Value at Risk and expected shortfall from a GARCH(1,1) volatility

SOURCE. Jorion, P. (2007), Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edition, McGraw-Hill, ISBN 978-0-07-146495-6. The parametric ("delta-normal") one-period VaR at confidence 1-alpha is the alpha-quantile of the return distribution with the sign flipped, so a loss is a positive number: VaR_alpha = -(mu + sigma z_alpha) = -mu + sigma z_{1-alpha}, with z_alpha = qnorm(alpha). The matching Gaussian expected shortfall is ES_alpha = -mu + sigma dnorm(z_alpha)/alpha.

Usage

Varatr(y, alpha = 0.05, a_grid = NULL, b_grid = NULL)

Arguments

y

Return series.

alpha

Tail probability in (0, 1); 0.05 is the 95 percent VaR.

a_grid, b_grid

Lattices for the ARCH and GARCH coefficients.

Value

List with var, es, mu, sigma, sigma2_next, omega, a, b, loglik, sigma2_path, z, alpha, n.

Details

The conditional sigma is the one-step-ahead GARCH(1,1) standard deviation of Bollerslev, T. (1986), Journal of Econometrics 31(3):307-327, doi:10.1016/0304-4076(86)90063-1, Eqs. (1)-(2): sigma^2_t = omega + a e^2_{t-1} + b sigma^2_{t-1}, e_t = y_t - mu.

VARIANCE TARGETING. omega is pinned to the sample unconditional variance s^2 by omega = s^2 (1 - a - b), the stationary-variance identity of Bollerslev Eq. (5) solved for omega. a and b are then chosen by exhaustive search over a fixed lattice maximising the Gaussian quasi-likelihood. A lattice rather than an optimiser is what makes both language arms land on identical numbers; it is this implementation's choice, not a quotation.

Jorion (2007) is not in the local corpus; the delta-normal formula is the standard statement, and the module is anchored on the degenerate a = b = 0 case where VaR collapses to a closed form in qnorm.

References

Jorion, P. (2007). Value at Risk, 3rd ed. McGraw-Hill. Bollerslev, T. (1986). Journal of Econometrics 31(3):307-327. doi:10.1016/0304-4076(86)90063-1.

Examples

Varatr(c(0.01, -0.02, 0.015, -0.005, 0.02, -0.03, 0.01, 0.005), 0.05)$var
#> [1] 0.0264769