Tabulación y procesamiento de la información

where Q∗_{V B}(r) and Q∗_EB(r) are the bootstrap empirical distribution function of [V aR∗(i)_{T +1}

and cES∗(i)_{T +1} respectively. The corresponding intervals are denoted as N R − F HS.

Alternatively, the quantile and expectation can also be estimated by using the Hill

estimators in (4.18) and (4.20) implemented to the bootstrap standardized residuals ∗_t.

The corresponding intervals will be denoted as N R − H. Finally, the GCCF expansion in (4.23) and (4.24) could be implemented and the intervals denoted as N R − GCCF .

4.3

Monte Carlo experiments

In this section, we carry out Monte Carlo experiments to analyze the finite sample performance of the proposed bootstrap procedure for constructing prediction intervals for the V aR and ES. We compare our procedure with those proposed by Christoffersen

and GonC¸ alves (2005) and Ho and Lee (2005).

We generate 1000 replicates from a GARCH (1, 1) model with parameters α0 =

0.002, α1 = 0.05 and β = 0.9. In order to take into account the potential presence of

leverage effect, we also simulate series using a EGARCH (1, 1) model with α0 = −0.17,

α1 = 0.15, γ = −0.13 and β = 0.92. We consider four sample sizes, T = 250, 500, 1000

and 3000 and three alternative distributions of the standardized observations, t, namely

Normal, Student-8 and a Skewed-Student distribution, with 10 degrees of freedom and a coefficient of asymmetry equal to −0.11; see Hansen (1994) for a complete descrip- tion of the Skewed distribution. Although T = 250 may seem a rather small value for real life applications, it is important to note that the Basel Commission requires to compute the V aR with at least one year of data which corresponds approximately to 250 daily observations. In each case, we estimate the parameters of the True Data Generating Process by M L by treating the degrees of freedom and asymmetry pa- rameters as unknown parameters. We compute 90% prediction intervals for the 1%

V aRT +1 and EST +1 based on B = 1000 bootstrap replicates by implementing the

2_{The Exponential GARCH (EGARCH) model of Nelson (1991) is given by}

ln σ2

4.3. Monte Carlo experiments 112

procedures proposes by Christoffersen and GonC¸ alves (2005) by assuming i) the true

distribution (C − G − D), ii) the F HS (C − G − F HS), iii) EV T (C − G − H) and iv) Gram-Charlier and Cornish-Fisher expansions (C − G − GCCF ). We also construct the prediction intervals by the bootstrap procedure proposed in this Chapter by i) F HS (N R − F HS), ii) EV T (N R − H) and iii) by the Gram-Charlier and Cornish-Fisher approximations (N R − GCCF ). Finally, the prediction intervals are computed by the procedure proposed by Ho and Lee (2005) (HL) using B = 1000 bootstrap replicates, and for the smoothed iterated, B = 1000 replicates for the outer level and C = 500 replicates for the inner level.

Table 4.1, that reports the coverages of the 90% intervals3 _{for the V aR}

T +1 and

EST +1 when the series are generated by the GARCH(1, 1) model, shows that regard-

less of the particular distribution, the coverages of the V aR computed by assuming the true error distribution are well under the nominal. The same can be said when using the Hill estimator regardless of whether the bootstrap replicates are obtained as

proposed by Christoffersen and GonC¸ alves (2005) or as proposed in this Chapter. It

seems that assuming a particular error distribution does not account for the uncer- tainty associated with the estimation of the V aR even when the assumed distribution is the true one. The coverages are closer to the nominal when using either F HS or the GCCF expansions and slightly better with the former. Furthermore note that the coverages are clearly closer to the nominal 90% when using the bootstrap procedure

proposed in this Chapter instead of that proposed by Christoffersen and GonC¸ alves

(2005). Therefore, with respect to the one-step ahead V aR, the bootstrap procedure proposed in this Chapter implemented to F HS gives coverages closer to the nominal for

all assumed distributions and sample sizes. Christoffersen and GonC¸ alves (2005) also

conclude that F HS gives the best results when implemented using their bootstrap pro- cedure. Furthermore, note that the coverages obtained using the procedure proposed in this Chapter are comparable to those obtained when the much more complicated procedure of Ho and Lee (2005) is implemented.

4.3. Monte Carlo experiments 113

Additionally, Table 4.1 also reports the average coverages for the ES. In this case, we can observe that the intervals constructed assuming the true distribution are once more well under nominal coverage. Furthermore, when comparing the coverages obtained when implementing F HS, Hill estimator or the GCCF expansions, it seems that the latter is, in general more appropriate regardless of whether one uses the bootstrap

procedure proposed by Christoffersen and GonC¸ alves (2005) or the one proposed in

this Chapter. Although this result seems to contradict the conclusions of Christoffersen

and GonC¸ alves (2005), notice that we are implementing the correction in the estimation

of the expectation of the tail proposed by Giamouridis (2006). As a consequence, the performance of the prediction intervals of one-step ahead ES constructed using the

GCCF expansion is clearly improved. When comparing the results obtained when

using our bootstrap procedure with those obtained using _b∗_t, we can observe that the

coverages are in general closer to the nominal when implementing the second bootstrap step proposed in this Chapter. In any case, our bootstrap intervals suffer from a slight overcoverage which is preferable from a conservative risk management strategy than to have undercoverage.

Table 4.2 reports the coverages of the 90% intervals for the V aRT +1and EST +1when

the series are generated by the EGARCH model. The first clear difference between the results obtained for the GARCH and the EGARCH models is that in the former, the coverages are smaller than the nominal while in the latter, there is an overcoverage. In any case, although the differences between alternative procedures are smaller than those observed in Table 4.1 for the GARCH model, we still observe that the coverages are closer to the nominal when using the two-step bootstrap proposed in this Chapter. In the context of the EGARCH model, F HS seems to work better for both V aR and ES prediction intervals.

On the other hand, last column of Table 4.1 reports the coverage of the procedure of Ho and Lee (2005) using the GARCH(1, 1) model. We can observe, that the cov- erages constructed with the smoothed bootstrap are better than those obtained with the percentile method, except when the data is generated using the Skewed-Student distribution and the sample size is 250. Similar results can be concluded from the last