An Extreme Value Theory Approach for Analyzing the Extreme Risk of the Gold Prices

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1 A Extreme Value Theory Approach for Aalyzig the Extreme Risk of the Gold Prices Jiah-Bag Jag Abstract Fiacial markets frequetly experiece extreme movemets i the egative side. Accurate computatio of value at risk ad expected shortfall are the mai tasks of the risk maagers or portfolio maagers. I this paper, gold prices (i US dollars) have bee examied to illustrate the mai idea of extreme value theory ad discuss the tail behaviour. GEV ad GPD are used to compute VaR ad ES. The results show that GPD model with threshold is a better choice. Keywords: value at risk, expected shortfall, geeralized extreme value distributio, geeralized Pareto distributio

2 Itroductio Risk maagers ad Portfolio maagers cocer extreme egative side movemets i the fiacial markets. A log list of research has posted o this topic. Ramaza ad Faruk (2006) examie the dyamics of extreme values of overight borrowig rates i a iter-bak moey market. Geeralized Pareto distributio has bee picked for its well fittig. Feradez (2005) usig extreme value theory to the Uited States, Europe, Asia, ad Lati America fiacial markets for computig value at risk (VaR). Oe of his fidigs is, o average, EVT provides the most accurate estimate of VaR. Byström (2005) applied extreme value theory to the case of extremely large electricity price chages ad declared a good fit with geeralized Pareto distributio (GPD). Bali (2003) determies the type of asymptotic distributio for the extreme chages i U.S. Treasury yields. I his paper, the thi-tailed Gumbel ad expoetial distributio are worse tha the fat-tailed Fréchet ad Pareto distributios. Neftci (2000) foud that the extreme distributio theory fit well for the extreme evets i fiacial markets. Geçay ad Selçuk (2004) ivestigate the extreme value theory to geerate VaR estimates ad study the tail forecasts of daily returs for stress testig. Maroh (2005) studies the tail idex i the case of geeralized order statistics, ad declares the asymptotic properties of the Fréchet distributio. Brooks, Clare, Dalle Molle ad Persad, G., (2005) apply a umber of differet extreme value models for computig the value at risk of three LIFFE futures cotracts. Based upo the log list of applicatios of extreme value theory ad theory basis, this paper will first discuss the geeralized extreme value methods ad the fit the model with gold prices (i US dollars) to compute the chace of ew record geerated i the ext period. Differet models are also bee fitted with gold data. The results show that GPD model with threshold is a better choice. 2. Methodology 2.1 Geeralized Extreme Value (GEV) Distributio Extreme movemets i gold prices ca be measured by the daily returs of the gold prices per troy ouce (reported o Lodo Gold Market). Losses or egative returs are the mai cocers i the field of fiacial risk maagemet, for examples stock market crashes. For simplicity, extreme movemets i the left tail of the distributio ca be characterized by the positive umbers of the right high quitiles. Let Xi of the ith daily retur of the gold prices betwee day i ad day i 1. Defie where Pi ad i 1 X (l P l P ) = i i i 1 be the egative P are the gold prices of day i ad day i 1. Suppose that X1, X 2,..., X be iid radom variables with a ukow cumulative distributio fuctio (CDF) F( x) = Pr( X x). Extreme values are defied as maxima (or miimum) of the i idepedetly ad idetically distributed radom variables X1, X 2,..., X. Let X ( ) the maximum egative side movemets i the daily gold prices returs, that is X X X X = ( ) max( 1, 2,..., ). Sice the extreme movemets are the focus of this study, the exact distributio of X ( ) ca be writte as be

3 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 99 Pr( X a) Pr( X a, X a,..., X a) = ( ) 1 2 I practice the paret distributio F = i = 1 F ( a) = F ( a ) (1) is usually ukow or ot precisely kow. The empirical estimatio of the distributio F ( a) is poor i this case. Fisher ad Tippet (1928) derived the asymptotic distributio of F ( a ). Suppose µ ( ) ad σ ( ) sequeces of real umber locatio ad scale measures of the maximum statistic X ( ). The the stadardized maximum statistic are Z ( X µ ) / σ * ( ) = ( ) ( ) ( ) Coverges to z = ( x µ ) / σ which has oe of three forms of o-degeerate distributio families such as H ( z ) e x p { e x p [ z ] }, z = < < H z z z ( ) 1 / ξ = e x p { }, > 0 = 0, e l s e H z z z ( ) 1 / ξ = e x p { [ ] }, < 0 = 1, e l s e (2) These forms go uder the ames of Gumbel, Fréchet, ad Weibull respectively. Here µ ad σ are the mea retur ad volatility of the extreme values x ad ξ is the shape parameter or called 1/ ξ the tail idex of the extreme statistic distributio. With ξ = 0, ξ > 0, ξ < 0 represet Gumbel, Fréchet, ad Weibull types of tail behavior respectively. ad short tails respectively. I fact Gumbel, Fréchet, ad Weibull types ca be fit for expoetial, log, Embrechts ad Mikosch (1997) proposed a geeralized extreme value (GEV) distributio which icluded those three types ad ca be used for the case statioary GARCH processes. GEV distributio has the followig form H ( x; µ, σ ) = exp{ exp[ ( x µ ) / σ ]}, x ; ξ = 0 ξ 2.2 Parameters Estimatio for GEV 1/ ξ = exp{ [1 + ξ ( µ ) / σ ], 1 + ξ ( µ ) / σ > 0; ξ 0 x Suppose that block maxima B1, B2,..., Bk are idepedet variables from a GEV distributio, the log-likelihood fuctio for the GEV, uder the case of ξ 0, ca be give x (3)

4 as l L k l (1 1 / ) l {1 ( B ) / } = σ + ξ + ξ µ σ i i = 1 k {1 ( B ) / } + ξ µ σ i i = 1 For the Gumbel type GEV form, the log-likelihood fuctio ca be writte as k = σ i µ σ i µ σ i 1 i 1 = = k l L k l ( B ) / e x p { ( B ) / } (5) As Smith (1985) declared that, for ξ > 0.5, the maximum likelihood estimators, for ξ, µ, ad σ, satisfy the regular coditios ad therefore havig asymptotic ad cosistet properties. The umber of blocks, k ad the block size form a crucial tradeoff betwee variace ad bias of parameters estimatio. 1 / k ξ (4) POT ad GPD Fittig models with more data is better tha less, so peaks over thresholds (POT) method utilizes data over a specified threshold. Defie the excess distributio as F ( x ) = P r ( X h < x X > h ) h = F ( x + h ) F ( h ) 1 F ( h ) where h is the threshold ad F is a ukow distributio such that the CDF of the maxima will coverge to a GEV type distributio. (6) For large value of threshold h, there exists a fuctio τ ( h) > 0 such that the excess distributio of equatio (6) will approximated by the geeralized Pareto distributio (GPD) with the followig form H ( x ) = 1 e x p ( x / τ ( h )), ξ = 0 ξ, τ ( h ) 1 / ξ = 1 (1 + ξ / τ ( )), ξ 0 where x > 0 for the case of ξ 0, ad 0 x τ ( h) / ξ for the case of ξ < 0. Defie x1, x2,..., xk as the extreme values which are positive values after subtractig threshold h. For large value of h, x1, x2,..., xk is a radom sample from a GPD, so the ukow parameters ξ ad τ ( h) log-likelihood fuctio. VaR ad ES derived as x ca be estimated with maximum likelihood estimatio o GPD Based o equatio (6) ad GPD distributio, the ukow distributio F F( y) (1 F( h)) H ( x) F( h) = ξ, τ ( h) + (8) h (7) ca be

5 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 101 where y = h + x. F( h) ca be estimated with o-parametric empirical estimator F( h) = ( k) / where k is the umber of extreme values exceed the threshold h. Therefore the estimator of (8) is where ξ ad τ ( h) F( y) (1 F( h)) H ( x; ξ, τ ( h)) F( h) are mle of GPD log-likelihood. shortfall ca be computed usig (9). First, defie F( VaRq ) distributio fuctio up to q th quatile VaR q. Next, give that as From (10), GPD distributio. VaRq = + (9) 1 VaR q = F ( q) Therefore High quatile VaR ad expected = q as the probability of ξ = h + τ ( h){[( / k)(1 q)] 1}/ ξ (10) is exceeded, defie the expected loss size, expected shortfall (ES), ES = E( X X > VaR ) ES q ca be computed usig VaR q Therefore, 3. Data ad Empirical Results 3.1 Data ad Data Exploratio q q VaR E( X VaR X VaR ) = + > q q q ES q = VaRq/(1 ξ ) + ( τ ( h) ξ h) /(1 ξ ) (11) ad the estimated mea excess fuctio of The upper pael of Figure 1 shows the daily gold prices per troy ouce (i US dollars) over the period of Jauary 1, 1985 through March 31, The lower pael of Figure 1 is the cotiuous percetage returs of the gold prices i the upper pael. Table 1 shows that daily gold returs have a positive skew (0.76>0) ad large kurtosis (14.16>3). It shows the distributio of returs has a fat tail.

6 Figure 1 Daily Gold Prices Per Troy Ouce (i US Dollars) ad Daily Gold Percetage Returs Daily Gold prices Daily Gold Percetage Chage Table 1 Descriptive Statistics of Daily Gold Percetage Returs Miimum st Quarter Mea Media 0 3rd Quarter 0.43 Maximum 9.64 Total N 5371 Std Deviatio 0.90 Skewess 0.76 Kurtosis Figure 2 shows that there is o autocorrelatio i the daily gold percetage retur series but the squared daily gold percetage returs appear sigificat autocorrelatio. Figure 2 suggest that the daily gold percetage returs ca be a GARCH time series model. It is reasoable to apply GEV distributio to model the maxima egative returs ad GPD distributio to model the excess distributio for egative daily gold percetage returs.

7 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 103 Figure2 ACF of Daily Gold Percetage Returs ad Squared Daily Gold Percetage Returs Series : gold.r ACF Lag Series : gold.r^2 ACF Parametric Maximum Likelihood Estimates Lag The block maxima of egative returs have bee fitted to GEV model with three differet block sizes (yearly, quarterly, ad mothly). The results, listed i Table 2, show the estimates of three parameters ξ, σ, ad µ with icreasig block umbers ad the estimates of stadard error of the parameters are listed i paretheses. All block umbers show that shape parameter is positive except yearly block size case is egative but ot sigificat. All the parameters have decreasig estimated stadard errors as the umber of blocks icreased. I geeral, the block maxima of the egative returs follows a Fréchet family of GEV for quarter ad moth block frames. The Fréchet type of GEV cofirms that the origial series has fat tail. Table 2 Parametric Maximum Likelihood Estimates with three block frames (block sizes: year, quarter, ad moth) ad chace of a ew record will occur durig the ext period Year Quarter Moth ξ (0.19) (0.10) (0.06) σ 0.97 (0.18) 0.75 (0.07) 0.59 (0.03) µ (0.24) (0.09) (0.04) New record 1.77% 0.86% 0.44%

8 Figure 3, from left to right is the crude residual plot ad quatile-versus-quatile plot with expoetial distributio as the referece distributio for quarter block maxima. The Q-Q plot looks liear ad shows that the GEV distributio model is a good choice. It shows that the Fréchet distributios are fitted well for the block maxima egative returs. Figure 3 Residual plots versus quarter frame block maxima Residuals Expoetial Quatiles Orderig Ordered Data With the above parameters ad correspodig Fréchet distributios, there is 1.77%, 0.86% ad 0.44% chace that a ew record will be occurred durig the ext period (year, quarter ad moth). The poit estimates ad 95% cofidece iterval of 10 ad 20 years retur levels are listed below. Give the same time iterval (10 or 20 years), the poit estimate is icreased with decreasig block sizes. With the icreasig time iterval, the poit estimate icreased ad cofidece iterval is wider. It reveals that time is a risk factor. Table 3 Retur levels for the three block frames with 95% cofidece bouds 10 year retur level 20 year retur level Year Quarter Moth (4.31, 6.91) (4.28, 7.17) (4.57, 7.29) 5.58 (4.75, 9.13) 5.96 (4.77, 9.08) 6.53 (5.18, 9.11)

9 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices Extreme Over Thresholds ad Risk Measures A suitable threshold should be specified to fid the GPD approximatio of the egative returs excess distributio. Try several thresholds to fit the excess GPD. Do QQ-plots with correspodig GPD distributios for the daily egative returs over the thresholds (show o Figure 4) ad compare some sigificat umbers (Table 4). Figure 4 Residual diagostic check for GPD ( with threshold 2) fit to Daily Gold egative returs Fu(x-u) F(x) (o log scale) x (o log scale) x (o log scale) Residuals Expoetial Quatiles Orderig Ordered Data Figure 4 show that the threshold 2 model fit the GPD well. O the top-left pael of Figure 4 is the GPD estimate of the excess distributio, top-right pael is the tail estimate of equatio (9), bottom-left pael is the scatter plot of residuals, ad bottom-right pael is the QQ-plot of residuals. Base o the diagostic plots show data fit GPD model well. Table 4 shows that the shape parameter is stable icreasig as threshold icreasig from 1.5 to 2, ad the shape parameter swigs to egative value. The details are displayed as Figure 5. Theoretically, shape parameter of GEV ad GPD should be the same. The shape parameters of threshold 2 ad the moth block size GEV are similar (see table 2). Accordig to the referred iformatio, the estimate value 2 for shape parameter is the best specified threshold. The GPD approximatio has shape parameter 0.15 ad scale parameter 0.56.

10 Table 4 GPD model fittig results with thresholds from 1 to 6 threshold # of exceed Probability less tha threshold ξ τ ( h) (0.05) 0.62 (0.04) (0.06) 0.66 (0.05) (0.07) 0.68 (0.06) (0.09) 0.60 (0.07) (0.12) 0.56 (0.09) (0.14) 0.72 (0.14) (0.18) 0.67 (0.16) Figure 5 Estimates of shape parameter for egative returs as a fuctio of the threshold values. Threshold Shape (xi) (CI, p = 0.95) Exceedaces To measure the risk, Value-at-Risk ad expected shortfall (ES) ca be calculated based o GPD approximatio with threshold 2. The results is listed i row 1 of Table 5. For compariso, the values of the GPD approximatio with threshold 2 ad ormal distributio approximatios are listed i the followig table. All the values of Normal Distributio approximatio is uder estimate. GPD model with threshold 2 is a better chose.

11 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 107 Table 5 Risk measures (VaR ad ES) for GPD model with thresholds 2 compared with ormal distributio VaR. 95 ES. 95 VaR. 99 ES. 99 Threshold= Normal Distr Cocludig Remarks Fisher ad Tippet theorem has a assumptio of iid, however, the GEV is suitable for statioary time series icludig statioary GARCH cases. Explorig the depedece properties o extreme values ad the joit-tail properties of multivariate extreme value cases are appeared ot log ago. Further ivestigatio is defiite welcome. Refereces Bali, T. G., (2003), A extreme value approach to estimatig volatility ad value at risk, Joural of Busiess, 76: Brooks, C., Clare, A. D., Dalle Molle, J.W., ad Persad, G., (2005), A compariso of extreme value theory approaches for determiig value at risk, Joural of Empirical Fiace, vol. 12, Byström, Has N.E., (2005), Extreme value theory ad extremely large electricity price chages, Iteratioal Review of Ecoomics & Fiace, 14:1, Chou, Heg-Chih ad Che, She-Yua, (2004), Price limits ad the applicatio of extreme value theory i margi settig, Joural of Risk Maagemet, 6:2, Coles, S., (2001), A itroductio to statistical modelig of extreme values, Lodo: Spriger. Embrechts, P., Klüppelberg, C, ad Mikosch, T., (1997),Modelig extremal evets for isurace ad fiace, Berli ad Heidelberg: Spriger. Feradez, V.,(2005), Risk Maagemet uder extreme evets, Iteratioal Review of Fiacial Aalysis, 14:2, Fisher, R. ad Tippett, L., (1928), Limitig forms of the frequecy distributio of the largest or smallest member of a sample, Proceegigs of the Cambridge Philosophical Society, 24,

12 Geçay, R. ad Selçuk, F., (2004), Extreme value theory ad value-at-risk: relative performace i emergig markets, Iteratioal Joural of Forecastig, 20:2, Jekiso, A. F., (1955), The frequecy distributio of the aual maximum (or miimum) values of meteorological elemets, Quarterly Joural of the Royal Meteorological Society, 87, Maroh, F., (2005), Tail idex estimatio i models of geeralized order statistics, Commuicatios i Statistics: Theory & Methods, 34:5, Neftci, S. N., (2000), Value at risk calculatios, extreme evets, ad tail Estimatio, Joural of Derivatives, 7:3, Ramaza, G. ad Faruk, S., (2006), Overight borrowig, iterest rates ad extreme value theory, Europea Ecoomic Review, Vol.50, Smith, R. L., (1985), Maximum likelihood estimatio i a class of oregular cases, Biometrika, 72,

13 A Extreme Value Theory Approach for Aalyzig The Extreme Risk of the Gold Prices 109

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.