Computers and Mathematics with Applications. Analytical VaR for international portfolios with common jumps

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1 Computers a Mathematics with Applicatios 6 (011) Cotets lists available at SciVerse ScieceDirect Computers a Mathematics with Applicatios joural homepage: Aalytical VaR for iteratioal portfolios with commo jumps Fe-Yig Che Departmet of Fiace, Shih Hsi Uiversity, #111, Sec. 1, Mu-Cha Rd., Taipei 116, Taiwa a r t i c l e i f o a b s t r a c t Article history: Received 10 March 010 Received i revised form 6 July 011 Accepted 4 August 011 Keywords: Iteratioal portfolios Exchage rate risk Jump diffusio Backtestig Out-of-sample fittig Iteratioal portfolios which are composed of domestic assets a foreig assets are popular ivestmet tools for fiacial istitutios i highly itegrated global fiacial markets. However, the focus of past studies had bee o either domestic assets or foreig assets, but ot both i the same cotext. They paid o attetio to the studies of cotrollig the market risk of the iteratioal portfolios i the risk maagemet literature. I cotrast to the existig literature i portfolios, this paper cosiders ot oly domestic assets but also foreig assets, a provides a aalytical value-at-risk (VaR) with commo jump risk a exchage rate risk to maage market risk of iteratioal portfolios with exchage rate risk a commo jumps over the subprime mortgage crisis. I geeral, the aalytical solutio ca be used to accurately calculate VaRs by the backtestig criterio i terms of i-sample a out-of-sample fittig for a iteratioal portfolio with commo jumps. 011 Elsevier Ltd. All rights reserved. 1. Itroductio Nowadays, the ivestmet i foreig currecy assets circulates rapidly arou the world. I Taiwa, the official mothly statistic reports offered by the Cetral Bak of Taiwa illustrate that the average percetage of ivestmet i foreig assets relative to domestic assets has bee approximately 46% at domestic commercial baks over the past te years. I Japa, the ratio is at least 5%, a i Korea it is arou 9%. O average, the percetage of overall portfolio allocatio to foreig assets is arou 0% at Asia baks, a the percetage is growig. Thus, cotrollig the market risk of portfolios composed of domestic assets a foreig assets is a icreasig cocer for fiacial istitutios. The VaR approach is a popular tool to maage market risk, which is defied as the maximum loss over a fixed target horizo with a give probability. Usig the VaR measure, Hofma a Plate [1] cosider the market risk of a large diversified portfolio i which the dyamic process of asset returs is distributed i ormal diffusio. Equally, the asset price follows a logormal distributio. However, substatial evidece exists i the empirical fiacial ecoomic literature of the existece of jumps i equity returs a foreig exchage rates such as [ 4]. Therefore, the logormal assumptio is, i actuality, cotrary to real life. Daily chages i may variables, especially i exchage rates, illustrate sigificat positive kurtosis. This meas that the probability distributios of asset returs have fat tails or discotiuity. Literature related to these studies has bee preseted by Stock a Watso [5], Hull a White [6], Hase [7], a Cosigli [8]. Besides them, Shag et al. [9] employ a jump diffusio process to price catastrophe mortality bos; Liu et al. [10] cosider a class of stochastic optimal parameter selectio problems described by liear stochastic differetial equatios with jumps to show that the costraied stochastic impulsive optimal parameter selectio problem is equivalet to a determiistic impulsive optimal parameter; Ma a Zhao [11] a Ti et al. [1] apply a jump diffusio process to a simulatio aalysis of earesteighbour rule uer stochastic dema a a web reliability rakig system. Alteratively, Gibso [13] demostrates that evet risk poses large jumps to fat tails i market prices, a icorporates evet risk ito VaR for a portfolio. Differig from Tel.: x63438; fax: address: fyiche@cc.shu.edu.tw /$ see frot matter 011 Elsevier Ltd. All rights reserved. doi: /j.camwa

2 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) the assumptio held by Hofma a Plate [1] a Gua et al. [14], Gibso cosiders jump diffusio asset returs to model large diversified portfolios. As stated above, the literature focuses o the portfolios oly valued i oe currecy. However, it is a commo pheomeo for istitutioal a iividual ivestors to ivest i the portfolios which iclude a umber of domestic-valued assets a foreig-valued assets i highly itegrated global fiacial markets, called iteratioal portfolios. Therefore, exchage rate risk should be cosidered i highly iteratioal ivestmet. This paper aims to preset a aalytical VaR formula for iteratioal portfolios. Usig the framework provided by Merto [15], we employ retur jumps at Poisso arrivals to avoid the assumptio of ormality of asset returs. Also, the Browia motios of betwee-jump returs are correlated. I geeral, the model solutio is more accurate tha that of the Mote Carlo simulatio techiques which are ofte adopted i fat-tail distributios i terms of the system ifrastructure a computatio time. I additio, this model ca be also applied to large portfolios. Compared with that of Hofma a Plate [1] a Gua et al. [14], the proposed model cosiders ot oly jumps but also exchage rate risk. It is more suitable to fit to real situatios i highly itegrated global fiacial markets. The rest of this paper is orgaized as follows. The ext sectio outlies the model, a a aalytic formula of the value at risk is derived. I the Sectio 3, we first employ a iteratioal portfolio icludig domestic assets a foreig assets to estimate model parameters. The, the oe-day VaRs at 99% sigificace level for the iteratioal portfolio are calculated, a a comparative static aalysis o the risk capital is provided. Usig the usual backtestig criterio, Sectio 4 ispects the model accuracy i terms of i-sample a out-of-sample fittig over the subprime mortgage crisis of August 007. The samples i this study spa from Jauary 1, 004 to November 7, 009, or 1367 daily log returs of a lie of domestic assets a foreig assets. The last sectio provides coclusios.. Model formulatio First, this paper assumes that (i) a value of a iteratioal portfolio is made up of the value of d kis of domestic assets with m i,t shares a f classes of foreig assets with g i,t shares for each i 1,,..., ; (ii) the capital market is a complete market with o trasactio cost or tax; (iii) there exists a riskless iterest rate for leers a borrowers; (iv) the dyamic processes of domestic asset returs, foreig asset returs a exchage rate returs follow Poisso jump diffusio over the iterval of iterest; (v) exchage rates are quoted at the price of oe uit of the foreig currecies i domestic dollars, a (vi) ivestmet strategies do ot vary over a ivestmet horizo. The dyamic processes of asset price a exchage rates are demostrated as follows, respectively. da di,t A di,t da fi,t A fi,t (µ di λv)dt + σ di dw 1,t + (π 1)dY t, (1) (µ fi λv)dt + σ fi dw,t + (π 1)dY t, () de i,t e i,t (µ ei λv)dt + σ ei dw 3,t + (π 1)dY t, (3) where µ di, µ fi, a µ ei deote the costat drift rates of domestic asset returs, foreig asset returs a exchage rate returs for each i 1,,...,, respectively; σ di, σ fi, a σ ei,t sta for the costat volatilities of domestic asset returs, foreig asset returs a exchage rate returs for each i 1,,...,, respectively. The W j,t are oe dimesioal staard Browia motios uer the origial probability measure, P for all j 1,, 3. Also, the correlatio coefficiets amog the three Browia motios are defied as corr(dw 1,t, dw,t ) ρ 1,, corr(dw,t, dw 3,t ) ρ,3, a corr(dw 1,t, dw 3,t ) ρ 1,3. 1 The, Y t is a iepeet Poisso process with the itesity λ at time t; dy t is iepeet of dw j,t for all j 1,, 3. The v represets E[π 1] where π 1 is the raom variable percetage i domestic assets or exchage rates resultig from a jump, a E(.) is the symbol of the expectatio operator over the raom variable Y t. Assume that the ature logarithm of π, which is the jump amplitude if Poisso evets occur, follows ormal distributios with the mea u π a variace σ π. That is also deoted as l π N(u π, σ π ), a v E[π 1] exp u π + 1 σ π 1. Now, cosider the potetial daily loss exposure to log tradig positios. Typically, the VaR is a specific left-ha critical value of a potetial loss distributio. Give covetios, oe ca defie the daily losses valued i domestic dollars relative to the e-of-period expected asset value (relative VaR) a the iitial asset value (absolute VaR), deoted by VaR(mea) a VaR(0) as follows, respectively: VaR(mea) V α E t (V T ), VaR(0) V α V 0, (4) 1 For simplicity, we assume that the depeece structure betwee exchage rates a equity returs is liear. However, there are some drawbacks. First, it is ot ivariat to trasformatios of the origial variables. Seco, coitioal correlatios are ot accouted for. Third, the proposed method caot be used i the case of portfolios that iclude assets with o-liear payoffs.

3 3068 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) i which the E t (.) is the expected value coitioal o iformatio at time t, the V α is the value of a iteratioal portfolio deomiated i domestic dollars give a percetile of α, a V T is the portfolio value at time T (ivestmet horizo). The iteratioal portfolio cosists of d kis of domestic assets a f kis of foreig assets, deoted as V T d m ia di,t + f g ie i,t A fi,t. Which defiitio of value at risk provides a more suitable measure of risk capital allocatio over ivestmet horizo? Kupiec [16, page 43] demostrates that the absolute VaR is a more appropriate measure of a asset s risk of postig losses. Thus, we adopt the measure throughout this article. Before the derivatio of the VaR aalytic formula for a iteratioal portfolio, it is ecessary to employ the followig propositios. Propositio 1. Give the dyamic processes of foreig currecy deomiated asset price a exchage rate followig the Geometric Browia motio, the dyamic process of f g ie i,t A fi,t ca be expressed as dx i,t X i,t f with X i,t f g ie i,t A fi,t. g i (µfi + µ ei λv + ρ,3 σ fi σ ei )dt + σ fi dw,t + σ ei dw 3,t + (π 1)dY t Appeix A provides a detailed proof of Propositio 1. Propositio. Give the dyamic processes of asset price a exchage rates, the dyamic process of ca be expressed as d f γ i,t (µ di λv) + β i,t (µ ei + µ fi λv + ρ,3 σ fi σ ei ) dt d + γ i,t σ di dw 1,t + f β i,t σ fi dw,t + f β i,t σ ei dw 3,t + (π 1)dY t with d m ia di,t + f g ie i,t A fi,t, γ i,t m ia di,t, a β i,t g ie i,t A fi,t. γ i,t a β i,t are also amed the weights (percetage) of the ivestmet i the ith ki of domestic asset a i the ith ki of foreig asset, respectively. Appeix B provides a detailed proof of Propositio. Coitioal o assumptio (vi), the weights ca be regarded as costat over the ivestmet horizo. Usig the previous propositios, oe ca quickly obtai the approximatio of the absolute VaR by utilizig the aalytic formula below: exp [λt ] [λt ] k Φ l [V 0 + VaR(0)] l V 0 µ t 1 σ t T kuξ α, (5) k! k0 σt T + kσ ξ i which the Z α stas for a critical value with a give probability α; V 0 represets the iitial value of a iteratioal portfolio; v E[π 1] exp u π + 1 σ π 1; uξ a σξ deote the expected value a the variace of the ature logarithm of π 1, respectively; d f µ t γ i,t (µ di λv) + β i,t (µ ei + µ fi λv + ρ,3 σ fi σ ei ); σ t f f f γ i,t σ di + β i,t σ fi + β i,t σ ei + ρ 1, γ i,t σ di β i,t σ fi f f f + ρ,3 β i,t σ fi β i,t σ ei + ρ 1,3 γ i,t σ di β i,t σ ei ; u ξ l e u π + 1 σ π 1 1 l 1 + 4eu π +σ π (e σ π 1) (e u π + 1 σ π 1) ; σ ξ l 1 + 4eu π +σ π (e σ π 1) (e u π + 1 σ π 1) Eq. (5) is derived i Appeix C. The derivatio of u ξ a σξ is show i Appeix D. By meas of Eq. (5), oe ca efficietly obtai the approximatio of the VaR capital allocatio for a iteratioal portfolio. The approximated aalytical VaR icludes some essetial elemets such as the volatility of uerlyig assets, the volatility of exchage rates, the correlatio coefficiets, the weights of the ivestmet i domestic assets a foreig assets,.

4 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Table 1 Summary statistics (Jauary 1, 004 November 7, 009). Statistics TSMC MSFT Exchage rate Mea Staard deviatio Skewess Kurtosis a the itesity of jumps. Also, Eq. (5) ca be reduced to the aalytic solutio of [16] as γ i,t 1, β i,t 0, d 1, λ 0, a dy t 0. This case represets that a firm value is oly composed of a ki of domestic asset with o jumps. Alteratively, Eq. (5) goes to the closed-form solutio of [17] 3 as γ i,t 0, β i,t 1, f 1, λ 0, a dy t 0, which meas that a firm value icludes oly a ki of foreig asset with o jumps. Also, the preseted model ca be regarded as the extesio of that of [16,17]. 3. Measuremet of value at risk a umerical aalysis For simplicity, this sectio cosiders the log tradig positios of a iteratioal portfolio with a ki of domestic asset a a ki of foreig asset. From the Taiwaese perspective, the iteratioal portfolio icludes oe domestic-issued stock valued i New Taiwa dollars a oe foreig-issued stock valued i US dollars. We the wat to kow the absolute VaR of the portfolio valued i New Taiwa dollars Source of the data Assume that the iteratioal portfolio icludes two specific domestic a foreig stocks which are TSMC a MSFT, respectively. The TSMC stocks, are issued by Taiwa Semicouctor Maufacturig Compay Limited a traded i Taiwa; the MSFT stocks are issued by Microsoft a traded i the USA. The daily log returs of TSMC a MSFT stocks are employed. Both of these securities are well-kow to istitutioal a iividual ivestors i the world. The time wiow legth is the period from Jauary 1, 004 to November 7, 009, so that the total of the daily log returs of each asset is All of the samples spa two periods, labelled Periods I a II. Period I is from Jauary 1, 004 to July 31, 007, durig which the subprime mortgage crisis had ot yet occurred a the daily log returs totalled 780. Alteratively, Period II is from August 1, 007 to November 7, 009 with a total of 587, which is through the subprime mortgage crisis of August 007. Table 1 provides some basic statistics o the daily log returs of TSMC, MSFT stocks a exchage rates quoted at the price of oe uit of US dollars i New Taiwa dollars from Jauary 1, 004 to November 7, 009. Obviously, the distributios of these stock returs a exchage rate returs have heavy tails. The log returs of TSMC a MSFT are egatively skewed. The volatility of TSMC returs is fewer tha that of MSFT returs. The volatility of exchage rate returs is the smallest. 3.. Estimatio of model parameters Before the VaR measuremet, it is ecessary to estimate a set of model parameters for various samples. Assume that the umber of jumps is te; u π 0.05, σπ 0.001, a λ 0.03 for Period I, a u π 0.055, σπ 0.00, a λ for Period II. From the data, the sample meas a staard deviatios of TSMC, MSFT a exchage rates i oe day are show i Table. Sice oe tradig day is equivalet to 1/5 year, oe ca obtai the sample meas a variace of these raom variables per aum, which are all multiplied by 5 from Pael A i Table, respectively. The results are stated i Pael B i Table. From Eqs. (1) (3), the dyamic processes of log returs of raom variables i domestic assets, foreig assets a exchage rates ca be derived as Eq. (6), respectively. d(l A d1,t) µ d1 1 σ d 1 λv dt + σ d1 dw 1,t + (π 1)dY t, d(l A f1,t) µ f1 1 σ f 1 λv dt + σ d1 dw,t + (π 1)dY t, d(l e 1,t ) µ e1 1 σ e 1 λv dt + σ e1 dw 3,t + (π 1)dY t. (6) ] Kupiec [16] shows the absolute VaR as follows: VaRk (0) A di,t 0 [exp µ di T 1 σ di + Z α σ d i T 1. [ 3 Che a Liao [17] derives the absolute VaR of foreig-issued assets as below: VaRc (0) A fi,t 0 e i,t0 exp µ fi + µ ei 1 σ f i 1 σ e i T + ] Z α (σ f i + σe i + ρ,3 σ fi σ ei )T 1.

5 3070 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Table Sample mea a staard deviatio of daily log returs of securities a exchage rates i various periods. Variables Period I Period II 004/1/1 007/7/31 007/8/1 009/11/7 E [d(l H t )] σ (d l H t ) E [d(l H t )] σ (d l H t ) Pael A: sample mea a staard deviatio of daily log returs TSMC MSFT NTD/USD Pael B: sample mea a variace of log returs per aum Variables E [d(l H t )] Variace E [d(l H t )] Variace TSMC MSFT NTD/USD Note that E [d(l H t )] a σ (d l H t ) represet the sample meas a staard deviatios of daily log returs of domestic assets, foreig assets, a exchage rates for all H t A d1,t, A f1,t, e 1,t, respectively. Pael B displays the sample meas a variaces of domestic assets, foreig assets, a exchage rates per aum all multiplied by 5 from Pael A, respectively. Table 3 Parameter estimatio of dyamic processes of asset returs a exchage rate returs i various periods. Period I Period II 004/1/1 007/7/31 007/8/1 009/11/7 Security a exchage rate µ i σ i µ i σ i TSMC MSFT NTD/USD Note that assume the umber of jumps is te. Give u π 0.05, σπ 0.001, a λ 0.03 for Period I, a u π 0.055, σπ 0.00, a λ for Period II, µ i a σ i demostrate the estimatio of drift terms a volatilities of asset returs a foreig exchage returs for i d 1, f 1, a e 1, respectively. Table 4 Estimatio of the correlatio coefficiets i various periods. Correlatio coefficiets Period I Period II 004/1/1 007/7/31 007/8/1 009/11/7 ρ 1, ρ, ρ 1, Note that ρ 1,, ρ,3, a ρ 1,3 deote the correlatio coefficiets betwee domestic assets (TSMC) a foreig assets (MSFT), foreig assets (MSFT) a exchage rates (NTD/USD), domestic assets (TSMC) a exchage rates (NTD/USD), respectively. Furthermore, the estimated results of µ d1, µ f1 a µ e1 ca be determied as E [d(l H t )] + 1 σ i + λv with v exp u π + 1 σ π 1 for all Ht A d1,t, e 1,t, a A f1,t, a i d 1, f 1, e 1, respectively. Similarly, σ d1, σ f1, a σ e1 ca be respectively estimated through the variaces of Eq. (6) because Var d(l A d1,t) σ d 1 dt + σπ λ, Var d(l A f1,t) σ f 1 dt + σ π λ, a Var d(l A e1,t) σ e 1 dt + σπ λ. Fially, the estimated results of µ d1, µ f1, µ e1, σ d1, σ f1 a σ e1 are preseted i Table 3. I additio, Table 4 reports the estimatios of the correlatio coefficiets betwee each asset a exchage rates i various periods Calculatio of VaR After the estimatio of model parameters, we ca quickly obtai a oe-day VaR at a 0.01 sigificace level for the iteratioal portfolio o TSMC a MSFT through Eq. (5). These results are summarized i Tables 5 a 6 give that the jump umber is 10, Z at a 0.01 quatile, T 1/5 a V 0 1 (iitial ivestmet); u π 0.05, σπ 0.001, a λ 0.03 for Period I, a u π 0.055, σπ 0.00, a λ for Period II. Clearly, there exists a commo pheomeo the maximum losses of iitial ivestmet of 1 New Taiwa dollar i Period I are fewer tha those i Period II, as the weights of foreig assets a correlatio coefficiets chage. This iicates that it is ecessary for a firm to maitai a sufficiet capital amout i order to prevet default risk durig the subprime mortgage crisis period. I additio, it ca decrease the losses of the portfolio for ivestors to declie weights of foreig assets durig the subprime mortgage crisis period.

6 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Table 5 Model accuracy usig backtestig i terms of i-sample fittig for alterative correlatio coefficiets. ρ 13 Period I Period II 004/1/1 007/7/31 007/8/1 009/11/7 samples VaR exceptios LR uc samples Pael A: correlatio coefficiet betwee domestic assets a foreig exchage rates chages VaR exceptios ρ 3 Pael B: correlatio coefficiet betwee foreig assets a foreig exchage rates chages * Note that this table displays backtestig i terms of i-sample fittig for alterative weights of domestic assets. The critical value is 3.84 at a sigificat level of 5%. The VaRs are the maximum losses of the iitial ivestmet of 1 New Taiwa dollar (NTD) over a oe-day horizo. * deotes the sigificace at a 5% level. LR uc Table 6 Model accuracy usig backtestig i terms of i-sample fittig for alterative weights of domestic assets. Weights of domestic assets (γ 1,t ) Period I Period II 004/1/1 007/7/31 007/8/1 009/11/7 samples VaR exceptios LR uc samples VaR exceptios * * Note that this table displays backtestig i terms of i-sample fittig for alterative weights of domestic assets. The critical value is 3.84 at a sigificat level of 5%. The VaRs are the maximum losses of the iitial ivestmet of 1 New Taiwa dollar (NTD) over a oe-day horizo. * deotes the sigificace at a 5% level. LR uc Alteratively, a ordiary Mote Carlo simulatio approach is employed to calculate the VaRs of the iteratioal portfolio uer o specific assumptios about the distributio of risk factors. The Mote Carlo simulatio is based o 4 time steps (represetig 4 quarter i a period) a trials. First, the followig variables are obtaied through Eq. (6) for simulatio l. [ A d1,t A d1,0 exp µ d1 1 ] σ d 1 λv t + σ d1 tεd,t Y(N), [ A f1,t A f1,0 exp µ f1 1 ] σ f 1 λv t + σ f1 tεf,t Y(N), e 1,t e 1,0 exp [ µ e1 1 ] σ e 1 λv t + σ e1 tεe,t Y(N), (7) where Y(N) N a1 Y a, a {Y a } is a iepeet Poisso series. ε d,t, ε f,t, a ε e,t iepeetly follow ormal distributio of zero mea a 1 variace. Furthermore, we use the Cholesky decompositio to obtai the correlatio matrix amog ε d,t, ε f,t, a ε e,t. The oe ca make ε d,t, ε f,t, a ε e,t be correlated through the correlatio matrix. Next, VaR j (0) is obtaied from Eq. (5) for simulatio l. We repeat the previous procedure times a sum the VaR j (0) for all l 1,,..., Fially, the mea of the sum of VaR j (0) is gaied, a we ca regard it as the amout of VaR i terms of ordiary Mote Carlo simulatios as illustrated i Table 8. Table 8 cosistetly demostrates that the losses valued by the aalytical VaR are higher tha those by the Mote Carlo simulatio approach i various domestic weights durig both Period I a Period II. If fiacial maagers adopt the historical simulatio approach to evaluate fiacial risk, the firm s fiacial ratio (such as ROE) is better. However, the default

7 307 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) VaR(0) Volatility Fig. 1. The impact of volatility o VaR. Note that the symbols o, a represet the impact of volatilities of domestic assets, foreig assets a exchage rates o absolute VaRs, respectively VaR(0) Correlatio Fig.. The impact of correlatio coefficiets o VaR. Note that the symbols o, a represet the impact of correlatio coefficiets, ρ 1,, ρ,3, ρ 1,3 o absolute VaRs, respectively. probability of the firm may icrease o the accout of a shortage of sufficiet capital requiremet. Hece the coservative policy of the aalytical VaR model would be suitable for fiacial istitutios to cotrol market risk Numerical aalysis Based o the estimatio of the model parameters show i Tables 3 a 4, this sectio provides the sesitivity aalyses of the impacts of importat parameters o VaR capital i terms of comparative statics. We start by assumig that (i) the value of a firm is made up of a lie of a domestic asset a a foreig asset, a the exchage rate is the ratio of the domestic currecy to the foreig currecy; (ii) the iitial value of a iteratioal portfolio is $100; (iii) the critical value is.33 at a give α of 0.01, a the ivestmet horizo is oe year (T 1); (iv) γ 1,t 0.3 a β 1,t 0.7. Accordig to Eq. (5), the effects of volatilities, correlatio coefficiets, a the itesities of jumps o the absolute VaR capital allocatio are show i Figs. 1 3, respectively. There is oe commo pheomeo exhibited i these figures: the loss amout icreases mootoically as volatilities, correlatio coefficiets a the itesities of jumps rise. As show i Figs. 1 a, the sesitivities of the volatility of foreig assets a the correlatio coefficiet betwee foreig assets a exchage rates are higher tha those of the others. Additioally, Fig. 4 illustrates the relatioship betwee the VaR a the weights of hump-shaped domestic assets shapes i hump. Also, the loss amout declies as the weights of foreig assets rise at arou Evaluatio of model accuracy Backtestig is a widely used method of evaluatig VaR accuracy. Moreover, we will compare the accuracy of the aalytical VaR derived from Eq. (5) with that of the Mote Carlo simulatio i terms of backtestig criterio for the iteratioal portfolio o the TSMC a MSFT stocks through i-sample a out-of-sample fittig. The usual backtestig techiques cosider the umber of violatios at which the losses are larger tha VaR. The proportio of times should be equal to oe mius the VaR cofidece level; i other words, the model should provide the correct ucoitioal coverage. I order to test the ull hypothesis that the ucoitioal coverage equals the sigificat level, Kupiec [18] presets a likelihood ratio statistic. Give a VaR at the 1% level left-tail over daily horizo for a total of D, oe ca cout how may times the actual loss exceeds oe day s VaR. Defie d as the umber of exceptios a d/d as the

8 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) VaR(0) Itesity of jumps Fig. 3. The impact of the itesity of jumps o VaR VaR(0) weight of domestic assets Fig. 4. The impact of weights of domestic assets o VaR. exceptio rate. The ull hypothesis is that a give cofidece level for losses is the true probability. Kupiec [18] approximates 95% cofidece regios, deoted by q for the test. The ucoitioal coverage is defied by the log-likelihood ratio: LR uc [ l (1 q) Dd q d + l 1 d ] Dd d d. (8) D D The LR uc statistic has a chi-square distributio with oe degree of freedom. Oe would reject the ull hypothesis if LR uc > 3.84 at a 95% cofidece level. The test procedure described above is called backtestig. Assume that the jump umber is 10, Z at a 0.01 quatile, T 1/5 a V 0 1 (iitial ivestmet); u π 0.05, σπ 0.001, a λ 0.03 for Period I, a u π 0.055, σπ 0.00, a λ for Period II. I isample fittig, the time wiow legth is the period from Jauary 1, 004 to November 7, 009, which is broke ito two periods, labelled Period I a Period II. Period I is from Jauary 1, 004 to July 31, 007, durig which the subprime mortgage crisis had ot yet occurred; Period II is from August 1, 007 to November 7, 009. Tables 5 a 6 demostrate that the ull hypothesis for the iteratioal portfolio ca almost ot be rejected at a sigificace level of 5% i Period I a Period II as domestic weights a correlatio coefficiets chage. I out-of-sample approach, we also split the data sample ito two parts. The first part is used to estimate the model from Jauary 1, 004 to July 31, 007 (estimated i Period I). The seco part is used to forecast VaRs from August 1, 007 to November 7, 009 (forecasted i Period II). The statistics of the ucoitioal coverage are illustrated i Table 7 showig how oe fails to reject the ull hypothesis that a give cofidece level for losses is the true probability i terms of out-ofsample fittig for alterative weights of domestic assets. I additio, Table 8 records the computatio time a VaRs by both Mote Carlo simulatio a the approximated solutio from Eq. (5). Obviously, ot oly are the VaRs from Eq. (5) early close to those of the Mote Carlo simulatio, but also the VaR solutio of Eq. (5) ca efficietly save computatioal cost. To summarize, the VaR model preseted by this paper ca be used to accurately calculate VaR for a iteratioal portfolio based o i-sample a out-of-sample approaches. 5. Coclusio Oe advatage of VaR is that it is a ituitively appealig measure of risk that ca be easily coveyed to a firm s seior maager. The measure most commoly used assumes that the probability distributio of daily asset returs is ormal.

9 3074 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Table 7 Model accuracy usig backtestig i terms of out-of-sample fittig for alterative weights of domestic assets. Weights of domestic assets (γ 1,t ) samples VaR exceptios LR uc * Note that this table displays backtestig i terms of out-of-sample fittig for alterative weights of domestic assets. The data samples are split ito two parts. The 1-day VaRs are estimated from Jauary 1, 004 to July 31, 007 (Period I). The seco part is used to forecast VaRs from August 1, 007 to November 7, 009 (Period II). The umber of exceptios iicates the times that the VaRs are exceeded i Period II. The critical value is 3.84 at a sigificat level of 5%. * deotes the sigificace at a 5% level. Table 8 Compariso with Mote Carlo simulatio for alterative weights of domestic assets. Domestic Period I Period II weights 004/1/1 007/7/31 007/8/1 009/11/7 VaR 1 CT 1 VaR SE CT LR 1,uc LR,uc VaR 1 CT 1 VaR SE CT LR 1,uc LR,uc * * * * Note that VaR 1 a VaR sta for oe-day VaRs at a 1% quatile, which are measured from Eq. (5) a the Mote Carlo simulatio, respectively. CT 1 a CT represet the computatio time based o Eq. (5) a the Mote Carlo simulatio, respectively. x y iicates that a simulatio takes x miutes a y secos. SE is the staard errors of the Mote Carlo estimates. The Mote Carlo simulatio is based o 4 time steps a trials. LR 1,uc a LR,uc sta for the ucoitioal coverage rates of the aalytical VaR a Mote Carlo simulatio i terms of i-sample fit, respectively. The critical value is 3.84 at a sigificat level of 5%. * deotes the sigificace at a 5% level. However, this assumptio is far from coitios i the actual world. This paper provides a mixed Poisso-jump model for a iteratioal portfolio to maage market risk, i particular the subprime mortgage crisis of August 007. Differig from past studies whose portfolios were oly valued i oe currecy, this model cosiders portfolios ot oly with jumps but also with exchage rate risk. It is vital for ivestors to cosider exchage rate risk i highly itegrated global fiacial markets. Additioally, through backtestig criterio, the fiig is that the model is more capable of accurately reflectig the loss probability of 1% i terms of i-sample a out-of-sample fittig. Hece, the proposed method i this paper is a more efficiet way i the presece of asymmetric a fat-tail portfolio returs durig periods of fiacial turbulece. Appeix A. The proof of Propositio 1 Let f g ie i,t A fi,t with g i,t foreig asset shares. Coitioal o self-fiacig strategy a by meas of Ito s lemma, oe ca obtai dx i,t X i,t f g i dafi,t A fi,t + de i,t + da f i,t de i,t. (A.1) e i,t A fi,t e i,t Substitutig the dyamic processes of foreig asset returs a exchage rates show i Eqs. () a (3) ito (A.1), Eq. (A.1) ca be expressed as dx i,t X i,t f g i (µfi + µ ei λv + ρ,3 σ fi σ ei )dt + σ fi dw,t + σ ei dw 3,t + (π 1)dY t.

10 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Appeix B. The proof of Propositio Suppose d m ia di,t + f g ie i,t A fi,t. Coitioal o assumptio (vi) a usig Ito s lemma, oe ca obtai d d γ i,t da di,t A Adi,t f + β i,t dx i,t X i,t, (B.1) with γ i,t m ia di,t, a β i,t g ie i,t A fi,t. Substitutig Propositio 1, a Eq. (1) ito (B.1), the result is as follows: d f γ i,t (µ di λv) + β i,t (µ ei + µ fi λv + ρ,3 σ fi σ ei ) dt d + γ i,t σ di dw 1,t + f β i,t σ fi dw,t + f β i,t σ ei dw 3,t + (π 1)dY. Appeix C. The derivatio of Eq. (5) Give a cofidece level of α, VaR ca be expressed as P r (V T V α ) α. Based o the absolute VaR beig deoted by VaR(0) V α V 0, Eq. (C.1) ca be trasformed ito P r (V T V 0 + VaR(0)) α. (C.1) (C.) Let σ t dw t d γ i,tσ di dw 1,t + l V T Y T k N l V 0 + f β i,tσ fi dw,t + f β i,tσ ei dw 3,t. From Propositio, oe ca obtai µ t 1 σ t T + ku ξ, σ t T + kσ ξ, (C.3) i which k stas for the umber of jumps a satisfies k 0, 1,..., ; N(.) represets a ormal distributio; u ξ a σ ξ deote the expected value a the variace of the ature logarithm of π 1, respectively; d f µ t γ i,t (µ di λv) + β i,t (µ ei + µ fi λv + ρ,3 σ fi σ ei ); σ t f f f γ i,t σ di + β i,t σ fi + β i,t σ ei + ρ 1, γ i,t σ di β i,t σ fi f f f + ρ,3 β i,t σ fi β i,t σ ei + ρ 1,3 γ i,t σ di β i,t σ ei. Assume σ B db t σ t dw t + l(π 1)dk. Thus, Eq. (C.) becomes k0 P r (Y T k) P r σ t W T + k l(π 1) l [V 0 + VaR(0)] l V 0 µ t 1 σ t T Y T k α. (C.4) Because jumps follow Poisso distributio, Eq. (C.4) ca be easily writte through (C.3) as (C.5) exp [λt ] [λt ] k P r σ tw T + k l(π 1) ku ξ l [V 0 + VaR(0)] l V 0 µ t 1 σ t T kuξ k! k0 σt T + kσ ξ σt T + kσ ξ α. Cosequetly, Eq. (5) ca be proved: exp [λt ] [λt ] k Φ l [V 0 + VaR(0)] l V 0 µ t 1 σ t T kuξ α. k! k0 σt T + kσ ξ (C.5)

11 3076 F.-Y. Che / Computers a Mathematics with Applicatios 6 (011) Appeix D. The derivatio of the expected value a variace of l(π 1) Give π follows a logormal distributio with parameters u π a σπ, the probability distributio of π 1 is also logormal. Let u ξ E[l(π 1)], a σπ Var[l(π 1)]. Because E[π] eu π + 1 σ π a Var[π] e u π +σπ (e σ π 1), the expected value a variace of π 1 are obtaied as follows: E[π 1] e u π + 1 σ π 1 e u ξ + 1 σ ξ, (D.1) Var[π 1] 4e u π +σ π (e σ π 1) e u ξ +σ ξ (e σ ξ 1). (D.) From Eqs. (D.1) a (D.), u ξ a σπ ca be solved as follows: u ξ l e u π + 1 σ π 1 1 l 1 + 4eu π +σ π (e σ π 1) e u π + 1, a σ σ ξ l 1 + 4eu π +σ π (e σ π 1) π 1 e u π + 1. σ π 1 Refereces [1] N. Hofma, E. Plate, Approximatig large diversified portfolios, Mathematical Fiace 10 (1) (000) [] S. Hesto, A closed form solutio for optios with stochastic volatility with applicatios to bo a currecy optios, Review of Fiacial Studies 6 (1993) [3] G. Bakshi, C. Cao, Z. Che, Empirical performace of alterative optio pricig models, Joural of Fiace 53 (1997) [4] M. Broadie, M. Cherov, M. Johaes, Model specificatio a risk premiums: the evidece from the futures optios, Joural of Fiace 6 (007) [5] J.H. Stock, M.W. Watso, Evidece o structural istability i macro-ecoomic time series relatios, Joural of Busiess a Ecoomic Statistics 14 (1996) [6] J. Hull, A. White, Value at risk whe daily chages i market variables are ot ormally distributed, Joural of Derivatives 5 (1998) [7] B.E. Hase, The ew ecoometrics of structural chage: datig breaks i US labour productivity, Joural of Ecoomic Perspectives 15 (001) [8] G. Cosigli, Tail estimatio a mea-var portfolio selectio i markets subject to fiacial istability, Joural of Bakig & Fiace 6 (00) [9] Q. Shag, X. Qi, Y. Wag, Desig of catastrophe mortality bos based o the comootoicity theory a jump diffusio process, Iteratioal Joural of Iovative Computig, Iformatio a Cotrol 5 (4) (009) [10] C.M. Liu, Z.G. Feg, K.L. Teo, O a class of stochastic impulsive optimal parameter selectio problems, Iteratioal Joural of Iovative Computig, Iformatio a Cotrol 5 (4) (009) [11] X. Ma, L. Zhao, A simulatio aalysis of earest-eighbor rule for AS/RS uer stochastic dema, ICIC Express Letters 4 () (010) [1] P. Ti, T.T. Zi, T. Toriu, H. Hama, A stochastic model for web reliability rakig system, ICIC Express Letters 4 (3) (010) [13] M. Gibso, Icorporatig evet risk ito value-at-risk, FEDS Papers , 001, pp [14] L.K. Gua, L. Xiaoqig, T.K. Chog, Asymptotic dyamics a value-at-risk of large diversified portfolios i a jump diffusio market, Quatitative Fiace 4 (004) [15] R. Merto, Optio pricig whe uerlyig stock returs are discotiuous, Joural of Fiacial Ecoomics 3 (1976) [16] P. Kupiec, Risk capital a VaR, Joural of Derivatives 7 (1999) [17] F.Y. Che, S.L. Liao, Modellig VaR for foreig-asset portfolios i cotiuous time, Ecoomic Modellig 6 (009) [18] P. Kupiec, Techiques for verifyig the accuracy of risk measuremet models, Joural of Derivatives 3 (1995)