Thinly Traded Securities and Risk Management

Transcription

1 Thinly Traded Securiies and Risk Managemen Alejandro Bernales, Dieher W. Beuermann and Gonzalo Corazar * Absrac Thinly raded securiies exis in boh emerging and well developed markes. However, plausible esimaions of marke risk measures for porfolios wih infrequenly raded securiies have no been explored in he lieraure. We propose a mehodology o calculae marke risk measures based on he Kalman filer which can be used on incomplee daases. We implemen our approach in a fixed-income porfolio wihin a hin rading environmen. However, a similar approach may be also applied o oher markes wih hinly raded securiies. Our mehodology provides reliable marke risk measures in porfolios wih infrequen rading. JEL codes: G11; G12; G32. Keywords: Incomplee Panels; Kalman Filer; Marke Risk; Risk Managemen; Thin Trading; Value-a-Risk. * Alejandro Bernales is a Banque de France (Financial Economics Research Divission, DGEI DEMFI RECFIN), and Universidad de Chile, e mail: alejandro.bernales@banque france.fr. Dieher W. Beuermann is a Iner American Developmen Bank, e mail: dieherbe@iadb.org. Gonzalo Corazar is a Ponificia Universidad Caólica de Chile, e mail: gcoraza@ing.puc.cl. The auhors hank Auguso Casillo, Juan Dixon, Nicolas Majluf, Jorge Vera, Marcela Valenzuela for heir commens on earlier versions of his paper. Addiionally, we would like o hank seminar/session paricipans a he Financial Managemen Associaion (Orlando), he Midwes Finance Associaion (Chicago) and he Annual Meeing of he Washingon Area Finance Associaion (Washingon DC USA). AN2 is par of he Briish Norh Wesern Grid). Gonzalo Corazar acknowledges parial financial suppor from Fondecy ( ) and from Grupo Securiy hrough Finance UC. The views expressed in his paper do no, necessarily, reflec he opinion of Banque de France, he Eurosysem or Sociéé Générale. All errors are ours. Elecronic copy available a: hp://ssrn.com/absrac=

2 Thinly Traded Securiies and Risk Managemen Absrac Thinly raded securiies exis in boh emerging and well developed markes. However, plausible esimaions of marke risk measures for porfolios wih infrequenly raded securiies have no been explored in he lieraure. We propose a mehodology o calculae marke risk measures based on he Kalman filer which can be used on incomplee daases. We implemen our approach in a fixed-income porfolio wihin a hin rading environmen. However, a similar approach may be also applied o oher markes wih hinly raded securiies. Our mehodology provides reliable marke risk measures in porfolios wih infrequen rading. JEL codes: G11; G12; G32. Keywords: Incomplee Panels; Kalman Filer; Marke Risk; Risk Managemen; Thin Trading; Value-a-Risk. 1 Elecronic copy available a: hp://ssrn.com/absrac=

3 1. Inroducion One of he iniial seps in any effecive risk managemen sraegy is o accuraely measure marke risks. Porfolio diversificaion, he ool mos ofen used o proec invesmens during financial crises, relies on i. Currenly, quaniaive mehods are commonly used in risk managemen; and here is a vas financial lieraure abou differen echniques o measure marke risks (e.g., Brien-Jones and Schaefer, 1999; Berkowiz and O Brien, 2002; and Guidolin and Timmermann, 2006). 2 However, all approaches assume a complee price daase; and hus hey do no ake ino accoun porfolios wih incomplee hisorical daa of prices due o infrequenly raded asses. 3 Infrequen (or hin) rading is pervasive and deeply affecs all financial markes worldwide. Infrequen rading is very common in emerging markes (e.g., Lim e al., 2009); bu i is also observable in some asses in well developed markes such as he NYSE (e.g., Roll e al., 2007), he Canadian sock marke (e.g., Boabang, 1996), or he sock opion marke a he CBOE (e.g. Chan e al., 2002). The infrequen rading problem has been ypically deal by praciioners hrough he replicaion of he price of las ransacion unil ha a new price appears (which is equivalen o assume ha he daily reurns are equal o zero for he days wihou prices). Neverheless his pracice could generae biases (e.g., Kallunki, 1997) or auocorrelaions which could affec he mean reversion processes of he asses (e.g, Mille e al., 1994). Surprisingly, however, he lieraure exploring marke risk measures for porfolios wihin an environmen of infrequen rading appears o be raher limied. Therefore, he main purpose of our sudy is o fill his gap by inroducing a mehodology o measure marke risks on porfolios wih hinly raded securiies. We use one of he mos popular marke risk measures in banking and finance: he Value-a- Risk (VaR). The VaR is a popular measure because i answers he following simple quesion: given he probabiliy α, wha is he expeced loss of an asse (or porfolio) over a ime inerval? The VaR has he propery ha risk is expressed in moneary unis, which is simple and easy o 2 See Duffie and Pan (1997); Manganelli and Engle (2001); or Chrisoffersen (2003) for a review of risk measures 3 Only risk measures based on opion implied volailiies do no use hisorical informaion (where he securiies ha need risk measures are he underlying asses of he opions). However, few asses have opions raded in opion markes. Moreover, he high volume raded on he underlying securiies (which is no a characerisic of asses wih infrequen rading) is one of he main selecion facors applied by opions exchanges o choose a securiy and hus o inroduce opion conracs using i as an underlying asse (see Mayhew and Mihov, 2004). Therefore, i is unlikely ha hinly raded asses have raded opions in opion exchanges.

4 undersand. In addiion, he VaR is no only used for porfolio managemen sraegies. The VaR is also paricularly criical in he financial secor where regulaory agencies periodically require financial insiuions o repor heir risk exposures, in order o se up heir minimum capial levels of reserves (e.g., Jackson e al., 1997; and Pérignona and Smih, 2010). 4 For insance, if he VaR is underesimaed, hen here are high chances of incurring large losses and penalies ha may come from regulaory agencies. Conversely, overesimaions of he VaR may lead financial insiuions o reain unnecessary reserves implying high coss of capial. We propose a mehodology based on hree sages. Firs, we obain a complee price daase using an asse pricing model esimaed by he Kalman filer wih he incomplee panel of prices. In his sage, we esimae he asse pricing model o characerize prices and hus o generae a complee hisorical daase of model prices and reurns. The Kalman filer is a well-known recursive mehod o esimae he parameers in dynamic models, which can also be used in environmens wih incomplee panels by a simple adjusmen. The mos imporan characerisic of he Kalman filer is ha we are able o obain an esimaion of a pricing model even for days wih very few price observaions. Second, we esimae he marke risk measures wih he complee panel daa of prices generaed in he firs sage. Since in he implemenaion of our mehodology we use VaR measures, we calculae hem by eigh differen mehods and in differen sub-periods as robusness checks. Third, we back-es he marke risk measures wih he original incomplee hisorical panel of observable prices o verify he reliabiliy of he esimaes. In our implemenaion, we should observe ha he percenage of imes in which losses exceed he calculaed VaR values o be close o he confidence level α a which he VaR measures were esimaed. We propose an ad-hoc procedure o make efficien use of all he available informaion in he back-esing process. The main advanage of he ad-hoc procedure is ha allows us o back-es he complee porfolio wih hinly raded asses; which is essenial for asse allocaion sraegies. 5 4 See Basel Commiee (1996a, 1996b) for a regulaory perspecive. 5 In our iniial aemps o find risk measures in a hinly raded environmen, we direcly used he Kalman filer o esimae he VaR measures wih he incomplee panel of marke prices (i.e. wihou using an asse pricing model o generae an iniial complee daase; or equivalenly wihou Sage I). However, our resuls improved significanly when we incorporaed an asse pricing model o obain he complee panel of prices (i.e. Sage I). This is due o he fac ha asse pricing models describe asses very well, since ha is heir objecive; which is very useful o characerize prices when here are no ransacions. The reason why asse pricing models are good o describe prices could be due o: he use of hese models by invesors (a self-fulfilling prophecy); because he prices are effecively well characerized by hem, or a combinaion of boh. 1

5 As an example of hinly raded securiies, o implemen our mehodology we use daa on Chilean governmenal bonds raded in he Saniago Sock Exchange o consruc a porfolio of 20 bonds wih differen mauriies. However, our approach can also be applied o oher markes where infrequen rading exiss, especially where pricing models can be esimaed by he Kalman filer. 6 In hose markes, risk managemen measures may be also calculaed for porfolios wih hinly raded asses by using he Kalman filer in a similar way o our approach. There are some sudies relaed o ours, bu hey do no invesigae marke risk measures in an environmen of hin rading. Moscadelli e al. (2005) and Chernobai e al. (2006) invesigae he problem of incomplee daa bu only from he perspecive of operaional VaR measures. Barholdy and Riding (1994), Marikainen e al. (1996), Boabang (1996), and Sercu e al. (2008) explore he influences of hin rading on asse pricing models. Anonios e al. (2002) and Lim e al. (2009) show how infrequen rading could affec marke efficiency. Corazar e al. (2007, 2012) sudy he erm-srucures of ineres raes and of corporae bond spreads, in scenarios of hin rading. Finally, Basse e al. (1991) and Jokivuolle (1995) presen economeric ools o calculae sock indexes in an environmen of incomplee panels of prices. Our mehodology offers reliable VaR measures for hinly raded markes using ou-of-sample (one-day ahead) ess. We obain comparable levels of VaR measures in relaion o previous sudies, which use similar markes o he one used in our implemenaion bu wih complee panel of prices (e.g., Kiesel e al., 2000; Fernandez, 2003; Bao e al., 2006; and Fernandes e al., 2008). Moreover, we show ha he bes VaR mehods o describe he marke risk in an environmen wih hinly raded asses are hose ha characerize he lef ail of he condiional disribuion modeling heeroskedasic financial reurn series (e.g. heeroskedasic exreme value mehods using he generalized Pareo disribuion). These resuls are also similar o he oucomes of earlier sudies wih complee panels of prices (e.g., Fernandez, 2003; Kueser e al., 2006; and Bao e al., 2006), which suppors he consisency of our mehodology. Consequenly, we conribue o he body of knowledge exploring marke risks in environmens wih porfolios including infrequenly raded asses. To he bes of our knowledge, here is no lieraure where marke risk measures have been invesigaed in he conex of hin rading. Therefore, he focus of our paper on he examinaion of asse risk managemen on porfolios wihin infrequen rading scenarios appears disincive. The paper is organized as 6 For insance, here are sudies abou pricing models ha use he sandard Kalman filer for sock markes (e.g. Brennan e al., 2005; and He e al., 2010); and for opion markes (e.g., Bedendo and Hodges, 2009). 2

6 follows. The daa used in he implemenaion of our mehodology is inroduced in Secion 2. Secion 3 presens he proposed mehodology and describes he implemenaion on a hinly raded porfolio. Secion 4 shows robusness checks evidencing he superioriy of he proposed mehodology wih respec o alernaive sraegies used o deal wih hin rading environmens. Lasly, conclusions are drawn in Secion The Daa We implemen our mehodology wih daily daa for he main governmenal bonds in he Chilean fixed income marke. We use semi-annual coupon bonds, called PRCs ( Pagare Reajusable con Cupones ) raded beween January 4, 1999 and December 30, 2005 (1749 rading days). PRCs are inflaion-proeced bonds issued by he Cenral Bank of Chile, which are raded a he Saniago Sock Exchange. 7 A porfolio comprising 20 PRC bonds wih differen mauriies ranging from one o weny years is creaed o implemen our mehodology. The porfolio is consruced wih he assumpion ha $10,000 is invesed in each of he 20 bonds, for a oal invesmen of $200,000. The porfolio is re-balanced daily so he $10,000 invesmen in each asse remains consan over ime. Table 1 illusraes he missing daa problem in his marke by presening he rading frequency of PRC bonds in our porfolio during he whole sample period. Trading frequency is defined as he number of days in which we have a leas one ransacion of a bond wih a specific mauriy over all available rading days. A rading frequency of 10% means ha a securiy is raded on average 25 days per year. In addiion, Table 2 presens a subsample of daily raded bond prices beween March 20, 2000 and May 15, 2000, where black spaces represen days on which he securiy was no raded. Table 2 provides a clear illusraion of he incomplee panel problem. [Inser Table 1 here] [Inser Table 2 here] 7 In pracice he inflaion adjusmens are achieved by expressing he coupons in a differen uni raher han he Chilean peso: he UF ( Unidad de Fomeno ), which is updaed daily using he previous monh s variaion of he Chilean consumer price index. 3

7 3. The Mehodology and Implemenaion on a Thinly Traded Porfolio Sage I: Generaion of a Complee Panel of Prices Our mehodology is based on hree sages which allow us o obain marke risk measures of porfolios in an environmen wih infrequenly or hinly raded asses. In he firs sage, we generae a complee hisorical daase of prices for he porfolio which are inpus necessary o calculae marke risks. Therefore, as a firs sep we esimae an asse pricing model using he Kalman filer o characerize prices. However, following Corazar e al. 2007, we adjus he sandard Kalman filer echnique o deal wih an incomplee panel of bond prices, as has also been done for oher markes like marke indices (Basse and Hodges, 2009) or commodiies (Corazar e al. 2006, 2008). In our implemenaion we use he Kalman filer o esimae on a daily basis a mulifacor dynamic erm-srucure model of he ineres rae using a rolling window. Tradiional esimaion approaches for models characerizing he erm-srucure of he ineres raes require a minimum number of differen observable bond prices a a given ime (i.e. bonds wih differen mauriies). 8 An example in which he uilizaion of radiional esimaion echniques (such as ordinary leas squares, OLS; generalized leas squares, GLS; among ohers) wih hinly raded securiies is precluded is when on a given day he number of parameers o be esimaed for an asse pricing model is larger han he number of he observable prices on he marke (i.e. we can fi infinie models o he daa). For example, he Svensson s parameric model (Svensson, 1994) for characerizing he erm-srucure of he ineres raes needs six parameers o be esimaed; herefore on days wih less han six differen raded bonds we canno ge an esimaion of his model. However, he Kalman filer allows an esimaion of he ermsrucure of he ineres raes even for days wih very few price observaions; since we use hisorical daa where new informaion is weighed more han older ones. 9 Once he ermsrucure of he ineres raes model is esimaed wih he incomplee daase, we use he daily 8 See, e.g., Nelson and Siegel (1987) and Svensson (1994) for parsimonious funcional srucures, Vasicek (1977), Cox e al. (1985), and Duffie and Kan (1996) for dynamics models; and McCulloch (1971) and Vasicek and Fong (1982) for spline curve-fiing models. 9 For a review of dynamic erm-srucure ineres rae models using he sandard Kalman filer (wihou infrequen rading) see Babbs and Nowman (1999); Geyer and Pichler (1999); and Chen and Sco (2003). 4

8 zero curves o compue discoun facors and hence prices of bonds in he invesmen porfolio generaing a complee panel of fair prices. 10 In he implemenaion, we use a hree-facor generalized Vasicek model o characerize he erm-srucure of he ineres raes, which is a mean-revering Gaussian specificaion of he insananeous spo ineres rae which exends he classic Vasicek s (1977) dynamic model. We selec his model for is simpliciy as a dynamic represenaion for he erm-srucure of he ineres raes, and for is flexibiliy given he hree facors ha allow diverse shape forms. The simpliciy of his model allows us o explain in a very simple way he Kalman filer echnique in an environmen of infrequen rading. 11 I is imporan o menion ha he Vasicek models assume a consan volailiy of he insananeous ineres rae, which could conradic he main purpose of finding risk measures ha depend of he volailiy of bond prices. However, we esimae each day his mulifacor dynamic erm-srucure model; and hus he volailiy of he ineres rae can change over ime. A reader can consider his procedure as an aemp of replicaing ypical pracices in he marke, such as he use of he Black and Scholes (1973) model ha assumes a consan volailiy of he underlying asse; neverheless praciioners obain hrough his opion model daily implied volailiies which change over ime, he srike price, and he ime-o-mauriy (see Goncalves and Guidolin, 2006). We esimae he model using a six-monh daily rolling window. Therefore, we obain a complee daa panel of fair prices ha sars on July 1999 because he firs 6 monhs of daa are used in he firs esimaion. To obain he fair prices, we calculae ineres raes for all mauriies on each day using he daily esimaed model. Therefore, using his esimaed yield curve, daily fair prices for all bonds in he porfolio are compued hrough he sum of he presen value of he coupons (See Appendix A for a mahemaical explanaion of he model and is esimaion). Sage II: Esimaion of Marke Risk Measures In he second sage, we calculae he marke risk measures. Risk measures are direcly relaed o he volailiy of gains and losses, herefore he relevan complee hisorical daase is he one 10 For a review of he Kalman filer see Harvey (1982), Davis (1982), and Simon (2006). 11 In addiion, our mehodology can be implemened wih oher dynamic models. For insance, here oher simple models for he erm-srucure of he ineres raes such as he model inroduced by Cox e al. (1985) or oher specificaions. In paricular, we did no use he Cox e al. (1985) model because he volailiy is muliplied by he squarer roo of he ineres rae ha implies ha ineres raes canno be negaive, which is a problem when using real insead of nominal ineres raes, as we are. 5

9 ha includes reurns (specifically, log-reurns). We use a hisorical panel of reurns calculaed wih only fair prices o esimae he marke risk measures. The use of daa from differen sources o calculae reurns (i.e. a mixed panel wih raded prices and fair prices on daes where asses are no raded) could imply addiional noise in he reurn values; and herefore a mixed daase could disurb marke risk esimaions. For insance, we show in Appendix B ha fair prices could be significanly biased in relaion o observable daa due o differen liquidiy premiums among securiies depending of liquidiy levels. 12 The reason is ha all asse pricing models, even hose esimaed dynamically like he one implemened in our sudy, on average replicae he behavior of prices. Therefore, consisenly liquid (illiquid) securiies migh be underesimaed (overesimaed) due o differences in he liquidiy premiums. For ha reason, reurns calculaed wih a mixed panel of fair and marke prices could generae addiional disurbances in he esimaions. However, in Appendix B we also show ha here are no biases beween reurns calculaed wih only fair prices in relaion o reurns calculaed wih only marke daa, since he levels issues are eliminaed. The inuiion behind unbiased reurns is simple. Even hough fair prices are biased, movemens or changes in fair and marke prices are going in he same direcion and following similar pahs (i.e. he dynamic model capures he marke movemens since i is esimaed daily). Consequenly, given ha he fair prices were biased bu fair reurns were no, hen i is beer o make use of reurns calculaed wih only fair prices o esimae he marke risk. 13 In he implemenaion of our mehodology we use he VaR as he marke risk measure. The VaR quanifies he risk of a loss in an invesmen wihin a ime inerval and for a given confidence level. More precisely, le w +, be he variaion in value of an invesmen resuling from he price variaion in he ime inerval, and f(w +, ) he unknown probabiliy densiy funcion of such variaions. The VaR of an asse (or porfolio of asses), is he amoun of money ha could be los from negaive evens which could occur wih a probabiliy α. Thus, he VaR can be compued as: P w VaR F (1) or, (,, ) 12 See Chen e al. (2007) for liquidiy effecs of bond premiums. 13 In Appendix B, we analyze he prices biases and he reurns unbiasness using he complee sample and subsamples as robusness checks. 6

10 VaR, f ( w, F ) dw (2) where F is he informaion se unil day. For example, if he VaR is calculaed a a confidence level of 5%, here is a chance of 5% ha an acual loss may exceed he value provided by he VaR. 14 There are hree main groups of mehods o calculae VaR. Firs, VaR mehods ha assume a known disribuion for he securiy reurns; second, he non-parameric VaR mehods; and hird, Mone Carlo simulaions. In he implemenaion of our mehodology, we use a leas one mehod from each group as a robusness check. 15 We use eigh differen mehods o esimae hese risk measures (he mehods are explained in deail in Appendix C): he VaR mehod of variancecovariance; he VaR mehod of exponenial decay (RiskMerics ); he VaR GARCH mehod; he VaR -suden disribuion mehod; he VaR exreme value heory mehod (saic version); he VaR exreme value heory mehod (dynamic version); he VaR hisorical simulaion mehod; and he VaR Mone Carlo simulaion mehod. Sage III: Back-Tesing wih an Incomplee Daa Se of Marke Prices In he hird sage, a back-esing procedure for he marke risk measures is required. However, a well performed back-es has o be based on he hisorical observable marke daa. In our implemenaion, he back-esing procedure for a VaR mehod lies in compuing he percenage of imes in which daily losses of an invesmen a ime have been larger han he one esimaed by VaR values a ime -1. This percenage should no be significanly differen from he confidence level α under which he VaR measures are calculaed. In our sudy, he backesing has wo main purposes. Firs, o observe he reliabiliy of he VaR measures obained wih he proposed mehodology. The analysis of he reliabiliy of our approach is a key issue since wih he back-esing we can analyze saisically wheher our mehodology can capure he marke risk. Moreover, a reader may inerpre he back-esing exercise as a way of esing our approach in a real environmen of infrequen rading. Secondly, he back-esing allows us o 14 For example, a VaR 5% = $-300,000 on an invesmen is equivalen o saying ha a loss of $300,000 (or more) can be expeced on five days ou of one hundred days. 15 See Duffie and Pan (1997), Manganelli and Engle (2001), and Jorion (2006) for addiional deails abou VaR measures. 7

11 deec he bes VaR mehod for characerizing he marke risk for he esed porfolio in an environmen wih hinly raded asses. In he back-esing, for example, a perfec VaR mehod calculaed daily wih 5% confidence level should repor ha 5% of he ime losses exceeded he values provided by he VaR esimaions in he previous day. Therefore, a proper back-esing procedure of he VaR implies an ou-of-sample (one-day ahead) esing process. We use he Kupiec (1995) es ha analyzes he hisoric percenage of losses exceeding he VaR. The null hypohesis saes ha, he proporion of losses beyond he VaR is equal o α. Consequenly, if he null hypohesis is no rejeced, he VaR mehod involved in he back-esing is inside a confidence inerval wih a confidence level β defined in he Kupiec (1995) es. In our implemenaion, as a firs sep, we back-es he VaR measures using he sandard approach inroduced in previous lieraure (e.g., Jorion, 2006; Prisker, 2006; and Kawaa and Kijima, 2007); and hen we propose an ad-hoc procedure ha makes use of all he available informaion. As we menioned previously, a well implemened back-esing has o be done wih hisorical prices raded in he marke. In he back-esing procedure for daily VaR measures, we need ses of wo successive days where prices are observed o accoun for daily gains or losses in he value of he securiy. However, in an environmen wih hinly raded asses his is unusual. For insance, an infrequenly raded securiy wih observable price on day bu nohing on day -1 implies ha in he ime series of reurns we do no have daa for days and -1 (i.e. no only a - 1). Consequenly, incomplee panels of hisorical prices generae more incomplee panels of hisorical reurns. This issue is even more complicaed in a porfolio wih various hinly raded asses because i is likely ha every day a leas one of hem does no have a raded price (see, e.g., Table 2). Therefore, wih incomplee panels of reurns and following radiional back-esing approaches, we could only es VaR mehods for each bond individually in he porfolio using he panel of observable daily reurns (i.e. reurns calculaed wih observable prices on days and +1). Neverheless, we propose an ad-hoc back-esing following a simple procedure ha makes efficien use of all available daa. Moreover, he main advanage of our ad-hoc procedure is ha a back-esing of he complee porfolio can be performed even wih a range of hinly raded asses. The back-es of he complee porfolio is an essenial par for he asse allocaion, since i is necessary o consider correlaions beween asses which are a fundamenal par of 8

12 diversificaion and risk managemen. This ad-hoc back-esing procedure is explained in he following paragraphs, and subsequenly we presen resuls applied o our implemenaion. 16 We assume ha an amoun M is invesed in a single asse on day, and during he subsequen d days he asse is no raded. Therefore, wo consecuive prices for he asse are observed a imes and +d+1 (i.e. P and P +d+1, respecively). The key issue is how o use his muli-day price reurn in order o compare i wih an esimae of a daily VaR measure. Le VaR +d,+d+1 be he one-day VaR measure under a confidence level α for he day +d+1 given all he informaion observable on he previous day F +d ; while w +d,+d+1 is he variaion in value beween +d and +d+1, hen: where P w VaR F d, d 1 w d, d 1 d (3) P d 1, d 1 M (4) P d d 1 We already have he value of VaR +d,+d+1 using he firs and second sages of our mehodology. However, we canno calculae w +d,+d+1 since on day +d he securiy does no have an observable price; and hence in he back-esing procedure we canno check if w +d,+d+1 is lower (or higher) han he VaR +1,+d+1. The basic idea of our ad-hoc back-esing procedure is o use he previous observable price P ogeher wih all he new informaion unil +d (F +d ) which is capured by he asse pricing model esimaed in he firs sage. Therefore, we calculae esimaes of P +d using he prior rade price P given F +d as: P Pˆ ~ d d d ln P d P d P ln Pˆ F (5) Pˆ Pˆ where Pˆ and P ˆ are fair prices. 17 Consequenly, d 16 See Hyung and de Vries (2007) for porfolio diversificaion effecs on VaR measures. 17 We empirically suppor his assumpion in Appendix B. 9

13 P d d M (6), 1 Pˆ d P Pˆ w~ 1 d 1 I is ineresing o observe ha w ~ d, d 1 is calculaed wih wo observable prices (P and P +d+1 ) joinly wih all he informaion available on he day +d given by F +d. This ad-hoc procedure allows us o compare daily VaR +d,+d+1 wih esimaes of he daily gains (or losses) w ~ which incorporae all he informaion a each insan. 18 d, d 1 Table 3 presens resuls of he back-esing for differen VaR mehods wih α=5% in our implemenaion using he porfolio of Chilean bonds for he complee sample. Table 3 Panel A shows he resuls of he back-esing for VaR mehods esimaed individually for each bond using daily reurns calculaed wih he original panel of observable prices (i.e. daily reurns are calculaed wih only successive prices ha ake place a and +1). Table 3 Panel B and Panel C repor he resuls of he back-esing procedure using our ad-hoc approach for each bond individually and for he complee porfolio, respecively. We also include hree addiional measures for back-esing purposes. We calculae he average VaR over he whole ime period. The average VaR is paricularly relevan for regulaed insiuions which are required o mainain capial levels ha dependen on heir repored VaR measures (e.g. commercial banks). For hese insiuions, a large average VaR implies high coss of capial; in conras, lower values could imply a rejecion of he null hypohesis in he Kupiec es and herefore penalies from regulaory agencies. Furhermore, we include he average excess over he VaR, which is similar o he Condiional Value-a-Risk (CVaR). The CVaR is he average loss condiional on losses greaer han he calculaed VaR; and hus he CVaR exceeds our measure (he average excess over VaR) by an amoun equal o he average VaR. 19 Addiionally, we include he maximum excess over VaR o see how caasrophic such even could be relaive o VaR esimaes. In Appendix D Table D.1 and Table D.2, we repor he same analysis as Table 3 bu using wo differen sub-samples as robusness checks. [Inser Table 3 here] 18 Of course, in he case of observing prices on wo successive rading days (i.e. d = 0), we calculae he w,+1 in he radiional way as w P / P 1]. d, d 1 [ 1 M 19 For a review of he CVaR see Rockafellar and Uryasev (2000, 2002). 10

14 Table 3 Panel A shows ha he resuls observed in ha back-esing are similar o hose presened using our ad-hoc procedure for individual bonds and he complee porfolio in Table 3 Panel B and Panel C, respecively. However, he ad-hoc procedure makes more efficien use of all he informaion available a each insan. Moreover, he ad-hoc procedure allows he backesing of he complee porfolio which is exremely imporan o evaluae he risk of all asses a he same ime; and hus aking ino accoun asse correlaions. For insance, he average VAR for he porfolio (see Table 3 Panel C) is lower han he average VAR for individual bonds muliplied by 20 (see Table 3 Panel A and Panel B) due o he effec of correlaions. In addiion, Table 3 shows ha he bes VaR mehods for a confidence level of 5% are he GARCH(1,1), he RiskMerics, and he dynamic Exreme Value Theory (EVT) mehods, in which he percenage of losses exceeding he VaR are very close o he 5% confidence level required. The variancecovariance mehod offers a poor performance in which he null hypohesis of no differences beween α and he proporion of losses under he VaR is rejeced by he Kupiec es. Furhermore, Table 3 shows ha he lowes average VaR values are provided by he EVT mehods; and hus hey should be aracive for insiuions ha are concerned abou effecive bu low VaR esimaions implying lower capial requiremens. Table 3 also repors ha he average excess over he VaR of he dynamic EVT, he Mone Carlo, and GARCH(1,1) mehods are slighly beer han ohers. The laer mehods provide relaively smaller excesses over he VaR showing ha hese have performed well in capuring marke dynamics. Table 4 repeas he back-esing analysis presened in Table 3 bu for VaR esimaions a he 1% confidence level wih our complee sample. Table D.3 and Table D.4 in Appendix D shows he same analysis as Table 4 bu using wo differen sub-samples o check for robusness. In Table 4 and similar o he resuls observed in Table 3, he back-esing calculaed by he sandard approach (Table 4 Panel A) repors comparable resuls o our ad-hoc procedure (Table 4 Panel A and Panel B). Addiionally, we can observe ha he only mehod which is no rejeced by he Kupiec es is he dynamic EVT. This means ha for pracically all he VaR mehods, he average percenage of excesses over he VaR is significanly differen from 1%. The poor performance of diverse VaR mehods, wih excepion of he dynamic EVT a he 1% level, suggess he imporance of adequaely modeling he lef ail of he reurn disribuions. However, he lef ail is no he unique imporan feaure given ha, in such case, he saic EVT or he hisorical simulaion mehods should also offer good resuls. Therefore, i appears ha i is also imporan 11

15 o accoun for he ime varying volailiy of reurns. For insance, Table 4 shows ha alhough he GARCH(1,1) mehod was rejeced by he Kupiec es, his mehod performs relaively beer han mehods such as he saic EVT and hisorical simulaion. [Inser Table 4 here] I is worh noing ha he poor performance of VaR mehods a he 1% confidence level has already been documened in several sudies for boh emerging and developed markes, in which (as is he case wih our sudy wih he dynamic EVT) he excepions are he mehods ha ake ino accoun boh he lef ail behavior and he heeroskedasiciy of he reurns. Fernandez (2003) finds ha he dynamic EVT mehods perform bes using Chilean marke daa (bu wih a complee panel of prices). Fernandes e al. (2008) analyze VaR measures for 41 counries, where hey show he superior performance of EVT mehods. In addiion, Kiesel e al. (2000) provide an analysis for emerging markes using Brady bonds reporing similar measures o hose presened in our sudy for VaR measures a 5% and 1% confidence levels. Finally, Bao e al. (2006) evaluae VaR models for Asian emerging markes (Korea, Indonesia, Malaysia, Taiwan, and Thailand). Their conclusions are also similar o ours for boh 5% and 1% confidence levels. 20 In summary, he back-esing procedure suggess ha our mehodology provides accurae measures o capure he marke risk, which was performed in a real environmen of infrequen rading using Chilean governmenal bonds. Moreover, alhough previous sudies did no address he problem of incomplee panels of prices, heir resuls regarding he alernaive VaR mehods a differen confidence levels are similar o hose presened in our research supporing he reliabiliy of our mehodology. 4. Robusness Checks The mehodology of repeaing he las price As menioned earlier, a common pracice in order o calculae VaR measures wih infrequen rading is o replicae he asse s las price unil a new ransacion is observed. Therefore, his 20 Also noice ha in environmens wih hin rading becomes harder o have exreme values and, herefore, ail esimaes (a he 1% level or lower) migh no be reliable by consrucion. 12

16 secion provides evidence on he reliabiliy of such pracice vis-à-vis our proposed mehodology. Table 5 replicaes he saisics (a he 5% confidence level) presened before bu using he las raded price for each asse unil a new ransacion is observed. 21 [Inser Table 5 here] Panel A of Table 5 evidences ha virually all mehods are rejeced by he Kupiec es. This conrass wih Table 3 where using our mehodology provided reliable VaR measures for all mehods wih he only excepion of he variance-covariance one. In addiion, average VaRs and excesses over he VaR are also larger han he ones produced wih he proposed mehodology. A similar picure emerges from Panels B and C. Therefore, i is apparen ha he common pracice of replicaing he las raded prices in order o deal wih infrequen rading problems is misleading. The evidence presened clearly suggess he superioriy of he proposed mehodology in order o deal wih missing daa owards calculaing VaR measures. 22 Changing he rolling window in he esimaion of he erm-srucure model So far we have been using a six monh rolling window for he esimaion of he dynamic erm-srucure model in Sage I. However, we now explore how our mehodology performs under differen lenghs of he rolling window. We firs apply he mehodology wih a narrower window of hree monhs. Noice ha by shorening he window we increase by hree monhs he panel of daily fair prices generaed via he dynamic model. This provides us wih more observaions for he VaR calculaions bu a he cos of less precise esimaes for he dynamic model. Table 6 displays he resuls for a 5% confidence level. The resuls are consisen wih Table 3 suggesing ha he GARCH(1,1), RiskMerics, and he dynamic EVT mehods perform relaively beer. In addiion, we also find ha he variance-covariance mehod offers he poores performance. However, Panel B also suggess rejecion of he saic EVT and he Mone Carlo mehod which performed relaively well when using a rolling window of six monhs (Table 3). 21 Noice ha we do no repor he resuls of he VaR mehod of variance-covariance (Var-Cov Marix) and he VaR Mone Carlo simulaion mehod (Mone Carlo) since in boh mehods we do no obain posiive-semidefinie marix due o he repeiion of prices 22 We also conduced a similar exercise using a 1% confidence level. The resuls rejeced all VaR mehods as well. 13

17 The laer observaion migh have resuled from he less precise esimaes obained for he ermsrucure model wihin a shorer rolling window for is esimaion. [Inser Table 6 here] Similarly, he rolling window could be exended. In ha case, we would gain precision in esimaing he dynamic erm-srucure model bu a he cos of having a smaller panel of fair prices o calculae he VaR measures. Therefore, we exend he rolling window o 12 monhs wih he resuls shown in Table 7. Overall we observe a similar paern favoring he GARCH(1,1), RiskMerics, and he dynamic EVT mehods. However, we also observe sronger rejecions for he res of mehods (Panel B). [Inser Table 7 here] Overall, varying he size of he rolling window used o esimae he dynamic erm-srucure model leads o resuls ha are consisen wih our baseline analysis. Mehods ha model boh he lef ail of he disribuion and he ime-varying volailiy like he GARCH(1,1), RiskMerics, and he dynamic EVT are consisen in reurning reliable VaR measures under differen formas of he rolling window. A Mone Carlo Analysis for differen levels of missing daa. The proposed mehodology has fared well wihin he Chilean bond marke. However, an imporan quesion is wheher he mehodology is exernally valid. To answer such quesion more research in differen rading markes is necessary. Bu o ake an iniial sep, we simulae a bond marke generaing daa hrough he esimaed erm-srucure model. Specifically, we simulae 100 years of daily daa wih forward vecors of he model s sae variables hrough is sochasic differenial equaion and using he average of he esimaed parameers (see Appendix A). In addiion, we also vary he rae of missing daa (i.e. non-rading days) considering wo scenarios, 60% and 30% raes of missing daa. 23 Tables 8 and 9 show he resuls for he 60% and 30% missing daa raes respecively. Boh ables are highly consisen suggesing a poor performance of he variance-covariance mehod 23 Noice ha our acual daa shows a 61.37% rae of missing daa. 14

18 and a relaively beer performance of he GARCH(1,1), RiskMerics, and he dynamic EVT mehods. 24 Overall, he resuls sugges ha our mehodology offers reliable VaR measures for a simulaed marke and for differen raes of infrequen rading. [Inser Table 8 here] [Inser Table 9 here] 5. Conclusion Infrequenly or hinly raded securiies exis in all markes around he world including boh emerging and developed markes. However, here is a lack of lieraure ha explores marke risk measures in porfolios wih hinly raded asses. We proposed a mehodology based on hree sages o calculae marke risks using incomplee hisorical panels of prices. Firs, we fi an asse pricing model esimaed by he Kalman filer using he incomplee daase o characerize all asses in a given porfolio and hus o obain a complee panel daa. Second, we esimae marke risk measures wih he new complee panel of prices. Third, we back-es he marke risk measures wih he observable incomplee hisorical panel of prices o check he reliabiliy of our mehodology. As an example of hinly raded securiies, we implemened our mehodology using Chilean governmenal bonds raded a he Saniago Sock Exchange. Neverheless, our approach can be applied o oher markes where infrequen rading is presen. Our mehodology o calculae marke risks in an environmen of hin rading is inuiive and flexible. We provide empirical evidence supporing ha our approach provides reliable VaR measures for infrequenly raded securiies. Our approach ouperformed he common pracice of replicaing he las raded price in order o calculae VaR measures. In addiion, we showed ha our mehodology was robus when varying he rolling window used o esimae he ermsrucure model and when applied o simulaed markes wih differen raes of non-rading days. Neverheless, here are oher imporan and ineresing issues ha we would like o invesigae in he fuure. For example, he esimaion of a complee porfolio managemen sraegy wih hinly 24 Noice ha he Mone Carlo VaR calculaion canno be direcly compared wih he one obained in Table 3 using he hisorical daa. This because in his exercise he arificial daa is being generaed wih he same dynamic model han he one used o calculae he VaR measure wih he Mone Carlo mehodology. 15

19 raded asses including marke risk measures ogeher wih asse allocaion mehods are beyond he scope of his paper. In addiion, possible sudies including oher measures of marke risk, implemenaions in oher markes, and he use of diverse securiies are lef for fuure research. Finally, we hope ha our sudy encourages furher research in hinly raded asses across differen areas of porfolio managemen. Appendix A: The Dynamic Term-Srucure Model and is Esimaion In he hree-facor generalized Vasicek model, hree sochasic unobservable mean-revering sae variables are defined and represened by he 3x1 vecor y. Then, he insananeous ineres rae, q, may be defined as: 1' y (A-1) q where δ is a consan. In addiion, he vecor of sae variables follows a mulifacor Vasicek-ype process governed by he sochasic differenial equaion: dy Ky d Σdw (A-2) in which K=diag(κ i ) and Σ=diag(σ i ) are 3x3 diagonal marices wih i represening he respecive sae variables; and dw is a 3x1 vecor of correlaed Brownian moion processes such ha: dw dw Ωd ' (A-3) Here, ρ i,j is one of he elemens of Ω which represens he insananeous correlaion beween he sae variables i and j. Under his specificaion, he sae variables have a mulivariae normal disribuion, and each of hem revers o zero wih a mean reversion rae given by κ i. Thus, from equaion (A-1) he insananeous ineres rae, q, revers o a long-run mean given by he consan δ. By assuming a consan 3x1 vecor of risk premiums, λ, he risk adjused process of he sae variables can be expressed as: dy λ Ky d Σdw (A-4) Addiionally, hrough sandard no-arbirage argumens we can obain he price of any purediscoun bond wih mauriy τ as: P y, exp u( )' y v( ) (A-5) 16

20 where and 1 exp( ki ) ui ( ) k N i 1exp( ki) v( ) i1 ki ki N N 1 i j ij 1exp( k ) 1 exp( kj) 1 exp( ( ki kj) ) i 2 i1 j1 kk i j ki kj ki k j i (A-6) (A-7) In addiion, he equivalen annualized spo rae, Q(y,τ), is: 1 Q( y, ) u( )' y v( ) (A-8) which is a linear funcion of he sae variables. Therefore, under he generalized Vasicek model spo raes also follow a Gaussian disribuion. Aferwards, we esimae his model by he Kalman filer, in which he unobservable sae variables conained in vecor y are calculaed recursively using all he informaion available unil ime. In he sae-space represenaion, he measuremen equaion ha relaes he vecor of observable variables b wih he vecor of unobservable sae variables y, is: b H y d v v ~ N( 0, Γ ) (A-9) We mus recall ha he sandard Kalman filer assumes a fixed number of observable variables a each ime. However, his assumpion can be relaxed in order o allow for missing observaions. Le m be he number of observaions available a ime, hen b is an m x1 vecor, H is an m x3 marix, d is an m x1 vecor, and v is an m x1 vecor of serially uncorrelaed Gaussian disurbances wih mean zero and covariance marix Γ wih dimensions m xm. The ransiion equaion, which describes he dynamics of he sae variables, may be wrien as: y A y 1 c ε ε ~ N(0, D ) (A-10) 17

21 where A is a 3x3 marix, c is a 3x1 vecor, and ε is a 3x1 vecor of uncorrelaed Gaussian noise wih mean zero and covariance marix D. 25 In he ieraive esimaion process of he Kalman filer, esimaes of he sae variables, ŷ, are obained recursively where J is he covariance marix of he esimaion errors: J ˆ ˆ ' E y y y y (A-11) Consequenly, in he ieraive process given all he informaion a -1, esimaes of he sae variables and he covariance marix of he esimaion errors a can be wrien as: yˆ c (A-12) Ayˆ 1 1 J ' 1 AJ 1A D (A-13) Equaions (A-12) and (A-13) are known as he predicion seps. In addiion when new informaion arrives a hrough he observable variables (b ), we can obain uncondiional esimaes of he sae variables and he covariance marix of he esimaion errors as follows: ' 1 J H F b H yˆ yˆ yˆ d (A-14) J J J H F H J (A-15) ' where F H J ' 1H Γ (A-16) As a final sep, he unknown model parameers represened by he vecor φ can be esimaed by maximizing he log-likelihood funcion of he error innovaions given by: 1 1 log L( φ) log F 1 (A-17) 2 2 ' 1 b H yˆ d F b H yˆ d 1 25 I is imporan o menion ha he Kalman filer needs ha a leas one securiy is raded; his issue is no a big consrain since he inexisence of a leas one ransacion is very unlikely (e.g. in our sample we have all days wih a leas one ransacion in he porfolio of 20 bonds). However, in he improbable case of a day wih zero ransacions i is possible o use he middle poins of he bid-ask spreads of he mos liquid securiies as he prices in he model. 18

22 We esimae he model using a six-monh daily rolling window. Therefore, we obain a complee daa panel from July 1999 because he firs 6 monhs are necessary for he firs esimaion. Table A.1 presens he averages of he mean value, he sandard deviaion, and he mean of sandard errors of parameer esimaes for he hree-facor generalized Vasicek ermsrucure dynamic model defined previously. The able shows average saisics of he daily mean reversion parameers (κ1, κ2, κ3); he volailiy parameers (σ1, σ2, σ3); he correlaion coefficiens of he sae variables (ρ1,2, ρ1,3, ρ2,3); he long-run mean of ineres raes δ; he marke prices of risk (λ1, λ2, λ3); and he sae variables conained in y. Even hough, Vasicek s (1977) dynamic model assumes a consan volailiy of he spo ineres rae and hus he volailiy parameers should be consan; he volailiy parameers change over ime which is refleced in heir sandard deviaions. The parameers of he model are no consan due o he fac ha we esimae each day his mulifacor dynamic erm-srucure model (and hence he spo volailiy of he ineres rae can also change over ime), which gives flexibiliy o our model o capure marke movemens ha are fundamenal for risk managemen. In addiion, i is imporan o observe ha on average he κ1 parameer esimae is nearly zero; his indicaes ha he firs facor is nearly a random walk and i explains why our esimae of 1 is close o zero. 26 [Inser Table A.1 here] Appendix B: Analysis of Prices and Reurns beween Fair and Marke Daa In his Appendix we presen relaionships beween fair and marke prices and beween fair and marke reurns. These analyses are useful o undersand how we should generae complee daa panels of hisorical prices. For insance, in he case ha fair prices and fair reurns are very close o marke daa, hen boh panels could be merged where fair prices are used when marke ransacions are no available; and hus laer o obain a complee daase of asse reurns. However, in he case ha fair prices are biased in relaion o he prices observed in he marke, bu fair reurns are no (as we will show in his Appendix), hen i is beer o make use of only fair prices o calculae he reurns insead of a mixed daase. 26 Noice ha we could have imposed a random walk facor. However, we allowed flexibiliy o he model as oher rading environmens migh no have shown he behavior found in our empirical applicaion. 19

23 Table B.1 Panel A provides summary saisics of measuremen errors beween fair and marke prices for he complee sample of 20 bonds. In addiion, Table B.1 Panel B repors summary saisics for errors beween fair and marke reurns calculaed wih only successive prices (i.e. o and +1). In Table B.1, we repor he mean of errors, he sandard deviaion of errors, -saisics, he mean of he absolue value of errors (MAE), and he roo mean of he squared errors (RMSE). Table B.1 Panel A show biases beween fair and marke prices; in which pracically all bonds have he mean of he errors beween fair and marke prices significanly differen from zero. The reason is ha he dynamic erm-srucure of he ineres raes model on average replicaes he behavior of opion prices. However, fair prices for liquid bonds (see Table 1), such as bonds wih mauriies of 07; 08; 19 and 20 years, are consisenly underesimaed by he model. By conras, illiquid bonds are overesimaed. Therefore, biases are due o he differences of liquidiy premiums among bonds. Neverheless, Table B.1 Panel B shows ha fair reurns are no saisically differen from he observed marke reurns. Table B.1 Panel B suggess ha, even hough fair prices are biased, he movemens or changes of he complee marke erm-srucure of he ineres raes are well represened by he movemens of he fair erm-srucure of he ineres raes esimaed hrough he hree-facor dynamic erm-srucure model. This is mainly due o he fac ha we esimae on a daily basis he mulifacor ermsrucure model of he ineres raes, which allow us o capure he movemens of he ineres raes over ime. Thus, he evidence suggess ha: P ln P 1 Pˆ ln Pˆ 1 (B-1) ˆ where Pˆ and P 1 are fair prices. We repea he same analysis repored in Table A.1 bu using wo sub-samples in Table B.2 and Table B.3. Table B.2 and Table B.3 shows he consisency of he resuls of Table B.1. Consequenly, in our implemenaion we assume ha fair reurns replicae very closely marke reurns. This assumpion is very imporan, given ha i is a necessary condiion o obain reliable VaR measures and is also useful for he ad-hoc back-esing procedure. [Inser Table B.1 here] [Inser Table B.2 here] 20

24 [Inser Table B.3 here] Appendix C: Value a Risk Measures Parameric Mehods Using Mulivariae Normal Disribuions The VaR mehod of variance-covariance The mehod of variance-covariance assumes a mulivariae normal disribuion of he log reurns, in which he variance-covariance marix is calculaed wih he hisorical reurns. In he case of a single asse, he VaR wih an α confidence level is: 2 ( u ) VaR e M 2, 1 1 (C-1) where u and σ 2 are he mean and variance of log-reurns, ψ is he inverse of a N(0,1) wih probabiliy α, and M is he amoun invesed. In addiion, equaion (C-1) is also used for he VaR of he porfolio, bu he porfolio variance is obained using he variance-covariance marix (i.e. σ 2 =ω Θω, where ω is he vecor of weighs for he differen asses of he porfolio, and Θ is he variance-covariance marix). In our implemenaion we use a rolling window of 252 days for he esimaions. The VaR mehod of exponenial decay (using he RiskMerics version) This mehod, which was popularized by J.P. Morgan in he nineies, is characerized by is simpliciy (see Longersaey and Spencer, 1996), where he elemens in he variance-covariance marix are calculaed using: 2 2 r r (C-2) RM i, 1 j, 1 i, j, RM i, j,