Specification:

As a function of confidence level c (in %), it is convenient to first define the exceedence limits $\chi_t^{+_{(k,P)}}(c)$ and $-\chi_t^{-_{(k,P)}}(c)$ so that:  

$$\int_{-\infty}^{\chi_t^{+_{(k,P)}}(c)}p_t^{_{(k,P)}}(x)d\!x ... ...i_t^{-_{(k,P)}}(c)}^\infty p_t^{_{(k,P)}}(x)d\!x = {c\over100}.$$ (22)

It should be clear from expression  that $\chi_t^{+_{(k,P)}}(c)$ is the loss that will be exceeded by a long (+) position on the US Dollar against series k or portfolio P with probability $(1-{c\over100})$. Similarly the exceedence limit $\chi_t^{-_{(k,P)}}(c)$ is the correspondingly probable loss for a short position on the US Dollar against series k or portfolio P. We note that for all the models except historical simulation - the conditional distribution p_t^(k,P) is symmetric around zero and $\chi_t^{+_{(k,P)}}(c) = \chi_t^{-_{(k,P)}}(c)$.

Next, we define the set of loss exceedence events $\Phi^{+_{(k,P)}}(c)$ and $\Phi^{-_{(k,P)}}(c)$ so that:  

$$\Phi^{\pm_{(k,P)}}(c) = \{t\vert\pm x_t^{_{(k,P)}}\gt\chi^{\pm_{(k,P)}}_{t-1}(c),t\in[2,1001]\}.$$ (23)

Using sets $\Phi^{\pm_{(k,P)}}(c)$ we can finally specify the exceedence ratio measure

$$r^{\pm_{(k,P)}}(c) = {{\char93 }(\Phi^{\pm_{(k,P)}}(c))\over1000(1-{c\over100})}.$$ (24)

Finally, as specified by expressions  and , we construct the univariate average exceedence ratio

$$r_u(c) = {1\over20}\left(\sum_{k=1}^{10}r^{+_{(k)}}(c) + \sum_{k=1}^{10}r^{-_{(k)}}(c)\right)$$ (25)

and the multivariate average exceedence ratio

$$r_m(c) = {1\over2}\left(r^{+_{(P)}}(c) + r^{-_{(P)}}(c)\right)$$ (26)

plotted in figures 1u and 1m respectively for all candidate models.