Estimating Expected Shortfall from an ARMA-GARCH Return Model

Summary

The document asks how to derive a 99% expected shortfall for a long position using a two-day forecast from an ARMA(1,1)-GARCH(1,1) return model. It outlines the model’s conditional mean and volatility equations, assumes independent standard normal innovations, and gives the general relationship between expected shortfall and the tail of the value-at-risk distribution.

No derivation or numerical result is provided; the text is a request for guidance. Its useful focus is the challenge of moving from a one-step conditional distribution to a multi-day loss distribution when volatility evolves over time. The question specifies a stationary model and a position size, but does not state parameter values or clarify whether the target is return loss or dollar loss, so an analytical calculation cannot be completed from the information given.

Key ideas

• Expected shortfall summarizes the average loss in the tail beyond a chosen value-at-risk threshold.
• The question concerns a two-day forecast under an ARMA(1,1)-GARCH(1,1) return process.
• Conditional volatility changes over the forecast horizon, complicating analytical tail calculations.
• A numerical expected-shortfall estimate would require parameter values and a precise definition of the loss measure.

Tags

# Expected Shortfall for ARMA-GARCH Model


# Expected Shortfall for ARMA-GARCH Model












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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