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I want to calculate the Value-at-Risk at date $t$ in such a way that I minimize the capital requirements given as \begin{align} \text{CR}_{\,t+1\,:\,t+250} = \sum_{h=0}^{249}\max\left( -(3+k_{t})\overline{\text{VaR}}^{60}_{t+h}, -\text{VaR}_{t+h}\right), \end{align} in which $\overline{\text{VaR}}^{60}_{t} = 1/60\sum_{\tau=1}^{60} \text{VaR}_{t-\tau+1}$ and $k_{t} = f(\sum_{\tau=1}^{250} H_{t-\tau+1})$ where $H_t = 1_{\left\{r_t < \text{VaR}_t\right\}}$.

The problem is that that $\text{VaR}_t$ minimizing the $\text{CR}_{t+1}$ does not necessarily minimize $\text{CR}_{\,t+1\,:\,t+250}$. Any idea how we can approach this problem is appreciated.

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