Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

Abstract

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.

Keywords:

• Copulas
• Expected shortfall
• Monte carlo
• Value-at-Risk

JEL Codes:

• C22
• C58
• G1

Acknowledgments

We are grateful to two anonymous referees for their helpful comments. Financial support from the Spanish Government under project PID2021-124860NB-I00 and from Generalitat Valenciana under project CIPROM/2021/060 is gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 All codes used for the simulations are available from the authors upon request.

2 The number of replications in this experiment is much lower than in the previous one because this is a much more time-consuming experiment, due to the rolling-window scheme that is applied in each replication.

3 Du and Escanciano (2017) showed that this asymptotic result holds under the assumption that the ratio between the number of observations in the out-of-sample period and the number of observations in the in-sample period converges to 0.

4 Note that these results differ from those reported in Du and Escanciano (2017), where a problem of over-rejection is detected. The difference in the results possibly stems from the fact that we use a rolling-window scheme, whereas their simulations are performed with a fixed in-sample estimation period.