Calendar Effects Analysis of Americas Indexes
Abstract
We apply an approach for estimating Value-at-Risk (VaR) describing the tail of the conditional distribution of a heteroscedastic financial return series. The method combines quasi-maximum-likelihood fitting of AR(1)-GARCH(1,1) model to estimate the current mean as well as volatility, and extreme value theory (EVT) to estimate the tail of the adjusted standardized return series. We employ the approach to investigate the existence and significance of the calendar anomalies: seasonal effect and day-of- the-week effect in Americas Indexes VaR. We also examine the statistical properties and made a comprehensive set of diagnostic checks on the one decade of considered Americas Indexes returns. Our results suggest that the lowest VaR of considered Americas Indexes negative log returns occurs on the fourth season among all seasons. Moreover, comparatively low Wednesday VaR is captured among all weekdays during the test period.
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Introduction
In today’s financial world, the large increase in the number of traded assets in the portfolio of most financial institutions has made the measurement of market risk a primary concern for regulators and for internal risk control. Following the Basle Accord on Market Risk (1996) most banks in more than 100 countries around the world have to calculate its risk exposure for every individual trading desk. Banks are also required to hold a certain amount of capital as a cushion against adverse market movements. Value-at-Risk has become the benchmark risk measure. From mathematics point of view, VaR is simply a quantile of the profit-and-Loss distribution of a given portfolio over a prescribed holding period. The importance of VaR is undoubted since regulators accept this model as a basis for setting capital requirements for market risk exposure.
In this paper, we discover the calendar anomalies in Americas equity market movements, which including the seasonal effect and the day-of-the-week effect on Americas Indexes returns. The calendar effect in stock market returns includes day-of-theweek effect, weekend effect, January effect, and holiday effect, etc. It has been widely studied and investigated in finance literature. Studies by Cross (1973), and Rogalski (1984) demonstrate that there are differences in distribution of stock returns for each day of the week. Studies by Baillie and DeGennaro (1990), Berument and Kiymaz (2001) posit that day-of-the-week effect has an impact on stock market volatility. In recent years, another stream of research has considered seasonality in stock returns and volatility, see Saunders (1993), Bouman and Jacobsen (2002), Hirshleifer and Shumway (2003), Kamstra, Kramer and Levi (2003), and Cao and Wei (2005), etc. These studies generally report that calendar anomalies are present in both returns and volatility equations in the stock market. None of these studies, however, test for the possible existence of day-of-the-week and seasonal variation in stock return VaR. Hence, the goal of this paper is to characterize the VaR of Americas Indexes returns. Based on investigations of the day-of-the-week effect and seasonal effect in extreme risk, we also provide valuable and applicable analysis for equity market investors. The major obstacle to this investigation is a viable measure of tail risk over time.
We are concerned with tail estimation for those considered financial return series. Our basic assumption, whose validation is examined in this paper, is that returns follow a stationary time series model with stochastic volatility structure. The presence of stochastic volatility implies that returns might dependent over time. Therefore, we consider to model the return distribution as the conditional return distribution where the conditioning is on the current volatility and mean. Although VaR only characterizes the extreme quantiles, disregarding the centre of the distribution, estimation of the extreme quantile is not an easy task. As one wants to make inferences about the extremal behavior of a portfolio, there is only a very small amount of data in the tail area of a sample set. Advanced methods and tools are needed to enable us to explore beyond the range of the limited data set. In this study, we use a semi-parametric method, based on extreme value theory (EVT), which is rather effective for obtaining reliable estimates.
Conclusion
With the empirical analysis of this paper we demonstrate how we can use a GARCH based dynamic EVT approach to model VaR for short term forecasting. The dynamic EVT method has the advantage of dynamically reacting to changing market conditions which is useful in getting better VaR forecasts. We apply the two stage approach on five Americas Indexes negative daily log return series. Empirical findings in this paper show that both seasonal effect and day-of-the-week effect are present in Americas Indexes returns. We captured the comparatively low VaR in Q4 and Wednesday for considered returns during the test period.
Overall, our findings have implications for investors, financial institutions, and futures exchanges. For example, for conservative investors who would prefer lower risk, they can choose to trade during the lower VaR period to avoid potential high loss. The two stage approach to VaR can also be used in other stock or asset returns. Finally, it has significant value for investors and regulators in terms of an in depth analysis of the equity market.