solution

Suppose a FI has invested in a long-short portfolio with the Fama-French three factors, namely, Mkt-RF, SMB and HML (in percentages). Assume the notional principal is one unit of wealth, 60 percent is invested in Mkt-RF, 20 percent invested in SMB and 20 percent invested in HML. The FI wants to examine the risk of this strategy by computing a one-day 99% value-at-risk (VAR) and expected shortfall (ES) by using historical simulations. Data provided is from 31/07/2007 to 23/07/2009.

a. Assume each historical scenario has equal probability. Estimate the VaR and ES. (2 marks)

b. Assume the probability of each historical scenario is given by the exponential- weighting formula in Section 13.3.1 of the textbook. Estimate the VaR and ES. (2 marks)

c. Apply volatility scaling (Section 13.3.2) to the market variables, namely, Mkt-RF, SMB and HML. Then, estimate the VaR and ES. You should estimate a GARCH (1,1) model for each market variable by using maximum likelihood and plot the estimated daily volatilities. Assume the next-day’s volatility sn+1 for each market variable is given by the 95th percentile of the estimated daily volatilities. (3 marks)

d. Apply volatility scaling (Section 13.3.3) to the portfolio, then estimate the VaR and ES. You should estimate a GARCH (1,1) model for the daily returns of the portfolio using maximum likelihood. Assume the next day’s volatility for the portfolio is given by the 95th percentile of the estimated volatilities