With omega = 0.000002, alpha = 0.08, beta = 0.90 and tomorrow's conditional variance at 0.00025, forecast the total variance of the 10-day cumulative return under GARCH(1,1), and compare it with scaling tomorrow's daily volatility by sqrt(10).

With omega = 0.000002, alpha = 0.08, beta = 0.90 and tomorrow's conditional variance at 0.00025, forecast the total variance of the 10-day cumulative return under GARCH(1,1), and compare it with scaling tomorrow's daily volatility by sqrt(10).

Approach: Iterate the variance recursion in expectation to get each daily forecast as the long-run level plus a decaying deviation, then sum the ten of them because the returns are uncorrelated.

0.002372. Iterating the recursion in expectation gives E[sigma_{t+h}^2] = V + (alpha + beta)^{h-1}*(sigma_{t+1}^2 - V), with V = omega/(1 - alpha - beta) = 0.0001 and persistence 0.98. Returns are uncorrelated, so the 10-day cumulative variance is the sum of the ten daily forecasts, 10*V + (sigma_{t+1}^2 - V)*(1 - 0.98^10)/0.02 = 0.001 + 0.00015*9.1465 = 0.002372. The 10-day volatility is 4.87%. Scaling tomorrow's daily volatility of 1.58% by sqrt(10) gives 5.00% and a variance of 0.0025, which is 5.4% too high because it holds variance at today's raised level while the model pulls it back toward 0.0001. Square-root-of-time scaling is exact only when the conditional variance already sits at the long-run level. Above it the rule overstates risk, and below it the rule understates risk, which is the dangerous direction because calm periods are when limits get set.

Follow-up: How does the 10-day forecast change if you need a 1% quantile rather than a variance, given that GARCH residuals are themselves fat-tailed?

Key concepts: GARCH forecast, mean reversion in variance, square-root-of-time scaling, persistence.