The optimal value-at-risk hedging strategy under bivariate regime switching ARCH framework

Abstract

Unlike the majority of other hedging literatures in which variance is taken as the risk indicator, this article uses the Value-at-Risk (VaR) as the risk management tool of the hedged portfolio. This article adopts a bivariate Markov regime Switching Autoregressive Conditional Heteroscedastic (SWARCH) model to formulate the optimal VaR hedging strategy and then compares it with the other dynamic futures hedging strategies mentioned in the literature in hedging performance. Using Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) futures data, the in-sample and out-of-sample results shows that when VaR is used as the criterion to measure the futures hedging effectiveness, the performance of the dynamic hedging strategy is superior to that of the static hedging strategy, and the performance of the optimal VaR hedging strategy is better than that of the minimum variance and mean-variance hedging strategies. Besides, from the standpoint that the volatility of hedge ratio and hedged portfolio variance decline, no matter what kind of hedging strategy is adopted, the regime switching model is better in in-sample and out-of-sample hedging effectiveness than the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model.

Acknowledgement

The author acknowledges the financial support from National Science Council of Taiwan (NSC 95-2416-H-035-011).

Notes

¹ For example, Pownall and Koedijk (1999), Billio and Pelizzon (2000), Giot and Laurent (2003), Haas et al. (2004a, b), Sarno and Valente (2005), Alexander and Lazar (2006) and Guidolin and Timmermann (2006).

² For the theory of formulating the hedging strategies and the estimation method, Lien and Tse (2002) and Chen et al. (2003) have made detailed classification and explanation. But they do not mention the applicability of regime switching model.

³ The empirical literatures (e.g. King et al., 1994; Longin and Solnik, 1995; Forbes and Rigobon, 2002; Sarno and Valente, 2000, 2005; Guidolin and Timmermann, 2006) have verified that the correlation among financial assets is not fixed. Moreover, because the correlation coefficient has a great effect on the skewness and kurtosis of portfolio returns (Harris and Shen, 2006), the importance of the time-varying correlation coefficient for the hedging performance must be considered.

⁴ Following Alizadeh and Nomikos (2004), Lee et al. (2006) also adopt the similar method to estimate the dynamic hedging strategy.

⁵ The conditional probabilities are constructed as follows:

where the transition probability P_{ii , t} is not constant over time.

⁶ In the researches on the applicability of information criteria, Francq et al. (2001) prove that in the regime switching ARCH model, the application of SIC to the model selection will not underestimate the number of states and that of ARCH terms. According to the SIC and the HQIC, I find that when the number of ARCH terms is 3 and the volatility is allowed to be asymmetric, this model fits better than the other models.

⁷ In the setting of Markov–switching model, whether the series has the phenomenon of skewness is closely related to whether the mean changes across regimes. When the means are the same under different states, the model does not produce skewness (Timmermann, 2000). Because the empirical data used in this article show that the distributions of spot and futures returns are slightly skewed, the means are still allowed to vary between regimes in the setting of model here.

⁸ Many researchers who study financial asset returns by the Markov switching model, such as Gray (1996), Dueker (1997), Maheu and McCurdy (2000), Edwards and Susmel (2001), Fong and See (2001) and Sarno and Valente (2005), record in their literatures that the returns have two obvious different volatility processes.

⁹ For the sake of saving space, this article does not show hedge ratios of hedging strategies. They are available upon request from the author.

¹⁰ Harris and Shen (2006) prove that the minimum variance hedging strategy can lower the variance of the hedged portfolio, but make the skewness coefficient decrease and kurtosis coefficient increase at the same time. When the returns are permitted to have the characteristic of regime switches, there is evidence showing that the empirical results in this article are the same as what Harris and Shen (2006) get.

¹¹ When Psaradakis and Sola (1998) test the finite sample characteristics of the maximum likelihood estimators in Markov switching model, they find that in small sample size, the skewness and kurtosis coefficients of the estimators and t-statistics are different from those in normal distribution. In addition, Alexander and Lazar (2006) also find that the bias of parameter estimates in the nonlinear model can be reduced through increasing the number of samples. The more complicated the model is, the more samples are needed.

¹² Because the VaR calculated under the assumption of conditional normal distribution cannot reflect the real downside risk of a portfolio, this article do not directly take the VaR as the performance measurement of the two econometric models.