![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
Abstract
In this work we propose Monte Carlo simulation models for dynamically computing MaxVaR for a financial return series. This dynamic MaxVaR takes into account the time-varying volatility as well as non-normality of returns or innovations. We apply this methodology to five stock market indices. To validate the proposed methods we compute the number of MaxVaR violations and compare them with the expected number. We also compute the MaxVaR-to-VaR ratio and find that, on average, dynamic MaxVaR exceeds dynamic VaR by 5–7% at the 1% significance level, and by 12–14% at the 5% significance level for the selected indices.
¶This work was carried out when Nityanand Misra and Bharat Kodase were graduate students at the Indian Institute of Management Bangalore.
Acknowledgements
The authors wish to thank all the anonymous reviewers in the two stages of the reviewing process for their valuable comments and suggestions.
Notes
¶This work was carried out when Nityanand Misra and Bharat Kodase were graduate students at the Indian Institute of Management Bangalore.
†In a heteroskedastic environment, their estimate may also overstate this risk in low volatility periods.
‡The expression is not a closed-form solution for an asset's MaxVaR, but a numerical approach like the Newton–Raphson method can be used to solve for MaxVaR.
§A financial institution would want to compute discrete MaxVaR in an end-of-day mark-to-market scenario.
¶Boudoukh et al. (Citation2004) use 50,000 Monte Carlo simulations to achieve this.
⊥The Johnson S U distribution can model a slightly wider range, see Yan (Citation2005).
†The expected value of is used to initialize ĥ
t
.
‡Note that these PMLEs for C, α0, α1 and β1 are the same as the MLEs for the respective parameters under the GARCH-N model.
§Also refer to equation (43) of Premaratne and Bera (Citation2001).
†Heinrich (Citation2004) notes that “…[the method of moments] is not really adequate in many cases, but may be used to provide starting values to a maximum likelihood fitter.”
‡Here the innovations , i = 1, 2, …, 10, are standardized with zero mean and unit variance for the GARCH-N model. For the GARCH-PIV (PML-MOM estimation) method, their mean and variance are equal to those of the empirical GARCH-N residuals. For the GARCH-PIV (ML estimation) method, they have unit variance while the mean is equal to
.
§However, conditional skewness may still be significant even if unconditional skewness is not (Grigoletto and Lisi Citation2006).
†If r ≤ −1, then and the Type IV PDF is not normalizable. The alternate hypothesis states that the PDF is normalizable.
‡The PMLEs for the GARCH-PIV (PML-MOM estimation) approach are not shown in ; they are equal to the respective MLEs for the GARCH-N model.
§Premaratne and Bera (Citation2001) report in their results for NYSE equal weighted returns that the empirical and implied skewness values are almost equal for the Type IV-AR-GARCH model, but the kurtosis values are not.
¶For every index, we use the same value of W across the models—GARCH-N, GARCH-PIV (PML-MOM estimation) and GARCH-PIV (ML estimation)—so that the results from these models can be compared.
†The system has a 1.6 GHz Intel Pentium M Processor and 512 MB RAM.
†The out-of-sample data consists of returns from November 2004 onwards (October 2002 onwards for the FTSE 100), while the dot com bubble (with the dot com crash) and the early 2000s' recession are a part of the in-sample data.
‡The pth percentile of the distribution of cumulative returns at the horizon is taken as the dynamic VaR.
§The expected number of violations is the same for VaR and MaxVaR at a particular significance level.
¶Boudoukh et al. (Citation2004) also find that the adjustment factor decreases with increasing standard deviation (σ).