Question 2 Instead of estimating the term structure of VaR or expected shortfalls, this exercise tries to estimate the term structure of volatility using Monte Carlo simulations. All the steps remain the same, except that in the very last step we estimate the volatility of simulated cumulative returns for each horizon. We will be using standard normal shocks throughout this exercise. The number of Monte Carlo simulations is denoted by MC. Use a MC of at least 50000 (you can use more simu- lations as long as your computer allows). The term structure horizon is denoted by T. Set T = 500. Suppose we have already estimated the one-period ahead volatility t+1 with historical data. The goal is to estimate the term structure of volatility t+1:t+k for k = 1, , T using Monte Carlo simulations. t+1:t+k is the estimated volatility of cumulative returns over the next k periods, i.e., Rt+1 + + Rt+k.

(A) Using Monte Carlo simulations, estimate the term structure of volatility with the RiskMetrics model. That said, you update future volatility using the RiskMetrics model. Start the simulation by setting t+1 = 0.01. Plot t+1:t+k t+1 k as a function of k.

(B) What is the theoretical value of t+1:t+k t+1 k in Part (A)? Is the plot in Part (A) close to the theoretical value? (Hint: The theoretical values of t+1:t+k are on the slides.)

(C) Using Mone Carlo simualations, estimate the term structure of volatility with the Garch(1,1) model. That said, you update future volatility using the Garch(1,1) model. Supposed you have estimated = 1e-6, = 0.05, = 0.9 based on historical data. Start the simulation by setting t+1 = 2 2 q 1 . Plot t+1:t+k t+1 k as a function of k.

(D) For Part (C), does the value of t+1 affect the pattern of t+1:t+k t+1 k ? You can answer this question by repeating Part (C) but perturbing the values of t+1, for example, resetting t+1 = 0.5 and t+1 = , respectively. (E) What is the theoretical value of t+1:t+k in Part (C)? Check how close the t+1:t+k esti- mated in Part (C) using Monte Carlo simulations is to the theoretical counterparts. An easy way to check this is to divide the estimated t+1:t+k by its theoretical counterpart, make a plot of the ratio, and compare with the horizon line which crosses the y-axis at (0, 1). (Hint: The theoretical values of t+1:t+k are on the slides.)