EF5123 Problem Set I
Objective
The goal is to compute increasingly re ned Value-at-Risk measures and evaluate
their Out-Of-Sample performance. Let IS and OOS be the In-Sample and Out-
Of-Sample periods, respectively. Consider the location-scale representation of the
log-returns rt:
rt = t + t zt with COV(zt; ztô€€€j) = 0; 8j 6= 0
Use the IS period data to estimate the following speci cations:
1: t = 0; 2
t = 2; zt N(0; 1)
2: t = ; 2
t = 2; zt N(0; 1)
3: t : ARMA(p; q); 2
t = 2; zt N(0; 1)
4: t = 0; 2
t : RiskMetrics; zt N(0; 1)
5: t = ; 2
t : RiskMetrics; zt N(0; 1)
6: t : ARMA(p; q); 2
t : RiskMetrics; zt N(0; 1)
7: t = 0; 2
t : GARCH(p; q); zt N(0; 1)
8: t = ; 2
t : GARCH(p; q); zt N(0; 1)
9: t : ARMA(p; q); 2
t : GARCH(p; q); zt N(0; 1)
GARCH indicates the best IS speci cation amongst the asymmetric EGARCH
and GJR-GARCH (Hint: rank the various models and their parametrizations using
Information Criteria). For all the above speci cations (1-9):
1. Produce OOS 1-period-ahead forecasts of t.
2. Produce OOS 1-period-ahead forecasts of 2
t .
3. Calculate OOS 1-period-ahead 5%, 1% and 0.1% VaRs.
4. Backtest your VaR forecasts: test whether the observed number of VaR vio-
lations is statistically di erent from the promised 5%, 1% and 0.1%. De ne:
N: number of OOS observations
N1: number of OOS VaR violations
b = 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
b
p
+ (N ô€€€ N1) ln
1 ô€€€ b
1 ô€€€ p
2
(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 speci ca-
tion/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 web-
page. Make sure to read the data description contained in the le. Your IS and
OOS periods are de ned as follows:
IS: 20080102-20131231
OOS: 20140102-20160129
Here you may nd 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