Stress-testing for portfolios of commodity futures Florentina Paraschiva,∗, Pierre-Antoine Mudrya , Alin Marius Andriesb a Institute
for Operations Research and Computational Finance, University of St. Gallen, Bodanstr. 6, CH-9000, Switzerland b Alexandru Ioan Cuza University of Iasi, Romania and University of St. Gallen, Switzerland
Abstract In this paper, we performed a stress-testing for a portfolio of commodity futures, which mimics the dynamics of the DJ-UBS index. We identified extreme events that impacted commodity prices over time, and looked at correlation structures in a dynamic way, with copula functions. In line with Basel III financial regulations, we derived baseline, historical, and hybrid scenarios and discussed their advantages and shortfalls. We found that the financialization of commodity markets led to an increase in correlations and in the probability for joint extremes. However, we identified structural breaks in commodity markets that temporarily led to a breakdown of expected statistical patterns and of traditional dependence structures among commodities. This fact shows the need for forward-looking stress testing techniques, like hybrid and hypothetical scenarios, as encouraged by financial regulators. Keywords: stress testing, commodity futures, risk measures, extreme value theory, copula functions
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1. Introduction
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1.1. Overview and motivation
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Investments in commodities have grown rapidly over the last years mainly via commodity futures and commodity index funds ([46]) and many institutional managers have embraced commodities as a profitable alternative asset ([22]). Recent literature has established that commodity futures can serve as diversification instruments in conventional portfolios because of their low correlations with equities and bonds ([27]). These characteristics could encourage investors to choose commodities as a refuge during periods of stress in traditional asset markets, especially if macroeconomic shocks tend to impact commodity and stock prices in opposite directions ([54]). Apart from their hedging function ∗ Corresponding
author Email address:
[email protected] (Florentina Paraschiv)
Preprint submitted to Economic Modeling
March 3, 2015
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for commercial traders, commodities are now regarded by investors as good alternative investments due to their historically low correlations with other asset classes. One of the consequences of the increasing financialization of commodities, especially through the introduction of commodity indices ([57]) is the increase of volatility shocks in these markets. Recent studies documented that volatility shocks in commodity markets have significant effects on the financial markets ([24] and [6]). In addition, the growing presence of index funds in commodity markets integrates the commodity- with the stock and bond markets ([54]). This high price volatility has led to a growing concern of the public and in policy circles as to whether financialization has distorted commodity prices, and whether more government regulation in these markets is warranted ([19], [8]). Given the exponential growth of investments in commodity indices by institutional investors, the question of adequate risk management tools for those indices in the context of a broader portfolio of financial securities is of great interest. Thus, a good knowledge of commodity futures pricing helps investors to understand the underlying risks and enables them to compose optimal portfolios ([16]). In the volatile world of commodity markets, quantifying and mitigating price risk presents a number of challenges due to the time dependence in volatility, non-linear dynamics and heavy tails in returns ([47], [43] [46] or [7]). Consequently, many forecasting models, risk measurement techniques and hedging tools have been developed during the last decades ([6]). Their main purpose is to provide financial institutions, risk managers and commodity traders with a technical approach to quantify financial and commodity risks. Value-at-risk (VaR) and stress testing have emerged as two of the most popular risk management tools. A stress test is a risk management tool used to evaluate the potential impact on portfolio values of unlikely, although plausible events or movements in a set of financial variables ([42]). It is further designed to explore the tails of the distribution of losses beyond the threshold (typically 99%) used in Value-at-Risk (VaR) analysis ([5]). Stress testing has become a major financial methodology in the risk management field, not only because it can assist financial institutions in understanding the effect of stress scenarios, but also because it can help with calculating risk measures that focus on extreme market conditions ([55]). Over the last two decades, many investors have suffered sizeable losses due to extreme events. Examples of such events include the 1987 crash, the Asian and Russian crises of 1997 and 1998, the burst of the dot-com bubble in 2000, the failure of Lehman Brothers in 2008, and the European debt crisis. Since the occurrence of these events, the importance of risk management has been extensively recognized by investors when deciding the amount of risk they are willing to bear ([4]). Under Basel II, risk managers performed stress testing based on historical scenarios, defined based on the losses experienced during a historical period of market stress. However, in this way, the maximum simulated loss is bounded to the historically observed loss and this stress testing technique cannot extrapolate beyond that. It became therefore obvious after the recent financial crisis that the 2
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stress testing techniques must be reviewed. Thus, Basel III proposes forwardlooking stress scenarios like hypothetical or hybrid scenarios. In this paper, we will discuss the advantages of forward-looking stress scenarios versus historical ones and discuss extensively implications for risk managers. The severity of the current global financial crisis determined a deep review of risk management practices by regulators in attempting to reduce the fragility of financial systems (see [13]). The current global financial crisis showed that the backward-looking historical data was of limited use in anticipating market turbulence ([9]). Another major flaw of traditional financial theory is the overreliance on correlations as a tool to track dependencies between different assets. An inappropriate model for dependence structures among the risk factors can lead to suboptimal portfolios and inaccurate assessments of risk exposures ([12], [38]). The historical data provide substantial evidence of extreme levels of co-movement in financial returns during episodes of financial turmoil and, therefore, the correlation matrix of the underlying asset returns of a portfolio can change dramatically when a financial crisis occurs. In adverse situations, correlations can move to unexpectedly extreme levels either upwards or downwards (see [10]). Traditionally, correlation is used to describe the dependency between random variables, but recent studies have ascertained the superiority of copulas, as they offer much more edibility than the correlation approach ([25], [58]). Copula models have become a major tool in statistics for modelling and analysing dependence structures between random variables and it has been often used for risk management purposes, and especially for stress testing, due to the fact that in contrast to linear correlation a copula captures the complete dependence structure inherent in a random vector. Copulas allow capturing stylized facts of financial returns such as fat tails and skewness. Moreover, copulas are not a number such as correlation, but functions, which allow them to accurately map varying levels of dependency between assets, and more specifically to capture tail dependency. Financial risk management typically deals with low-probability events in the tails of return distributions. It is therefore important to be able to model the extreme events accurately. Traditional risk management models, however, might fail to give us accurate estimates and forecasts of the tails because they usually focus on the whole distribution, of which the tails are just small portions. Extreme value theory based risk management, on the other hand, focuses directly on the tails and could therefore potentially give us better estimates and forecasts of risk ([14]). The modelling of extreme events is the central issue in Extreme Value Theory (EVT), and the main purpose of the theory is to provide asymptotic models for the tails of a distribution. EVT helps forecasting much more accurately the potential returns of a portfolio through robust stress-testing methods [2]. Despite the rise of commodity indices, EVT and copulas applications for stress testing purposes have rarely been used for commodities. The vast majority of papers applying EVT to finance employ time series from the stock market. The second most frequent source of data is probably exchange rates, but there are articles dealing with almost any kind of data, from equity returns and interest rates to energy and commodity market data and up to credit derivatives 3
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data ([50]). To our knowledge, EVT and copulas have been extensively applied to equities and currency portfolios (see [28], [34], [49]), but rarely to portfolios of commodity futures. In this paper, we will show how EVT can be used to properly estimate the distribution of commodity futures, by focusing on a portfolio composed of the ten most important commodity futures in the DJ-UBS index. We apply a combined approach of extreme value theory (EVT) for modeling the risk factors and we look at the dependency structures in a dynamic way, with copula functions. We further perform stress testing in accordance with Basel III regulations. Our study contributes to the empirical literature on the risk management in several ways. First, we focus on one portfolio of commodities that are in the focus of financial investors over the last years. We show the importance of identifying extreme events in the evolution of risk factors and compute their impact on the final profit and loss distribution. Second, we show clear examples of the value added of hybrid and hypothetical forward-looking scenarios versus historical ones. This is of great importance for practitioners, since often financial regulations are restricted to theoretical discussions and lack in empirical support. Third, we bring evidence that the overreliance on historical simple correlations can induce misleading stress testing results. 1.2. Literature overview on stress testing In this subsection, we give an overview of the existing literature on stress testing methodologies for futures prices. One comprehensive example of complex stress testing techniques for portfolios of futures can be found in [1]. The authors built an intuitive stress testing tool that helps risk managers to perform forwardlooking scenario analysis, in line with the requirements of the post-crisis financial regulations. Examples of studies that applied EVT and copulas to commodity prices are [2], [14], [33] and [44]. These studies compared the performance of EVT to conventional models such as GARCH, Historical Simulation and Filtered Historical Simulation on oil markets and their results indicate that conditional Extreme Value Theory and Filtered Historical Simulation procedures offer a major improvement over the traditional methods. [31] assessed the performance of RiskMetrics, skewed Student APARCH and skewed student ARCH models to measure VaR of returns in commodity spot and futures markets. The skewed student APARCH model provides the best performance in all cases. [30] provided a comparative study of the predictive ability of energy two- and three-dimensional portfolio VaR estimates by employing various estimation techniques. Their results show that Filtered Historical Simulation, conditional EVT and copula methods are the most appropriate riskmanagement techniques for the majority of cases. [17] made comparisons between the performance of the three VaR calculation methods (Historical Simulation, Historical Simulation with ARMA process and Variancecovariance Method) for oil price risk quantification. Results indicate that the Historical Simulation with ARMA forecasts provides a flexible VaR quantification, which fits the continuous oil price movements well and provides 4
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an efficient risk quantification. [51] used several different statistical models to estimate forecasts of daily volatility in petroleum futures returns. From the risk management perspective, non-parametric models provide better VaR estimations than the parametric models. [47] examined the performance of Regime Switching models in forecasting volatility and Value at Risk in the energy markets. Results show that the augmented GARCH-X model is the most consistent one passing all the tests for all examined markets. Also, conditional EVT, as a conservative alternative to VaR forecasts, is more appropriate to risk averse investors. [21] showed that GARCH with historical simulation is an effective technique for estimating VaR, especially when the second moment of financial series is correlated. [35] investigated the influence of fat-tailed innovation process on the performance of one-day-ahead VaR estimates using three GARCH models and highlighted the importance of selecting the appropriate distribution in a GARCH context. Their findings suggest that the heavy-tailed distribution is more suitable for energy commodities, particularly for VaR calculation. [35] addressed the importance of fat-tailed property in energy commodities and showed that the GARCH model with heavy-tailed distribution has a good accuracy at both low and high confidence levels. [20] examined comparatively the predictive performance of various Valueat-Risk models in the energy market. Forecasting techniques considered here include the stable density, Kernel density, Hull and White, GARCH-GPD, plus composite forecasts from linearly combining two or more of the competing models above. Findings show Hull and White to be the most powerful approach for capturing downside risk in the energy market. [6] evaluated the VaR for some major crude oil and gas commodities for both short and long trading positions using three GARCH models including FIGARCH, FIAPARCH and HYGARCH. Their results show that considering for long-range memory, fat-tails and asymmetry performs better in predicting a one-day ahead VaR for both short and long trading positions. Moreover, the FIAPARCH model outperforms the other models in the VaRs prediction. They showed that considering for long-range memory, fat-tails and symmetry performs better in predicting a one-day-ahead VaR for both short and long trading positions. The rest of the paper is structured as follows: Section 2 explains the selection of the data. Section 3 motivates the choice of the methods. Estimations results are shown and interpreted in section 4. Finally, section 5 concludes.
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2. Data selection
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The two main commodity indices are the DJ-UBS index, and the S&P-GSCI index. Both indices incorporate the most liquid, and widely traded commodities, but the weighting systems are different. The S&P-GSCI uses the world production of each commodity to determine its weights, while the DJ-UBS index is based on the trading activity of major commodities. Consequently, using the weights announced for 2013, the S&P-GSCI index is heavily geared towards 5
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energy, with a weight of 69.7%, compared to only 32.7% for its counterpart. For this reason, we focused on the DJ-UBS index, as its weighting is much more balanced across all types of commodities. It would be, however, impractical to perform our analysis on the 22 commodities part of the DJ-UBS index. Consequently, we will focus on a basket of ten commodity futures: WTI Light Sweet Crude Oil, Brent Crude Oil, Natural Gas, Corn, Wheat, Soybeans, Live Cattle, Gold, Aluminium, Copper. In total, these ten commodities represent 69% of the total weights, making the chosen portfolio a good proxy for the index. In order to determine the weight of each commodity in the test portfolio, we simply divided each weight in the index by the combined weight of 69%. The results are summarized in Table 1. We used daily logarithmic returns, from 01 January 1998 to 31 December 2011. In Figure 1 we display the evolution of commodity prices that are chosen in our portfolio. In the case of the reference index, the provider usually rolls the futures contracts over four times a year, depending on the most liquid contracts trading on a particular commodity. The provider therefore buys relatively short termed contracts. For simplicity, it would have been very cumbersome to roll the contracts over in the same way as the index provider. We therefore limited ourselves to a bi-annual roll. The months of rollover were chosen given the availability and liquidity of the various futures contracts tradable on the markets, based on information given by the Chicago Mercantile Exchange (CME). Moreover, to soften the jumps linked to rolling over contracts, we used contracts with approximately one year maturity, that are rolled over six months before expiration. This way of proceeding means a certain loss of liquidity, but leads to a softening of the negative (positive) returns associated with contango (backwardation), since basis linked to contango tends to tighten the fastest with maturity closing on. The months of rollover (as shown in Table 2) were chosen given the availability and liquidity of the various futures contracts tradable on the markets, based on information given by the Chicago Mercantile Exchange (CME).
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3. Methodology
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To perform adequate stress-tests, a realistic model for the risk factors is required. In a fist step, we perform a descriptive analysis of the data and based on this we motivate the choice of the methods.
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3.1. Preliminary tests
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A descriptive analysis of logarithmic commodity returns is shown in Table 3. Financial returns often display fat tails and volatility clustering. The graphical inspection of the commodity returns of our ten risk-factors point towards that direction. The negative skewness of financial returns has been proven in many studies (see for example [36]). In the case of a stock, this phenomenon can be explained by increased investors’ nervousness in case of a string of negative returns, and potentially also by an increase in the leverage of the company
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Commodity WTI Brent Natural Gas Corn Wheat Soybean Live Cattle Gold Aluminium Copper TOTAL
Weight in index 9.21% 5.76% 10.42% 7.05% 4.75% 5.49% 3.28% 10.82% 4.91% 7.28% 68.99%
Weight in portfolio 13.34% 8.36% 15.11% 10.22% 7.96% 6.89% 4.76% 15.68% 7.12% 10.55% 100.00%
Table 1: Portfolio weights
Commodity WTI Brent Natural Gas Corn Wheat Soybean Live Cattle Gold Aluminium Copper
Month of rollover December, June December, June Front month December, July December, July January, July December, June December, June December, June December, June
Table 2: Rollover months
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(due to a decrease of equity relatively to debt). Increased leverage is very often synonymous with increased volatility. It is not clear, however, whether these explanations hold for commodities. To investigate this fact we first computed the skewness and kurtosis of our ten risk factors. Results show that all included commodities exhibit fat tails. Five of the ten commodities exhibit positive skewness, while the others exhibit a negative skew. One of the major reasons of positive skewness is that commodity prices tend to react most strongly to shortages in supply, which can trigger large spikes upward, a so called “inventory effect” (see [18]). However, all commodity returns exhibit a higher kurtosis than in the case of normal distribution. The Jarque-Bera test rejects the normality assumption for all investigated returns. The Augmented Dickey-Fuller analysis shows that all investigated time series are stationary (see Table 4). The ARCH-LM test for conditional heteroscedasticity rejects the null hypothesis of no ARCH effect for all commodities and for all lags (5, 10, 15, 20), which shows the volatility clustering effect. Commodity WTI Gold Copper Live Cattle Corn Wheat Aluminium Natural Gas Brent Soybean
Mean(%) 0.048 0.047 0.041 0.015 0.024 0.018 0.007 0.009 0.052 0.017
St. Dev. (%) 1.900 1.166 1.798 0.742 1.683 1.771 1.280 3.684 1.894 1.481
Skewness -0.353 0.107 -0.317 0.036 0.322 0.366 -0.394 0.621 -0.227 -0.837
Kurtosis 7.242 8.439 7.9911 21.657 9.406 7.789 6.183 8.545 7.697 12.844
Jarque-Bera (p-value) 2689(0.001) 4311(0.001) 3684(0.001) 50683(0.001) 6034(0.001) 3416(0.001) 1564(0.001) 4699(0.001) 3241(0.001) 14514(0.001)
Table 3: Descriptive statistics of sample returns. The skewness for a normal distribution is 0 and kurtosis 3, and any symmetric data should have a skewness near 0 and kurtosis near 3.
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3.2. Modeling approach The descriptive statistics show asymmetric distributions of commodity returns, stationarity, and conditional heteroscedasticity. Many scholars provide evidence of volatility asymmetries given that negative news have a larger impact on volatility than positive news (see, e.g., [53], [15], [40]). We take all these Commodity WTI Gold Copper Live Cattle Corn Wheat Aluminium Natural Gas Brent Soybean
ADF test -61.73 (0.001) -57.90 (0.001) -63.36 (0.001) -59.67 (0.001) -58.22 (0.001) -60.56 (0.001) -61.14 (0.001) -63.72 (0.001) -62.88 (0.001) -59.43 (0.001)
ARCH (L5) 247.45 (0.00) 195.66 (0.00) 425.23 (0.00) 100.19 (0.00) 84.63 (0.00) 102.91 (0.00) 189.76 (0.00) 115.53 (0.00) 182.16 (0.00) 44.35 (0.00)
ARCH (L10) 295.06 (0.00) 221.22 (0.00) 484.33 (0.00) 102.59 (0.00) 106.33 (0.00) 132.01 (0.00) 272.07 (0.00) 137.81 (0.00) 230.18 (0.00) 66.43 (0.00)
ARCH (L15) 327.77 (0.00) 265.20 (0.00) 524.05 (0.00) 108.48 (0.00) 115.16 (0.00) 141.10 (0.00) 311.59 (0.00) 152.25 (0.00) 253.93 (0.00) 58.31 (0.00)
ARCH (L20) 363.45 (0.00) 277.85 (0.00) 528.87 (0.00) 111.83 (0.00) 121.40 (0.00) 155.14 (0.00) 338.34 (0.00) 188.67 (0.00) 275.84 (0.00) 74.95 (0.00)
Table 4: ADF and ARCH LM Statistics (p-value between brackets)
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characteristics into account by fitting an ARMA(1,1)-GARCH(1,1) GJR asymmetric model (see [32]) for the risk factors. The lags were found by performing the Akaike (AIC) and Bayesian (BIC) information criteria. The GARCH process for the risk factors ensures that the conditional variances of the univariate distributions are time-varying (see [2]). The model reads:
yt = µ + φ1 yt−1 + ξεt−1 + εt h2t 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282
= a0 +
a1 ε2t−1
+
a2 h2t−1
+
(1) bψ(εt−1 )ε2t−1
(2)
where h2t is the conditional variance of εt , zt = εt /ht , with zt N (0, 1) or Student’s t-distributed (scaled to have variance 1) IID innovations with mean = 0, variance = 1, and degree of freedom parameter, ν. Additionally, an indicator function is introduced: ψ(εt−1 ) = 1 if εt−1 (or zt−1 ) is negative, or 0 if εt−1 (or zt−1 ) is positive. As there is no restriction on the sign of b, the model can be applied to describe both negatively or positively skewed data. We determine the model parameters by maximum likelihood estimation. Two versions are tested: asymmetric AR-GARCH with normal and with tinnovations. A likelihood ratio test shows the superiority of the latter model version1 . This is not surprising, given the fat tails of commodity returns. We further produced a probability plot and compared the returns to the standard normal distribution and the fitted AR-GARCH(1,1) model with t− innovations. We observed that the model strongly underestimates extreme events (see Figure 4). However, for a rigorous stress testing, exactly the extremely large returns are of importance. [45], [25] prove evidence for a good performance of a combined approach GARCH with parametric tails based on extreme value theory (EVT). For the center of the distribution, where most of the data are concentrated, kernel smooth interior is used for the estimation. For the tails, where usually data is scarce, a parametric approach based on extreme value theory is selected, whereas the generalized Pareto distribution is able to asymptotically describe the behavior of the tails (see [48]). We will therefore apply this approach to model the standardized residuals zt , in Equation (3.2). The notation for a generalized Pareto (GP) distribution is introduced for any ξ ∈ R, β ∈ R+ ([45]): − ξ1 z ,z ∈ R GPξ,β (z) = 1 − 1 + ξ β +
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where item 1/ξ is known as the tail index and β a scaling parameter. We fit GR to the zt standardized residual series that exceed a certain high threshold u. In general, the threshold u is chosen within reasonable limits of 5–13% of the data (see [45], [25], [48]). We look at the most extreme 10% upper and lower 1 Results
are available upon request.
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tail of the standardized residuals zt and fit the GPD to the excesses over the threshold. EVT theory can be split in two methods when it comes to modeling extreme events. One is the Peak Over Threshold method (POT), the other is the block maxima model. In the Block maxima method, the extremes are defined as the maximum data point in successive periods, and the limiting distribution of these extremes is the generalized extreme value distribution (GEV) [26]. The POT method on the other hand focuses only on the observations in a sample that exceeds a certain threshold. The POT method has been shown to be more efficient, and is therefore the preferred method (see [25]). Thus, the POT method focuses on modeling the tails of the distribution directly, rather than the behavior of extrema of random variables observations stacked in non overlapping blocks. Moreover, since the block maxima method only focuses on the maxima of a sequence of samples, large observations in those samples that are smaller than the local maximum are eliminated from the analysis. We therefore choose to apply POT to describe the extreme tails of commodity returns. The GP parameters are determined by maximum likelihood estimation. So far, we showed how we modeled the risk factors individually. However, for a realistic portfolio stress testing, the evolution of dependency structures among the considered commodities is of great importance. Given the contagion effect, it is expected and empirically observed that in times of market stress, joint extreme returns occur in commodity markets. Empirical literature shows that copula functions are a realistic choice for describing joint dynamics of portfolio risk factors (see, e.g., [11], [2]). Following the insights from the literature, we fitted to our commodity returns both one normal and one Student-t copulas. The likelihood ratio test and the information criteria show however a clear superior fit of the t-copula2 . This result is not surprising, given that t-copulas can better describe heavy tails of joint dynamics among the returns in times of market stress. Furthermore, this is in line with the literature. Among the static copulas, the literature shows a better performance of the Student-t copulas over normal copulas to describe joint dynamics of portfolio risk factors. In addition, in a recent study of [2] it has been shown that the Student-t copula yields a superior prognosis of the expected shortfall than other static copulas. We therefore model joint positive or negative returns with a t-copula. In the case of t-distributions the d-dimensional t-copula with ν degrees of freedom is given by: t −1 Cν,Σ (u) = tν,Σ (t−1 ν (u1 ), ..., tν (ud ))
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(3)
where Σ is a correlation matrix, tν is the cumulative distribution function of the one dimensional tν distribution and tν,Σ is the cumulative distribution function of the multivariate tν,Σ distribution. We apply the semi-parametric maximum likelihood estimation to determine the parameters of the t-copula (see [29]). 2 Results
are available upon request
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4. Estimation results We have fitted the ARMA(1,1)-GARCH(1,1) GJR model to the commodity returns. As shown in Figures 2 and 3, the model filters out the volatility clustering discussed in descriptive statistics. Figure 5 shows the semi-parametric distribution for WTI resulted by fitting GP to the extreme tails of standardized residuals. Once the extreme tails of the commodity returns have been identified and modeled, we can further proceed with stress testing our portfolio. Scenarios for multivariate stress tests can be constructed as historical, hybrid, or hypothetical scenarios ([12]). While historical scenarios assume a repetition of past crises, in hybrid scenarios the historical market movements are only used to calibrate the (conditional or unconditional) process of risk factors evolution and to evaluate the general market conditions during a shock. Hypothetical scenarios are not restricted to a repetition of the past, but allow a more flexible formulation of potential events. In this study, we show the limitations of historical scenarios and the importance of a forward looking analysis in the context of hybrid scenarios. The first stress test we consider is based on the derivation of historical scenarios. Creating scenarios with historical data is probably the most intuitive approach, since the events did happen in reality and are thus plausible to reappear. We construct the P&L of our portfolio for the next 22 days horizon starting at 1st January 2012, based on the returns of the risk factors empirically observed during the financial crisis from 28 March 2008 to 31 March 2010. In this case, we want to assess the portfolio losses in case of a repetition of a financial stress situation. The P&L of the portfolio under the simulated historical scenario is simply given by the empirical distribution of past gains and losses on this portfolio, during the financial crisis. The implementation of this non-parametric method is simple, since it does neither require a statistical estimation of the multivariate distribution of risk factor changes, nor an assumption of their dependence structure. The second stress test to be considered is based on hybrid scenarios. The parameters and residuals of the AR-GARCH with EVT processes of the different commodity returns and the t- copula are calibrated on the financial crisis data ranging from 28 March 2008 to 31 March 2010. Based on these parameters, the risk factors are simulated for the next 22 days, 10’000 scenarios, and the P&L at the end of the planning horizon is constructed using the weights shown in Table 1. Figure 6 shows comparatively the P&L for the historical and the hybrid scenarios. The P&L for the historical scenario obviously displays the characteristic stepwise pattern. The more extreme the returns, the more the two distributions drift apart. However, the hybrid scenario overestimates positive returns significantly. One explanation for this lies in the symmetry of the t-copula, which struggles to account for skewed portfolio returns, despite its many merits (23, p. 9; 56, pp. 17–18) The lower tail of the historical scenario distribution is truncated at −32.93%, while the maximum simulated loss with the hybrid scenario is −64.08%. Thus, for extreme tail quantiles, we observe that the historical 11
Metric Degrees of Freedom Max. Simulated Loss Max. Simulated Gain Simulated 90% VaR Simulated 95% VaR Simulated 99% VaR Simulated 90% ES Simulated 95% ES Simulated 99% ES Simulated 99.9% ES Simulated 99.99% ES
Baseline 12.79 -37.86% 30.57% -4.67% -6.35% -9.64% -6.92% -8.40% -11.58% -16.93% -29.38%
Hybrid 15.25 -64.08% 58.45% -13.31% -17.96% -29.83% -19.73% -24.10% -34.10% -49.26% -63.78%
Historical N/A -32.93% 14.49% -12.19% -16.56% -28.31% -18.06% -21.86% -31.17% N/A N/A
Table 5: Metrics for hybrid and historical stress scenarios
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scenario signals a much lower simulated loss than the hybrid scenario. This underlines the main drawbacks of the overreliance on the historical simulation method, as discussed in [12]: this method is unconditional and it neglects the time-varying nature of financial time series and furthermore the dependence structure. Furthermore, based on a limited time span, extreme quantiles are difficult to estimate. This example shows additionally that the hybrid scenario is able to extrapolate beyond the historical data, which, from the view point of financial regulations ([12]), is a major feature of a realistic stress testing technique. To show the importance of stress scenarios, we compare the P&L values derived from historical and hybrid scenarios with the P&L derived from the baseline scenarios. The latter aims at estimating the portfolio performance at the end of the 22 day period, without the impact of stress. We therefore calibrate the AR-GARCH model with EVT for the risk factors and the t-copula for interdependencies to the entire data sample: 01 January 1998 to 31 December 2011. With the simulations over 22 days for each risk factor, we recompute the P&L. Comparative statistics over risk measures are offered in Table 5. We observe that with the baseline scenarios, the risk of the portfolio, expressed by the VaR and Expected Shortfall (ES) for tail quantiles above 90%, is significantly underestimated. Well identified stress scenarios are of great important for portfolio risk managers, as they quantify the magnitude of losses that might be expected in case of market stress. In Table 5 we observe that the estimated degrees of freedom of the copula function for the hybrid scenario are higher than in the case of baseline scenario. This is surprising, since it indicates lower tendency of joint extremes in commodities during the financial crisis than in the overall investigated period. To better understand the cause for these results, we recalibrated our AR-GARCH with EVT and t-copula on a rolling time window, collected the degrees of freedom and additionally computed average correlations. We focused on three-year 12
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windows, and repeatedly performed the computations by moving the window forward by one year. The results are plotted in Figure 7, and compiled in Table 6. Overall we observe that correlations among commodity returns increased, while the degrees of freedom show some oscillating patterns. Until 2006, we conclude that commodity returns became more correlated with each other, and joint extremes are more likely. However, during the boom and bust cycle of 2007–2009, and further during the 2008–2010 window, although correlations increased among commodities, we observe an increase in the degree of freedom as well. This confirms our previous results, that the tail dependence structures among commodities weakened during the financial crisis. A possible explanation for this are the different dynamics among commodity prices during the financial crisis: some underwent a relatively moderate growth and fall (agricultural commodities), while others, (oil, gas, copper) went through a massive boom and bust cycle.
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5. Conclusion
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In this study, we performed stress testing for a portfolio of commodity futures. The risk factors of our portfolio are the most important commodities in the DJ-UBS index. In line with the financial regulations from Basel III, we identified the extreme tails of the risk factors and modeled them with EVT. Furthermore, the dynamic dependence structures are modeled with a t-copula function. Our results are three fold: First, we show the importance of identifying extreme events that impacted the evolution of risk factors over time. We show that classical time series have serious limitations in modeling accurately extreme tails. Instead, as discussed in the recent financial regulations, EVT is a very useful tool to describe the distribution of commodity prices in times of market stress. This is of a major importance for stress testing. Second, we bring empirical evidence that reveals the drawbacks of historical scenarios: extreme losses are truncated to the historically observed ones and often extreme tail quantiles cannot be computed, due to too limited number of historical observations. We show instead the importance of forward-looking analysis, like the derivation of hybrid scenarios, that help to correct the mentioned drawbacks of historical scenarios. Third, we bring evidentiary support for the critics in Basel III concerning the overreliance on historical correlations. Our results show that the reliance on standard assumptions like the increase in the probability of joint extremes among financial assets in times of market stress is not always realistic. By contrary, we show that there have been structural breaks in commodity markets that temporarily led to a breakdown of expected statistical patterns, like tail dependence structures. This fact should be explored by risk managers in performing hypothetical scenarios, by shocking arbitrary combinations of market factors, volatilities, and dependence structures. The pure reliance on historical assumptions can have serious limitations for stress testing.
13
Relative Daily Index Closings
9 8 7 Index Value
6 5 4 3
WTI Gold Copper Live Cattle Corn Wheat Aluminium Natural Gas Brent Soybean
2 1 0 1998
2000
2002
2004 2006 Date
2008
2010
2012
Figure 1: Historical daily price movements in relative value
Dates 1998-2000 1999-2001 2000-2002 2001-2003 2002-2004 2003-2005 2004-2006 2005-2007 2006-2008 2007-2009 2008-2010 2009-2011
Correlation 0.13 0.11 0.10 0.12 0.15 0.18 0.21 0.23 0.33 0.38 0.40 0.36
df 24.37 19.6 18.85 15.66 12.63 11.70 11.95 Student Version of MATLAB 14.82 14.06 16.93 21.06 12.52
Table 6: Correlation and Degrees of Freedom
14
WTI Filtered Conditional Standard Deviations Volatility
0.05
0 1998
2000
2002 2004 2006 2008 Gold Filtered Conditional Standard Deviations
2010
2012
2000
2002 2004 2006 2008 Copper Filtered Conditional Standard Deviations
2010
2012
2000
2002 2004 2006 2008 Live Cattle Filtered Conditional Standard Deviations
2010
2012
2000
2002 2004 2006 2008 Corn Filtered Conditional Standard Deviations
2010
2012
2000
2002 2004 2006 2008 Wheat Filtered Conditional Standard Deviations
2010
2012
2010
2012
Volatility
0.04 0.02 0 1998 Volatility
0.1 0.05 0 1998 Volatility
0.04 0.02 0 1998 Volatility
0.1 0.05 0 1998 Volatility
0.04 0.02 0 1998
2000
2002
2004
2006
2008
Figure 2: Filtered Conditional Standard Deviations part 1
Student Version of MATLAB
15
Aluminium Filtered Conditional Standard Deviations
0.02 0.01 0 1998
2000
2002
2004
2006 2008 2010 Date Natural Gas Filtered Conditional Standard Deviations
2000
2002
2004
2002
2004
2002
2004
Volatility
0.1
0 1998
Volatility
0.06
2006 2008 Date Brent Filtered Conditional Standard Deviations
2010
2012
2006 2008 2010 Date Soybean Filtered Conditional Standard Deviations
2012
0.04 0.02 0 1998
2000
0.04
0.02
0 1998
2000
2006
2008
2010
2012
Date
Figure 3: Filtered Conditional Standard Deviations part 2
Probability Plot (Copper) 0.9999 0.999 0.99
Probability
Volatility
2012
0.05
Probability Plot (Gold) 0.9999
Normal returns t
0.999 0.99
0.9 0.75
Probability
Volatility
0.03
0.5 0.25 0.1
0.9 0.75 0.25 0.1
0.01
0.01 0.001
−0.2
Student Version of MATLAB
0.5
0.001 0.0001
Normal returns t
0.0001 −0.1
0 Data ta
0.1
0.2
−0.1
−0.05
0 Data
0.05
Figure 4: Probability Plot (Copper & Gold)
16
Student Version of MATLAB
0.1
WTI Semi−Parametric Empirical CDF
1
Pareto Lower Tail Kernel Smoothed Interior Pareto Upper Tail
0.9 0.8
Probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −8
−6
−4
−2 0 Standardized Residual
2
4
6
Figure 5: Semi-parametric empirical cumulative distribution function (WTI)
Student Version of MATLAB
Simulated One−Month Global Portfolio Returns CDF vs Historical Scenario 1 Hybrid 0.9 Historical 0.8
Probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.8
−0.6
−0.4
−0.2 0 Logarithmic Return
0.2
0.4
0.6
Figure 6: Hybrid vs historical stress scenarios
17 Student Version of MATLAB
Correlation vs Degrees of Freedom! 30
0.4 0.35
25
df!
20
0.25
15
0.2 0.15
10 5
df
0.1
Correla1on
0.05
9
01 1
8
20
09
-2
00
20
07
-2
00
7 -2 06 20
20
05
-2
00
6
5 20
04
-2
00
4 20
03
-2
00
3 20
02
-2
00
2
00
20
01
-2
00
20
00
-2
00 -2 99
19
19
98
-2
00
1
0 0
0
Correlation!
0.3
Figure 7: Rank Correlations vs Degrees of Freedom
444 445
446 447 448
449 450
451 452 453
454 455
456 457 458
459 460
461 462 463
464 465
466 467
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