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A Wavelet Based Multi Scale VaR Model for Agricultural Market Kaijian He1,2 , Kin Keung Lai1 , Sy-Ming Guu3 , and Jinlong Zhang4 1

4

Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong [email protected], [email protected] 2 College of Business Administration, Hunan University, Changsha, Hunan, 410082, P.R. China 3 College of Management, Yuan-Ze University, 135 Yuan-Tung Road, Chung-Li, Taoyuan, 32003, Taiwan [email protected] College of Management, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China [email protected]

Abstract. Participants in the agricultural industries are subject to significant market risks due to long production lags. Traditional methodology analyzes the risk evolution following a time invariant approach. However, this paper analyzes and proposes wavelet analysis to track risk evolution in a time variant fashion. A wavelet-econometric hybrid model is further proposed for VaR estimates. The proposed wavelet decomposed VaR (WDVaR) is ex-ante in nature and is capable of estimating risks that are multi-scale structured. Empirical studies in major agricultural markets are conducted for both the hybrid ARMA-GARCH VaR and the proposed WDVaR. Experiment results confirm significant performance improvement. Besides, incorporation of time variant risks tracking capability offers additional flexibility for adaptability of the proposed hybrid algorithm to different market environments. WDVaR can be tailored to specific market characteristics to capture unique investment styles, time horizons, etc. Keywords: financial, risk management, time series analysis, wavelets and fractals, Value at Risk.

1

Introduction

Risks are an inherent part of agricultural production process due to the complexities in the surrounding physical and economic environment. Proper measurement and management of agricultural market risks are essential due to the following reasons: Firstly, the past and present risk levels shape expectations about future risk evolutions and influence production decisions. Secondly, risk levels affect important operational decisions concerning the cost of capital and revenue targets, etc. Thirdly, agricultural industries are capital intensive, with majority of H.A. Le Thi, P. Bouvry, and T. Pham Dinh (Eds.): MCO 2008, CCIS 14, pp. 429–438, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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capital deployment concentrated on farm real estate and machinery. Fourthly, agricultural risks increase continuously as there are rising levels of uncertainties during the production cycle. Thus, the industries are increasingly vulnerable to devastating consequences of unexpected market risks [1]. Value at Risk (VaR) is one popular approach to measure market risk. Despite its significance, there are only a handful of research methodologies concerning quantitative measurement and management of risks in agricultural industries - e.g. Giot measures risks in commodities markets, including metals, agricultural commodities and oil markets, using the VaR methodology. VaR estimated by using the APARCH model provides the highest reliability. VaR estimates based on implied volatility are also found to provide comparable performance [2,3]. Ani and Peter applied two popular credit risk models to risk measurement in agricultural loans and calculated the required VaR to protect investors’ interests [4]. At the same time, measurement of the multi-scale heterogeneous structure of agricultural risk evolution remains the unexplored area. Therefore, this paper proposes an ex-ante decomposition based approach for risk measurement in agricultural markets, in contrast with the traditional approaches. Wavelet analysis is proposed to conduct multi-resolution analysis of the heterogeneous market structure of the risk evolution process in agricultural markets. Wavelet analysis has been used extensively in different fields of economics and finance, such as the economic relationship identification and wavelet decomposed forecasting methodology, etc [5,6,7]. Despite the apparent need for multi-scale risk structure analysis, there have been only a handful of researches identified in the literature. These approaches focus on multi-resolution analysis of historical market risk structure and its distribution [8]. However, they are more of a historical simulation approach during their modeling attempts and offer little insights into the evolution of these structures. The proposed Wavelet based approach for VaR estimates allows the flexibility of combining the power of different econometric models in the time scale domain, which reflects different investment strategies over various investment time horizons in the agricultural markets. Empirical studies have been conducted in major US agricultural markets to evaluate and compare the performance of the proposed wavelet based approach against the traditional ARMA-GARCH approach for VaR estimates. Experiment results confirm improved reliability and accuracy offered by WDVaR due to its ability to analyze multi-scale heterogeneous structures and its processing power. The rest of the paper is organized as follows: the second section briefly reviews the relevant theories, including wavelet analysis and different approaches to VaR estimates. The status quo of applications of wavelet analysis in risk management is also reviewed. The third section proposes the wavelet based VaR algorithm. The fourth section conducts empirical studies in major US agricultural markets. Performance evaluation and comparison of models tested are based on Kupiec backtesting procedures. The fifth section concludes.

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2 2.1

431

Relevant Theories Value at Risk

Value at Risk is the dominant risk measure that has received endorsement from both, academics and industries, recently [9]. Given the confidence level α, VaR is defined as the p-quantile of the portfolio’s profit/loss distribution over certain holding period at time t as in (1). V aRt = −qp,t = Ft−1 (α) = μt + σt G−1 t (α)

(1)

Where qp,t refers to the pth conditional quantile of the portfolio distribution. Ft−1 (α) and G−1 t (α) refer to the inverse of the portfolio distribution function. μt is the conditional mean, while σt is the conditional variance. VaR is used to compress and give approximate estimates as to the maximal possible losses. Estimation of VaR can be classified into three groups, depending on the degree of assumptions made - parametric approach, non-parametric approach and semi-parametric approach [9]. The parametric approach fits the curve into the risk evolution process and derives analytical forms. The advantage of the parametric approach is its intuitive appeal and simplicity to understand and track. It is especially useful in the tranquile environments. However, when the market gets volatile with more extreme events occurring, assumptions in parametric approaches are easily violated and lead to biased estimates. Also, the current parametric approaches lack the essential ability to analyze the multi-scale nonlinear dynamics in the markets. The non-parametric approach takes a different route by imposing weak assumptions during the estimation process. It includes techniques such as Monte Carlo simulation methods and more recently neural network, etc. The advantage lies in its adaptability to non-linear environments, where Data Generating Process (DGP) is unknown. However, These approaches are mostly black box in nature and offer little insights into the underlying risk evolutions. The semi-parametric approach strikes the balance between the previous two approaches. It relaxes to some extent, assumptions made in parametric approaches while providing more insights. Techniques used include extreme value theory (EVT) and wavelet analysis, etc. Backtesting procedures are formal statistical methods to verify whether the projected losses are in line with the actual losses observed in the market. Over the years, different approaches have been developed. This paper uses the unconditional coverage tests as the basis for the model evaluation and comparison. The VaR exceedances are Bernoulli random variable, which is equal to 1 when VaR is exceeded by losses; and is 0 otherwise. The null hypothesis for unconditional coverage test is the acceptance of the model at the given confidence level. Kupiec develops the likelihood ratio test statistics as in (2). LR = −2ln[(1 − ρ)n−x ρx ] + 2ln[(1 − x)/n)n−x (x/n)x ]

(2)

Where x is the number of exceedances, n is the total number of observations and p is the confidence level chosen for the VaR estimates. The Kupiec likelihood ratio test statistics is distributed as χ2 (1) .

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Wavelet Analysis

Wavelet analysis with time-frequency localization capability is introduced as the advancement to the traditional band limited Fourier transform [10]. The wavelet functions utilized during wavelet analysis are mathematically defined as functions that satisfy the admissibility condition as in (3).  ∞ |ϕ(f )| df < ∞ (3) Cψ = f 0 Where ϕ is the Fourier transform of the frequency f . ψ is the wavelet transform. Location parameter u and scale parameter s can be used to translate and dilate the original function during wavelet analysis as in (4) [10].  ∞ 1 t−u )dt (4) x(t) √ ψ( W (u, s) = s s −∞ Where s ∈ R+ , μ ∈ R. Thus, the transformed wavelet function convolves with the market return series to obtain wavelet coefficients as in (4) An inverse operation could also be performed, as in (5), which is referred to as the wavelet synthesizing process.  ∞ ∞ ds 1 W (u, s)ψu,s (t)du 2 (5) x(t) = Cψ 0 s −∞ By design, wavelets are dilated shorter at higher frequency, which provides shorter time windows to capture time sensitive information. Wavelets are also dilated longer at lower frequency to emphasize frequency level information with longer time windows. Typical wavelet families include Haar, Daubechies, Symlet and Coiflet, etc.

3

Wavelet Decomposed Value at Risk

Markets are heterogeneous in nature. A typical financial market (e.g. agricultural, energy market, etc.) consists of the following participants: market makers, intraday traders, daily traders, short term traders and long term traders. Since different types of traders have different investment strategies, determined by their investment horizon and financial health, their contributions to the market price formation process vary in terms of both time horizon and frequency level [11]. Previous parametric approaches, including single and hybrid models based approaches to VaR estimates, are categorized as ex-post approaches. They would lead to significant biases in estimates in heterogeneous markets. In heterogeneous markets, prices are formed with influences from different types of investors, characterized by different investment strategies and time horizons, which change over time. Since most current single parametric approaches are based on stationary assumptions and focus on frequency domain, their performances are unstable in the volatile market environment, i.e. the violation of the stationary assumptions

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invalidates the model during periods of intense fluctuations with investment strategies changing with time horizons. The ensemble approaches also share the same problem with the single model approach, although it improves the performance by nonlinearly ensembling different individual forecasts. Besides, it offers little insights into the multi-scale market risk structure. Meanwhile, the current hybrid algorithm linearly filters the data through different models to extract maximal possible information and minimize the residuals. However, the bias introduced in the first stage filtering process would not only be carried forward, but would also distort the next filtering process and lead to an increasing level of biases in estimates. If the distribution can be described with location and scale parameters, then the VaR is estimated parametrically as in (6) V aRt = μt + σt G−1 (α)

(6)

Where G−1 (α) refers to the inverse of the cumulative normal distribution. Estimation of the conditional mean μt follows the ARMA-GARCH process. Estimation of the conditional standard deviation σt follows a multi-scale framework based on wavelet analysis. Firstly , the original data series are projected into the time scale domain with the chosen wavelet families as in (7) f (t) = fAJ (t) +

J 

fDj (t)

(7)

j=1

Where f (t) refers to the original time series. fAJ (t) refers to the decomposed time series using scaling function at scale J. fDj (t) refers to the decomposed time series using wavelet function at scales j, up to scale J. Secondly econometric or time series models serve as individual volatility forecasters at each scale. Parameters are estimated by fitting different models to decomposed data at each scale. Then volatilities are forecasted, using the estimated model specifications. Thirdly, according to the preservation of energy property, estimates of volatility are reconstructed from volatility estimates at each scale, using wavelet synthesis techniques as in (8). σ  = V aR((f (t)) = var(fAJ (t)) + 2

J 

V ar(fDJ (t))

j=1

=

1  2λJ N

N 2J −1

 t=2

2 ωJ,t +

J 

1

j=1

 2λj N

N 2j −1



ϕ2j,t

(8)

t=2

 refers to the Where N = 2J refers to the length of the dyadic data series. N wavelet coeeficients at scale λj .

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Empirical Studies Data and Experiment Design

The data examined in this paper are daily aggregated spot prices in two major US agricultural markets: cotton and live hog. These markets are selected, based on their significant market shares and data availability. The market shares for both cotton and live hog are 2.20% and 5.90% respectively. The data set for cotton covers the time period from 27 March, 1980 to 14 June, 2006 while the data set for live hog covers the time period from 2 January, 1980 to 14 June, 2006. The total sample size is 6613 daily observations. The data are divided into two parts, i.e., the first 60% of the data set forms the training set while the rest 40% serves as the test set. Table 1. Descriptive Statistics and Statistical Tests Agricultural Commodities Cotton Live Hog Mean

0.0000

0.0000

Maximum

0.1292

0.5125

Minimum

-0.9049

-0.5316

Medium

0.0000

0.0000

Standard Deviation

0.0196

0.0248

-24.7588

-0.3208

Skewness Kurtosis

1150.6933 108.1585

Jarque-Bera Test (P value)

0

0

BDS Test (P value)

0

0

Table 1 reports the descriptive statistics for daily returns in both markets. The agricultural market represents a volatile environment, as indicated by the high volatility level. Investors face significant losses, as suggested by the negative and skewnesses. The market environment is considerably risky, as indicated by the high degree of excess kurtosis, which suggests the prevalence of extreme events. Thus, proper measurement and management of risks are crucial to both, investors and governments, in agricultural markets. Rejection of Jarque-bera test of normality and BDS (Brock-Dechert-Scheinkman) test of independence suggests the existence of nonlinear dynamics in the data [12]. Further performance improvement upon traditional approaches demands innovative techniques to account for the multi-scale heterogeneous structure of the markets. A portfolio of one asset position worth 1 USD is assumed during each experiment. Geometric returns rt are calculated assuming continuous compoundt ing as ln PPt−1 , where Pt refers to the price at time t. Based on the analysis of the autocorrelation and partial autocorrelation function, the model order for ARMA-GARCH is determined as ARMA(2,2) and GARCH(1,1). The length of the moving windows during the one step ahead forecasts is set at 3967 to cover the most relevant information set.

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Empirical Results

ARMA-GARCH VaR. As suggested by results in table 2, the ARMA-GARCH approach don’t offer sufficient reliability for VaR estimates. The ARMA-GARCH approach is rejected uniformly in both markets, across all confidence levels. Generally, the estimates are too conservative. The poor performance stems from the linear hybrid approach adopted, which lacks the ability to extract information concerning the multi-scale risk structures. WDVaR(X,1). When wavelet analysis is applied to multi-resolution analysis of the risk evolution, two new parameters are introduced in the notion WDVaR(X,i), i.e., the wavelet families chosen X and the decomposition level i. The sensitivity of the model’s performance to the wavelet families chosen is investigated by estimating VaRs based on different wavelet families at the decomposition level 1. Experiment results are listed in table 3. Taking VaRs estimated at 95% confidence level, experiment results in table 3 confirm that the wavelet families chosen affect the perspectives taken during the analysis of the risk evolution and, as a consequence, affect the VaR estimated. The wavelet families could be treated as a pattern recognition tool since different families would lead to the extraction of different data patterns. Convolution of wavelets to the original data series is a process of searching for the relevant data patterns across time horizons and scales. Meanwhile, further experiment results confirm that Take symlet 2 for example, experiment results in table 4 show that changing wavelet families do lead to significant performance improvement. VaRs estimated are accepted at both 97.5% and 99% confidence level in the live hog market and are accepted at both 95% and 97.5% confidence levels in the cotton market. WDVaR(Haar, i). The sensitivity of the model’s performance to the selection of decomposition level is further investigated. Decomposition level is set to 3. Increases in the decomposition level improve the model’s performance significantly. Firstly, the reliability of the proposed VaRs estimates are accepted at Table 2. Experiment Results for ARMA-GARCH VaR in Two Agricultural Commodities Markets Across All Confidence Levels Agricultural Confidence ARMA-GARCH MSE Kupiec Test P-value Commodities Cotton

Live Hog

Level

VaR Exceedance

99.0%

8

Statistics 0.0018

17.8826

0

97.5%

17

0.0013

52.9588

0

95.0%

31

0.0010

116.5033

0

99.0%

15

0.0027

5.9125

0.0149

97.5%

33

0.0020

20.7756

0

95.0%

71

0.0015

35.6069

0

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Table 3. Experiment Results for WDVaR(X,1) at 95% Confidence Level in Cotton Market Wavelet WDVaR(x,1) MSE Kupiec Test P-Value Family

Exceedance

Haar(db1)

45

0.0007

Statistics 80.4040

0.0000

db2

113

0.0005

3.0806

0.0792

db3

141

0.0005

0.6041

0.4370

db4

131

0.0005

3.0806

0.0792

db5

143

0.0004

0.9057

0.3413

db6

151

0.0004

2.6962

0.1006

dmey

138

0.0004

0.2642

0.6072

sym2

113

0.0005

3.0806

0.0792

sym3

141

0.0005

0.6041

0.4370

sym4

135

0.0005

0.0620

0.8033

sym5

132

0.0004

0.0003

0.9858

coif1

148

0.0005

1.9169

0.1662

Table 4. Experiment Results for WDVaR(Sym2,1) in Two Agricultural Commodities Markets Across All Confidence Levels Agricultural Confidence ARMA-GARCH MSE Kupiec Test P-value Commodities Cotton

Live Hog

Level

VaR Exceedance

99.0%

57

Statistics 0.0008

26.8099

0

97.5%

79

0.0006

2.4328

0.1188

95.0%

113

0.0005

3.0806

0.0792

99.0%

25

0.0027

0.0807

0.7764

97.5%

51

0.0020

3.8353

0.0502

95.0%

97

0.0016

10.8276

0

99% confidence level in the cotton market and are accepted at both 97.5% and 99% confidence levels in the live hog market. Secondly, the accuracy of the estimates improves as the size of the exceedances measured by Mean Square Error (MSE) decreases uniformly. This performance improvement results from finer modeling of details at higher decomposition levels using the wavelet analysis. As the decomposition level increases, market structures are projected into the higher dimension domain to reveal more subtle details, i.e. investors with longer investment horizons are separated out for further analysis. Thus, the attempted ARMA-ARCH model could be estimated with more suitable parameters at the individual scale, which results in the more accurate description of risk evolutions. Aggregated together, it will result in the closer tracking of risk evolutions and,

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Table 5. Experiment Result for WDVaR(Haar,3) in Two Agricultural Commodities Markets Across All Confidence Levels Agricultural Confidence ARMA-GARCH MSE Kupiec Test P-value Commodities Cotton

Live Hog

Level

VaR Exceedance

99.0%

21

Statistics 0.0011

1.2164

0.2701

97.5%

36

0.0008

16.7993

0

95.0%

58

0.0006

55.0014

0

99.0%

29

0.0026

0.2427

0.6222

97.5%

55

0.0020

2.0258

0.1546

95.0%

94

0.0015

12.8661

0

thus, more accurate and reliable estimates of risk measurements - VaR. Besides, by tuning the two new parameters, i.e. the wavelet families and the decomposition level, reliability and accuracy of the VaR estimates improves significantly. The wavelet based approach offers considerably more flexibility during the VaR estimation process. However, the increased performance doesn’t come without costs. More subtleties are revealed with the exponential growth of computational complexities, which are not always desirable.

5

Conclusion

Given the long production cycle and unexpected factors involved, proper measurement of risks has a significant impact on agricultural production decisions and the revenue generated. This paper proposes the wavelet based hybrid approach to measure agricultural risks using the VaR methodology, due to its long production cycle. The contribution of this paper is two fold. Firstly, multiresolution analysis is conducted to investigate the heterogeneous market structures using the wavelet analysis. Agricultural data are projected into the time scale domain to reveal its the composition factors. Secondly, the ex-ante based methodology is proposed for hybrid algorithm design. Wavelet analysis is used as an example of ex-ante based hybrid algorithm. The combination methodology is based on time scale decomposition in contrast with the traditional linear filtering process. Experiments conducted in two major US agricultural markets show that the proposed WDVaR outperforms the traditional ARMA-GARCH VaR. The advantage of this model is that the estimates unify different models with different parameter settings in a given time scale domain. Besides, this model also offers additional insights into the multi-scale structure of risk evolution.

Acknowledgement The work described in this paper was supported by a grant from the National Social Science Foundation of China (SSFC No.07AJL005) and a Research Grant of City University of Hong Kong (No. 9610058).

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