The Ten Commandments for Optimizing Value-at-Risk and Daily Capital Charges*

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1 The Ten Commandmens for Opimizing Value-a-Risk and Daily Capial Charges* Michael McAleer Deparmen of Quaniaive Economics Compluense Universiy of Madrid and Economeric Insiue Erasmus Universiy Roerdam Revised: February 2009 * The auhor wishes o acknowledge deailed commens from he Edior and a referee, and helpful discussions wih Dave Allen, Manabu Asai, Federico Bandi, Peer Boswijk, Massimiliano Caporin, Felix Chan, Chialin Chang, Philip Hans Franses, Kees Jan van Garderen, Abdul Hakim, Shawka Hammoudeh, Peer Reinhard Hansen, Eric Hillebrand, Juan-Ángel Jiménez-Marin, Masahio Kobayashi, Offer Lieberman, Shiqing Ling, Essie Maasoumi, Colin McKenzie, Marcelo Medeiros, Les Oxley, Teodosio Peréz-Amaral, Hashem Pesaran, Bernardo da Veiga and Alan Wong, and he financial suppor of he Ausralian Research Council. Previous versions of his paper have been presened a he Bank of Thailand, Caholic Universiy of Brasilia, Chiang Mai Universiy, Compluense Universiy of Madrid, Erasmus Universiy Roerdam, Feng Chia Universiy, Taiwan, Hong Kong Bapis Universiy, Hong Kong Universiy of Science and Technology, Keio Universiy, Kyoo Universiy, LaTrobe Universiy, Naional Chung Hsing Universiy, Taiwan, Ponifical Caholic Universiy of Rio de Janeiro, Soka Universiy, Japan, Sock Exchange of Thailand, Yokohama Naional Universiy, he Universiies of Amserdam, Balearic Islands, Canerbury, Groningen, Padova, Palermo, Tokyo, Venice, Vigo and Zaragoza, and conferences in Chiang Mai, Thailand, Canerbury, New Zealand, Yokohama, Japan, Xiamen, China, and Taichung, Taiwan. 1

2 Absrac Credi risk is he mos imporan ype of risk in erms of moneary value. Anoher key risk measure is marke risk, which is concerned wih socks and bonds, and relaed financial derivaives, as well as exchange raes and ineres raes. This paper is concerned wih marke risk managemen and monioring under he Basel II Accord, and presens Ten Commandmens for opimizing Value-a-Risk (VaR) and daily capial charges, based on choosing wisely from: (1) condiional, sochasic and realized volailiy; (2) symmery, asymmery and leverage; (3) dynamic correlaions and dynamic covariances; (4) single index and porfolio models; (5) parameric, semiparameric and nonparameric models; (6) esimaion, simulaion and calibraion of parameers; (7) assumpions, regulariy condiions and saisical properies; (8) accuracy in calculaing momens and forecass; (9) opimizing hreshold violaions and economic benefis; and (10) opimizing privae and public benefis of risk managemen. For pracical purposes, i is found ha he Basel II Accord would seem o encourage excessive risk aking a he expense of providing accurae measures and forecass of risk and VaR. Keywords and phrases: Dail;y capial charges; excessive risk aking; marke risk; risk managemen; value-a-risk; violaions. JEL Classificaions: G32, G11, G17, C53. 2

3 We regulaors are ofen perceived as consraining excessive risk-aking more effecively han is demonsrably possible in pracice. Excep where marke discipline is undermined by moral hazard, owing, for example, o federal guaranees of privae deb, privae regulaion generally is far beer a consraining excessive risk-aking han is governmen regulaion. Alan Greenspan, Conference on Bank Srucure and Compeiion, 8 May 2003 Elizabeh Turner: Wai! You have o ake me o shore. According o he Code of he Order of he Brehren... Capain Barbossa: Firs, your reurn o shore was no par of our negoiaions nor our agreemen so I mus do nohing. And secondly, you mus be a pirae for he pirae s code o apply and you re no. And hirdly, he code is more wha you d call guidelines han acual rules. Piraes of he Caribbean: Curse of he Black Pearl 1. Inroducion The caaclysmic financial meldown worldwide ha seems o have sared in Sepember 2008 has made i manifesly obvious ha self-regulaion in he finance and banking indusry has been far from adequae. A period of soul searching is likely o be followed by much needed regulaory changes in he indusry. Some changes o regulaions governing he finance indusry, especially as regards he monioring and managemen of excessive risk, overseeing new financial insrumens, and increased regulaion of banks, are likely o be warraned, whereas ohers may have lile or no effec. Following he 1995 amendmen o he Basel Accord (see Basel Commiee on Banking Supervision (1988, 1995, 1996)), banks were permied o use inernal 3

4 models o calculae heir Value-a-Risk (VaR) hresholds (see Jorion (2000) for a deailed discussion of VaR ). This amendmen was in response o widespread criicism ha he sandard approach o calculae and forecas VaR hresholds led o excessively conservaive forecass and higher mean daily capial charges ha are associaed wih higher perceived risk. In shor, he Basel II Accord was inended o encourage risk aking hrough self regulaion. Despie he well known anipahy of Alan Greenspan, he former Federal Reserve Chairman, o governmen regulaion of excessive risk aking, i is no longer possible o argue logically ha self regulaion alone is a viable opion in he banking and finance indusry. The primary purpose of his paper is o evaluae how bes o forecas VaR and o opimize daily capial charges in an aemp o manage excessive risk aking as efficienly as possible, and o offer some idiosyncraic suggesions and guidelines regarding how improvemens migh be made regarding pracical sraegies for risk monioring and managemen, especially when here is a srong and undersandable movemen ino holding and managing cash (circa lae-2008) han in dealing wih risky financial invesmens. The plan of he remainder of he paper is as follows. Secion 2 presens he opimizaion problem facing auhorized deposi-aking insiuions. The daa for a brief empirical analysis of VaR and daily capial charges are illusraed in Secion 3. Secion 4 provides a brief discussion of regression models and volailiy models. Ten reasons for modelling ime-varying variances, covariances and correlaions using high and ulra high frequency daa are given in Secion 5. The Ten Commandmens for 4

5 opimizing VaR and daily capial charges are discussed in Secion 6. Some concluding commens are given in Secion The Opimizaion Problem for Auhorized Deposi-aking Insiuions Value-a-Risk (VaR) may be defined as a wors case scenario on a ypical day. As such, i is concerned wih relaively unlikely, or exreme, evens. Given he financial urmoil in 2008, especially afer Sepember 2008, exreme evens have become more commonplace, such ha an exreme even would probably now need o be caasrophic o qualify as such. Insofar as financial meldowns end o encourage Auhorized Deposi-aking Insiuions (ADIs) o shif financial asses ino cash, VaR should sill be opimized. However, ADIs may decide no o enerain he excessive risk associaed wih financial asses by holding a higher proporion of heir porfolio in cash and/or relaively low risk asses. Under he Basel II Accord, VaR forecass need o be provided o he appropriae regulaory auhoriy (ypically, a cenral bank) a he beginning of he day, and is hen compared wih he acual reurns a he end of he day. For purposes of he Basel II Accord penaly srucure for violaions arising from excessive risk aking, a violaion is penalized according o is cumulaive frequency of occurrence in 250 working days, and is given in Table 1. A violaion is defined as follows: 5

6 Definiion: A violaion occurs when VaR > negaive reurns a ime. As encouraged by he Basel II Accord, he opimizaion problem facing ADIs, wih he number of violaions and forecass of risk as endogenous choice variables, is as follows: Minimize max ( 3 + k) { } k,var DCC = VaR 60, VaR 1, (1) where DCC = daily capial charges, VAR = Value-a-Risk for day, VAR = Yˆ z σˆ, 60 VaR = mean VaR over he previous 60 working days, Yˆ = esimaed reurn a ime, z = 1% criical value a ime, σˆ = esimaed volailiy a ime, 0 k 1 is a violaion penaly (see Table 1). As Ŷ is ypically difficul o predic, and as z is unlikely o change significanly especially on a daily basis), changes in σˆ are crucial for modelling VaR. Subsanial 6

7 Table 1: Basel Accord Penaly Zones Zone Number of Violaions k Green 0 o Yellow Red Noe: The number of violaions is given for 250 business days. Noe: The penaly srucure under he Basel II Accord is specified for he number of penalies and no heir magniude, eiher individually or cumulaively. 7

8 Figure 10: Realized Reurns and VARMA-GARCH VaR Forecass Porfolio Reurns VARMA-GARCH Noe: This is Figure 10 in McAleer and da Veiga (2008a). 8

9 research has been underaken in recen years o develop univariae and mulivariae models of volailiy, under he condiional, sochasic and realized volailiy frameworks, in order o esimae σˆ. Alhough VaR may also be esimaed direcly using regression quaniles (see, for example, Engle and Manganelli (2002)), his approach is no as popular as modelling volailiy and hen calculaing VaR. Alhough considerable research has been underaken on higher-order momens, especially in he conex of condiional volailiy models, hese are no considered in he paper. Model uncerainy for univariae and mulivariae processes is also no considered (see Pesaran, Schleicher and Zaffaroni (2008) for an analysis of model averaging echniques for porfolio managemen). As discussed above, he amendmen o he Basel Accord was designed o reward insiuions wih superior risk managemen sysems. For esing performance, a backesing procedure, whereby he realized reurns are compared wih he VaR forecass, was inroduced o assess he qualiy of a bank s inernal model. In cases where he inernal models lead o a greaer number of violaions han could reasonably be expeced, given he confidence level, he bank is required o hold a higher level of capial (see Table 1 for he penalies imposed under he Basel Accord). If a bank s VaR forecass are violaed more han nine imes in any financial year (see Table 1), he bank may be required o adop he sandard approach. As discussed in McAleer and da Veiga (2008b), he imposiion of such a penaly is severe as i affecs he profiabiliy of he bank direcly hrough higher daily capial charges, has a damaging effec on he bank s repuaion, and may lead o he imposiion of a more 9

10 sringen exernal model o forecas VaR hresholds, which would have he resul of increasing daily capial charges for ADIs. I should be noed ha DCC is o be minimized, wih k and VAR as endogenous choice variables. [The acronym DCC should be disinguished from a widely used mulivariae condiional volailiy model, which will be discussed below.] I is worh giving a cavea wih respec o he minimizaion of daily capial charges in imes of exreme financial flucuaions, such as he period saring in Sepember When excessive risk is very high, and is changing dramaically on a daily basis, daily capial charges should sill be minimized, bu sensible porfolio managemen may involve greaer access o cash afer VaR and daily capial charges have been deermined. 3. Daa The daa used in he empirical applicaions in McAleer and da Veiga (2008a, 2008b), and given as porfolio reurns in Figure 10 above, are daily prices measured a 16:00 Greenwich Mean Time (GMT) for four inernaional sock marke indexes, namely S&P500 (USA), FTSE100 (UK), CAC40 (France), and SMI (Swizerland). All prices are expressed in US dollars. The daa were obained from DaaSream for he period 3 Augus 1990 o 5 November A he ime he daa were colleced, his period was he longes for which daa on all four variables were available. The synchronous reurns for each marke i a ime ( R i ) are defined as: 10

11 Ri = log( Pi, / Pi, 1), where P i, is he price in marke i a ime, as recorded a 16:00 GMT. For he empirical analysis, i was assumed ha he porfolio weighs are equal and consan over ime, alhough boh of hese assumpions can easily be relaxed. The various condiional volailiy models were used o esimae he variance of he porfolio direcly for he single index model, and o esimae he condiional variances and correlaions of all asses and asse pairs o calculae he variance of he porfolio for he porfolio model. Apar from he Sandardized Normal and Riskmerics TM models, all he condiional volailiy models were esimaed under he assumpion ha he disribuion of he uncondiional shocks was (1) normal; and (2), wih 10 degrees of freedom. The empirical resuls will be discussed furher in he conex of he las wo Commandmens, bu he following poins should be highlighed: (i) There is a rade-off beween he number of violaions and daily capial charges, wih a higher number of violaions leading o a higher penaly and lower daily capial charges hrough lower VaR. (ii) Apar from Sandardized Normal, which does no esimae any parameers, and Riskmerics TM, where he parameers are calibraed raher han esimaed, he number of violaions is higher for single index models han for heir porfolio model counerpars, and he mean daily capial charges are correspondingly higher. 11

12 (iii) The use of he disribuion wih 10 degrees of freedom always leads o fewer violaions, and hence higher mean daily capial charges, for boh he single index and porfolio models. The wo curves in Figure 10, as reproduced below, represen daily reurns on he porfolio of he four aggregae indexes, as well as he forecas VaR hresholds using he VARMA-GARCH model of Ling and McAleer (2003). From he figure, i can be seen clearly ha here are very few violaions, which could mean one or more of he following: (1) he volailiy model provides an accurae measure of risk and VaR; (2) he volailiy model provides a conservaive measure of risk and VaR; (3) he number of violaions is no being used as an endogenous choice variable. 4. Regression Models and Volailiy Models The purpose of a (linear) regression model is o explain he condiional mean, or firs momen, of Y : Y = α + β + ε, (2) X such ha E ( Y X ) α + βx =, 12

13 where α and β are aken o be scalar parameers for convenience. The purpose of a (univariae) volailiy model is o explain he variance, or second momen, of Y in (2): ε = η h, η ~ iid(0,1) (3) where ε is he uncondiional shock o he variable of ineres, Y (ypically, a sock reurn in empirical finance), which hereby has a risk componen, η is he sandardized residual (namely, a riskless shock, or fundamenal), and h denoes volailiy (or risk). One of he primary purposes in modeling volailiy is o deermine h o enable η o be calculaed from he observable reurns shock, ε. The volailiy, h, may be laen or observed (hough i is also likely o be subjec o measuremen error). In he class of condiional volailiy models, which are widely used for analysing monhly, weekly and daily daa (hough hey can also be used for hourly daa), h is modelled as: h 2 = + αε 1 + βh 1 ω, (4) where here are (ypically sufficien) condiions on he parameers ω, α, β, o ensure ha condiional volailiy, h, is posiive. The specificaion for h in (4) is he widely used generalized auoregressive condiional heeroskedasiciy, or GARCH(1,1), model of Engle (1986) and Bollerslev (1986). As he model is 13

14 condiional on he informaion se a ime -1, curren shocks do no affec h. For recen surveys of univariae and mulivariae condiional volailiy models, see Li, Ling and McAleer (2002), McAleer (2005a), and Bauwens, Lauren and Rombous (2006). Sochasic volailiy models can incorporae leverage direcly hrough he negaive correlaion beween he reurns and subsequen volailiy shocks (see, in paricular, Zhang, Mykland and Aï-Sahalia (2005), Aï-Sahalia, Mykland and Zhang (2006), and Barndorff-Nielsen, Hansen, Lunde Shephard (2008)). For a recen review of a wide range of sochasic volailiy models, see Asai, McAleer and Yu (2006). Realized volailiy models can incorporae leverage as his is no explicily excluded in he specificaion. However, realized volailiy models are ypically no specified o incorporae leverage. A he mulivariae level, i is no enirely clear how o define leverage in he conex of condiional, sochasic or realized volailiy models (for furher deails, see McAleer and Medeiros (2008a)). 5. Ten Reasons for Modelling Time-varying Variances, Covariances and Correlaions Using High and Ulra High Frequency Daa Regression models are used o explain he condiional firs momen of Y when i is no consan. Similarly, volailiy models are used o explain he second momen ofy when i is no consan. I is well known ha neiher he firs nor second momens of Y is consan, especially in he case of financial reurns daa. 14

15 Ten reasons for modelling ime-varying variances, covariances and correlaions using high frequency (namely, monhly, weekly, daily and hourly) and ulra high frequency (namely, minue and second) daa are as follows: (1) Volailiy from high frequency daa can be aggregaed, whereas aggregaed daa a low frequencies ypically display no volailiy; (2) Enables predicion of uncerainy regarding he imposiion of ourism axes on inernaional ouris arrivals; (3) Enables predicion of uncerainy regarding he imposiion of environmenal axes on polluers; (4) Enables predicion of risk in finance; (5) Enables he predicion of dynamic correlaions for consrucing financial porfolios; (6) Enables he compuaion of dynamic confidence inervals; (7) Enables he compuaion of dynamic Value-a-Risk (VaR) hresholds; (8) Enables he compuaion of dynamic confidence inervals for dynamic VaR hresholds; (9) Enables he predicion of dynamic variances and covariances for consrucing dynamic VaR hresholds for financial porfolios; (10) Enables he derivaion of a sraegy for opimizing dynamic VaR. 6. The Ten Commandmens for Opimizing Value-a-Risk and Daily Capial Charges Credi risk is he mos imporan ype of risk in erms of moneary value, while marke 15

16 risk is ypically concerned wih socks and bonds, and relaed financial derivaives, as well as exchange raes and ineres raes. Operaional risk involves credi and marke risk, as well as oher operaional aspecs or risk. For marke, credi and operaional risk, choose wisely from: (C1) Condiional, sochasic and realized volailiy. (C2) Symmery, asymmery and leverage. (C3) Dynamic correlaions and dynamic covariances. (C4) Single index and porfolio models. (C5) Parameric, semiparameric and nonparameric models. (C6) Esimaion, simulaion and calibraion of parameers. (C7) Assumpions, regulariy condiions and saisical properies. (C8) Accuracy in calculaing momens and forecass. (C9) Opimizing hreshold violaions and economic benefis. (C10) Opimizing privae and public benefis of risk managemen. The commandmens progress from he heoreical o he pracical. The remainder of his secion briefly discusses each self-explanaory commandmen. (C1) Choose Wisely from Condiional, Sochasic and Realized Volailiy Differen volailiy models provide differen esimaes and forecass of risk, while differen daa frequencies lead o differen choices of volailiy model. (i) Condiional volailiy, which is ypically used o model monhly and daily 16

17 daa, is laen (see Li, Ling and McAleer (2002) and Bauwens, Lauren and Rombous (2006) for recen reviews). The ease of compuaion, even for some mulivariae models, as well as is availabiliy in several widely used economeric sofware packages, has made his class of models very popular. (ii) Sochasic volailiy is ypically use o model and forecas daily daa, and is also laen (see Asai, McAleer and Yu (2006) for a recen review of mulivariae sochasic volailiy models). A disinc advanage of sochasic volailiy models is he incorporaion of leverage, a leas a he univariae level. However, he compuaional burden can be quie severe, especially for mulivariae processes. Moreover, sandard economeric sofware packages do no ye seem o have incorporaed sochasic volailiy algorihms. (iii) Realized volailiy is observable, bu is subjec o measuremen error (see McAleer and Medeiros (2008a) for a recen review). Such models are used o calculae observed volailiy using ick daa. The generaed realized daily volailiy measures are hen ypically modelled using a wide variey of long memory or fracionally inegraed ime series processes. I should be emphasized ha microsrucure noise (or measuremen error) is a sandard problem ha arises where realized volailiy is used as an esimae of daily inegraed volailiy. (C2) Choose Wisely from Symmery, Asymmery and Leverage Disinguish carefully beween asymmery and leverage, and models ha incorporae asymmery and leverage, eiher by consrucion or hrough he use of parameric 17

18 resricions. Asymmery is a sraighforward concep, bu leverage seems o be he subjec of much misundersanding in pracice. Definiion: Asymmery capures he differen impacs of posiive and negaive shocks of equal magniude on volailiy. Definiion: Leverage capures he effecs of negaive (posiive) shocks of equal magniude on increasing (decreasing) he deb-equiy raio, hereby increasing (decreasing) subsequen volailiy and risk (see Black (1976)). Thus, leverage is a special case of asymmery, wih volailiy decreasing progressively as reurns shocks change progressively from negaive o posiive. I follows ha symmery is he absence of asymmery, including leverage. The widely used ARCH and GARCH models are symmeric. Several popular models of volailiy display asymmery, hough no necessarily leverage. For example, he GJR model of Glosen, Jagannahan and Runkle (1992) is asymmeric bu does no display leverage (see McAleer, Hoi and Chan (2008) for a mulivariae exension of he asymmeric GJR model, VARMA-AGARCH), while he EGARCH model of Nelson (1991) is asymmeric and can display leverage, depending on he signs of he coefficiens relaing o he size and sign effecs. In he conex of sochasic volailiy models, leverage is imposed hrough a negaive 18

19 correlaion beween reurns and subsequen volailiy shocks. There are many alernaive ypes of asymmery a he univariae and mulivariae levels (see, for example, Asai and McAleer (2005, 2008c) and McAleer and Medeiros (2008b)). Alhough realized volailiy models may be modified o incorporae boh inra- and iner-day leverage, his does no seem o have been accomplished o dae. Exensions of leverage o mulivariae processes are somewha more difficul. (C3) Choose Wisely beween Dynamic Correlaions and Dynamic Covariances Covariances and correlaions are used o model relaionships beween pairs of asses for porfolio risk managemen: Correlaions are used in he consrucion of a porfolio. Consider wo financial asses, X and Z: (a) correlaion (X,Z) +1 specialize on he asse wih he higher hisorical and/or expeced reurns; (b) correlaion (X,Z) -1 hedge (or diversify); (c) correlaion (X,Z) 0 need o balance risk and reurns. On he oher hand, dynamic variances and covariances are used o calculae he VaR of a given porfolio. As a mulivariae exension of equaion (3), consider he relaionship beween 19

20 covariances and correlaions, as follows: Q = D Γ D, (5) in which Q is he condiional covariance marix, Γ is he condiional correlaion marix, and D is he diagonal marix of condiional sandard deviaions, namely h i, where h i = i αiε i, 1 βihi, 1 ω, and i= 1,,m is he number of asses in he porfolio. The marix Q is used o calculae VaR forecass, while he marix Γ is used o consruc and updae porfolios. I should be emphasized ha Q can be modelled direcly, as in he BEKK model of Engle and Kroner (1995), or indirecly hrough modelling Γ, and using (5), such as using he dynamic condiional correlaion (DCC) model of Engle (1992), of which he consan condiional correlaion (CCC) model of Bollerslev (1990) is a special case (for an applicaion of he scalar BEKK versus indirec DCC models, see Caporin and McAleer (2008)). The BEKK model is given as follows: 20

21 Q ' = QQ + Aε ε A' BQ ', (6) ' B where Q, Q, A, B are m dimensional marices. I is clear ha he specificaion in (6) guaranees a posiive definie covariance marix. However, as he dimensions of he hree parameer marices, namely Q, A, B, are he same as for Q, a compuaional difficuly can arise frequenly in he BEKK model of Q as i suffers from he so-called curse of dimensionaliy in having far oo many parameers. I follows from (5) ha Γ = D Q D, (7) 1 1 so ha Γ can be modelled direcly, or indirecly hrough modelling Q. A parsimonious model of Γ is given in he DCC model, as follows: Γ = ' ( 1 1 θ2) + θ1η i, 1η i, 1 + θ2γ 1 θ, (8) where θ, θ 1 2 are scalar parameers. As he specificaion does no guaranee ha he elemens along he main diagonal are all uniy, and all of he off-diagonal erms lie in he range [-1, 1], Engle (2002) sandardizes he marix in (8) so ha he elemens saisfy he definiion of a condiional correlaion. 21

22 Alhough he DCC model is parsimonious in erms of parameers, a common empirical finding, especially for sock indexes, is ha he long run condiional correlaion marix is consan (namely, θ =, θ 1), wih he oucome being ha = news has lile pracical effec in changing he purporedly dynamic condiional correlaions. This is no alogeher surprising as, apar from he posiive diagonal elemens in a marix version of DCC (see, for example, he GARCC model of McAleer, Chan, Hoi and Lieberman (2008)), he off-diagonal erms of he coefficien marix can be posiive or negaive. The imposiion of an unrealisic consrain ha all he elemens of he marix are he same consan has he effec of making θ 1 very close o zero and θ 2 very close o uniy, especially for a porfolio wih a large numbers of asses. This pracical problem associaed wih DCC is alleviaed wih various exensions of he he model, such as GARCC. Dynamic correlaions have recenly been developed for mulivariae sochasic volailiy models using he Wishar disribuion (see Asai and McAleer (2008b)), bu he compuaional burden can be quie severe. To dae here does no seem o have been any research underaken on modelling he dynamic correlaions for pairs of sandardized residuals for mulivariae realized volailiy models. (C4) Choose Wisely beween Index and Porfolio Models Esimaion and forecasing of VaR hresholds of a porfolio requires esimaion and forecasing he variance and covariances of porfolio reurns. Volailiy models can be used o esimae he variance of porfolio reurns eiher by (1) fiing a univariae 22

23 volailiy model o he porfolio reurns (hereafer called he single index model (see McAleer and da Veiga (2008a, 2008b)); or (2) using a mulivariae volailiy model o forecas he condiional variance of each asse in he porfolio o calculae he forecased porfolio variance (hereafer called he porfolio model). Asai and McAleer (2008b) have exended he idea of a porfolio index model o a mulivariae GARCH process. The cenral issue is he signal o noise raio. Single index models require only a univariae model o calculae he variance of he single index, and hence require no covariances or correlaions. There is lile signal, bu here is also lile noise. Porfolio models have a lo of signal because of all he pairs of covariances and correlaions, bu here is also a lo of noise in he esimaed parameers or coefficiens. As an illusraion in he case of wo asses, le y = λx + ( 1 λ) z, where y = single index reurn o he porfolio of wo asses, x = reurn o financial asse, X z = reurn o financial asse, Z λx + ( 1 λ) = porfolio reurn on wo asses, z h y h = 2 ω + + y α yε y 1 β y y 1 23

24 h x 2 = ω x + α xε x 1 + β xhx 1 h h z xz 2 = ω z + α zε z 1 + β zhz 1 = xz + α xzε x 1ε z 1 + β xzhxz 1 ω. As he wo condiional variances and single covariance are esimaed, i follows ha: 2 2 h λ h + (1 λ) h + 2λ(1 λ) h, y x z xz even hough 2 2 var( y ) = λ var( x ) + (1 λ) var( z ) + 2λ(1 λ)cov( x, z ). Therefore, he single index and porfolio approaches can lead o differen resuls when he ime-varying variances and covariances are esimaed. The number of covariances increases dramaically wih m, he number of asses in he porfolio. Thus, for m = 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, he number of covariances is 1, 3, 6, 10, 45, 190, 435, 780, 1225 and 4950, respecively. This increases he compuaional burden significanly, unless some srucure can be imposed o increase parsimony. (C5) Choose Wisely from Parameric, Semiparameric and Nonparameric Models Condiional volailiy is laen, and condiional volailiy models are ypically 24

25 semiparameric or parameric. Sochasic volailiy is laen, and sochasic volailiy models are ypically parameric. Realized volailiy is observable, and realized volailiy models are nonparameric. Each ype of model ypically has an opimal mehod of esimaion, as follows: (1) Condiional volailiy models are ypically esimaed by he maximum likelihood (ML) mehod when he sandardized residuals are normally disribued, or by he quasi-maximum likelihood (QML) mehod when hey are no normal (for furher deails, see Li, Ling and McAleer (2002) or Bauwens, Lauren and Rombous (2006); (2) Sochasic volailiy models are ypically esimaed by he Bayesian Markov Chain Mone Carlo (MCMC), Mone Carlo Likelihood (MCL), Empirical Likelihood (EL), or Efficien Mehod of Momens (EMM) mehods (for furher deails, see Asai, McAleer and Yu (2006)); (3) Realized volailiy models are ypically esimaed by nonparameric mehods (for deailed analyses, see Zhang, Mykland and Aï-Sahalia (2005), Aï-Sahalia, Mykland and Zhang (2006), and Barndorff-Nielsen, Hansen, Lunde Shephard (2008); for recen reviews of he lieraure, see Bandi and Russell (2007) and McAleer and Medeiros (2008a)). (C6) Choose Wisely from Esimaion, Simulaion and Calibraion of Parameers 25

26 Riskmerics TM calibraes he parameers in boh univariae and mulivariae models, and hence is concerned wih forecasing volailiy. As here are no esimaed parameers, here are no sandard errors, and here can be no inference. Condiional volailiy models are concerned wih esimaion and forecasing. The saisical properies of consisency and asympoic normaliy for univariae and mulivariae condiional volailiy models are now well esablished in he lieraure (see, for example, Ling and McAleer (2002a, 2002b, 2003)). Sochasic volailiy models are concerned wih simulaion and forecasing. As he Bayesian MCMC, MCL, EL or EMM mehods are ypically used for esimaion, he small sample or asympoic properies of he esimaors are well known. Realized volailiy models are based on nonparameric esimaion mehods, and heir asympoic properies are well known. Large samples are ypically required for he use of nonparameric mehods. In analyzing ulra high frequency ick daa on financial reurns, very large samples are usually available. (C7) Choose Wisely from Assumpions, Regulariy Condiions and Saisical Properies I is essenial o disinguish beween assumpions and regulariy condiions ha are derived from he assumpions of he underlining model. Assumpions are required o obain he momen condiions (oherwise known as regulariy condiions), especially he second and fourh momens, as well as 26

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