EF5123 Problem Set I Objective The goal is to compute increasingly refined Value-at-Risk measures and evaluate their Out-Of-Sample performance. Let IS and OOS be the In-Sample and OutOf-Sample periods, respectively. Consider the location-scale representation of the log-returns rt: rt = µt + σt · zt with COV(zt; zt−j) = 0; 8j 6= 0 Use the IS period data to estimate the following specifications: 1: µt = 0; σt2 = σ2; zt ∼ N(0; 1) 2: µt = µ; σt2 = σ2; zt ∼ N(0; 1) 3: µt : ARMA(p; q); σt2 = σ2; zt ∼ N(0; 1) 4: µt = 0; σt2 : RiskMetrics; zt ∼ N(0; 1) 5: µt = µ; σt2 : RiskMetrics; zt ∼ N(0; 1) 6: µt : ARMA(p; q); σt2 : RiskMetrics; zt ∼ N(0; 1) 7: µt = 0; σt2 : GARCH∗(p; q); zt ∼ N(0; 1) 8: µt = µ; σt2 : GARCH∗(p; q); zt ∼ N(0; 1) 9: µt : ARMA(p; q); σt2 : GARCH∗(p; q); zt ∼ N(0; 1) GARCH∗ indicates the best IS specification amongst the asymmetric EGARCH and GJR-GARCH (Hint: rank the various models and their parametrizations using Information Criteria). For all the above specifications (1-9): 1. Produce OOS 1-period-ahead forecasts of µt. 2. Produce OOS 1-period-ahead forecasts of σt2. 3. Calculate OOS 1-period-ahead 5%, 1% and 0.1% VaRs. 4. Backtest your VaR forecasts: test whether the observed number of VaR violations is statistically different from the promised 5%, 1% and 0.1%. Define: • N: number of OOS observations • N1: number of OOS VaR violations • πb = N1=N: observed fraction of violations • p: promised fraction of violations The Likelihood-Ratio-Test (LRT) for the Unconditional Coverage (UC) of the VaR should be performed using the following test statistic: LRTUC = 2 · N1 ln πbp  + (N − N1) ln 11 −− πbp  ∼ χ2 (1)EF5123 Problem Set I Analysis Discuss: • the performance of the various approaches to VaR • the performance of RiskMetrics vs GARCH: expected or unexpected? • the sensitivity of variance and VaR estimates and forecasts to the specification/modeling of the conditional mean Data You are to perform your investigations on the returns of the assigned Fama-French 25 portfolio. The data 25 Portfolios 5x5 Daily is posted on the course webpage. Make sure to read the data description contained in the file. Your IS and OOS periods are defined as follows: • IS: 20080102-20131231 • OOS: 20140102-20160129 Here you may find your assigned portfolio: Duaa Maad Small-Low Anthony Small-3 Ang Small-High Tao 2-2 Manasa 2-4 Elisa 3-Low Michael 3-3 Kanika 3-High Amodu-Rufai Big-Low Lin Big-3 Yunjing Big-High