مدیریت پرتفوی تحت تغییرات ناگهانی در نوسانات و افق های سرمایه گذاری ناهمگن

We analyze the implications for portfolio management of accounting for conditional heteroskedasticity and sudden changes in volatility, based on a sample of weekly data of the Dow Jones Country Titans, the CBT-municipal bond, spot and futures prices of commodities for the period 1992–2005. To that end, we first proceed to utilize the ICSS algorithm to detect long-term volatility shifts, and incorporate that information into PGARCH models fitted to the returns series. At the next stage, we simulate returns series and compute a wavelet-based value at risk, which takes into consideration the investor's time horizon. We repeat the same procedure for artificial data generated from semi-parametric estimates of the distribution functions of returns, which account for fat tails. Our estimation results show that neglecting GARCH effects and volatility shifts may lead to an overestimation of financial risk at different time horizons. In addition, we conclude that investors benefit from holding commodities as their low or even negative correlation with stock and bond indices contribute to portfolio diversification.

To date, there is an extensive literature on the behavior of volatility of assets returns and the effect of this on both the value and composition of investors portfolios. Indeed, the GARCH model and its numerous extensions have been widely used to account for the existence of conditional heteroskedasticity in financial time series (see, for instance, the survey by Poon and Granger [1]).1 However, less attention has been paid to the detection of multiple shifts in unconditional variance over time. For example Ref. [2] et seq. conclude that persistence in variance may be overstated by not accounting for deterministic structural breakpoints in the variance model. A relatively recent approach to testing for volatility shifts is the Iterative Cumulative Sums of Squares (ICSS) approach of Ref. [3]. This algorithm allows for detecting multiple breakpoints in variance in a time series. Examples of this approach to equity markets include [4], [5] and [6]. Another subject, which has received attention in recent research and that also has important implications for portfolio management, is the existence of heterogeneous investors. Ref. [7] points out that, for the specific case of commodity markets, long-horizon traders will essentially focus on price fundamentals that drive overall trends, whereas short-term traders react to incoming information within a short-term horizon. Hence, market dynamics in the aggregate will be the result of the interaction of agents with heterogeneous time horizons. In order to model the behavior of financial series at different time spans, researchers have resorted to wavelet analysis. Wavelet analysis is a refinement of Fourier analysis that allows for decomposing a time series into its high frequency or noisy components and its low frequency or trend components, among many other applications. See Refs. [7], [8] and [9] for commodity and derivative markets, for interest and foreign exchange rates see Refs. [10] and [11], and for equity markets see Refs. [12], [13], [14], [15], [16], [17] and [18]. Finally, one of the main issues in the analysis of portfolios is that of what the likelihood is of a loss of a particular magnitude. This Value at Risk (VaR) analysis has attracted very significant attention in the economics and finance literature (see for example Refs. [19], [20] and [21]) but relatively little in econophysics (see Refs. [22], [23] and [24] as exceptions). In essence the VaR approach provides an integrated approach to examine and assess the probability of a given percentage loss of wealth over a given time period. The aim of this article is two-fold. First, we analyze whether accounting for conditional heteroskedasticity and long-term volatility shifts in asset returns really matters when comes to quantifying the potential market risk an investor faces. In doing so, we consider different time horizons by resorting to a wavelet-based decomposition of VaR. Second, we look at the potential diversification gains in terms of the VaR decrease obtained by adding commodities to a portfolio. This article is organized as follows. Section 2 presents the main methodological tools utilized in the empirical section of the article. Section 3 presents some descriptive statistics of the data used in the simulations carried out later on. Section 4 presents the simulation exercises involving a portfolio primarily composed of stock indices and a portfolio that also include spot and futures positions in commodities. We discuss the implications of not accounting for correlated volatility and volatility shifts for risk quantification. In addition, we focus on the benefits of holding commodities for portfolio diversification. Section 5 concludes.

In this study, we primarily focus on two themes that have received attention in the finance literature in recent years: structural shifts in variance and heterogeneous investment horizons. These two subjects are of particular importance to portfolio managers at the time of quantifying accurately the risk of their positions. Indeed, an incorrect model specification of returns volatility will pervade the quantification of financial risk, as it will yield biased estimates of long-term volatility. On the other hand, market dynamics will be the result of the interaction of heterogeneous investors: long-term investors will focus on price fundamentals, while short-term investors will be more sensitive to incoming information within a short time span. Acknowledging such heterogeneity will allow us to have a proper risk measure depending on the time horizon under consideration. A third subject we tackle is the diversification gains arising from commodities investment. Specifically, we formulate a statistical specification that enables us to quantify the extent to which accounting for conditional heteroskedasticity and long-term volatility shifts has a quantitatively significant impact on a portfolio value at risk. In order to accommodate for heterogeneous investment horizons, we utilize a time-scale decomposition of value at risk based on wavelet analysis. Our simulation results, based on weekly data of the Dow Jones Country Titans and spot and futures prices of commodities for the period 1992–2005, show that neglecting GARCH effects and volatility shifts may lead us to overestimate financial risk considerably, at various investment horizons. In addition, we conclude that investors benefit from holding commodities—particularly futures—as their low or even negative correlation with stock and bond indices contribute to portfolio diversification. Our results have important policy implications for portfolio managers. First of all, in order to forecast future volatility accurately, a model built on historical data must incorporate volatility shifts observed in the past. Otherwise, estimates of potential portfolio losses, at a given confidence level, can exhibit a severe upward bias. Second, an investor's time horizon is also a key element to determine the value at risk of his/her position. This issue has been discussed in recent financial applications of wavelet analysis. Third, commodities are a risk-reduction source, which investors should resort to when considering financial diversification. A potential extension of this research would be to account for cross correlations of assets returns when carrying out the simulations. To that end, the statistical technique of copulas should be suitable. In essence, copulas enable us to extract the dependence structure from the joint distribution function of a set of random variables, and simultaneously to separate the dependence structure from the univariate marginal behavior. Recent applications of this technique in the financial field have shown that the Gaussian and t-Student's copulas are suitable choices. It is likely that we would have to consider a smaller number of assets in order to make the estimation process computationally tractable.