GJR-GARCH model in value-at-risk of financial holdings

Abstract

In this study, we introduce an asymmetric Generalized Autoregressive Conditional Heteroscedastic (GARCH) model, Glosten, Jagannathan and Runkle-GARCH (GJR-GARCH), in Value-at-Risk (VaR) to examine whether or not GJR-GARCH is a good method to evaluate the market risk of financial holdings. Because of lacking the actual daily Profit and Loss (P&L) data, portfolios A and B, representing FuBon and Cathay financial holdings are simulated. We take 400 observations as sample group to do the backward test and use the rest of the observations to forecast the change of VaR. We find GJR-GARCH works very well in VaR forecasting. Nonetheless, it also performs very well under the symmetric GARCH-in-Mean (GARCH-M) model, suggesting no leverage effect exists. Further, a 5-day moving window is opened to update parameter estimates. Comparing the results under different models, we find that the model is more accurate by updating parameter estimates. It is a trade-off between violations and capital charges.

Keywords:

• market risk
• value-at-risk
• GARCH
• GJR GARCH
• financial holdings

JEL Classification:

• G2
• G21

Notes

¹ Nevertheless, Yamai and Yoshiba (2005) propose the ‘tail risk’ to document that VaR models are useless for measuring market risk when the events are in the ‘tails’ of distributions. Wong (2008) use a risk measure that takes into account the extreme losses beyond VaR, and expected shortfall proposed by Artzner et al. (1997, 1999) to remedy the shortcoming of VaR.

² Pérignon and Smith (2010) find that historical simulation is the most popular VaR method in the world, as 73% of banks that disclose their VaR method report using historical simulation. Moreover, Pérignon et al. (2008) find that commercial banks exhibit a systematic excess of conservatism when setting their VaR, which contradicts the common wisdom that banks intentionally understate their market risk to reduce their market risk capital charges.

³ In response to the lack of ready-to-use data, Pérignon et al. (2008) develop a data extraction technique to extract the data from the graph included in the banks’ annual reports. Therefore, the daily VaR and P&L data are not anonymous.

⁴ The leverage effect means that the negative shock generally has a greater impact on stock return volatility than the positive shock does.

⁵ While many models of the accuracy of VaR are available in the literature, little is known on the accuracy of disclosed VaRs. Pérignon and Smith (2010) contribute to fill this gap. They find that although there is an overall upward trend in the quantity of information released to the public, the quality of VaR disclosure shows no sign of improvement over time.

⁶ Since financial returns exhibit three widely reported stylized facts, which are volatility clustering, substantial kurtosis and mild skewness of the returns, ‘standard’ methods, based on the assumption of iid-ness and normality, tend not to suffice, which has led to various alternatives for VaR prediction (for example, GARCH model).

⁷ Back testing a VaR model represents assessing its ex post performance using a sample of actual historical VaR and P&L data.

⁸ Cuoco and Liu (2006) document the financial institution chooses the VaR by trading off the cost of higher capital requirements in the current period resulting from a higher reported VaR against the benefit of a lower probability of higher capital requirements in the future as a result of a loss exceeding the reported VaR. Moreover, the minimum capital requirement is then equal to the sum of a charge to cover credit risk and a charge to cover general market risk.

⁹ The reasons are as follows. First, the majority of the retail loans are prefixed at a ‘floated’ foundation in Taiwan. That means banks are allowed to adjust the rates whenever the primary rate are adjusted. Therefore, banks are always protected by a spread and there is less price volatility. Second, retail funding is more related in credit risk management area based in Basle II amendments.

¹⁰ There may be different expression of conditional variance

 Still the news impact curve is steeper for negative shocks than it is for positive shocks when C(2) is positive (Hentschel, 1995). There is no difference to change the sign expression of C(2).

¹¹ There is a minus sign before MA in our models. If MA is positive, good news leads to the downward revise of return and bad news leads to the upward revise of return. If MA is negative, good news leads return enlarging and bad news leads return narrowing.

¹² The VaR there is not a value-at-risk value. In order to compare with the P&L return, we discuss it in the percentage form. It is calculated as at 99% confidence level.