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Seasonality in Portfolio Risk Calculations

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Seasonality in Portfolio Risk Calculations UMEÅUNIVERSITET Department of Physics Master’s Thesis in Engineering Physics HT 2019, 30 Credits Seasonality in Portfolio Risk Calculations Empirical study on Value at Risk of agricultural commodities using conditional volatility models v1.0 Author Emil Söderberg Nasdaq Umeå University Supervisor: Anders Stäring Supervisor: Joakim Ekspong Examiner: Markus Ådahl Seasonality in Portfolio Risk Calculations Abstract In this thesis, we examine if the conditional volatility models GARCH(1,1) and eGARCH(1,1) can give an improved Value-at-Risk (VaR) forecast by considering seasonal behaviours. It has been shown that some markets have a shifting volatility that is dependent on the seasons. By including sinusoidal seasonal components to the original conditional models, we try to capture this seasonal dependency. The examined data sets are from the agricultural commodity market, corn and soybean, which are known to have seasonal dependencies. This study validates the volatility performance by in-sample fit and out-of-sample fit. Value-at-Risk is estimated using two methods; quantile estimation (QE) of the return series and filtered historical simulation (FHS) from the standardized return series. The methods are validated by a backtesting process which analyse the number of outliers from each model by the Kupiec’s POF-test (proportional of failures) and the Christoffersen’s IF-test (interval forecast). The results indicates that the seasonal models has an overall better volatility validation than the non-seasonal models but the effect from the distributional assumption of standardized returns varies depending on model. Both VaR methods, for non-seasonal and seasonal models, seems however to benefit from assuming that the standardized return series experience kurtosis. The QE method is greatly dependable on the distributional assumption and almost all models assuming normal distribution fails the POF-test. The seasonal effects on VaR is in general the same for both commodities. Years with more dominant seasonal trends show an increased volatility during the summer season and generally a lower volatility during winter season. It is shown that the seasonal models have fewer violations during summer and is closer to the expected value of violation. It is hard to determine which model has the best overall performance, but the seasonal models have a better VaR estimate during summer. 1 Contents 1 Introduction1 2 Model specification3 2.1 GARCH...................................... 3 2.2 eGARCH ..................................... 3 2.3 Seasonal components............................... 4 2.4 Conditional distributions............................. 5 2.4.1 Normal distribution ........................... 5 2.4.2 Student’s t ................................ 5 3 Value at Risk6 3.1 Historical simulation............................... 6 3.2 Quantile estimation................................ 7 3.3 Filtered historical simulation .......................... 7 4 Estimation8 4.1 Maximum likelihood estimation......................... 8 4.1.1 Normal distribution ........................... 8 4.1.2 Student’s t distribution ......................... 9 4.2 Information criterion............................... 9 5 Forecast 10 5.1 Volatility validation ............................... 10 5.2 Backtesting Methods............................... 11 5.2.1 Kupiec’s POF-Tests ........................... 12 5.2.2 Christoffersen’s Interval Forecast Test ................. 12 6 Data 14 7 Empirical study 16 7.1 Corn........................................ 16 7.1.1 Forecast.................................. 18 7.1.2 Out-of-sample fit............................. 20 7.1.3 Backtesting................................ 21 7.1.4 Seasonal behaviour of VaR ....................... 25 7.2 Soybean...................................... 28 7.2.1 Forecast.................................. 29 7.2.2 Out-of-sample fit............................. 32 7.2.3 Backtesting................................ 32 7.2.4 Seasonal behaviour of VaR ....................... 36 8 Conclusions 39 A Thomas Reuters 43 A.1 Data extraction: Settlement prices of Corn .................. 43 B Tables 43 B.1 Corn........................................ 43 i Seasonality in Portfolio Risk Calculations B.2 Soybean...................................... 45 C Figures 47 C.1 Corn........................................ 47 C.2 Soybean...................................... 49 ii Seasonality in Portfolio Risk Calculations 1 Introduction Volatility forecasting has an essential role on the financial market and is a big part of risk management. One method that incorporates volatility to measure the market risk of a portfolio of assets is Value-at-Risk (VaR), which has become one of the most popular risk of loss measurement techniques. VaR can be described as the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval. Many VaR models use distributional assumptions of the return series which involve numerical calculations of volatility. One way to calculate volatility is by using conditional volatility models. In 1982 Engle introduced the autoregressive conditional heteroscedasticity model (ARCH) that tries to capture the time-varying volatility and volatility clustering, that is commonly seen in financial time series.[1] The model estimates the variance term by using the past values of the squared return series, which is a normal volatility proxy. This model was further generalized by Bollerslev into the generalized ARCH model (GARCH) which also include the past volatility calculation in the model.[2] The GARCH model has been used to a wide extent and the GARCH(1,1) model is by far the most popular GARCH model.[3] However, the volatility estimate from the family of ARCH-models may not give sufficient results when the volatility of financial time series has seasonal patterns. The future commodity market, especially the agricultural sector, is well-known for its seasonal behaviour due to harvest seasons and the periods of the year. There are two different kinds of seasonality however, price and volatility. Price tends to change according to the supply which typically leads to higher prices of future contracts with delivery prior to harvest and low prices after harvest when the supply is full.[4] The volatility changes mainly by the effect from information. An example using agricultural commodities is the weather. The months preceding harvest hold a lot of important information since it will have a direct impact on the final yield. This type of information has minor impact on the harvest season and little or no after harvest. Other factors to consider is large information releases like monthly or yearly reports. Earlier work by Goodwin et al. evaluated determinants of price variability in U.S. corn and wheat futures markets and revealed important seasonal and autoregressive effects in the volatility.[5] They further showed that conditional heteroscedasticity models can estimate the volatility in a effective way. Another study by Wang et al. examined seasonality, long-term memory and structural change of corn future market by using the Fractional Integrated GARCH model (FIGARCH).[6] They concluded that seasonality, structural change and long memory plays an important role in volatility estimation. The importance of long memory was not as important at lower time horizons (1-day ahead) in the presence of seasonality and structural change. In this study we examine if the 1-day ahead VaR estimate of agricultural futures can further be improved by including seasonal components to the conditional volatility model. The chosen conditional models are the GARCH model and the exponential GARCH model (eGARCH) introduced by Nelson.[7] The GARCH model is preferred over the FIGARCH model due to simplicity and that Wang et al. showed that the two models had very similar results at the 1-day ahead forecast. The eGARCH model is chosen since it can account for asymmetry in time series. 1 Seasonality in Portfolio Risk Calculations It is a popular idea that the return series of commodity futures is positively skewed since commodities have positive exposure to supply shocks. Regarding the statistical assumption of the return series, it either follows normal distribution or student’s t distribution which has heavier tails than normal distribution. The student’s t distribution is motivated by a study from Kat et al. which concludes that the distribution of commodity futures returns has fat tails and less significant skewness.[8] The VaR estimate will be confirmed by the number of violations and two VaR tests; Kupiec’s proportion of failure test and Christoffersen’s interval forecast test.[9, 10] In our analysis, we will also compare how well the seasonal and non-seasonal VaR models are behaving during the year. 2 Seasonality in Portfolio Risk Calculations 2 Model specification This section introduces the selected models and the continuous distributions used throughout this report. The chosen financial time series models are extensions of the auto-regressive conditional heteroscedasticity (ARCH) model.[1] The fundamental 2 idea for the family of ARCH models is that the conditional variance σt (also known as volatility) of the financial process rt has a time dependency and a relation to the process {rs, s < t}. The volatility is modelled as, rt = σtet, {et} ∼ iid N(0, 1) (1) 2 where the volatility is related to the past values of rt according to, p 2 X 2 σt = ω + αirt−i i for some positive integer p and ω, αi > 0 for all i. The model originally
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