Suppose that daily volatility changes of your stock(Pick a stock and it's weekly returns) are not independent on successive days and often bunched together and/or that the return distribution of your stock is leptokurtic (i.e. exhibits excess peakedness and fat tails). Under these circumstances, it may be suitable to utilize an estimate of volatility that does not equally weight historical variances. You consider two alternative weighting schemes. One weighting scheme is an Exponentially Weighted Moving Average (EWMA) which reflects the previous days' estimated return variance as well as the information in the previous day's returns. The alternative weighting scheme is the GARCH(1,1) which also allows for the possibility of some reversion to the long-run mean of the historical variances. Section I 1. Assume that RiskMetric's characterization of the Exponentially Weighted Moving Average (EWMA) model provides a good description of the data. Compute a forecast for the next day's volatility using the EWMA model and compute one-day Value-at-Risk with 99% confidence level using the next day's volatility estimated from the EWMA model. 2. Compare this estimate of VaR to an estimate of a 99%, one-day, Value-at-Risk constructed using an estimate of volatility derived from the equally weighted historical variance of the return distribution. 3. Discuss the difference between these two estimates or forecasts of VaR. Back-test the acceptability of a parametric VaR model (i.e., 99%, one-day, Value-at-Risk) calculated using RiskMetric's EWMA volatility. Use Kupiec's Proportion of Failures Test at a 5% significance level (i.e. the cutoff point for a 5% significance level (=.05) is LR=3.84). Please see the page 20 of VaR Back Testing (3) PowerPoint for Kupiec's Proportion of Failures Test. Discuss the results of back-test. Note that the significance level of the statistical back-test is different from the confidence level used in the VaR calculation. In this case, VaR confidence level is 99% while the significance level in statistical tests is 5%. Section II 1

1. Use maximum likelihood estimation to estimate the parameters of a GARCH(1,1) model for your stock. Find the values of parameters of a GARCH(1,1) that provide a good description of the data using Excel Solver tool. 2. Compute a forecast for the day n's volatility using the GARCH(1,1) volatility model and use it to compute an estimate of the 99%, one-day, Value-at-Risk. Interpret the resulting %VaR estimate and compare this estimate of VaR to an estimate of a 99%, one-day, Value-at-Risk constructed using an estimate of volatility derived from the equally weighted historical variance of the return distribution. 3. Discuss the difference between these two estimates or forecasts of VaR. Back-test the acceptability of a parametric VaR model (i.e., 99%, one-day, Value-at-Risk) calculated using GARCH(1,1) volatility. Use Kupiec's Proportion of Failures Test at a 5% significance level (i.e. the cutoff point for a 5% significance level (=.05) is LR=3.84). Please see the page 20 of VaR Back Testing (3) PowerPoint for Kupiec's Proportion of Failures Test. Discuss the results of back-test.