MEASURING HISTORICAL VOLATILITY Close-to-Close, Exponentially Weighted, Parkinson, Garman-Klass, Rogers-Satchell and Yang-Zhang Volatility

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1 Equty Dervatves Europe Madrd, February 3, 01 MEASURIG HISTORICAL VOLATILITY Close-to-Close, Exponentally Weghted, Parknson, Garman-Klass, Rogers-Satchell and Yang-Zhang Volatlty Coln Bennett US nvestors enqures should be drected to Santander Investment Securtes Inc. (SIS at ( US recpents should note that ths research was produced by a non-member afflate of SIS and, n accordance wth ASD Rule 711, lmted dsclosures can be found on the back cover. Mguel A. Gl Equty Dervatves Strategy Head of Dervatves Strategy ( ( cdbennett@gruposantander.com mgl@gruposantander.com The mpled volatlty of an opton s usually compared aganst hstorcal volatlty to see f t s cheap or not. However, whle there s only one mpled volatlty there are many dfferent measures of hstorcal volatlty whch can use some or all of the open (O, hgh (H, low (L and close (C. Generally, for small sample szes the Yang-Zhang measure s best overall, and for large sample szes the standard close to close measure s best. CLOSE-TO-CLOSE (C: The smplest and most common type of calculaton that benefts from only usng relable prces from closng auctons. We note that the volatlty should be the standard devaton multpled by /(-1 to take nto account the fact we are samplng the populaton. EXPOETIALLY WEIGHTED (C: Exponentally weghted volatltes are rarely used, partly due to the fact they do not handle regular volatlty drvng events such as earnngs very well. Prevous earnngs jumps wll have the least weght just before an earnngs date, and the most weght just after earnngs. It could, however, be of some use for ndces. PARKISO (HL: The frst advanced volatlty estmator was created by Parknson n 1980, and nstead of usng closng prces t uses the hgh and low prce. One drawback of ths estmator s that t assumes contnuous tradng, hence t underestmates the volatlty as potental movements when the market s shut are gnored. Whle other measures are more effcent based on smulated data, some studes have shown ths to be the best measure for actual emprcal data. GARMA-KLASS (OHLC: Later n 1980 the Garman-Klass volatlty estmator was created. It s an extenson of Parknson whch ncludes openng and closng prces. As overnght jumps are gnored, the measure underestmates volatlty. Yang-Zhang modfed the Garman-Klass volatlty measure n order to enable t to handle jumps. ROGERS-SATCHELL (OHLC: The Rogers-Satchell volatlty created n the early 1990s s able to properly measure the volatlty for securtes wth non-zero mean. It does not, however, handle jumps (hence t underestmates the volatlty. YAG-ZHAG (OHLC: In 000 Yang-Zhang created the most powerful volatlty measure that handles both openng jumps and drft. It s the sum of the overnght volatlty (close to open volatlty and a weghted average of the Rogers-Satchell volatlty and the open to close volatlty. The assumpton of contnuous prces does mean the measure tends to slghtly underestmate the volatlty.

2 MEASURIG HISTORICAL VOLATILITY The mpled volatlty for a certan strke and expry has a fxed value. There s, however, no sngle calculaton for hstorcal volatlty. The number of hstorcal days for the hstorcal volatlty calculaton changes the calculaton, n addton to the estmate of the drft (or average amount stocks are assumed to rse. There should, however, be no dfference between the average daly or weekly hstorcal volatlty. We also examne dfferent methods of hstorcal volatlty calculaton, ncludng close-to-close volatlty and exponentally weghted volatlty, n addton to advanced volatlty measures such as Parknson, Garman-Klass (ncludng Yang-Zhang extenson, Rogers and Satchell and Yang-Zhang. CLOSE TO CLOSE HISTORICAL VOLATILITY IS THE MOST COMMO Volatlty s defned as the annualsed standard devaton of log returns. For hstorcal volatlty the usual measure s close-to-close volatlty, whch s shown below. c d Log return = x = Ln where d = ordnary (not adjusted dvdend and c s close prce c1 Volatlty 1 (not annualsed = σ x 1 ( x x 1 where x = drft = Average (x For relatvely short tme perods (daly, weekly, the drft should be close to zero and can be gnored BEST TO ASSUME ZERO DRIFT FOR VOLATILITY CALCULATIO The calculaton for standard devaton calculates the devaton from the average log return (or drft. Ths average log return has to be estmated from the sample, whch can cause problems f the return over the perod sampled s very hgh or negatve. As over the long term very hgh or negatve returns are not realstc, the calculaton of volatlty can be corrupted by usng the sample log return as the expected future return. For example, f an underlyng rses 10% a day for 10 days, the volatlty of the stock s zero (as there s zero devaton from the 10% average return. Ths s why volatlty calculatons are normally more relable f a zero return s assumed. In theory, the expected average value of an underlyng at a future date should be the value of the forward at that date. As for all normal nterest rates (and dvdends, borrow cost the forward return should be close to 100% (for any reasonable samplng frequency.e. daly/weekly/monthly. Hence for smplcty reasons t s easer to assume a zero log return as Ln(100% = 0. 1 We take the defnton of volatlty of John Hull n Optons, futures and other dervatves n whch n day volatlty uses n returns and n+1 prces. We note Bloomberg uses n prces and n-1 returns.

3 LOG RETURS CA BE APPROXIMATED BY PERCETAGE RETURS As returns are normally close to 1 (=100% the log of returns s very smlar to return 1 (whch s the percentage change of the prce. If the return over the perod s assumed to be the same for all perods, and f the mean return s assumed to be zero (t s normally very close to zero, the standard devaton of the percentage change s smply the absolute value of the percentage return. Hence an underlyng whch moves 1% has a volatlty of 1% for that perod. As volatlty s usually quoted on an annualsed bass, ths volatlty has be multpled by the square root of the number of samples n a year (.e. 5 for daly returns, 5 for weekly returns and 1 for monthly returns. umber of tradng days n year = 5 => Multply daly returns by 5 16 umber of weeks n year = 5 => Multply weekly returns by 5 7 umber of months n year = 1 => Multply monthly returns by WHICH HISTORICAL VOLATILITY SHOULD I USE? When examnng how attractve the mpled volatlty of an opton s, nvestors wll often compare t to hstorcal volatlty. However, hstorcal volatlty needs two parameters. Length of tme (e.g. number of days/weeks/months Frequency of measurement (e.g. daly/weekly Hstorcal volatlty should be a multple of 3 months to have a constant number of quarterly reportng perods LEGTH OF TIME FOR HISTORICAL VOLATILITY Choosng the hstorcal volatlty number of days s not a trval choce. Some nvestors beleve the best number of days of hstorcal volatlty to look at s the same as the mpled volatlty of nterest. For example, 1 month mpled should be compared to 1 tradng day hstorcal volatlty (and 3 month mpled should be compared to 63 day hstorcal volatlty, etc. Whle an dentcal duraton hstorcal volatlty s useful to arrve at a realstc mnmum and maxmum value over a long perod of tme, t s not always the best perod of tme to determne the far level of long dated mpleds. Ths s because volatlty mean reverts over a perod of c8 months. Usng hstorcal volatlty for perods longer than c8 months s not lkely to be the best estmate of future volatlty (as t could nclude volatlty caused by earler events, whose effect on the market has passed. Arguably a multple of 3 months should be used, to ensure that there s always the same number of quarterly reportng dates n the hstorcal volatlty measure. Addtonally, f there has been a recent jump n the share prce that s not expected to reoccur, the perod of tme chosen should try to exclude that jump. The Best Hstorcal Volatlty Perod Does ot Have to be the Most Recent If there has been a rare event whch caused a volatlty spke, the best estmate of future volatlty s not necessary the current hstorcal volatlty. A better estmate could be the past hstorcal volatlty when an event whch caused a smlar volatlty spke occurred. For example, the volatlty post credt crunch could be compared to the volatlty spke after the Great Depresson, or durng the burstng of the tech bubble. 3

4 FREQUECY OF HISTORICAL VOLATILITY Whle hstorcal volatlty can be measured monthly, quarterly or yearly t s usually measured daly or weekly. ormally, daly volatlty s preferable to weekly volatlty as 5 tmes as many data ponts are avalable. However, f volatlty over a long perod of tme s beng examned between two dfferent markets, weekly volatlty could be the best measure to reduce the nfluence of dfferent publc holdays (and tradng hours. If stock prce returns are ndependent, then the daly and weekly hstorcal volatlty should on average be the same. If stock prce returns are not ndependent, there could be a dfference. Autocorrelaton s the correlaton between two dfferent returns so ndependent returns have an autocorrelaton of 0%. Trendng Markets Imply Weekly Volatlty s Greater Than Daly Volatlty Wth 100% autocorrelaton, returns are perfectly correlated (a postve return s followed by a postve return,.e. trendng markets. Should autocorrelaton be -100% correlated then a postve return s followed by a negatve return (mean revertng or range tradng markets. If we assume markets are 100% daly correlated wth a 1% daly return, ths means the weekly return s 5%. The daly volatlty s therefore c16% (1% 5 whle the weekly volatlty of c35% (5% 5 s more than twce as large. Fgure 1. Stock Prce wth 100% Daly Autocorrelaton Stock Prce wth -100% Daly Autocorrelaton % autocorrelaton (weekly vol > x daly vol % autocorrelaton (weekly / monthly vol Days Days Source: Santander Investment Bolsa estmates. Hgh Market Share of Hgh Frequency Tradng Should Prevent Autocorrelaton Hstorcally (decades ago, there could have been postve autocorrelaton due to momentum buyng, but once ths became understood ths effect s lkely to have faded. Gven the current hgh market share of hgh frequency tradng (accountng for up to three-quarters of US equty tradng volume, t appears unlkely that a smple tradng strategy such as buy f securty goes up, sell f t goes down wll provde above average returns over a sgnfcant perod of tme. Pancked Markets Could Cause Temporary egatve Autocorrelaton Whle postve autocorrelaton s lkely to be arbtraged out of the market, there s evdence that markets can overreact at tmes of stress as market panc (rare statstcal events can occur under the weak form of effcent market hypotheses. Durng these events human traders and some automated tradng systems are lkely to stop tradng (as the event s rare, the correct response s unknown, or potentally exaggerate the trend (as postons get stopped out or to follow the momentum of the move. A strategy that s long daly volatlty and short weekly volatlty wll therefore usually gve relatvely flat returns, but occasonally gve a postve return. 4 Advanced volatlty measures could be used to remove part of the effect of dfferent tradng hours

5 Advanced volatlty measures should be used by traders wshng to take nto account ntraday prces ITRADAY VOLATILITY IS OT COSTAT For most markets, ntraday volatlty s greatest just after the open (as results are often announced around the open and just before the close (performance s often based upon closng prces. Intraday volatlty tends to sag n the mddle of the day, due to the combnaton of a lack of announcements and reduced volumes/lqudty due to lunch breaks. For ths reason usng an estmate of volatlty more frequent than daly tends to be very nosy. Traders who wsh to take nto account ntraday prces should nstead use an advanced volatlty measure. Fgure. Intraday Volatlty Intraday volatlty 30% 8% 6% 4% % 0% Volatlty tends to be greatest at the open, but also rses nto the close 18% 16% Tme Source: Santander Investment Bolsa. Exponentally weghted movng average can be used to reduce effect of spkes n volatlty dsappearng EXPOETIALLY WEIGHTED VOLATILITIES ARE RARELY USED An alternate measure could be to use an exponentally weghted movng average model, whch s shown below. The parameter λ s between zero (effectvely 1 day volatlty and 1 (gnore current vol and keep vol constant. ormally, values of c0.9 are used. Exponentally weghted volatltes are rarely used, partly due to the fact they do not handle regular volatlty drvng events such as earnngs very well. Prevous earnngs jumps wll have least weght just before an earnngs date (when future volatlty s most lkely to be hgh, and most weght just after earnngs (when future volatlty s most lkely to be low. It could, however, be of some use for ndces. 1 ( 1 x Exponentally Weghted Volatlty Avods Volatlty Collapse of Hstorc Volatlty Exponental volatlty has the advantage over standard hstorcal volatlty n that the effect of a spke n volatlty gradually fades (as opposed to suddenly dsappearng causng a collapse n hstorc volatlty. For example, f we are lookng at the hstorcal volatlty over the past month and a spke n realsed volatlty suddenly occurs the hstorcal volatlty wll be hgh for a month, then collapse. Exponentally weghted volatlty wll rse at the same tme as hstorcal volatlty, and then gradually declne to lower levels (arguably n a smlar way to how mpled volatlty spkes, then mean reverts. 5

6 Volatlty measures can use open, hgh and low prces n addton to closng prce ADVACED VOLATILITY MEASURES Close-to-close volatlty s usually used as t has the beneft of usng the closng aucton prces only. Should other prces be used, then they could be vulnerable to manpulaton or a fat fngered trade. However, a large number of samples need to be used to get a good estmate of hstorcal volatlty, and usng a large number of closng values can obscure short-term changes n volatlty. There are, however, dfferent methods of calculatng volatlty usng some or all of the open (O, hgh (H, low (L and close (C. The methods are lsted n order of ther maxmum effcency (close to close varance dvded by alternatve measure varance. Close to close (C: The most common type of calculaton that benefts from only usng relable prces from closng auctons. By defnton ts effcency s 1 at all tmes. Parknson (HL: As ths estmate only uses the hgh and low prce for an underlyng, t s less senstve to dfferences n tradng hours. For example, as the tme of the EU and US closes are approxmately half a tradng day apart, they can gve very dfferent returns. Usng the hgh and low means the tradng over the whole day s examned, and the days overlap. As t does not handle jumps, on average t underestmates the volatlty, as t does not take nto account hghs and lows when tradng does not occur (weekends, between close and open. Although t does not handle drft, ths s usually small. The Parknson estmate s up to 5. tmes more effcent than the close to close estmate. Whle other measures are more effcent based on smulated data, some studes have shown t to be the best measure for actual emprcal data. Garman-Klass (OHLC: Ths estmate s the most powerful for stocks wth Brownan moton, zero drft and no openng jumps (.e. openng prce s equal to closng prce of prevous perod. Whle t s up to 7.4 tmes as effcent as the close to close estmate, t also underestmates the volatlty (as lke Parknson t assumes no jumps. Rogers-Satchell (OHLC: The effcency of the Rogers-Satchell estmate s smlar to that for Garman-Klass, however t benefts from beng able to handle non-zero drft. Openng jumps are, however, not handled well, whch means t underestmates the volatlty. Garman-Klass Yang-Zhang extenson (OHLC: Yang-Zhang extended the Garman- Klass method that allows for openng jumps hence t s a far estmate, but does assume zero drft. It has an effcency of 8 tmes the close to close estmate. Yang-Zhang (OHLC: The most powerful volatlty estmator whch has mnmum estmaton error. It s a weghted average of Rogers-Satchell, the close-open volatlty and the open-close volatlty. It s up to a maxmum of 14 tmes as effcent (for days of data as the close to close estmate. Fgure 3. Summary of Advanced Volatlty Estmates Estmate Prces Taken Handle Drft? Handle Overnght Jumps? Effcency (max Close to close C o o 1 Parknson HL o o 5. Garman-Klass OHLC o o 7.4 Rogers-Satchell OHLC Yes o 8 Garman-Klass Yang-Zhang ext. OHLC o Yes 8 Yang-Zhang OHLC Yes Yes 14 Source: Santander Investment Bolsa. 6

7 Standard close-toclose s best for large samples, Yang-Zhang s best for small samples EFFICIECY AD BIAS DETERMIE BEST VOLATILITY MEASURE There are two measures whch can be used to determne the qualty of a volatlty measure: effcency and bas. Generally, for small sample szes the Yang-Zhang measure s best overall, and for large sample szes the standard close to close measure s best. Effcency: Effcency cc ( x where x s the volatlty of the estmate and x the volatlty of the standard close to close estmate. cc s Bas: Dfference between the estmated varance and the average (.e. ntegrated volatlty. Effcency Measures the Volatlty of the Estmate The effcency descrbes the varance, or volatlty of the estmate. The effcency s dependent on the number of samples, wth effcency decreasng the more samples there are (as close to close wll converge and become less volatle wth more samples. The effcency s the theoretcal maxmum performance aganst an dealsed dstrbuton, and wth real emprcal data a far smaller beneft s usually seen (especally for long tme seres. For example, whle the Yang-Zhang based estmators deal wth overnght jumps f the jumps are large compared to the daly volatlty the estmate wll converge wth the close-to-close volatlty and have an effcency close to 1. Close To Close Volatlty Should Use At Least 5 Samples (and Ideally 0 or More The varance of the close-to-close volatlty can be estmated as a percentage of the actual varance by the formula 1/( where s the number of samples. Ths s shown n Fgure 4 below, and demonstrates that at least 5 samples are needed (or the estmate has a varance of over 10% and that only margnal extra accuracy s ganed for each addtonal sample above 0. Fgure 4. Varance of Close-To-Close Volatlty/Actual Varance Varance sample / true varance 10% 9% 8% 7% 6% 5% 4% 3% % 1% 0% Sample sze Source: Santander Investment Bolsa. 7

8 Bas can be postve or negatve dependng on the dstrbuton Bas Depends on the Type of Dstrbuton of the Underlyng Whle effcency (how volatle the measure s s mportant, so too s bas (s the measure on average too hgh or low. Bas depends on the sample sze, and the type of dstrbuton the underlyng securty has. Generally, the close-to-close volatlty estmator s too bg 3 (as t does not model overnght jumps whle alternatve estmators are too small (as they assume contnuous tradng, and dscrete tradng wll have a smaller dfference between the maxmum and mnmum. The key varables whch determne the bas are: Sample sze: As the standard close-to-close volatlty measure suffers wth small sample szes, ths s where alternatve measures perform best (the hghest effcency s reached for only days of data. Volatlty of volatlty: Whle the close-to-close volatlty estmate s relatvely nsenstve to a changng volatlty (vol of vol, the alternatve estmates are far more senstve. Ths bas ncreases the more vol of vol ncreases (.e. more vol of vol means a greater underestmate of volatlty. Approxmately 1/6 of total volatlty occurs overnght Overnght jumps between close and open: Approxmately one-sxth of equty volatlty occurs outsde the tradng day (and approxmately twce that amount for ADRs. Overnght jumps cause the standard close-to-close estmate to overestmate the volatlty, as jumps are not modelled. Alternatve estmates whch do not model jumps (Parknson, Garman Klass and Rogers-Satchell underestmate the volatlty. Yang-Zhang estmates (both Yang- Zhang extenson of Garman Klass and the Yang-Zhang measure tself wll converge wth standard close-to-close volatlty f the jumps are large compared to the overnght volatlty. Drft of underlyng: If the drft of the underlyng s gnored as t s for Parknson and Garman Klass (and the Yang Zhang extenson of Garman Glass, then the measure wll overestmate the volatlty. Ths effect s small for any reasonable drfts (.e. f we are lookng at daly, weekly or monthly data. Correlaton daly volatlty and overnght volatlty: Whle Yang-Zhang measures deal wth overnght volatlty, there s the assumpton that overnght volatlty and daly volatlty are uncorrelated. Yang-Zhang measures wll underestmate volatlty when there s a correlaton between daly return and overnght return (and vce versa, but ths effect s small. Varance, Volatlty and Gamma Swaps Should Look at Standard Volatlty (or Varance As the payout of varance, volatlty and gamma swaps are based on close-to-close prces, the standard close-to-close volatlty (or varance should be used for comparng ther prce aganst realsed. Addtonally, f a trader only hedges at the close (potentally for lqudty reasons then agan the standard close-to-close volatlty measure should be used. 3 Compared to ntegrated volatlty 8

9 As the average s taken from the sample, close-toclose volatlty has -1 degrees of freedom CLOSE-TO-CLOSE The smplest volatlty measure s the standard close-to-close volatlty. We note that the volatlty should be the standard devaton multpled by /(-1 to take nto account the fact we are samplng the populaton (or take standard devaton of the sample 4. We gnored ths n the earler defnton as for reasonably large n t /(-1 s roughly equal to zero. Standard dev of x = s x = F ( x x 1 As = = ( s E < E ( s = E(s by Jensens s nequalty Volatlty = σ x = s x 1 Volatlty close to close = σ cc = F 1 1 ( x x F c = Ln( assumng zero drft 1 c 1 1 PARKISO The frst advanced volatlty estmator was created by Parknson n 1980, and nstead of usng closng prces t uses the hgh and low prce. One drawback of ths estmator s that t assumes contnuous tradng, hence t underestmates the volatlty as potental movements when the market s shut are gnored. Volatlty Parknson = σ P = F 1 4 Ln( 1 h Ln( l GARMA-KLASS Later n 1980 the Garman-Klass volatlty estmator was created. It s an extenson of Parknson whch ncludes openng and closng prces (f openng prces are not avalable the close from the prevous day can be used nstead. As overnght jumps are gnored the measure underestmates the volatlty. Volatlty Garman-Klass = σ GK = F 1 1 h ( Ln l c (Ln( 1 Ln( o 4 As the formula for standard devaton has -1 degrees of freedom (as we subtract the sample average from each value of x 9

10 ROGERS-SATCHELL All of the prevous advanced volatlty measures assume the average return (or drft s zero. Securtes whch have a drft, or non-zero mean, requre a more sophstcated measure of volatlty. The Rogers-Satchell volatlty created n the early 1990s s able to properly measure the volatlty for securtes wth non-zero mean. It does not, however, handle jumps, hence t underestmates the volatlty. F h h l l Volatlty Rogers-Satchell = σ RS = Ln( Ln( Ln( Ln( c o c o 1 GARMA-KLASS YAG-ZHAG EXTESIO Yang-Zhang modfed the Garman-Klass volatlty measure n order to let t handle jumps. The measure does assume a zero drft, hence t wll overestmate the volatlty f a securty has a non-zero mean return. As the effect of drft s small, the fact contnuous prces are not avalable usually means t underestmates the volatlty (but by a smaller amount than the prevous alternatve measures. Volatlty GKYZ = σ GKYZ = YAG-ZHAG F o Ln( c 1 h ( Ln l 1 1 c (Ln( 1 Ln( o Yang-Zhang s the sum of overnght volatlty, and a weghted average of Rogers-Satchell and open-to-close volatlty In 000 Yang-Zhang created a volatlty measure that handles both openng jumps and drft. It s the sum of the overnght volatlty (close-to-open volatlty and a weghted average of the Rogers-Satchell volatlty and the open-to-close volatlty. The assumpton of contnuous prces does mean the measure tends to slghtly underestmate the volatlty. Volatlty Yang-Zhang = σ YZ = F overnght volatlty k open to close volatlty (1 k RS where k = overnght volatlty Ln( Ln( 1 1 c 1 c 1 o o 1 open to close volatlty Ln( Ln( 1 1 o o c c 10

11 The products and strateges addressed n ths report are complex, typcally nvolve a hgh degree of rsk and are ntended for sale only to sophstcated nvestors who are capable of understandng and assumng the rsks nvolved. The market value of any structured securty may be affected by changes n economc, fnancal and poltcal factors (ncludng, but not lmted to, spot and forward nterest and exchange rates, tme to maturty, market condtons and volatlty, and the credt qualty of any ssuer or reference ssuer. Any nvestor nterested n purchasng a structured product should conduct ther own nvestgaton and analyss of the product and consult wth ther own professonal advsers as to the rsks nvolved n makng such a purchase. The opnons and recommendatons ncluded n ths report are not necessarly those of the Equty Research Department of Santander Investment Bolsa or of ts afflates. A Tradng Places ratng on a specfc company equatng to that assocated wth a conventonal Buy, Hold or Underweght recommendaton should not be construed as a fundamental or offcal ratng of a Santander Investment Bolsa analyst. Furthermore, the opnons and strateges contaned n ths report are completely ndependent of those that the Equty Research and Sales/Tradng Departments of Santander Investment Bolsa may have from tme to tme. Some nvestments dscussed n ths report may have a hgh level of volatlty. Hgh volatlty nvestments may experence sudden and large falls n ther value causng losses when that nvestment s realsed. Those losses may equal your orgnal nvestment. Indeed, n the case of some nvestments the potental losses may exceed the amount of ntal nvestment and, n such crcumstances, you may be requred to pay more money to support those losses. Income yelds from nvestments may fluctuate and, n consequence, ntal captal pad to make the nvestment may be used as part of that ncome yeld. Some nvestments may not be readly realsable and t may be dffcult to sell or realse those nvestments, smlarly t may prove dffcult for you to obtan relable nformaton about the value, or rsks, to whch such an nvestment s exposed. Past performance should not be taken as an ndcaton or guarantee of future performance, and no representaton or warranty, express or mpled, s made regardng future performance. Informaton, opnons and estmates contaned n ths report reflect a judgement at ts orgnal date of publcaton by Grupo Santander and are subject to change wthout notce. The prce, value of and ncome from any of the securtes or fnancal nstruments mentoned n ths report can fall as well as rse. The value of securtes and fnancal nstruments s subject to exchange rate fluctuaton that may have a postve or adverse effect on the prce or ncome of such securtes or fnancal nstruments. Informaton and opnons presented n ths report have been obtaned or derved from sources beleved by Grupo Santander to be relable, but Grupo Santander makes no representaton as to ther accuracy or completeness. Grupo Santander accepts no lablty for loss arsng from the use of the materal presented n ths report, except that ths excluson of lablty does not apply to the extent that such lablty arses under specfc statutes or regulatons applcable to Grupo Santander. Ths report s not to be reled upon n substtuton for the exercse of ndependent judgment. Grupo Santander may have ssued, and may n the future ssue, other reports that are nconsstent wth, and reach dfferent conclusons from, the nformaton presented n ths report. Those reports reflect the dfferent assumptons, vews and analytcal methods of the analysts who prepared them and Grupo Santander s under no oblgaton to ensure that such other reports are brought to the attenton of any recpent of ths report. See back cover of ths report for further dsclamer dsclosures. 11

12 Important Dsclosures AALYST CERTIFICATIO: I, Coln Bennett, hereby certfy that the vews expressed n ths research report accurately reflect my personal vews about the subject company and ts securtes. I also certfy that I have not been promsed compensaton ether drectly or ndrectly for expressng the recommendaton n ths report. Ths report has been prepared by Santander Investment Bolsa, Socedad de Valores, S.A. ( Santander Investment Bolsa and s provded for nformaton purposes only. Ths document must not be consdered as an offer to sell or a solctaton of an offer to buy. Any decson by the recpent to buy should be based on publcly avalable nformaton on the related securty and, where approprate, should take nto account the content of the related prospectus fled wth the CMV (Spansh atonal Securtes Market Commsson and avalable from the CMV, the company governng the related market (Socedad Rectora de la Bolsa and the company ssung the securty. Ths report s ssued n the Unted States by Santander Investment Securtes, Inc. ( SIS, n Span by Santander Investment Bolsa and n the Unted Kngdom by Banco Santander, S.A., London Branch ( Santander London. Santander London s authorsed by the Bank of Span. SIS, Santander Investment Bolsa and Santander London are members of Grupo Santander. Ths report s not beng ssued to prvate customers. The nformaton contaned heren has been compled from sources beleved to be relable, but whle all reasonable care has been taken to ensure that the nformaton contaned heren s not untrue or msleadng at the tme of publcaton, we make no representaton that t s accurate or complete and t should not be reled upon as such. All opnons and estmates ncluded heren consttute our judgement as at the date of ths report and are subject to change wthout notce. Santander Investment Bolsa may change the recommendaton t has on a stock at any gven tme. It may also cease to cover a stock or ntate coverage of new stocks. There s no specfc calendar for any such actons. From tme to tme, Grupo Santander and/or any of ts offcers or drectors may have a poston, or otherwse be nterested n, transactons n securtes whch are drectly or ndrectly the subject of ths report. Santander Investment Bolsa has nternal rules of conduct that contan, among other thngs, procedures to prevent conflcts of nterest wth respect to recommendatons, ncludng: the consderaton of ts Research Department as a separate area, Chnese Walls, and the possblty of establshng specfc restrctons on research actvty where approprate. Santander Investment Bolsa s research reports contan a certfcaton statng that they reflect the authors own opnons. Grupo Santander may from tme to tme perform servces for or solct busness from any company mentoned n ths report. ether Grupo Santander nor any other person accepts any lablty whatsoever for any drect or consequental loss arsng from any use of ths report or ts contents. Ths report may not be reproduced, dstrbuted or publshed by any recpent for any purpose. Santander Investment Bolsa s under the supervson of the CMV. Any US recpent of ths report (other than a regstered broker-dealer or a bank actng n a broker-dealer capacty that would lke to effect any transacton n any securty dscussed heren should contact and place orders n the Unted States wth the company dstrbutng the research, SIS at ( , whch, wthout n any way lmtng the foregong, accepts responsblty (solely for purposes of and wthn the meanng of Rule 15a-6 under the US Securtes Exchange Act of 1934 under ths report and ts dssemnaton n the Unted States. US recpents of ths report should be advsed that ths research has been produced by a non-member afflate of SIS and, therefore, by rule, not all dsclosures requred under ASD Rule 711 apply. Santander Investment Bolsa, Socedad de Valores, S.A., 01. All Rghts Reserved. KEY TO IVESTMET CODES* % of Companes Ratng Defnton Covered wth Ths Ratng Provded wth Investment Bankng Servces n Past 1M Buy Upsde of more than 15% Hold Upsde of 10%-15% Underweght Upsde of less than 10% Under Revew OTE: Gven the recent volatlty seen n the fnancal markets, the recommendaton defntons are only ndcatve untl further notce. (* Target prces set from January to June are for December 31 of the current year. Target prces set from July to December are for December 31 of the followng year. LOCAL OFFICES Madrd Lsbon London ew York Tel: Tel: Tel: Tel: Fax: Fax: Fax: Fax: Bogotá Buenos Ares Caracas Lma Tel: Tel: Tel: Tel: Fax: Fax: Fax: Fax: Mexco Cty Santago São Paulo Tokyo Tel: Tel: Tel: Tel: Fax: Fax: Fax: Fax: