Bias in the Estimation of Non-Linear Transformations of the Integrated Variance of Returns. Richard D.F. Harris and Cherif Guermat

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1 Bias in he Esimaion of Non-Linear Transformaions of he Inegraed Variance of Reurns Richard D.F. Harris and Cherif Guerma Xfi Cenre for Finance and Invesmen Universiy of Exeer June 005 Absrac Volailiy models such as GARCH, alhough misspecified wih respec o he daa generaing process, may well generae volailiy forecass ha are uncondiionally unbiased. In oher words, hey generae variance forecass ha, on average, are equal o he inegraed variance. However, many applicaions in finance require a measure of reurn volailiy ha is a non-linear funcion of he variance of reurns, raher han of he variance iself. Even if a volailiy model generaes forecass of he inegraed variance ha are unbiased, non-linear ransformaions of hese forecass will be biased esimaors of he same non-linear ransformaions of he inegraed variance because of Jensen s inequaliy. In his paper, we derive an analyical approximaion for he uncondiional bias of esimaors of non-linear ransformaions of he inegraed variance. This bias is a funcion of he volailiy of he forecas variance and he volailiy of he inegraed variance, and depends on he concaviy of he non-linear ransformaion. In order o esimae he volailiy of he unobserved inegraed variance, we employ recen resuls from he realized volailiy lieraure. As an illusraion, we esimae he uncondiional bias for boh in-sample and ou-of-sample forecass of hree non-linear ransformaions of he inegraed sandard deviaion of reurns for hree exchange rae reurn series, where a GARCH(1,1) model is used o forecas he inegraed variance. Our esimaion resuls sugges ha, in pracice, he bias can be subsanial. Keywords: Inegraed volailiy; Realized volailiy; Non-linear ransformaion; Small sample bias; Generalised auoregressive condiional heeroscedasiciy. Address for correspondence: Professor Richard D. F. Harris, Xfi Cenre for Finance and Invesmen, Universiy of Exeer, Exeer EX4 4ST, UK. Tel: +44 (0) , Fax: +44 (0) R.D.F.Harris@exeer.ac.uk. The auhors would like o hank he edior and a referee, whose commens have undoubedly helped o improve he paper subsanially. We would also like o hank seminar paricipans a he Universiy of Exeer, he London School of Economics, Universiy of Reading, Singapore Managemen Universiy, Beijing Universiy, Hong Kong Bapis Universiy, he Chinese Universiy of Hong Kong and he Ciy Universiy of Hong Kong.

2 1. Inroducion One of he mos well esablished feaures of financial markes is ha hey display ime-varying volailiy. The ime-varying inegraed volailiy of financial asse reurns is inherenly unobservable. However, a range of volailiy models have been proposed in order o forecas inegraed volailiy and incorporae i ino boh heoreical and applied research. Economeric models of volailiy include he GARCH class of models, iniially proposed by Engle (198) and Bollerslev (1986), and subsequenly developed by many oher auhors, and Sochasic Volailiy. 1 More ad hoc approaches o forecasing inegraed volailiy include rolling esimaors of he sample variance and exponenially weighed moving average (EWMA) variance esimaors (see, for example, Alexander and Leigh, 1997). None of hese models is likely o capure he rue daa generaing process of financial asse reurns, and so hey can, a bes, be assumed o be ad hoc approximaions ha are essenially misspecified wih respec o he rue daa generaing process (see Nelson and Foser, 1994). Neverheless, volailiy models may well generae volailiy forecass ha are uncondiionally unbiased. In oher words, hey generae variance forecass ha are, on average, equal o he inegraed variance. This will be rue by consrucion for he rolling sample variance and EWMA models, and can be easily imposed on GARCH and Sochasic Volailiy models. This arises because volailiy models are usually specified as a funcion of squared (de-meaned) reurns, and so expecaions of he forecass ha hey generae are funcions of he uncondiional variance of reurns, which, by he law of ieraed expecaions, is equal o he expecaion of he inegraed variance. However, many applicaions in finance require a measure of reurn volailiy ha is a non-linear funcion of he variance of reurns, raher han of he variance iself. For example, esimaion of Value a Risk uses he square roo of he variance (see, for example, Jorion, 000), calculaion of he Sharpe raio uses he reciprocal of he square roo of he variance (see, for example, Sharpe, 1994), and he Black-Scholes 1 For a review of he GARCH lieraure, see Bollerslev, Engle and Nelson (1994). For a comprehensive empirical evaluaion of a wide range of GARCH models, see Hansen and Lunde (001). For a review of he Sochasic Volailiy lieraure, see, for example, Taylor (1994).

3 opion pricing model uses he cumulaive sandard normal funcion of he variance (see, for example, Black and Scholes, 1973). The usual approach in pracice is o forecas he inegraed variance of reurns using a volailiy model such as EWMA or GARCH, and o apply he same non-linear ransformaion o he forecas variance from his model. Even if a volailiy model generaes forecass of he inegraed variance ha are unbiased, non-linear ransformaions of hese forecass will be biased esimaors of he same non-linear ransformaions of he inegraed variance because of Jensen s inequaliy. The bias resuling from non-linear ransformaions of esimaors in general is well known in he academic lieraure, and in he case of he sandard deviaion of he normal disribuion, can be raced back o Gauss (Cureon, 1968; Markowiz, 1968). In he conex of forecasing, Granger and Newbold (1976) have analysed his problem and provide opimal bias correcions for a variey of non-linear ransformaions when he forecased variables are normally disribued. However, bias correcion for non-linear ransformaions of he inegraed volailiy of reurns poses special problems because he inegraed volailiy is unobservable. When esimaing non-linear ransformaions of he uncondiional variance, using, for example, he sample variance, he bias due o Jensen s inequaliy is no likely o be a problem in pracice, since samples sizes used in finance are ypically large. One can herefore rely on he fac ha while biased, non-linear ransformaions of he sample variance are consisen. However, models of ime-varying volailiy ypically use sample sizes ha are effecively much smaller, eiher explicily, as in he case of, say, a rolling sample variance esimaor, or implicily, hrough an exponenial weighing of he observaions, as in he case of EWMA or GARCH. Consequenly, when esimaing non-linear ransformaions of he ime-varying inegraed variance, he effec of Jensen s inequaliy is likely o be more imporan. One soluion, consisen wih he idea of maching he loss funcions a he esimaion and predicion sages (see Weiss, 1996; Chrisoffersen and Diebold, 1996; Chrisoffersen and Jacobs, 004), is o direcly forecas he ransformed volailiy iself. For example, he inegraed sandard deviaion can be forecased using a GARCH-based esimaor ha is specified in erms of he absolue value of lagged 3

4 reurns, raher han he square of lagged reurns (for example, he absolue value GARCH model of Taylor (1986) and Schwer (1989), and he NARCH model of Higgins and Bera (199)). However, he saisical properies of such forecass depend heavily on disribuional assumpions abou he underlying reurn series and hence he forecass of hese models will no in general be unbiased or consisen (see Nelson and Foser, 1994). In his paper, we derive an analyical approximaion for he uncondiional bias of esimaors of non-linear ransformaions of he inegraed variance. The bias is a funcion of he volailiy of he forecas variance and he volailiy of he inegraed variance, and depends on he concaviy of he non-linear ransformaion. For an opimally specified volailiy model, namely one ha generaes forecass of he inegraed variance ha are boh uncondiionally and condiionally unbiased, esimaes of non-linear ransformaions of he inegraed variance will be biased wih a sign ha is uniquely deermined by he non-linear ransformaion used. For volailiy models ha are uncondiionally unbiased, bu condiionally biased, he bias of esimaes of non-linear ransformaions of he inegraed variance can be eiher posiive or negaive, depending on he condiional bias of he volailiy model. In order o implemen he bias approximaion, we require an esimae of he volailiy of he unobserved inegraed variance, which can be obained using recen resuls from he realized volailiy lieraure. As an illusraion, we esimae he uncondiional bias for forecass of hree non-linear ransformaions of he inegraed sandard deviaion of reurns he sandard deviaion, he reciprocal of he sandard deviaion and he naural logarihm of he sandard deviaion for hree exchange rae reurn series, where a GARCH(1,1) model is used o forecas he inegraed variance. We consider boh in-sample and ouof-sample forecass, and boh one-day and en-day forecasing horizons. We employ he realized volailiy esimaes of Andersen, Bollerslev, Diebold and Labys (003) in order o esimae he variance of inegraed volailiy, and adjus hese using he correcion of Andersen, Bollerslev and Meddahi (005) o allow for he discree sampling frequency used o consruc he realized volailiy series. Our empirical resuls sugges ha, in pracice, he bias can be subsanial. For example, for insample forecass of he log sandard deviaion, we find ha he average bias across he 4

5 hree currencies is abou 6% for one-day forecass and 0% for en-day forecass. For he ou-of sample forecass, he esimaed bias is even higher, alhough his is parially aribuable o a single once-in-a-generaion move in he Japanese Yen. Excluding his observaion, he esimaed bias is abou 9% for one-day and 11% for en-day forecass. The ouline of his paper is as follows. In Secion we derive an expression for he uncondiional bias of esimaors of non-linear ransformaions of he inegraed volailiy and describe how i can be implemened using esimaes of realized volailiy. In Secion 3, we repor he resuls of he empirical illusraion. Secion 4 concludes.. Theoreical Framework Consider an asse wih a coninuous logarihmic price, p (), whose sample pah is governed by he sochasic differenial equaion, dp( ) = µ ( ) d + σ ( ) dw ( ) (1) where µ () is he insananeous drif, σ () is he insananeous sandard deviaion and W () follows a geomeric Brownian moion process wih cov( dw ( ), σ ( )) = 0. Suppose ha prices are observed a discree inervals = 1, K, T. The sochasic process governing he discreely observed logarihmic reurn, defined as r = p( ) p( 1), is given by r = µ + σ () z where z is a sandard normally disribued, serially uncorrelaed random variable and σ is he inegraed variance, defined by σ = σ ( s) ds (3) 1 5

6 (see, for example, Andersen and Bollerslev, 1998). The uncondiional variance of r is given by E( r ) = E( z σ ) = E( z = σ ) E( σ ) (4) Suppose ha a volailiy model such as EWMA or GARCH is used o generae forecass of he inegraed volailiy, σ, condiional on 1, he informaion se a ime 1. These forecass are denoed ˆ σ. We assume ha while he volailiy model does no represen he rue daa generaing process, i is well specified in he sense ha he variance forecass ha i generaes are unbiased, a leas uncondiionally. For ˆ σ o be uncondiionally unbiased, we mus have Ω E( ˆ σ ) E( = σ = σ ) (5) As noed in Secion 1, his condiion holds by consrucion for many condiional variance models, such as rolling window models and he EWMA model, and can be easily imposed on ohers, such as GARCH and SV models. A sronger requiremen of a volailiy model is ha i generaes forecass ha are also condiionally unbiased. A variance forecas, ˆ σ, is condiionally unbiased if E ˆ ( σ 1) = σ Ω (6) If σ were observable, he bias of σ could be analysed using he following regression: ˆ σ = a + bσˆ + v (7) 6

7 where E ( ) = 0 and E( ) σ ) = 0. A necessary and sufficien condiion for ˆ σ o be v v uncondiionally unbiased is ha a = ( 1 b) E( σ ) = (1 b) σ (see, for example, Taylor, 1999). A necessary condiion for ˆ σ be condiionally unbiased is ha, in addiion, a = 0 and b = 1. This is no a sufficien condiion because for σ o be ˆ condiionally unbiased we mus have E σ ˆ σ Ω ) = 0 ( 1. The (hypoheical) regression given by (7) ess wheher E( σ ˆ σ ˆ σ ) = 0 bu ˆ σ is only a subse of Ω 1. The propery of condiional unbiasedness is known as efficiency in he forecasing lieraure, and in paricular, when he only condiioning variable is he forecas iself, i is known as weak efficiency (see Nordhaus, 1987). The use of regression (7) for evaluaing economic forecass in general is due o Theil (1966) and Mincer and Zarnoviz (1969). In order o esimae a non-linear ransformaion of he inegraed volailiy, g( σ ), we assume ha he same non-linear ransformaion is applied o he forecas variance, g( ˆ σ ). The uncondiional bias of g( ˆ σ ) is given by E( g( ˆ σ ) g( σ )). This will no in general be equal o zero because of Jensen s inequaliy. However, he uncondiional bias of g( ˆ σ ) can be approximaed using second order Taylor series expansions for boh g( ˆ σ ) and g( σ ), around σ, which yield ˆ ˆ g ( σ ) g( σ ) + g'( σ )( σ σ ) + g''( σ )( σ σ ) (8) 1 ˆ and g ( σ ) g( σ ) + g'( σ )( σ σ ) + g''( σ )( σ σ ) (9) 1 where g '(.) and g ''(.) are he firs and second derivaives of g ( σ ) wih respec o σ. Taking he uncondiional expecaion of (8) and (9), we can herefore approximae he uncondiional bias of g( ˆ σ ) as 7

8 B = E[ g( ˆ σ ) g( σ )] 1 g''( σ ) { var( ˆ σ ) var( σ )} (10) For a well-specified volailiy model (namely one for which a = 0 and b = 1 ), var( ˆ σ ) var( ) and he sign of he uncondiional bias of g( ˆ σ ) is he opposie of σ ha of g ''( σ ). Thus, for example, he esimaed sandard deviaion, for which g ''( σ ) = (1/ 4) σ 3, will be biased upwards, while he esimaed reciprocal of he sandard deviaion, for which g ''( σ ) = (3/ 4) σ 5, will be biased downwards. For volailiy models ha generae forecass of he inegraed volailiy ha are uncondiionally unbiased bu condiionally biased (i.e. hose for which a = ( 1 b) σ bu wih b 1), he bias may be posiive or negaive, depending on he populaion values of he slope coefficien, b, and he R coefficien from regression (7). In paricular, if b R he bias would have he same sign as g ''( σ ), while if b R, he bias would have he sign opposie o ha of g ''( σ ). Expression (10) allows us o esablish he sign of he uncondiional bias of any paricular forecas of a non-linear ransformaion of g( σ ) bu does no readily provide a poin esimae of he bias. The uncondiional variance of ˆ σ can be esimaed by is sample variance, and g ''( σ ) can be esimaed using g ''( ˆ σ ), where ˆ σ is he sample variance of r. However, he inegraed volailiy, σ, is no observable, and so i is no possible o direcly esimae var( σ ). Forunaely, an esimae of σ can be based on a measure of realized volailiy, which is defined as 1 / h h, 1+ jh j= 1 RV h, = r (11) This can be seen by noing ha R = b var( ˆ σ ) / var( σ ) and hence expression (10) can be wrien as B (1/ ) g' ' ( ){ var( σ )(1 R / b )} σ. 8

9 where 1 / h is an ineger ha represens he sampling frequency of inra-period reurns and = p( ) p( ). Many sudies have used a regression analogous o equaion r h, h (7) in order o evaluae volailiy forecasing models in erms of heir condiional bias, employing RV 1, (namely he squared per-period reurn) as a measure of σ (see, for example, Pagan and Schwer, 1990; Day and Lewis, 199; Cumby, Figlewski and Hasbrouk, 1993; Jorion, 1995). Leas squares esimaion of his regression will yield consisen esimaes of he parameers a and b, since, by definiion, r σ + w, = where E( σ ) = 0, and so he error erm in his regression is uncorrelaed wih ˆ w σ. However, since var( r ) = var( σ ) + var( w ), using var( r ) in place of var( σ ) in he expression for he bias given by (10) will yield only an upper bound for he bias, and given he variabiliy of w, his is likely o be a relaively loose upper bound. Wih he growing availabiliy of inra-day daa on securiy prices, increasingly precise esimaes of inegraed volailiy, and hence increasingly igh bounds on he bias given by expression (10), can be obained using finer measures of RV h,. Moreover, recen significan heoreical advances have shown ha under very general condiions, RV, converges uniformly in probabiliy o σ as h 0 (see Andersen, Bollerslev, h Diebold and Labys, 003), and so, in principle, arbirarily precise esimaes of inegraed volailiy (and hence arbirarily igh bounds for he bias) can be obained by using inra-period reurns recorded a increasingly small inervals. However, he accuracy of such an approach is limied by he fac ha marke microsrucure effecs disor he measuremen of reurns a high frequencies in such a way ha measured reurns no longer saisfy he regulariy condiions ha are required for he consisency properies of realised volailiy. Moreover, recen research has shown ha reducing he inerval over which reurns are measured increases he amoun of noise in recorded reurns relaive o he informaion ha hey conain abou he underlying reurn volailiy (see Ai-Sahalia, Mykland and Zhang, 005; Zhang, Mykland and Ai- Sahalia; 003; Bandi and Russell, 003). Consequenly, some auhors have suggesed feasible esimaion of inegraed volailiy by sampling reurns a non-negligible ime inervals, wih inervals beween five minues and half an hour being common in pracice (Andersen, Bollerslev, Diebold, and Labys, 001; Barndorff-Nielsen and 9

10 Shephard, 00; Andersen, Bollerslev, Diebold, and Labys, 003). Such esimaes of inegraed volailiy, while considerably more accurae han daily squared reurns, however, sill conain an elemen of measuremen error. Ai-Sahalia, Mykland and Zhang (005) provide opimal sampling frequencies when microsrucure effecs are no accouned for, bu advocae using he smalles available inerval while explicily modelling he microsrucure noise. However, an alernaive soluion ha does no require he explici modelling of he microsrucure noise has been suggesed by Andersen, Bollerslev and Meddahi (005). In paricular, hey provide an approximaion for he variance of he unobserved inegraed volailiy based on he asympoic resuls of Barndorff-Nielsen and Shephard (003). They show ha for small h, var( σ ) var( RV, ) he[ RQ, ] (1) h h where RQ h, is he realized quariciy given by RQ h, 1 = h 3 1/ h j= 1 r 4 h, 1+ j. h (13) Thus, he approximaion for he bias given by equaion (10) can be wrien as { var( ˆ σ ) var( RV ) he( RQ )} 1 B g' ' ( σ ) h, + h, (14) or, as a proporion of he uncondiional expecaion of he ransformed inegraed variance, B 1 g' ' ( σ ) g( σ ) + { var( ˆ σ ) var( RVh, ) + he( RQh, )} g' ' ( σ ){ var( RV ) he( RQ )} 1 h, h, (15) Expression (15) herefore provides an approximaion for he uncondiional bias of forecass of non-linear ransformaions of he inegraed variance ha can be 10

11 esimaed using daa on inra-period reurns. This esimaed bias could be used o produce, for example, a bias-correced forecas of he non-linear ransformaion of he inegraed variance for a paricular volailiy forecasing model for a paricular series. In he following secion, we use expression (15) o evaluae he bias empirically. 3. Empirical Applicaion In his secion, we use expression (15) o esimae he uncondiional bias of esimaes of hree non-linear ransformaions of he inegraed variance he sandard deviaion, he reciprocal of he sandard deviaion and he naural logarihm of he sandard deviaion for hree exchange rae reurn series, where he commonly employed GARCH(1,1) model is used o forecas he inegraed variance. The sandard deviaion of reurns is imporan for a number of applicaions in finance, including he esimaion of Value a Risk (VaR). A common approach o he calculaion of VaR for a porfolio is o assume ha reurns are normally disribued condiional on daily volailiy, and o muliply he esimaed sandard deviaion of reurns by he appropriae quanile of he sandard normal disribuion (see, for insance, Jorion, 000). Clearly, if he esimaed sandard deviaion is biased, hen so oo will be he esimaed VaR, which under he Basle Commiee rules, will lead o eiher oo lile or oo much capial being held on average (see Basle Commiee, 1996). The reciprocal of he sandard deviaion is used in calculaion of he Sharpe raio, which is used for porfolio performance evaluaion (see, for example, Sharpe, 1994). Bias in he esimaed reciprocal of he sandard deviaion will resul in biased esimaes of he Sharpe raio and hence incorrec inferences abou porfolio performance. The daa ha we consider are he DEM/USD, JPY/USD and JPY/DEM exchange raes ha are analysed by Andersen, Bollerslev, Diebold and Labys (003) and Andersen, Bollerslev and Meddahi (005). Andersen, Bollerslev, Diebold and Labys have made available realized volailiy daa for hese series for boh one-day and enday reurns, for he period December 1986 o 30 December They have also made available he variance forecass from a range of volailiy models for hese series, including he GARCH(1,1) model. These forecass are divided ino an insample period and an ou-of-sample period. For he in-sample period, he GARCH(1,1) model is esimaed for he period 1 December 1987 o 1 December

12 and he esimaed model used o calculae in-sample variance forecass for his period (a oal of 3 forecass). The esimaed model from he firs period is hen used o generae ou-of-sample variance forecass for he period December 1996 o 30 June 1999 (a oal of 596 forecass). 3 In his paper, we consider boh one-day and en-day reurns, and use he same wo sub-samples. In order o implemen he bias approximaion derived in his paper, we also require daa on he underlying exchange rae reurns. We obained daily mid-raes for he DEM/USD, JPY/USD and JPY/DEM from Daasream for he period 30 November 1987 o 30 June From hese, we calculaed coninuously compounded reurns as he one-day difference in log raes for he one-day reurn, and he en-day difference in log raes for he en-day reurn. The reurn series was hen mached by dae wih he realized volailiy and GARCH(1,1) forecas series from Andersen, Bollerslev, Diebold and Labys (003). 4 Table 1 presens summary saisics for he reurn daa (Panel A), he realized volailiy daa (Panel B) and he GARCH(1,1) forecas daa (Panel C) for he one-day and en-day reurn series for he in-sample period, while Table repors he same saisics for he ou-of-sample period. Reurns for he hree series are lepokuric, and for he USD/JPY and DEM/JPY, are also significanly negaively skewed. The mean reurn in each case is very close o zero. For boh he insample period and he ou-of sample period, he realized variance series are highly lepokuric and posiively skewed. The forecas variance series from he GARCH(1,1) are also lepokuric and posiively skewed, bu wih much lower variabiliy han he realized variance series, and wih much lower kurosis. [Tables 1 and ] In order o esimae he uncondiional bias given by (14), we require esimaes of g ( σ ), g ' ' ( σ ), var( ˆ σ ), var( RV h, ) and E ( RQ h, ). For he sandard deviaion, g ( σ ) = σ and g ''( σ ) = (1/ 4) σ 3. For he reciprocal of he sandard deviaion, 1 g ( σ ) = σ and g ''( σ ) = (3/ 4) σ 5. For he naural logarihm of he sandard 3 See Andersen, Bollerslev, Diebold and Labys (003) for full deails of he calculaion of realized volailiy and he esimaion of he GARCH(1,1) model. 4 Andersen, Bollerslev, Diebold and Labys (003) omi a number of observaions from he full sample, including holiday periods and oher inacive days. 1

13 deviaion, g ( σ ) = ln( σ ) and g ''( σ ) = (1/ ) σ 4. These six erms are esimaed using he corresponding sample esimaors, which, alhough biased (owing o Jensen s inequaliy), are consisen. Moreover, any small sample bias should be negligible given he sample size. To esimae var( RV h, ), we use he sample variance of he daily RV h, series. The daa for RQ h, are no readily available, bu an esimae of E ( RQ h, ) can be impued from Andersen, Bollerslev and Meddahi (005), who analyse he same hree RV h, series. In paricular, Andersen, Bollerslev and Meddahi repor he R coefficien from a regression of model for RV h, on ˆ σ (where ˆ σ is generaed by an AR(4) RV h, ), and he R coefficien adjused for he approximaion given by equaion (1) above. Noing ha R var( ˆ unadjused = b σ ) / var( RVh, ) and R adjused = b var( ˆ σ ) /(var( RVh, ) he( RQh, )), E ( RQ h, ) can be calculaed as var( RV = h, ) Runadjused E ( RQh, ) 1 (16) h Radjused Noe ha his is independen of he model ha generaes ˆ σ. Panel D of Tables 1 and repor R unadjused, R adjused, he impued value of E ( RQ h, ) and he esimae of var( σ ) for each series, for he in-sample and ou-of-sample periods, respecively. [Tables 1 and ] Table 3 repors he esimaed uncondiional bias for he one-day and en-day insample forecass of he hree non-linear ransformaions of he inegraed variance, σˆ, ˆ σ 1 and ln( ˆ σ ), for he hree exchange rae reurn series. The esimaed uncondiional bias is in every case posiive for σˆ and ln( ˆ σ ) and negaive for ˆ σ 1, and he magniude of he bias is larger for one-day forecass han for en-day forecass. Generally, he bias is larges for ln( ˆ σ ) and smalles for σˆ, reflecing he relaive concaviy of he hree non-linear ransformaions. For σˆ, he esimaed bias ranges from 0.6% for he DEM/JPY en-day forecass o 1.3% for he USD/JPY one- 13

14 day forecass, while for ln( ˆ σ ), i ranges beween.0% for he DEM/JPY en-day forecass o 34.4% for he USD/JPY one-day forecass. [Table 3] Table 4 repors he esimaed bias for he one-day and en-day ou-of-sample forecass of σˆ, ˆ σ 1 and ln( ˆ σ ). For he USD/DEM, he ou-of-sample bias is similar in magniude o he in-sample bias, for boh he one-day and en-day forecass, bu for he USD/JPY and he DEM/JPY, he bias in he ou-of-sample one-day forecass is considerably higher han for he in-sample forecass. For example, for σˆ, he ou-ofsample bias is 61.7% for he USD/JPY and 6.0% for he DEM/JPY, while for ln( ˆ σ ), he bias is 84.9% and 6.5%, respecively. However, his is aribuable o a single massive oulier in realized volailiy, which Andersen, Bollerslev and Meddahi describe as a once-in-a-generaion move in he Japanese Yen, a day on which he realized volailiy increased o 41 imes is average value for he USD/JPY and 35 imes is average value for he DEM/JPY. 5 Excluding his observaion and one day eiher side yields much smaller esimaes of he bias. For example, for σˆ, he bias is reduced o 7.9% for he USD/JPY and 0.% for he DEM/JPY, while for ln( ˆ σ ), he bias is reduced o 16.0% and.0%, respecively. [Table 4] 4. Conclusion The ime-varying inegraed volailiy of financial asse reurns is inherenly unobservable. However, a range of volailiy models have been proposed in order o forecas inegraed volailiy, including GARCH, Sochasic Volailiy, rolling window and EWMA esimaors. These models, alhough almos cerainly misspecified wih respec o he daa generaing process, may well generae volailiy forecass ha are uncondiionally unbiased. In oher words, hey generae variance forecass ha, on 5 For background o his once-in-a-generaion move in he Japanese Yen, see Tokyo Socks Surge, Then Plunge Again; Yen Rally Hurs U.S. Treasury Bond Prices, Sandra Sugawara, The Washingon Pos, Washingon, D.C., 8 Ocober, 1998, page E.0. 14

15 average, are equal o he inegraed variance. However, many applicaions in finance require a measure of reurn volailiy ha is a non-linear funcion of he variance of reurns, raher han he variance iself. The usual approach in pracice is o forecas he inegraed variance of reurns using a volailiy model such as GARCH, and o apply he same non-linear ransformaion o he forecas variance. Even if a volailiy model generaes forecass of he inegraed volailiy ha are unbiased, non-linear ransformaions of hese forecass will, in general, be biased esimaors of he same non-linear ransformaions of he inegraed volailiy because of Jensen s inequaliy. In his paper, we derive an analyical approximaion for he uncondiional bias of esimaors of non-linear ransformaions of he inegraed variance. We show ha he bias is a funcion of he volailiy of he forecas variance and he volailiy of he inegraed variance, and depends on he concaviy of he nonlinear ransformaion. The volailiy of he inegraed variance is no observable, and so he approximaion canno be direcly implemened. However, we use recen resuls from he realized volailiy lieraure o esimae he volailiy of he inegraed variance. As an illusraion, we esimae he bias funcion for esimaes of hree nonlinear ransformaions of he inegraed variance of daily reurns for hree exchange rae reurn series, using a GARCH(1,1) model o esimae he inegraed variance. We find ha, in pracice, he bias can be subsanial. The resuls of his paper have imporan implicaions for any applicaion ha uses esimaes of a non-linear ransformaion of he inegraed variance. As noed above, for example, he posiive uncondiional bias in he esimaed sandard deviaion of reurns will lead o VaR esimaes ha are, on average, oo high, and hence invesmen banks will end o hold oo much risk capial. Similarly, he negaive bias in he reciprocal of he sandard deviaion will lead o Sharpe raio esimaes ha are oo high, on average, and hence an oversaemen of porfolio performance. An ineresing avenue for fuure research would be o evaluae he economic impac of he bias idenified in his paper, and o idenify he possible improvemens ha migh be achieved by correcing for his bias. 15

16 References Ai-Sahalia Y, Mykland P, Zhang L How Ofen o Sample a Coninuous-Time Process in he Presence of Marke Microsrucure Noise. Review of Financial Sudies 18: Alexander C, Leigh C On he Covariance Marices Used in Value a Risk Models. Journal of Derivaives 4: Andersen. TG, Bollerslev T Answering he Skepics: Yes. Sandard Volailiy Models Do Provide Accurae Forecass. Inernaional Economic Review Andersen. TG, Bollerslev T, Diebold FX, Labys P The Disribuion of Exchange Rae Realized Volailiy. Journal of he American Saisical Associaion 96: Andersen. TG, Bollerslev T, Meddahi N Correcing he Errors: Volailiy Forecas Evaluaion Using High-Frequency Daa and Realized Volailiies. Economerica 73: Andersen. TG, Bollerslev T, Diebold FX, Labys P Modeling and Forecasing Realized Volailiy. Economerica 71: Bandi F, Russell J Microsrucure Noise, Realized Volailiy, and Opimal Sampling. Working Paper. Universiy of Chicago. Graduae School of Business. Barndorff-Nielsen OE, Shephard N. 00. Economeric Analysis of Realized Volailiy and Is Use in Esimaing Sochasic Volailiy Models. Journal of he Royal Saisical Sociey B 64: Basle Commiee on Banking Supervision Overview of he Amendmen o he Capial Accord o Incorporae Marke Risks. January. Black F, Scholes M The Pricing of Opions and Corporae liabiliies. Journal of Poliical Economy 81: Bollerslev T Generalised Auoregressive Condiional Heeroskedasiciy. Journal of Economerics 31: Chrisoffersen P, Diebold F Furher Resuls on Forecasing and Model Selecion Under Asymmeric Loss. Journal of Applied Economerics 11: Chrisoffersen P, Jacobs K The Imporance of he Loss Funcion in Opion Valuaion. Journal of Financial Economics 7: Cumby R, Figlewski S, Hasbrouk J Forecasing Volailiy and Correlaions wih EGARCH Models. Journal of Derivaives. Winer Cureon EE Prioriy correcion o: Unbiased esimaion of he sandard deviaion American Saisician : 7. 16

17 Day TE, Lewis CM Sock Marke Volailiy and he Informaion Conen of Sock Index Opions. Journal of Economerics 5: Engle R Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion Economerica 50: Figlewski S Forecasing Volailiy. Financial Markes, Insiuions and Insrumens 6: Granger C, Newbold P Forecasing Transformed Series. Journal of he Royal Saisical Sociey B 38: Hansen P, Lunde A A forecas comparison of volailiy models: does anyhing bea a GARCH(1.1)? Journal of Applied Economerics. forhcoming. Hasbrouck J, Seppi D Common Facors in Prices, Order Flows, and Liquidiy. Journal of Financial Economics 59: Higgins M, Bera A A Class of Nonlinear ARCH Models. Inernaional Economic Review 33: Jorion P Predicing Volailiy in he Foreign Exchange Marke. Journal of Finance 50: Jorion P Value a Risk. McGraw Hill: New York. JP Morgan RiskMerics TM Technical Documen, fourh ediion. New York. Markowiz E Minimum Mean-Square-Error Esimaion of he Sandard Deviaion of he Normal Disribuion. American Saisician : 6. Meddahi N. 00. A Theoreical Comparison beween Inegraed and Realized Volailiy Journal of Applied Economerics 17: Mincer J, Zarnoviz V The Valuaion of Economic Forecass. In Economic Forecass and Expecaions. Mincer J (ed); Naional Bureau of Economic Research. Nelson D, Foser D Asympoic Filering Theory for Univariae ARCH Models. Economerica 6: Nordhaus W Forecasing Efficiency: Conceps and Applicaions. Review of Economics and Saisics 69: O Hara M Presidenial Address: Liquidiy and Price Discovery. Journal of Finance 58: Pagan A, Schwer GW Alernaive Models for Condiional Sock Volailiy. Journal of Economerics 45:

18 Schwer GW Sock Volailiy and he Crash of 87. Review of Financial Sudies 3: Sharpe W The Sharpe Raio. Journal of Porfolio Managemen. Fall Taylor J Evaluaing Volailiy and Inerval Forecass. Journal of Forecasing 18: Taylor S Modelling Financial Time Series. Wiley. Taylor S Modelling Sochasic Volailiy. Mahemaical Finance 4: Theil H Applied Economic Forecasing. Norh Holland. Weiss A Esimaing Time Series Models Using he Relevan Cos Funcion. Journal of Applied Economerics 11: Zhang L, Mykland P, Ai-Sahalia Y A Tale of Two Time Scales: Deermining Inegraed Volailiy wih Noisy High Frequency Daa. Working Paper. Carnegie Mellon Universiy. 18

19 Table 1 Summary Saisics: In-Sample Panel A: Reurns Mean Variance S deviaion Skewness Kurosis Panel B: Realized Variance Mean Variance S deviaion Skewness Kurosis Panel C: GARCH(1,1) Forecas Variance Mean Variance S deviaion Skewness Kurosis Panel D: Realised Quariciy R unadjused 1.9%.9% 36.1% 37.4% 35.5% 51.3% R adjused 31.4% 31.5% 45.0% 41.1% 38.6% 53.6% E ( RQ h, ) var( σ ) Noes: The sample period is December 1986 o 1 December The able repors he mean, variance, sandard deviaion, skewness and excess kurosis coefficiens of coninuously compounded daily reurns (Panel A), realized variance (Panel B) and in-sample GARCH(1,1) forecass (Panel C) for each series a he 1-day and 10-day horizons. Panel D repors he R and unadjused R saisics from adjused Andersen e al (005), he impued esimae of he expeced realized quariciy, E ( RQ h, ), and he esimaed variance of he inegraed variance, var( σ ). 19

20 Table Summary Saisics: Ou-of-sample Panel A: Reurns Mean Variance S deviaion Skewness Kurosis Panel B: Realized Variance Mean Variance S deviaion Skewness Kurosis Panel C: GARCH(1,1) Forecas Variance Mean Variance S deviaion Skewness Kurosis Panel D: Realised Quariciy R unadjused 15.8% 19.7% 18.9% 16.8% 18.7% 17.8% R adjused 0.0% 3.0% 1.5% 18.% 19.7% 18.6% E ( RQ h, ) var( σ ) Noes: The sample period is December 1996 o 30 June The able repors he mean, variance, sandard deviaion, skewness and excess kurosis coefficiens of coninuously compounded daily reurns (Panel A), realized variance (Panel B) and ou-of-sample GARCH(1,1) forecass (Panel C) for each series a he 1-day and 10-day horizons. Panel D repors he R unadjused and R adjused saisics from Andersen e al (005), he impued esimae of he expeced realized quariciy, E ( RQ h, ), and he esimaed variance of he inegraed variance, var( σ ). 0

21 Table 3 Esimaed Uncondiional Bias: In-Sample Forecass Panel A: σˆ σˆ (1/ 4) ˆ σ Bias E( σ ) Bias / E( σ ) 9.055% 1.335% 4.658% 3.11% 5.005% 0.608% Panel B: ˆ σ 1 ˆ σ (3 / 4) ˆ σ Bias E( σ ) Bias / E( σ ) % -.790% % % % % Panel C: ln( ˆ σ ) ln(σ ˆ) (1/ ) ˆ 4 σ Bias E(ln( σ )) Bias / E(ln( σ )) 9.931% % % 9.08% %.00% Noes: The able he esimaed uncondiional bias of in-sample forecass of σ (Panel A), ˆ σ 1 (Panel B) and ln( ˆ σ ) (Panel C). Each panel repors he sample esimaes of g ( σ ) and g ''( σ ), he esimaed bias using expression (14), he esimaed value of g ( σ ) and he bias as a proporion of his using expression (15). 1

22 Table 4 Esimaed Uncondiional Bias: Ou-of-Sample Forecass Panel A: σˆ σˆ (1/ 4) ˆ σ Bias E( σ ) Bias / E( σ ) 9.941% % 6.011% 3.89% 9.307% 1.756% Panel B: ˆ σ 1 ˆ σ (3 / 4) ˆ σ Bias E( σ ) Bias / E( σ ) % % % % % % Panel C: ln( ˆ σ ) ln(σ ˆ) (1/ ) ˆ 4 σ Bias E(ln( σ )) Bias / E(ln( σ )) 4.899% % 6.494% % 18.81% 3.563% Noes: The able he esimaed uncondiional bias of ou-of-sample forecass of σ (Panel A), ˆ σ 1 (Panel B) and ln( ˆ σ ) (Panel C). Each panel repors he sample esimaes of g ( σ ) and g ''( σ ), he esimaed bias using expression (14), he esimaed value of g ( σ ) and he bias as a proporion of his using expression (15).