = 50)$ is based on all 1000 out-of-sample events x_t^(P) $t\in[2,1001]$$(t\in[2,1001])$ .