Time Series Modeling for Risk of Stock. Price with Value at Risk Computation

Transcription

1 Applied Mahemaical Sciences, Vol 9, 015, no 56, HIKARI Ld, wwwm-hikaricom hp://dxdoiorg/101988/ams Time Series Modeling for Risk of Sock Price wih Value a Risk Compuaion Dodi Deviano, Maiyasri and Dian Rezki Fadhilla Deparmen of Mahemaics Faculy of Mahemaics and Naural Sciences Andalas Universiy, Limau Manis Campus Padang, Indonesia, 5163 Copyrigh 015 Dodi Deviano, Maiyasri and Dian Rezki Fadhilla This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied Absrac Risk of sock price invesmen is defined as an unexpeced oucome of ha asse s value in he fuure Value a Risk (VaR) is one of measuremen in marke risk, i is measured loss ha associaed wih a rare even under normal marke condiions or he maximum loss of financial posiions during a given ime periods for a given probabiliy The maximum loss in cerain ime and level of confidence can be compued by forecasing volailiy ha is represened by sandard deviaion The model o be used o represen volailiy as risk of sock prices is Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) In his sudy, GARCH(1,1) is he bes suied model o forecas volailiy of sock price Unilever Indonesia The refinemen of his volailiy model can be used o compue VaR as consideraion o help invesors for heir invesmen Mahemaics Subjec Classificaion: 6M10, 91B84, 37M10 Keywords: ime series, generalized auoregressive condiional heeroscedasiciy, value a risk, volailiy, sock price 1 Inroducion Risk and reurn are he main invesmen principles Reurn is he profi from an invesmen while risk is he possibiliy of experiencing loss in he invesmen The mos of relaionship beween reurn and risk is said o be linear Generally, he lower

2 780 Dodi Deviano e al risk aken, lower reurn gained and vice versa These principles are imporan o invesors before making he invesmen deal The amoun of he risk could be approximaed by using Value a Risk (VaR) defined as VaR W ˆ Z where W is he amoun of invesmen, ˆ is volailiy a he period of ime and Zα is criical value of given confidence level The mehod of VaR is commonly used in risk managemen as a financial conrol o measure financial risk of many banks, broker firms, invesmen funds and even non financial corporaion The mean model for risky asses such as sock prices is usally explained by using Auoregressive Moving Average (ARMA), and while he variance model is firs developed by Engle [6] o be Auoregressive Condiional Heeroscedasiciy (ARCH) This model is exended by Bollerslev [1,] included he ARMA srucure and is exension became Generalized ARCH (GARCH) Furhermore, Engle [5] also used VaR mehods o measure he risk of porofolio ha consiss of Nasdaq and Dow Jones bonds by esimaing ime series models for volailiy as ime varying variance which GARCH(1,1) as he bes suied model This evidence shows ha GARCH models are able o capure eniraly he ime varying volailiy for sock price daa The refinemen of ARCH/GARCH ime series models is ineresing o learn when i is commonly used in financial o characerize volailiy properies of risk of sock prices for esablished company porofolio However, i is sill unknown he naure ime varying risk of sok prices in shares of developing counries The BBC (Briish Broadcasing Company) repored on February 01 ha growh of Indonesia s economy reached is fases pace in las 15 years by invesmen increasing and domesic demand growing Furhermore, Indonesia is argeed he developed saus of he counry in 030, hen his paper devoed o give he mahemaical ime series model applied o economic growh because no oo many mahemaical models on Indonesia s sock reurn while is economy keeps growing Daa and Mehods In his paper we will make he model of mean and variance for reurn of sock price as risky asse We use hisorical prices of Unilever Indonesia sock (PT Unilever Indonesia Tbk) from January 4h 008 unil January 4h 014 ha is 1509 daa The ime series analysis is used o obain he model of his hisorical price The analysis follows by daa exploraion, making ime series plo, calculaing reurn of sock prices and hen deermining he models Mean model for reurn of sock price {r} follows ARMA model if r 1 r 1 r pr p 1 1 q q

3 Time series modeling for risk of sock price 781 where r is reurn a he ime, μ=e(r), ϕ and θ are parameers, ε is residual and ε ~ N(0,σ ) The model of non consan variance of residuals (heeroscedasiciy) of reurn of sock prices was inroduced by Engle [6] follows he equaion 1 1 h 0 q q where h is defined as h = E[ε ε-1, ε-, ] and α0 > 0, 0 < αi <1, ε ~ N(0, h) and i=1,,,q while q is he order number, he model is noaed as ARCH(q) Bollerslev [5] exended Engle s work abou ARCH by large order numbers become GARCH(p,q) formulaed as follows q q 1h 1 h h ph p where α0 > 0, αi, βi 0 and αi + βi 0 where i=1,,, q and j=1,,, p Saisical properies and mehodology of mean model in his research is adaped from Brockwell and Davis [3] and Enders [4], while variance model is adaped from Engle [6] and Bollerslev [] The seps o deermine he models are explaining as follows: (i) Making plo of reurn (ii) Saionary es using Augmened Dicky Fuller (ADF) es Suppose a regression model r r p 1 1 i1 r where Δr = r - r-1 and γ is regression coefficien The ADF es is execued under he null hypohesis γ = 0 agains he alernaive hypohesis γ < 0 Tes saisics compued using he formula ˆ ADF SE( ˆ) (iii) Making plo of auocovariance funcion (ACF) and parial auocovariance funcion (PACF) of reurn o idenify mean model by analyzing plo s paern where mean models are prediced based on he propery of ACF and PACF (iv) Esimaing mean model, and diagnosic es using Ljung-Box Q es wih null hypohesis is auocorrelaion of daa is zero (ρk=0) while he alernaive hypohesis is no zero The saisic es is K rk QLB n( n ) k1 n k where n is he sample size, rk is he esimaor of ρk, and K is he number of lag ha is esed Null hypohesis rejeced if QLB ( k pq) (v) Calculaing squared error or square of residual (vi) Idenifying and esimaing ARCH/GARCH model (vii) Selecing he bes ARCH/GARCH model which has lowes AIC and BIC, where value of he crieria as follows AIC n ln ( RSS ) k and BIC n ln ( RSS ) k ln ( n) i i

4 78 Dodi Deviano e al (viii) Evaluaing seleced ARCH/GARCH model wih saisical es ARCH linear model es is deermined by making he regression model of he residuals (ε) in he form k The es is carried ou by compuing he saisic es LM = nr and compare he value wih criical value of χ (α) If LM > χ (α) hen he null hypohesis is rejeced The diagnosic Ljung-Box Q es wih null hypohesis he auocorrelaion of GARCH residuals is zero (ρk=0) while he alernaive hypohesis ρ is no equal o zero Normally es defined as follows n K 3 JB S 6 4 where n is sample size, S is skewness and K is kurosis The null hypohesis is residuals are normally disribued agains he alernaive hypohesis he residuals are no normally disribued Null hypohesis is rejeced if JB >χ (α) (ix) Compuing value a risk by using ime series models for risk of sock prices and is inerpreaionthe VaR is compued by using volailiy model a he period of ime, ˆ h where h is coming from GARCH model 3 Proposed Model and Analysis This secion will explain modeling for risk of sock prices wih VaR compuaion The early sage is idenifying mean models by making plo of daa The Figure 1 presens plo of observaions ha used in his discussion The observaion plo shows ha i has monoonous rends which are increasing in his period, so we can conclude ha sock price daa has a posiive rend This posiive rend indicaes a good performance of he Unilever Indonesia Figure 1: Sock price of Unilever Indonesia Figure : Reurn of he sock price Unilever Indonesia The reurn of he sock price is calculaed by using logarihm reurn follows he formula

5 Time series modeling for risk of sock price 783 r log x x1 where r is reurn a he ime, x is price a ime and x-1 is price a ime -1 The imporan nex sep is ploing he reurn series o check he series saionary and i is shown by he Figure Graphically, he plo a Figure shows a saionary process bu i is necessary o run a saiscial es, wheher i is a saionary or no, he reurn series is saionary process wih es saisic s value -046 and Dicky-Fuller s criical value More informaion on he plo ha is also o show he volailiy clusering where large volailiy ends o be followed by large change, and small changes end o be folowed by small changes The presence of volailiy clusering is a signal of heeroscedasiciy The main model of reurn series will be esimaed by using correlogram (plo ACF and PACF) Figure 3: Correlogram of sock price reurn of Unilever Indonesia I is seen on Figure 3 ha he correlogram has exponenially he ail off paerns so i is expeced o be an ARMA model Orders of ARMA are idenified by judging significan lags from appearance of ACF and PACF plo From Figure 3, he significan lags are 1, 3 and 4 The possibiliy of all ARMA models is summarized in Table 1 alongside wih he AIC and BIC values The values have been ranked from he lowes o highes The bes mean model is ARMA(1,3) wih lowes AIC and BIC value, r r e e 3

6 784 Dodi Deviano e al Table 1: Summary of ARMA model esimaion Model AIC BIC Rank ARMA(1,3) ARMA(3,1) ARMA(1,1) ARMA(1,4) ARMA(4,4) ARMA(3,3) ARMA(4,1) ARMA(3,4) ARMA(4,3) The nex sep of his ime series modeling is o model ime varying variance from residuals series of ARMA(1,3) o be idenified in order o find he suied ARCH/GARCH model The orders of esimaed ARCH/GARCH model are seleced a lags of 1 dan The AIC and BIC values from esimaed models are presened by Table Table : Summary of ARCH/GARCH model esimaion Model AIC BIC Rank GARCH(1,1) GARCH(1,) GARCH(,) ARCH() ARCH(1) GARCH(,1) * The bes suied volailiy model is GARCH(1,1) which has he lowes AIC and BIC values, where he sarred GARCH models can no be used because i has negaive coefficien so he models do no mee he properies of ARCH/GARCH The GARCH(1,1) is expressed as follows h h e Furhermore, o make sure he seleced model is adequae, we need o do he diagnosic checking using saisical es and described as follows (i) Lagrange Muliplier Tes The purpose of his es is o check he presence of ARCH effec in GARCH(1,1) model If here is no ARCH effec, hen i means he GARCH(1,1) is good o describe he daa Null hypohesis is no more ARCH effec in GARCH(1,1) residuals and he alernaive hypohesis here is ARCH effec in GARCH(1,1) The p-value from LM es displayed >5%, hen we can conclude ha here is no more effec in GARCH(1,1) residuals (ii) Ljung-Box Q Tes In his es we need he sandardized-squarred residuaal of GARCH(1,1) o 1

7 Time series modeling for risk of sock price 785 idenify wheher auocorrelaion in squarred residuals is sill presens Figure 4 shows ha here is no auocorrelaion in residuals series Figure 4: Correlogram of sandardized residuals (iii) Jarque-Berra Tes The null hypohesis of his es is GARCH(1,1) residuals normally disribued Jarque-Berra es value is and we conclude ha residuals of GRACH(1,1) are no normally disribued Non-normaliy residuals needs o be fixed by using robus esimaion wih he resul as follows h h e The appropriae model GARCH(1,1) in his sudy has obained, i can be used o forecas volailiy in cerain periods The volailiy a he period of ime defined as squares roo of h from GARCH(1,1) model, where his forecas volailiy is used o compue value a risk 1 Table 3: Summary of ARCH/GARCH Model Esimaion Periods VaR Volailiy 1 day $15, days $16, days $17, days $18, days $18, In his case we predic volailiy for 30 seps ahead or 30 days displayed in Table 3 For insan predicion, we discuss VaR for he nex 5 rading days (nex week)

8 786 Dodi Deviano e al The VaR a he 5% level for nex week is $16,368, his predicion means ha if someone invess $1 million in he Unilever Indonesia in 4 April 014, and some exraordinary evens happened, he maximum loss for he nex 5 rading days occurred wih probabiliy 95% is $16,368 I is seen from Table 3 ha longer duraion of invesmen hen higher possibiliy of maximum loss These compuaion and predicion of VaR are likely can help invesors o consider he invesmen on he company 4 Conclusion The ime series can be used o express mean and volailiy models for sock price of Unilever Indonesia which he bes suied mean model is ARMA(1,3) and he volailiy model is GARCH(1,1) These are he mos adequae model for he Unilever Indonesia sock price during he period January 4h 008 o January 4h 014 I is necessary o noe ha he models are limied for shor erm forecasing The model should be coninually esimaed a he recen observaions Furhermore, VaR mehod is used o esimae he risk value of cerain invesmen in Unilever Indonesia The resul is used in financial conrol and risk managemen and also could be someone s consideraion of making deal in invesmen Acknowledgemens The auhors would like o hank he referees and he edior for helpful suggesions which improved he qualiy of he paper References [1] T Bollerslev, A condiionally heeroscedasic ime series model for speculaive prices and raes of reurn, Review of Economics and Saisics, 69, , 1987 hp://dxdoiorg/10307/ [] T Bollerslev, Generalized auoregressive condiional heeroscedasiciy, Journal of Economerics, 31, 307-7, 1986 hp://dxdoiorg/101016/ (86) [3] P J Brockwell and R A Davis, Inroducion o Time Series and Forecasing, Spring Verlag, New York, 00 hp://dxdoiorg/101007/b97391 [4] W Enders, Applied Economerics Time Series, John Wiley & Son, New York, 1995 [5] R Engle, The use of ARCH/GARCH models in applied economerics, Journal Economeric Prespecive, 15, , 001 hp://dxdoiorg/10157/jep154157

9 Time series modeling for risk of sock price 787 [6] R Engle, Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, , 198 hp://dxdoiorg/10307/ Received: March 4, 015; Published: April 1, 015