Evaluating the Performance of Forecasting Models for Portfolio Allocation Purposes with Generalized GRACH Method

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1 Advances in mahemaical finance & applicaions, 2 (1), (2017), 1-7 Published by IA Universiy of Arak, Iran Homepage: Evaluaing he Performance of Forecasing Models for Porfolio Allocaion Purposes wih Generalized GRACH Mehod Adel Azar a, Mohsen Hamidian b, Maryam Saberi a*, Mohammad Norozi b a Faculy of Managemen & Economics, Universiy of Tarbia Modares, Tehran, Iran b Faculy of Economics & Accouning, Universiy of Islamic Azad souh Tehran, Tehran, Iran ARTICLE INFO Aricle hisory: Received 11 Ocober 2016 Acceped 8 March 2017 Keywords: Loss funcions Porfolio allocaion Evaluaing forecass ABSTRACT Porfolio heory assumes ha invesors accep risk. This means ha in he equal rae of reurn on he wo asses, he asses were chosen ha have a lower risk level. Modern porfolio heory is acceped by invesors who believe ha hey are no cope wih he marke. So hey keep many differen ypes of securiies in order o access he opimum efficiency rae ha is close o he rae of reurn on marke. One way o conrol invesmen risk is esablishing he porfolio shares. There are many ways o choose he opimal porfolio shares. Among hese mehods in his sudy we use loss funcions. For his, we choose all firms from he year 2011 o he end of 2015 ha had been a member in he Tehran Sock Exchange. The resuls of his research show ha he likelihood funcions have he bes performance in Forecasing he opimal porfolio allocaion problem. 1. Inroducion A special feaure of economic forecasing compared o general economic modelling is ha we can measure a model s performance by comparing is forecass o he oucomes when hey become available. Generally, several forecasing models are available for he same variable and forecasing performances are evaluaed by means of a loss funcion (Ellio and Timmermann [9]). Model selecion generally involves he evaluaion of forecass of volailiy wihin loss funcions, which are classified as eiher direc or indirec by Paon and Sheppard [15]. Direc loss funcions are measures of he forecas accuracy based on radiional saisical measures of precision. Alhough i would appear ha * Corresponding auhor. address: accsaberi@gmail.com All righs reserved. Hosing by IA Universiy of Arak Press

2 Evaluaing he Performance of Forecasing Models for Porfolio Allocaion direc loss funcions should be easy o implemen and inerpre, he fac ha volailiy is unobservable, hereby necessiaing he use of an observable proxy for volailiy, confounds he issue. Indeed, Hansen and Lunde [13] and Paon [16] demonsrae ha noise in he volailiy proxy renders cerain direc loss funcions incapable of ranking forecass consisenly in he univariae seing. Subsequen sudies by Lauren e al. [14] and Paon and Sheppard [15] have repored equivalen resuls in he mulivariae seing. On he oher hand, indirec measures of he volailiy forecasing performance evaluae forecas efficacy in he conex of he applicaion for which he forecas is required, for example he porfolio allocaion problem. One appealing aribue of his ype of evaluaion is ha i is specifically relaed o he economic decision from which he forecas derives is value (Ellio & Timmermann, [9]). Danielsson [6] argues ha forecass should be evaluaed and seleced on he basis of heir inended applicaion. Many sudies have used indirec measures o evaluae volailiy forecasing models. For example, Engle and Colacio [8] evaluae he forecasing performance in erms of porfolio reurn variance, while Fleming e al. [10, 11] apply a quadraic uiliy funcion ha values one forecas relaive o anoher. Despie he srong economic appeal of measures ha combine risk and reurn, especially hose ha repor a measure of relaive economic value, i is easy o show ha hese measures can favour incorrec forecass of volailiy. One noable excepion is he porfolio variance, which does no display his problem. Engle and Colacio [8] and Paon and Sheppard [15] have demonsraed ha he porfolio variance is minimised when he correc forecas is applied; a resul ha links he porfolio variance wih robus saisical loss funcions (Berker [4]). 2. Lieraure review Provide an excellen survey on he sae of he ar of forecasing in economics. Deails on volailiy and correlaion forecasing can be found in Andersen e al. [2]. The evaluaion of volailiy forecass raises he problem ha he variable of ineres is laen. This problem can be solved by replacing he laen condiional variance by a proxy; see e.g. he realized variance esimaor of Andersen and Bollerslev [1]. However, as noed by Andersen e al. [2], he evaluaion of volailiy forecass using a proxy may no lead, asympoically, o he same ordering ha would be obained if he rue volailiy was observed. In a general framework, Hansen and Lunde [13] show ha when he evaluaion is based on a arge observed wih error, he choice of he loss funcion becomes criical in order o avoid a disored oucome. They also provide condiions on he funcional form of he loss funcion which ensure consisency of he proxy based ordering. For univariae volailiy, Paon [16] derives a class of loss funcions which are able o order consisenly in he presence of noise. Building on hese resuls, Paon and Sheppard [15] give a direc mulivariae analogue bu wihou providing any general ex- [2] Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaions

3 Azar e al. pression. The above resuls have imporan implicaions on esing procedures for superior predicive abiliy, see among ohers Diebold and Mariano [7], Clark and McCracken [5] and he conribuions of Hansen and Lunde [13] wih he superior predicive abiliy es and Hansen e al. [12] wih he model confidence se es. In fac, when he arge variable is unobservable, an unforunae choice of he loss funcion may deliver uninended resuls even when he esing procedure is formally valid. Wih respec o he evaluaion of mulivariae volailiy forecass lile is known abou he properies of he loss funcions. This is he firs paper ha addresses his issue. This paper exends he previous lieraure by considering he role played by loss funcions in ex-ane mulivariae volailiy model selecion, where forecass from hese models will subsequenly be used in mean variance porfolio opimisaion. In doing so, i will assess he abiliy of a range of loss funcions o discriminae beween volailiy forecasing models where he inended use of he forecass is a porfolio opimisaion problem. While his paper focuses on mean variance porfolio opimisaion, here is nohing o preven he consideraion of higher momens of reurns, esimaion error or porfolio consrains. However, he main focus here is no he final applicaion iself, bu raher he way in which he loss funcions perform in erms of model selecion wih a given applicaion in mind. This is achieved in par by a simulaion sudy ha considers he relaive powers of a range of saisical loss funcions and porfolio variances. Power is imporan in he conex of his problem because i reflecs he abiliy of loss funcions o discriminae beween forecass. A subsequen empirical sudy hen assesses he consisency beween he various loss funcions and he final porfolio applicaion. I will gauge wheher he bes models seleced from he evaluaion period coninue o be he bes performers in he applicaion period, he opimal oucome in erms of model selecion. This differs from he radiional forecas evaluaion lieraure in ha i considers he use of saisical measures o discriminae beween models, he performances of which are hen measured based on an economic crierion in a subsequen period (Berker, [4]). 3. Mehodology This secion describes he inended porfolio applicaion and he loss funcions employed o evaluae he volailiy forecass. Begin by considering a sysem of N asse excess reurns r, (0,) F Where r is an N 1 vecor, µ is an N 1 vecor of expeced excess reurns and ε is an N 1 vecor of disurbances following he mulivariae disribuion F. In his conex, he opimizaion problem of an invesor who seeks o minimise he variance of a porfolio of N risky asses and a risk-free asse is min s.., 0 Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaion [3]

4 Evaluaing he Performance of Forecasing Models for Porfolio Allocaion The specific loss funcions employed are now described. Mean square error (MSE) The MSE crierion is simply he mean squared disance beween he volailiy forecas, H, and he proxy ˆ 1 ( H,) abs(). H ˆ N MSE 2 Mean absolue error (MAE) An alernaive o squared errors is o measure he absolue error, MAE ˆ 1 (,) abs(). ˆ H H N Quasi-likelihood funcion (QLK) Given a forecas of he condiional volailiy, H, he value of he quasi log-likelihood funcion of he asse reurns assuming a mulivariae normal likelihood is ( H,) ˆ ln() Habs(). ˆ H QLK 1 Porfolio variance (MVP) To evaluae he forecass wihin he given porfolio applicaion, he porfolio opimisaion problem is solved by MVP ( ˆ,) ˆ ˆ ˆ. ˆ To examine he performance of he loss funcions, a range of models for producing one-sep-ahead mulivariae volailiy forecass, H, are required. A number of models have been chosen for his purpose. While his is clearly no an exhausive lis, each model is able o generae volailiy forecass for moderaely sized covariance marices, wih he qualiy of heir forecass being expeced o vary widely. The Orhogonal GARCH (OGARCH) model of Alexander and Chibumba [3] is also considered. To obain he condiional covariance marix of he original reurn series, he following calculaion is required o reverse he ransformaion and sandardizaion ha was performed. K 2 k k k, k 1 OGARCH: H AV A a a a 4. Empirical Resuls By solving he porfolio applicaion wih Malab sofware, he Opimize risk efficien fronier porfolio was as Fig. 1. [4] Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaions

5 Azar e al. Fig. 1: Opimize risk efficien fronier As we can see, wih he increase in he minimum expeced reurn, variance porfolio has increased. I is noable ha here he variance is considered as a measure of risk. Also porfolios hisograms obained by his mehod are as Fig. 2. Fig. 2: Porfolios hisogram The performances of MSE, MAE and QLK are examined firs, and he resuls are presened in Table 1. These resuls show ha QLK ouperforms MSE and MAE, and MAE is no a robus loss funcion Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaion [5]

6 Evaluaing he Performance of Forecasing Models for Porfolio Allocaion for mulivariae volailiy forecas evaluaions. They are also consisen wih he univariae resuls of Paon [16]. Table 1: Obained Resuls Evaluaing he Performance MSE MAE QLK MVP Value for OGRACH mehod 0/2499 0/2500 0/2605 0/ Conclusion Techniques for evaluaing univariae volailiy forecass are well undersood, he lieraure relaing o mulivariae volailiy forecass is less developed. Many financial applicaions employ mulivariae volailiy fore- cass, and hus i may be appealing o evaluae and selec forecasing models on he basis of such an applicaion. This paper has invesigaed he performances of a range of loss funcions for selecing models o be applied in a subsequen porfolio allocaion problem. Two issues underlie his problem: he relaive power of he loss funcions, and he consisency wih he final porfolio applicaion. Simulaion resuls demonsrae ha a saisical loss funcion based on he mulivariae normal likelihood (QLK) and has more power han any oher loss funcion considered. References [1]. Andersen, T.G., Bollerslev, T., Encyclopedia of Saisical Sciences, John Wiley & Sons, Ld, [2]. Andersen, T. G., Bollerslev, T., Chrisoffersen, P., & Diebold, F. X., Pracical volailiy and correlaion modeling for financial marke risk managemen The risks of financial insiuions, Universiy of Chicago Press, [3]. Alexander, C. and Chibumba, A., Mulivariae orhogonal facor GARCH, Working Paper, Mahemaics Deparmen, Universiy of Sussex, [4]. Becker, R., Clemens, A.E., Doolan, M.B., Hurn, A.S., Selecing volailiy forecasing models for porfolio allocaion purposes, Inernaional Journal of Forecasing, 2015, 31, (3), P [5]. Clark, T. and McCracken, M., Tess of equal forecas accuracy and encompassing for nesed models, Journal of Economerics, 2001, 105, (1), P [6]. Danielsson, J., Financial Risk Forecasing, Wiley [7]. Diebold, F. and Mariano, R., Comparing Predicive Accuracy, Journal of Business & Economic Saisics, 1995, 13, (3), P [8]. Engle, R, and Colacio, R., Tesing and Valuing Dynamic Correlaions for Asse Allocaion, Journal of Business & Economic Saisics, 2006, 24, P [9]. Ellio, G. and Timmermann, A., Economic Forecasing, Princeon Universiy Press, [10]. Fleming, J., (2001), The Economic Value of Volailiy Timing, Journal of Finance, 2001, 56, (1), P [11]. Fleming, J., Kirby, C. and Osdiek, B., The economic value of volailiy iming using "realized" volaili- [6] Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaions

7 Azar e al. y, Journal of Financial Economics, 2003, 67, (3), P [12]. Hansen, P. R., Lunde, A., & Nason, J. M., The model confidence se. Economerica, 2011, 79, (2), P [13]. Hansen, PR., Lunde, A., Realized variance and marke microsrucure noise, Journal of Business and Economic Saisics, 2006, 24 (2), P [14]. Lauren SFJA, Rombous JVK, Violane F., On loss funcions and ranking forecasing performances of mulivariae volailiy models. Journal of Economerics. 2013, 173 (1), P [15]. Paon, AJ, Sheppard, K., Evaluaing volailiy and correlaion forecass, Handbook of financial ime series, [16]. Paon, AJ, Volailiy forecas comparison using imperfec volailiy proxies, Journal of Economerics, 2011, 160, P Vol. 2, Issue 1, (2017), Advances in mahemaical finance and applicaion [7]