Value-at-Risk Forecasts in Gold Market Under Oil Shocks

Middle Eastern Finance and Economics ISSN: 1450-2889 Issue 4 (2009) © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/MEFE.htm

Value-at-Risk Forecasts in Gold Market Under Oil Shocks Wan-Hsiu Cheng Assistant Professor, Department of Banking and Finance, Tamkang University 151 Ying-Chuan Road, Tamsui, Taipei County, 25137, Taiwan E-mail: [email protected] Tel: 886-2-26215656 ext. 2518 Jung-Bin Su Associate Professor, Department of Finance China University of Science and Technology Yi-Pin Tzou Ph.D. Candidate, Department of Banking and Finance Tamkang University Abstract This paper investigates the out-of-sample value-at-risk (VaR) forecasts in gold markets by considering both oil volatilities and the flexible model construction. We used the combined BHK (Brenner, Harjes, and Kroner, 1996) and power GARCH (PGARCH) models to consider not only the effect of spot prices, but also the endogenized power term. The empirical results indicate that the PGARCH-HV model with its flexibility in power term for data transformation and the high volatility of crude oil is the best model for VaR forecasting. The findings have implications for investors, financial institutions, and futures exchanges. Keywords: Gold; Oil volatility; PGARCH; BHK; Value-at-risk JEL Classification Codes: G00

Introduction One of the most powerful historical commodity interrelationships exists between oil and gold. Oil is the most important commodity on earth because almost everything tangible that we physically move burns oil in the process, while gold is the only one of the many commodities that over the years has served as money in international trade and financial transactions. Gold and crude oil prices tend to rise and fall in synchronicity with each other. One reason is that, historically, oil purchases were paid for in gold. Even today, a sizable percentage of oil revenue ends up being invested in gold. As oil prices rise, much of the increased revenue, considered as surplus to current needs, is invested, and much of this is invested in gold or other hard assets. Another reason is that rising oil prices place upward pressure on inflation and this enhances the appeal of gold because it acts as an inflation hedge. The relationship also appears in the Gold-Oil Ratio which has traded within a well-defined range since just after World War II a period of more than 60 years. With stabilized crude oil market in the background, gold traders were free to focus on the plight of the credit-ridden US dollar, the speculative fever in China for gold and red-chip stocks, the central bank diversification out of the US dollar, and so on. Once the oil price

Middle Eastern Finance and Economics - Issue 4 (2009)

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fluctuates, gold prices react immediately. For example, in the beginning of 2007, the sudden and unexpected plunge in the prices of crude oil and other base metals resulted in gold prices tumbling to its lowest levels in more than two months. However, it is surprising to note that little research has been carried out on the influence of oil price to the gold market. Melvin and Sultan (1990) and Cai et al. (2001) are the two papers in this area. In Melvin and Sultan (1990), the political unrest in South Africa in addition to oil price volatility are significant factors that constitute the gold spot price forecast errors. Cai et al. (2001) undertook an analysis of the effect of 23 macroeconomic announcements on the gold market using high-frequency data and found that one of the occurrences that cause the largest returns on the gold market is oil prices. More than that, none of these studies measures the risk in the gold market while considering oil volatilities. This problem should be taken seriously by market participants while the relationship between gold and oil prices is in such close proximity. Therefore, this paper investigates the value-atrisk (VaR) in gold markets by considering both oil volatilities and the flexible model construction. Literature review of the gold market Gold is the only one among many commodities that, over the years, has served as money in both international trade and financial transactions. Gold is a liquid metal. As one of history’s longest-valued commodities, it has a unique 24-hours-a-day market for purchase or sale. Gold helps in hedging inflation. Governments, banks, and Wall Street brokerages hold physical gold to limit their exposure from losses in paper assets. Household investors should also consider using a similar successful hedge strategy to protect their own portfolios. A number of studies have reported on the relationship between gold and inflation. Sherman (1983) and Baker and Van-Tassel (1985) documented that the gold prices depend on future inflation rates. Kaufmann and Winters (1989) reported that the price of gold is based on changes in the US rate of inflation. Likewise, Sjaastad and Scacciavillani (1996) reported that gold is a store of value against inflation. Lawrence (2003) indicated that no significant correlations exist between gold returns and changes in certain macroeconomic variables such as inflation, gross domestic products (GDP), and interest rates. Gold also plays a useful role in diversifying risk (see Sherman, 1983; Kaufman and Winters, 1989; Chua, et al. , 1990; Davidson et al., 2003), as well as being an attractive investment in its own right. On the other hand, Christie-David et al. (2000) and Cai et al. (2001) are two recent studies that examined the macroeconomic news releases on gold prices. Christie-David et al. (2000) found that gold responds strongly to CPI news announcements, in addition to the unemployment rate, GDP, and PPI. Future gold prices responded strongly to the release of capacity utilization information but do not respond strongly to federal deficit announcements. Cai et al. (2001) indicated that the largest returns were due to the following occurrences: central bank sales, interest rates, oil prices, inflation rates, US unemployment rates, Asian financial crisis, and political tension in South Africa. Recently, gold has been showing signs of becoming a more mainstream asset in many portfolios around the world. This could result in a higher offtake for gold products, but on the negative side, it could also imply that its correlation with other financial assets has increased. Johnson and Soenen (1997) and Egan and Peters (2001) show that gold has a high negative correlation with equity indices. However, Lawrence (2003) pointed out that gold returns are less correlated with equity returns and bond indices than returns of other commodities. Volatility measurement in the oil market Energy markets are characterized by extremely high levels of price volatility. Fluctuations in energy prices are caused by supply and demand imbalances arising from events like wars, changes in political regimes, economic crises, formation/ breakdown of trade agreements, unexpected weather patterns, etc. For example, from August 1990 to January 1991, the spot prices of crude oil climbed up to historically high levels resulting from the First Gulf War. The Israeli-Arab conflicts in 2000, the Second Gulf War in 2003, and OPEC’s strategies for controlling the supply of oil all led to significant price changes.

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However, changes in oil prices affect economic activities in ways that are not entirely reflected in transactions in the oil market (Koopmann, 1989; Sadorsky, 1999; Bahmani-Oskooee and Brown, 2004; Jiménez-Rodríguez and Sánchez, 2005). Sudden and large hikes in its price through actual or envisioned supply interruptions have far-reaching implications on all economies. For instance, during the period of the first oil shock (1973–1974) and in the period of the second oil shock (1979), the United States experienced extremely high inflation and unemployment rates. Kim and Loungani (1992) even supported the view that these oil shocks have been an important source of economic fluctuations over the past three decades. Therefore, analyzing oil volatility is of practical importance to financial market participants. Traditionally, it is assumed that the time series data follows a smooth and continuous process, however the presence of jumps implies that diffusion models are erroneously specified statistically. Following that, the dynamic jump model has most recently been considered as a good one to provide a value for the volatility of financial asset returns (Chang and Kim, 2001; Pan, 2002; Eraker, Johannes and Polson, 2003; Chan and Maheu, 2002; Johannes, 2004; Maheu and McCurdy, 2004). However, little attention has been given to petroleum volatility because most papers center on financial assets. By observing the trend of crude oil, the jump events are easily found and cannot be ignored in the measurement of volatility. Therefore, the oil volatility is estimated using Chan and Maheu’s (2002) dynamic jump model, with the volatility further distinguished into stochastic and jump volatility. There are several purposes in this paper. First, this paper investigates the value-at-risk in gold markets by considering both the stochastic and jump volatility of oil. Then, we assume that the gold volatility is related not only to unexpected shocks to the markets, but also to the level of spot prices. Third, we used the power GARCH (PGARCH) model with an endogenized power term to capture the data transformations. The PGARCH structure is flexible enough to nest both the conditional variance and conditional standard deviation models as particular cases. Finally, we adopt the rolling window approach with various models to obtain the appropriate out-of-sample VaR estimates. The remainder of this paper proceeds as follows. Section 2 presents the methodology, including the jump model for crude oil volatility and the variety economic models for the gold market. Section 3 provides the descriptions of the VaR definition and evaluation method. Section 4 describes the data and descriptive statistics. Section 5 presents empirical evidence, and the final section offers a conclusion.

Methodology Autoregressive Conditional Jump Intensity (ARJI) model for crude oil volatility The purpose of this paper is to investigate the relationship between gold return and oil volatility. Crude oil markets are characterized by high levels of volatility and occasional jumps appear in the time series data of crude oil as seen during the Gulf War period. Therefore, the Autoregressive Conditional Jump Intensity (ARJI) model proposed by Chan and Maheu (2002) is adopted in this paper to measure the volatility of crude oil. Given an information set of returns at time t − 1 , Ω t −1 = {rt −1 ,.....,r 1 } , and the two independent stochastic innovations, ε1, t and ε 2, t , the jump model can be expressed as follows:

R t = μ + ε1, t + ε 2, t

(1)

where ε1,t is a mean-zero innovation with a normal stochastic process and it is assumed as, ε1, t = h t Z t , Z t ~ NID(0,1)

(2)

Let the smooth conditional variance h t follows a GARCH(1,1) process, that is h t = ω + αε 2t −1 + βh t −1 , and ε t −1 =ε1, t −1 +ε 2, t −1

(3)

Where ε 2, t is a jump innovation assigned to be a conditional zero-mean value, and ε 2, t = J t − E[J t Ω t −1 ] = ∑k =t 1 π t ,k − θλ t n

(4)

51


where J t is the jump component affecting returns from t − 1 to t , and equals to

∑

nt k =1

π t ,k . The jump

size, π t ,k , is assumed to follow a normal distribution with a mean θ and a variance δ 2 . The variable n t denotes the discrete counting process governing the number of jumps that arrive between t-1 and t, which follows the Poisson distribution with parameter λ t > 0 . Its probability density function is e − λ t λjt , j = 0,1,2,......... (5) j! Both the mean and variance for the Poisson random variable are λ t , called the jump intensity, and is assumed to follow ARMA process as λ t = λ 0 + ρλ t −1 + γξ t −1 , (6) P(n t = j Ω t −1 ) =

where ξ t−1 represents the innovation to λ t−1 , which measures the unexpected forecast of n t −1 as the information set is updated. That is, ∞

ξ t −1 ≡ E[n t −1 Ω t −1 ] − λ t −1 = [∑ j P(n t −1 = j Ω t −1 )] − λ t −1

(7)

j= 0

where P(n t −1 = j Ω t −1 ) , called the filter, is the ex post inference on n t −1 given the information set Ω t−1 , and E(n t −1 Ω t −1 ) is the ex post judgment of the expected number of jumps that occurred from t − 2 to t − 1 . Note from this definition that ξ t is a martingale difference sequence with respect to information

set Ω t−1 . Therefore, E[ξ t ] = 0 and Cov(ξ t , ξ t −i ) = 0 , i > 0 . Hence, the jump intensity can be rewritten as follows: (8) λ t = λ 0 + (ρ − γ )λ t −1 + γE[n t −1 Ω t −1 ] where λ t > 0 , and λ 0 > 0 , ρ ≥ γ , γ ≥ 0 . The conditional variance of returns is decomposed into two separate components: a smoothly developing conditional variance component related to the diffusion of past news impacts and the conditional variance component associated with the heterogeneous information arrival process which generates jumps. The conditional variance of returns is Var (R t Ω t −1 ) = Var (ε1, t −1 Ω t −1 ) + Var (ε 2, t −1 Ω t −1 ) = h t + (θ 2 + δ 2 )λ t (9) We further construct the likelihood function as follows. Conditional on j jumps occurring, the conditional density of returns is normal, that is f (R t n t = j, Ω t −1 ) =

1 2 π( h t + j δ 2 )

× exp[ −

( R t − μ + θλ t − jθ) 2 ] 2( h t + j δ 2 )

(10)

Maheu and McCurdy (2004) proposed and provided an ex post distribution for the number of jumps, n t . The filter is constructed as P(n t = j Ω t ) =

f (R t n t = j, Ω t −1 )P(n t = j Ω t −1 ) P(R t Ω t −1 )

j = 0,1,2,....

(11)

After integrating out all the jumps during one unit interval, the conditional probability density function can be expressed as ∞

f (R t Ω t −1 ) = ∑ f (R t n t = j, Ω t −1 )P(n t = j Ω t −1 )

(12)

j= 0

Therefore, the likelihood function can be written as: T

L(ψ ) = ∑ log f (R t Ω t −1 ; ψ ) t =1

(13)

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where ψ = (μ, ω, α, β, θ, δ 2 , λ 0 , ρ, γ ) is the vector of parameters to be estimated. The variety of BHK model

By integrating the feature in the interest rate, this paper considers the idea of whether or not gold volatility is related not only to unexpected shocks to the market, but also to the level of spot prices. The interest rate model proposed by Brenner, Harjes, and Kroner (1996) (BHK, hereafter) is applied here to investigate gold volatility. Furthermore, GARCH models have become common tools for the time series heteroskedastic models; however, the data transformations involved the use of a squared term. Ding, Granger, and Engle (1993) suggested a new class of GARCH models, called the power GARCH (PGARCH) model, where the power term is endogenized rather than fixed arbitrarily. The PGARCH structure is flexible enough to nest both the conditional variance (Bollerslev, 1986) and the conditional standard deviation (Taylor, 1986) models as particular cases. It thus provides an encompassing framework for model analysis and selection. Therefore, this paper adopts the power GARCH model. The BHK-PGARCH model is constructed as follows. rt = φ 0 + φ1 rt −1 + ϕ1 Var (ε1oil, t ) + ϕ 2 Var (ε oil (14) 2, t ) + e t e t = σ tκ Ptγ−1 Z t , Z t ~ N(0,1)

(15)

E(e t ) = 0 , E(e 2t ) = σ tκ Pt2−γ1 κ

σ κt = c + a e t −1 + bσ κt −1 + ϕ 3 Var(ε1oil, t ) + ϕ 4 Var (ε oil 2, t )

(16)

where rt is the gold return at time t. Var (ε1oil, t ) and Var (ε oil 2 , t ) are the diffusion-induced and jumpinduced variances of the crude oil, respectively. The residual e t is both related to gold volatility ( σ ) and gold price level ( P ), and the mean and variance of the residual e t are zero and σ κt Pt2−γ1 , respectively. The parameter γ allows the volatility to depend on the level of the gold prices, called the level effect. At a higher γ , the volatility is more sensitive to gold price levels. In contrast, if γ = 0 , then the level effect disappears and we return to the PGARCH framework. However, parameter κ is the power term from which the data are transformed. In addition, the GARCH model is restricted by κ = 2. Extending the original model, the high volatility characteristics of crude oil are emphasized and called the BHK-PGARCH-HV model. The variables Var (ε1oil, t ) and Var (ε oil 2 , t ) are replaced by _ HV Var (ε1oil, t _ HV ) and , Var (ε oil ) respectively. High volatility is defined as higher than average, that is 2, t

⎧Var(ε ioil, t ) if Var(ε ioil, t ) ≥ E[Var(ε ioil,t )] for i = 1, 2 (17) Var(ε ioil, t _ HV ) = ⎨ 0 , otherwise ⎩ Furthermore, the BHK-PGARCH-HV-A model is constructed additionally as compared with the above two models. In this model, the oil volatility does not separate into diffusion-induced or jumpinduced, but the total sum is instead considered. The mean and variance equations are _ HV rt = φ 0 + φ1 rt −1 + ϕ1 Var(ε oil ) + et (18) t e t = σ tκ Ptγ−1 Z t , Z t ~ N(0,1) κ

_ HV σ tκ = c + a e t −1 + bσ tκ−1 + ϕ 3 Var(ε oil ) t

(19) (20)

Other definitions and assumptions are identical to former models. Finally, the likelihood function is constructed as follows. The conditional density of returns is ⎫ ⎧ e 2t 1 (21) f (rt Ω t −1 ) = × exp⎨− ⎬, κ 2 γ 2πσ κt Pt2−γ1 ⎩ 2(σ t Pt −1 ) ⎭ and the likelihood function can be written as:

Middle Eastern Finance and Economics - Issue 4 (2009) L(ψ ) = ln f (rt Ω t −1 ; Ψ )

53 (22)

where Ψ = (φ 0 , φ1 ,ϕ1 , ϕ 2 ,ϕ 3 , ϕ 4 , c, a , b, κ, γ ) is the vector of parameters to be estimated. Relative to the three PGARCH models, three GARCH models restricted to κ = 2 are also examined in this paper, namely, the BHK-GARCH, BHK-GARCH-HV, and BHK-GARCH-HV-A models. Therefore, six alternative models are estimated for a detailed analysis of the gold market. In addition, for a more compact description, we omit the name ‘BHK’ in later portions of this paper.

Model-Based VaR Estimates and Evaluation Method Value-at-Risk (VaR) definition

This paper uses the VaR for further comparison of the alternative volatility models. Since 1998, financial institutions have been required to hold capital against their defined market risk exposure using VaR estimates. A VaR model measures market risk for a portfolio of financial assets and measures the potential loss that a portfolio could experience at a given degree of confidence level α over a given period h. It is, (23) Pr[rt + h < VaR t (h )] = α. . where rt + h is the portfolio return at time t + h . Apparently, VaR is simply a specific quantile of a portfolio’s potential loss distribution over a given holding time horizon. Mathematically, the function can be expressed as:

∫

VaR ( h , α )

−∞

f t + h (r )dr = 1 − α ,

(24)

where f(r) represents the probability density function of r. The length of holding period is short term, usually one day to a few days, and the confidence levels α are generally 90%, 95%, and 99%. Thus, the VaR is useful for an investor’s portfolio in forecasting the maximum amount of money that can be lost in a short period of time. Evaluation method

The performance of VaR models will be assessed from two different perspectives: accuracy and efficiency (Engle and Gizycki, 1999). Based on Kupiec (1995) and Lopes (1998), we consider a binary loss function test, a quadratic loss function test, and the LR test of unconditional coverage to measure the accuracy of VaR models in this paper. (1). General loss functions The tests did not yield any results on the relative accuracy of competing VaR models until Lopez (1998). The general form of these loss functions is defined below: ⎧ f (rt +1 , VaR t ) if rt +1 < VaR t , (25) L t +1 = ⎨ ⎩ g (rt +1 , VaR t ) if rt +1 ≥ VaR t .

The numerical scores are constructed with a negative orientation: f ( x, y) and g( x , y) are functions such that f ( x, y) ≥ g( x, y) for any given y. If a return larger than the VaR is observed, that is termed an “exception”. Hence, lower values of L t +1 are preferred because exceptions are given higher scores than non-exceptions. Under general conditions, the best VaR estimate will thus generate the lowest numerical scores. In this study, we use two different loss functions: a binary loss function and a quadratic loss function. (2). Binary loss functions (BLF) The binary loss function is merely the reflection of the LR test of unconditional coverage test and gives a penalty of one to each exception of the VaR. Here the major concern is the number of violations

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rather than the magnitude of these violations. Equal weight is accorded to each loss that exceeds the VaR estimate and all other profits and losses have a zero weight. That is, ⎧ 1 if rt +1 < VaR t . (26) L t +1 = ⎨ ⎩ 0 if rt +1 ≥ VaR t If a VaR model truly provides the level of coverage defined by its confidence level, then the average binary loss function (ABLF) over the full sample will equal α for the (1 − α) th percentile VaR. The average binary loss function provides a point estimate of the probability of observing a loss greater than the VaR amount. (3). Quadratic loss functions (QLF) The quadratic loss function takes into account the magnitude of the exceptions. Lopez (1998) found that the quadratic loss function's use of the additional information embodied in the size of the violation provided a more powerful measure of model accuracy than the binary loss function. The loss function is defined as ⎧ 1 + (rt +1 − VaR t ) 2 if rt +1 < VaR t , L t +1 = ⎨ (27) if rt +1 ≥ VaR t . ⎩ 0

As before, a score of one is imposed when a violation occurs, but this time, an additional term based on the magnitude of the exception is also introduced. (4). LR test of unconditional coverage (LRUC ) Kupiec (1995) proposed a likelihood ratio test (LR) for testing the model accuracy which is identical to a test of the null hypothesis that the probability of failure for each trial ( αˆ ) equals the specified model probability ( α ). The likelihood ratio test statistics is given by: ⎡ α n1 (1 − α) n 0 ⎤ (28) LR = −2 ln ⎢ n ~ χ 2 (1) n0 ⎥ 1 ˆ ˆ α ( 1 − α ) ⎣ ⎦ where the failure rate αˆ = n 1 /(n 0 + n 1 ) , the total sample of which is ( n 0 + n 1 ) where n 1 denotes a Bernoulli random variable representing the total number of observed failures. The LRUC test can be employed to test whether or not the sample point estimate is statistically consistent with the VaR model’s prescribed confidence level.

Data and Descriptive Statistics The data consists of the daily prices of gold ($/Ounce) and crude oil ($/Barrel) over from April 29, 1998 to July 20, 2006, with a total of 2,055 observations. All daily data are obtained from Datastream. The gold and crude oil returns are the logarithm of daily gold and crude oil prices, respectively. The rolling window method with 1,800 days observation period is used to obtain the out-of-sample VaR estimates for each day. The descriptive statistics of gold and crude oil returns are reported in Table 1. The mean and standard deviation of the crude oil returns are 0.0757 and 2.4968, almost twice than the gold returns with a mean of 0.0351 and a standard deviation of 1.0086. Clearly, the volatility of oil returns is more violent than gold returns. Both returns present a significant asymmetry as the skewness is significant at the 1% level; however, it is interesting to observe that the gold presents a positive skewness while the oil exhibits a negative skewness. The excess kurtosis is significantly larger than zero, thereby indicating a leptokurtic characteristic. Thus, both returns fail to pass the Jarque-Bera normality test due to the feature of fat-tail. The time series plot of gold and crude oil prices are drawn in Figure 1. We can find that the oil and gold trends move in symmetry except for some temporary anomalies.

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Middle Eastern Finance and Economics - Issue 4 (2009) Table 1:

Descriptive statistics of daily return

Mean S.D. Max. Min. Skewness Excess Kurtosis J-B Crude oil 0.0757 2.4968 12.6991 -16.5514 -0.4475*** 2.6314*** 661.5028*** Gold 0.0351 1.0086 8.8707 -7.2397 0.2969*** 7.3238*** 4622.3741*** Notes: *** denotes significantly at the 1% level. S.D. is the standard deviation. J-B is the Jarque-Bera normality test statistics (1987).

THE PRICE LEVELS OVER 1998/4/29 -2006/7/20 80

700 650

70

Oil Price per Barrel

550

50

500

40

450 400

30

350 20

Gold Price per Ounce

600 60

300

10

250 1998

1999

2000

2001

2002

2003

2004

2005

2006

Date CRUDE OIL

GOLD

Empirical Results The empirical results of ARJI model for crude oil volatility

The volatility of crude oil is estimated using the ARJI model, and the result is listed in Table 2. First, the Ljung-Box Q statistics is not significant at the 1 % level and it indicates that the serial correlation does not exist in standard residuals, that is, the model construction is appropriate. Then, the GARCH component in ARJI model exhibits a strong persistence in the volatility parameter, as the α + β is approximately one. As to the jump component, the mean ( θ ) and variance ( δ 2 ) of the jump size are 1.1402 and 10.6274, respectively, and both are significant at the 1% level. The results imply that the jump behavior is significantly exhibited in crude oil return, which is the main reaction to the positive information, due to the significantly rising price trend since 2001. In the following, all coefficients ( λ 0 , ρ and γ ) in the jump intensity are significant at the 1% level and agree with the argument of timevarying feature in Bates (1991). Hence, we cannot ignore the jump in modeling the crude oil returns. The conditional variances combining the GARCH and jump components are plotted in Figure 2. Clearly, the GARCH variances describe the main body of the daily volatility with a relatively smooth behavior, while jumps capture unusual events of the volatility. We assume the GARCH variance as oil Var (ε1oil ) , the jump variance as Var (ε oil 2 ) , and the combined conditional variance as Var (ε ) to further bring them into the various models for analyzing gold returns.

56 Table 2: μ

Middle Eastern Finance and Economics - Issue 4 (2009) Empirical results of the ARJI model for crude oil GARCH component 0.2568***

ω

θ

Jump component 1.1402***

Q(6)

11.2542

Q (6) 5.6166 10.6274*** δ α λ0 0.0568*** 0.0134*** ρ β 0.9652*** 0.6572*** Log-likelihood -4703.0067 γ 0.4885*** Notes: *** denotes significantly at the 1% level. Q(6) and Q2(6) are the Ljung-Box Q statistics for the standardized residuals and squared standardized residuals with six lags, respectively. 2

2

0.0629***

Figure 2: Conditional Variance Components, Crude Oil (1998/4/29~2006/7/20) Conditional Variance Components, Crude oil 17.5

17.5

15.0

15.0

12.5

12.5

10.0

10.0

7.5

7.5

5.0

5.0

2.5

2.5

0.0

0.0 1998

1999

2000

2001

2002

2003

2004

2005

2006

Date GARCH component

Jump component

The empirical results of six different models for gold returns

The empirical results of all six models during the overall sample period are reported in Table 3. First, the effects of crude oil volatility on gold return and volatility are emphasized by observing the coefficients ϕ1 to ϕ 4 . Most of them are significantly negative and it means that the crude oil volatility is negative relative to either gold return or volatility. The higher the crude oil volatility, the lower the gold return and volatility. Omitting the GARCH-HV-A and PGARCH-HV-A models at this stage, we find that only the jump volatility of crude oil exhibits a negative relationship to the gold return, while the GARCH volatility of the crude oil does not. However, as to the gold volatility, both crude oil volatilities are significantly negative related to the gold volatility. It is worth noting that the absolute value of the coefficient ϕ 4 is two times higher than ϕ 3 in the GARCH-HV and PGARCH-HV models, rather than having almost the same value in the GARCH and PGARCH models. The influence of high jump shocks is much higher than the high GARCH volatility, and it points out the importance of an analysis on HV-type models. Turning back to the GARCH-HV-A and PGARCH-HV-A models, which did not distinguish the volatility into jump and GARCH component except for one, we find that the coefficients ϕ1 and ϕ 3 are also significantly negative and that the results are consistent with the other four models. Next, in terms of the variance coefficients c, a, and b, all of them are positive and significant at the 1% level. The γs are all significant at the 1% level. Notice that the gold volatility tends to depend on the level of the gold prices, that is, the level effects exist in the gold market. Among them, the level effects are higher in HV-type models than in non HV-type models, as well as in the PGARCH-type models than the GARCH-type models. Also, the κs are between 1.0458 to 1.0887. The following

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forecasting of VaR is though a rolling window approach by shortening the estimation window for determining the practical value of the PGARCH-type models further. Table 3: φ0 φ1 ϕ1 ϕ2 ϕ3 ϕ4 c a b κ γ L Notes:

Empirical results of various models GARCH 0.0211 -0.0167 0.0053 -0.0209*** -0.0059*** -0.0057 0.0824*** 0.0501*** 0.8360

GARCH-HV 0.0191 -0.0158 -0.0005 -0.0174** -0.0017*** -0.0036*** 0.0485*** 0.0462*** 0.8383***

GARCH-HV-A 0.0228 -0.0176 -0.0060*

PGARCH 0.0202 -0.0164 0.0055 -0.0204*** -0.0051*** -0.0048*** 0.0718*** 0.0372*** 0.8386*** 1.0645*** 0.0541***

PGARCH-HV PGARCH-HV-A 0.0189 0.0241 -0.0156 -0.0172 -0.0002 -0.0060* -0.0174** -0.0031*** -0.0015*** -0.0025*** -0.0031*** 0.0602*** 0.0440*** 0.0504*** 0.0518*** 0.0370*** 0.0345*** 0.8274*** 0.8408*** 0.8303*** 1.0458*** 1.0887*** 0.0406*** 0.0472*** 0.0386*** 0.0572*** 0.0574*** -2337.02 -2336.99 *, **, and *** denote significantly at the 10%, 5%, and 1% level. L stands for Log-likelihood value.

The measurements of out-of-sample VaR forecasting

The VaR estimations with 90%, 95%, 99%, and 99.5% confidence levels are reported in Table 4 and drawn in Figures 3 and 4. Taking a look at the volatility of VaR in the beginning, we find that the highest VaR variance appears in the HV-type models and the lowest is in the HV-A-type models. Moreover, the fluctuation of the forecasting VaR is heavier in the PGARCH-type models than in the GARCH-type models. For example, under the 95% VaR confidence level, the highest standard deviation of VaR is 0.7937 in the PGARCH-HV model, 0.7734 in the PGARCH model, and 0.7718, the lowest, in the PGARCH-HV-A model. All the standard deviations are higher than the GARCHtype model, 0.7156 in the GARCH-HV model, 0.6946 in the GARCH model, and 0.6836 in the GARCH-HV-A model. Turning into the column of LR UC , we find that all the statistics are not significant. This indicates that the estimations are statistically consistent with the VaR model’s prescribed confidence level. A further examination of the AQLF column shows that the lower the value, the higher the accuracy of forecasting VaR. Three interesting results are observed in this paper: (1) HV-type models are the best while the HV-A-type models are the worst model in the gold market. Taking the 95% VaR confidence level as an example, the lowest AQLF is 0.2171 in the PGARCH-HV model, 0.2218 in the PGARCH model, and 0.2280, the highest, in the PGARCH-HV-A model. The same ranking appears in GARCH-type models, as the lowest 0.2227 is in the GARCH-HV model, 0.2261 in the GARCH model, and the highest, 0.2286, is in the GARCH-HV-A model. In brief, the importance of jump volatility stands out in this paper. The performances of HV-A-type models through the merging of the GARCH and jump volatility are the poorest for offsetting the impact of jump volatility. Furthermore, the high volatilities of crude oil are substantial to gold returns. The HV-type models, which not only distinguish the crude oil volatility but also consider the high volatilities, perform best in this paper. That is to say, the findings of the relationships between gold and crude oil prices are consistent with Cai et. al (2001) and Melvin and Sultan (1990). However, this paper stresses the importance of the presence of high volatility and jump volatility in forecasting gold VaR. (2) PGARCH-type models outperform than the GARCH-type models. Similarly, taking the 95% VaR confidence level as an example, the AQLF in PGARCH model is 0.2218, which is lower than the 0.2261 in the GARCH model. The AQLF in the PGARCH-HV and PGARCH-HV-A models are 0.2171 and 0.2280, and both values are also lower than the 0.2227 and 0.2286 in the GARCH-HV and GARCH-HV-A models, respectively. The results reveal that the additional flexibility brought by

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the PGARCH model provides an improvement in forecasting. (3) However, under higher confidence levels (eq, 99% and 99.5% in this paper), the PGARCH models do not perform better than the GARCH-HV models. The result reveals that the high volatility of crude oil plays a key role with regard to gold returns. The same results are observed on the column of unexpected loss. Altogether, the PGARCH-HV model is considered the best model in VaR forecasting, that is, both the flexibility in power term for data transformation and the high volatility of crude oil are essential to out-of-sample VaR forecasting in gold returns. Table 4:

The out-of-sample forecasting of VaR

Part A. 90% VaR confidence level S.D. ABLF LRuc GARCH 0.5396 0.1000 0.0000 GARCH-HV 0.5565 0.0960 0.0449 GARCH-HV-A 0.5319 0.1000 0.0000 PGARCH 0.6007 0.0920 0.1821 PGARCH-HV 0.6171 0.0920 0.1821 PGARCH-HV-A 0.6004 0.0960 0.0449 Part B. 95% VaR confidence level Mean S.D. ABLF LRuc GARCH -2.0649 0.6946 0.0520 0.0207 GARCH-HV -2.0820 0.7156 0.0520 0.0207 GARCH-HV-A -2.0585 0.6836 0.0520 0.0207 PGARCH -2.1105 0.7734 0.0520 0.0207 PGARCH-HV -2.1273 0.7937 0.0520 0.0207 PGARCH-HV-A -2.1060 0.7718 0.0560 0.1826 Part C. 99% VaR confidence level Mean S.D. ABLF LRuc GARCH -2.9346 0.9858 0.0200 1.9568 GARCH-HV -2.9594 1.0145 0.0200 1.9568 GARCH-HV-A -2.9238 0.9681 0.0200 1.9568 PGARCH -2.9994 1.0975 0.0200 1.9568 PGARCH-HV -3.0236 1.1250 0.0200 1.9568 PGARCH-HV-A -2.9917 1.0934 0.0200 1.9568 Part D. 99.5% VaR confidence level Mean S.D. ABLF LRuc GARCH -3.2530 1.0924 0.0160 3.8357 GARCH-HV -3.2806 1.1239 0.0160 3.8357 GARCH-HV-A -3.2405 1.0724 0.0160 3.8357 PGARCH -3.3248 1.2161 0.0160 3.8357 PGARCH-HV -3.3517 1.2464 0.0160 3.8357 PGARCH-HV-A -3.3159 1.2112 0.0160 3.8357 Note: The critical value of the LRuc statistic at the 5% significance level is 3.8414.

Mean -1.6013 -1.6142 -1.5973 -1.6366 -1.6495 -1.6338

AQLF 0.3529 0.3450 0.3556 0.3386 0.3334 0.3448

Unexpected loss -0.0961 -0.0949 -0.0972 -0.0933 -0.0920 -0.0942

AQLF 0.2261 0.2227 0.2286 0.2218 0.2171 0.2280

Unexpected loss -0.0621 -0.0609 -0.0632 -0.0594 -0.0579 -0.0603

AQLF 0.1094 0.1067 0.1116 0.1068 0.1025 0.1088

Unexpected loss -0.0344 -0.0341 -0.0349 -0.0338 -0.0331 -0.0343

AQLF 0.0837 0.0810 0.0858 0.0813 0.0772 0.0833

Unexpected loss -0.0287 -0.0284 -0.0291 -0.0280 -0.0272 -0.0284

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Figure 3: Returns and VaR forecasts at different confidence levels with a sequence of GARCH, GARCH-HV, GARCH-HV-A, PGARCH, PGARCH-HV, and PGARCH-HV-A model.

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Figure 4: Returns and VaR forecasts under each model at a sequence of 90%, 95%, 99%, and 99.5%confidence levels.

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Conclusion This paper investigates the value-at-risk in gold markets by considering both oil volatilities and the flexible model construction. The oil volatility is estimated using the dynamic jump model, and the volatility is distinguished further into stochastic and jump volatility. The flexible models include the BHK and PGARCH models. The former is considered the effect of spot prices and the latter is assumed as an endogenized power term rather than the fixed square term in the GARCH model. Finally, by combining the data with the rolling window approach, the appropriate out-of-sample VaR estimates are clearly obtained in this paper. The empirical results indicate that the importance of oil jump volatility stands out in this paper. The performances of HV-A-type models merged with the GARCH and jump volatility are the poorest and are offset by the impacts of jump volatility. The HVtype models, which distinguish both the crude oil volatility and focus on the high volatilities, perform best in this paper. That is to say, the high volatility and jump volatility cannot be absent in forecasting gold VaR. In the following, the PGARCH-type models outperform than the GARCH-type models. However, under a higher confidence level, the PGARCH models do not perform better than the GARCH-HV models. The result reveals that the high volatility of crude oil plays a key role with regard to gold returns. In summary, the PGARCH-HV model is considered the best model for VaR forecasting, that is, both the flexibility in power term for data transformation and the high volatility of crude oil are essential to out-of-sample VaR forecasting in gold returns. Therefore, for any participants in the market, the PGARCH-HV model is the most appropriate model to use in measuring and forecasting the value-at-risk. Conservative investors may prefer to choose the PGARCH-HV model to measure and forecast their investment risk. The futures exchanges can apply the PGARCH-HV model to compute the optimal margin levels. These findings have implications for investors, financial institutions, and futures exchanges.


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