Consider historical data for the S&P 500 index.

1. Use the analytical method to calculate the 1% significance level, 1-day VaR on a $10 million portfolio assuming normally distributed returns based on the historical mean and standard deviation of returns.
2. The VIX index estimates the forward looking S&P 500 return volatility from the implied volatility of a number of options. It reports the annual volatility measured in units of percent. Suppose the VIX index is 23.10. Calculate the 1% significance level, 1-day VaR on a $10 million portfolio assuming normally distributed returns using the VIX index value to estimate the volatility.
3. Recognizing that liquidity dries up during time of crisis, repeat Problem 2 to find the 1% significance level, 2-day VaR on a $10 million portfolio assuming normally distributed returns using the VIX index value to estimate the volatility.
4. Use the historical method for calculating the 1% significance, 1-day VaR on a $10 million portfolio. Hint: Sort the daily returns from lowest to highest and choose the observation that corresponds to 1% of the total sample.