Session 8 Value-at-Risk and correlation modelling

8.1 Exercises

Exercise 1 (Value-at-Risk using GARCH models)

Consider the problem of estimating the Value-at-Risk \(-\text{VaR}_t^p = Q_p(r_t | I_{t-1})\) using a GARCH model \(r_t = \sigma_t z_t\) and \(\sigma_t\) known given the information set \(I_{t-1}\).

• Discuss how to estimate \(Q_p(r_t | I_{t-1})\) under the assumption that \(z_t \sim \mathcal{N}(0,1)\). How could you estimate the VaR without assuming the true distribution of \(z_t\)?

• How would you evaluate the VaR forecasts out of sample? Discuss how to apply the unconditional coverage test and the dynamic quantile test in this context.

Exercise 2 (Covariance matrix of factor ARCH model)

Consider the factor ARCH model

\[ \begin{aligned} r_{it} &= \lambda_i r_{mt} + \sigma_{it} z_{it}, &\quad z_{it} \sim \mathcal{D}(0,1) \\ r_{it} &= \sigma_{mt} z_{mt}, &\quad z_{mt} \sim \mathcal{D}(0,1) \end{aligned} \]

where \(r_{mt}\) is the market return, \(\sigma_{mt}\) is the market volatility and \(\lambda_i\) is the loading of stock \(i\) on the market return. Assume that \(\text{cov}(z_{mt}, z_{it}) = 0\) (the common shock and the idiosyncratic shock are uncorrelated), \(\text{cov}(z_{it}, z_{jt}) = 0\) (the idiosyncratic shocks are uncorrelated) and \(\sigma_{mt}, \sigma_{it}\) are known given \(I_{t-1}\) (ARCH/GARCH dynamics).

Derive the covariance matrix of the returns. In particular, show that:

• \(\text{Var}_{t-1}(r_{it}) = \lambda_i^2 \sigma^2_{mt} + \sigma_{it}^2\)
• \(\text{cov}_{t-1}(r_{it}, r_{jt}) = \lambda_i \lambda_j \sigma_{mt}^2\)