I need to find an analytical solution for the 99% confidence expected shortfall (CVaR) for a long position of 100 dollars at time $t$ for an asset with returns modeled by an ARMA(1,1)-GARCH(1,1) model with $r_t = θr_{t−1} + u_t + ψu_{t−1}$, $u_t = σ_t\epsilon_t$, $σ^2_t = ω + αu^2_{t−1} + βσ^2_{t−1}$,where $\epsilon_t$ are independent and identically distributed standard normal random variables, and $θ,ψ,α,β$ satisfy conditions that make the market stationery.

This is a question from an old exam that was posted online by the department (which you can see here, this question is problem 10) for students to study from to prepare for an upcoming exam, but no solutions were given and while I know expected shortfall/CVaR for confidence level $\alpha$ is the average value of the worst $1-\alpha$ percent of returns, given by $ES_\alpha(X) = \frac{1}{1 - \alpha} \int^1_\alpha VaR_u(X)du$ where the Value at Risk is given by $VaR_\alpha(X) = \{ Y|P(Y\leq X) = 1 - \alpha\}$, I have no idea how to apply that to a two-day forecast for a time series analytically. Any advice would be greatly appreciated, thanks!


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