"Frontmatter". In: Analysis of Financial Time Series

386 MULTIVARIATE VOLATILITY MODELSLet VaR 1 be the value at risk for holding the position on Cisco Systems stock andVaR 2 for holding Intel stock. Results of Chapter 7 show that the overall daily VaRfor the investor is√VaR = VaR 2 1 + VaR2 2 + 2ρVaR 1VaR 2 .In this illustration, we consider three approaches to volatility modeling for calculatingVaR. For simplicity, we do not report standard errors for the parameters involvedor model checking statistics. Yet all of the estimates are statistically significant at the5% level and the models are adequate based on the Ljung–Box statistics of the standardizedresidual series and their squared series. The log returns are in percentagesso that the quantiles are divided by 100 in VaR calculation. Let r 1t be the return ofCisco stock and r 2t the return of Intel stock.(A) Univariate ModelsThis approach uses a univariate volatility model for each stock return and uses thesample correlation coefficient of the stock returns to estimate ρ. The univariatevolatility models for the two stock returns arer 1t = 0.380 + 0.034r 1,t−1 − 0.061r 1,t−2 − 0.055r 1,t−3 + a 1tσ 2 1t = 0.599 + 0.117a2 1,t−1 + 0.814σ 2 1,t−1r 2t = 0.187 + a 2tσ 2 2t = 0.310 + 0.032a2 2,t−1 + 0.918σ 2 2,t−1 .The sample correlation coefficient is 0.473. The 1-step ahead forecasts needed inVaR calculation at the forecast origin t = 2275 areˆr 1 = 0.626, ˆσ 2 1 = 4.152, ˆr 2 = 0.187, ˆσ 2 2= 6.087, ˆρ = 0.473.The 5% quantiles for both daily returns areq 1 = 0.626 − 1.65 √ 4.152 =−2.736,q 2 = 0.187 − 1.65 √ 6.087 =−3.884,where the negative sign denotes loss. The VaR for the individual stocks are VaR 1 =$1000000q 1 /100 = $27360 and VaR 2 = $1000000q 2 /100 = $38840. Consequently,the overall VaR for the investor is VaR = $57117.(B) Constant Correlation Bivariate ModelThis approach employs a bivariate GARCH(1, 1) model for the stock returns. Thecorrelation coefficient is assumed to be constant over time, but it is estimated jointlywith other parameters. The model isr 1t = 0.385 + 0.038r 1,t−1 − 0.060r 1,t−2 − 0.047r 1,t−3 + a 1tr 2t = 0.222 + a 2t