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Concerning the weighted portfolio returns. If you have weights $w_i$ and individual returns $r_i$ of your assets then it is only precisely true that the portfolio return $r$ is given by the scalar product $$ r = \sum_{i=1}^n w_i r_i $$ if $r_i$ is the usual arithmetic/simple return (not logreturn). Thereby I mean the simple return $$ r = P_{t+1}/P_t - 1 $$ ...


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Note that \begin{align*} \mathbb{E}\big(L \mid L\geq q_\alpha(L)\big) &= \frac{\mathbb{E}\big(\pmb{1}_{\{L\geq q_\alpha(L)\}} L\big)}{\mathbb{P}\big(L\geq q_\alpha(L) \big)}. \end{align*} The formula follows immediately.


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I tried using mgarchBEKK (or mgarch) but it seems like the package firstly estimate the VECM model, then use the residuals (Epsilon t) of the VECM (and their variances) in estimating the BEKK-GARCH model. I believe the correct method is to run the two models as a system, but I do not know how to proceed! Can anyone give me a hint please?


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The "right" thing to do is to treat the options as derivative contracts. Let's say for simplicity that you are using Monte Carlo to compute VaR. Then you would simulate the equity prices on each iteration, and then apply an option-pricing formula to get the corresponding option prices on that iteration. This lets you obtain an accurate simulated portfolio ...



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