EAID 2011 Wednesday, April 6, 8:30 10:00

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2 EAID 2011 Wednesday, April 6, 8:30 10:00 Inroducing a New Form of Volailiy Index: he Cross-Secional Volailiy Index (CVIX ) Lionel Marellini Professor of Finance, EDHEC Business School Scienific Direcor, EDHEC-Risk Insiue Scienific Advisor, EDHEC-Risk Indices & Benchmarks lionel.marellini@edhec.edu 2 Research suppored bv he Fédéraion Bancaire Française

3 Ouline Volailiy: Definiions and Uses Benefis and Limis of Exising Volailiy Indices A New Form of Volailiy Index: he CVIX Approach Properies of he CVIX Index 3

4 Volailiy: Definiions and Uses Benefis and Limis of Exising Volailiy Indices A New Form of Volailiy Index: he CVIX Approach Properies of he CVIX Index 4

5 Defining and Measuring Volailiy Volailiy is defined as he sandard-deviaion of reurns. Hisorical volailiy can be esimaed from ime-series of sock index reurns; forward-looking esimaes are available hrough GARCH-ype models. Implied volailiy can be obained from ime-series of opion prices: e.g., he VIX index, (now) a model-free quaniy. For any given sock, anoher imporan disincion exiss beween sysemaic volailiy, driven by he sock exposure wih respec o sysemaic risk facors, and specific volailiy, which is driven by he uncerainy impacing a paricular company. 5

6 Volailiy as an Indicaor and as an Asse Class Informaion abou volailiy is very useful in invesmen managemen, from boh an asse pricing and a porfolio selecion perspecives. More recenly, invesors have shown an increased ineres in volailiy as an asse class: one of he main moivaions for rading in volailiy is o diversify equiy risk hrough long volailiy exposure (Szado (2009)), wih changes in volailiy negaively correlaed wih sock index reurns. In addiion, negaive correlaion and high h volailiy are paricularly l pronounced in sock marke downurns, offering proecion agains sock marke losses when i is needed mos and when oher forms of diversificaion are no very effecive. 6

7 Inverse Relaionship (I) Volailiy ends o rise when equiy markes fall

8 Inverse Relaionship (II) Negaive relaionship is sronger in recessions

9 Inverse Relaionship (III) Negaive relaionship is sronger in imes of crisis

10 Explanaions for Inverse Relaionship Explanaions behind he inverse relaionship beween volailiy and equiy reurns: Leverage effec from an ex-pos perspecive (*): Black (1976), Chrisie (1982), Schwer (1989); Volailiy feedback effec from an ex-ane perspecive (**): French e al. (1987), Bekaer e Wu (2000), Wu (2001), Kim e al. (2004). The presence of profound economic reasons behind he inverse relaionship beween equiy reurn and volailiy is a comforing indicaion of robusness of expeced diversificaion benefis, (*) The leverage effec : a decrease (respecively, an increase) in equiy prices increases (respecively, decreases) he company s leverage, hereby increasing i (respecively, decreasing) he risk o equiy holders and increasing i (respecively, decreasing) equiy volailiy. (**) The "volailiy feedback effec assuming ha volailiy is incorporaed in sock prices, a posiive volailiy shock increases he fuure required reurn on equiy and sock prices are expeced o fall simulaneously. 10

11 Oher Moivaions for Trading in Volailiy Hedging demand: Implici volailiy exposure is anoher moivaion for long volailiy posiions by hedge funds muual funds who are ofen implicily shor volailiy. In paricular, benchmarked equiy fund managers are shor volailiy because porfolio racking error and rebalancing coss increase wih an increase in he volailiy of equiy markes (Hill (2004)). Speculaive/arbirage demand: Direcional bes on volailiy changes; Non direcional bes (e.g., implied versus hisorical, 3-monhs implied vs. 1 monh-implied, ec.). 11

12 Volailiy: Definiions and Uses Benefis and Limis of Exising Volailiy Indices A New Form of Volailiy Index: he CVIX Approach Properies of he CVIX Index 12

13 Volailiy Indices Because informaion abou changes in volailiy is of a criical imporance o marke paricipans, a number of iniiaives have been launched so as o make volailiy measures available o invesors in form of volailiy indices. Such indices are calculaed based on opion prices, and as such are based on implici as opposed o acual volailiy measures. The mos popular volailiy index is he VIX, which is buil from prices of equiy index opions on he S&P 500, firs inroduced by he CBOE in 1993, wih a mehodology ha has been revised o derive hee volailiy index from he implied risk-neural disribuion ha can be exraced from call and pu opions wih differen srike prices. 13

14 Available (Implied) Volailiy Indices Currenly available volailiy indices are implied volailiy indices based on opion prices Index Ticker Underlying Index Provider AMEX Volailiy Index QQV QQQ AMEX CBOE Volailiy Index VIX SPX CBOE CBOE DJIA Volailiy Index VXD DJX CBOE CBOE NASDAQ-100 Volailiy Index VXN NDX CBOE CBOE Russell 2000 Volailiy Index RVX RUT CBOE CBOE S&P 100 Volailiy Index VXO OEX CBOE CBOE S&P Monh hvolailiy Index VXV SPX CBOE CBOE VIX Premium Sraegy Index VPD VIX CBOE CBOE Capped VIX Premium Sraegy Index VPN VIX CBOE CBOE Crude Oil Volailiy Index OVX USO CBOE CBOE Gold Volailiy Index GVZ GLD CBOE CBOE EuroCurrency Volailiy Index EVZ FXE CBOE AEX Volailiy VAEX AEX Euronex BEL 20 Volailiy VBEL BEL 20 Euronex CAC 40 Volailiy VCAC CAC 40 Euronex FTSE 100 Volailiy VFTSE FTSE 100 Euronex uo DAX Volailiy VDAX-NEW DAX Deusche Borse AG SMI Volailiy VSMI SMI SIX Swiss Exchange AG EURO STOXX 50 Volailiy VSTOXX EURO STOXX 50 STOXX Limied NIKKEI Volailiy Index VNKY NIKKEI 225 Nikkei Inc. India NSE VIX INVIXN NIFTY 50 India NSE KOSPI 200 Volailiy Index VKOSPI KOSPI 200 Korea Exchange Mexico Volailiy Index VIMEX IPC MexDer 14

15 Available Derivaives Conracs Trading in opions is no a clean be on volailiy alone (vega bu also dela exposure); volailiy can now be direcly raded hrough OTC swaps, and more recenly hrough exchange- raded fuures and opions on volailiy indices. Insiuion Index Ticker (Owner) Fuures Ticker VX (VIX Fuures), VM CBOE Volailiy Index VIX CBOE (Mini-VIX Fuures) CBOE DJIA Volailiy Index VXD CBOE DV CBOE NASDAQ-100 Volailiy Index VXN CBOE VN CBOE Russell 2000 Volailiy Index RVX CBOE VR VDAX- Deusche Borse DAX Volailiy NEW AG FVDX (delised) SIX Swiss SMI Volailiy VSMI Exchange AG FVSM (delised) EURO STOXX 50 VSTOX STOXX FVSX (delised), FVS Volailiy X Limied (Mini-Fuures) Opions on Fuures Available - Available Available

16 Issues wih Volailiy Indices Volailiy indices are no available for an exensive se of markes, because hey require he presence of a liquid opion marke; for example, very few volailiy indices exiss for Asian markes, or oher emerging markes. In developed markes, hey only exis a he broad aggregae level: no volailiy index available for small cap socks, growh/value socks, secors, ec. Beside, where and when hey exis, implied volailiy esimaes are plagued by opion-marke problems ha have lile o do wih underlying equiy markes. 16

17 Volailiy: Definiions and Uses Benefis and Limis of Exising Volailiy Indices A New Form of Volailiy Index: he CVIX Approach Properies of he CVIX Index 17

18 Inroducing a New Volailiy Index Series EDHEC-Risk Insiue has inroduced a new se of volailiy indices, which are mean o be : Model-free; Based on equiy marke daa; Available for all markes/secors a all frequencies. Useful references: The concepual framework has been developed in a paper eniled Idiosyncraic Risk and he Cross-Secion of Sock Reurns, by René Garcia, Daniel Manilla and Lionel Marellini. A subsequen paper eniled Inroducing a New Form of Volailiy Index: he Cross-Secional Volailiy Index (CVIX ), by Felix Golz, Renaa Guobuzaie and Lionel Marellini, furher explores he properies of his index and compares i wih exising volailiy indices. 18

19 On he Cross-Secional Disribuion of Sock Reurns This index is based on exploiing informaional conen in he cross-secional disribuion of sock reurns, i.e., he dispersion of sock reurns on a given dae. When he S&P 500 index goes down by 5% on a given dae, one may be looking a very differen siuaions: All socks in he S&P500 universe wen down by 5%. Half of he socks wen down by 20% and half wen up by 10%. The performance of an index is nohing bu he average value of a disribuion, and as such a very imperfec proxy for he whole disribuion. 19

20 Cross-Secion of Sock Reurns Low Dispersion 50% Cross-Secional Disribuion of Reurns (Minimum Value) 45% 40% 35% 30% Percenage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Reurns Cross-secional disribuion ib i of daily sock reurn daa for he S&P500 universe on December 28, 2008 (corresponding o he lowes dispersion for he sample period ranging from January 1990 o November 2010). 20

21 Cross-Secion of Sock Reurns Average Dispersion 50% Cross-Secional Disribuion of Reurns (Median Value) 45% 40% 35% 30% Percenage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Reurns Cross-secional disribuion ib i of daily sock reurn daa for he S&P500 universe on Sepember 24, 1997 (corresponding o he median dispersion for he sample period ranging from January 1990 o November 2010). 21

22 Cross-Secion of Sock Reurns High Dispersion 50% Cross Secional Disribuion of Reurns (Maximum Value) 45% 40% 35% 30% Percenage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Reurns Cross-secional disribuion ib i of daily sock reurn daa for he S&P500 universe on Nov 24, 2008 (corresponding o he highes dispersion for he sample period ranging from January 1990 o November 2010). 22

23 Inroducing CVIX Broadly speaking, we propose o use he cross-secional dispersion of reurns a any given dae and frequency as an observable, model-free, proxy for volailiy. Formally, define for a given universe of N socks he reurn on a porfolio weighing scheme w as (e.g., CW or preferably EW): ( w ) r N = w The CVIX index is defined as he square-roo roo of he crosssecional variance (CSV), a.k.a. variey (*): N ( ) = w ( w CSV w r r ) i= 1 i= 1 i i r i ( ) i 2 (*) Robus quanile-based esimaes are used in empirical applicaions. 23

24 Asympoic Properies of CSV Assume wihou any loss of generaliy he following single facor model for (excess) sock reurns: r i = β F + ε Furher make wo assumpions (o be discussed laer): i Homogeneous bea assumpion: β i = β for all i; Homogeneous residual variance assumpion: E(ε 2 i)=σ 2 ε() for all i. i Under hese assumpions, we have (for all weighing schemes) ha: N ( w CSV ) w r r ( w ) 2 2 = i= 1 i ( ) () i N σ ε 24

25 Finie Disance Properies of CSV Under he same assumpions, we have ha: N [ ( w )] = ( ) 2 E CSV σ ε 1 i= 1 w ( ) 2 N N N w Var[ CSV ] = ( ) + 2σ ε wi wi 2 wi i= 1 i= 1 i= 1 We formally define CVIX as he cross-secional volailiy, which is heoreically (and empirically) shown o be an asympoically unbiased, efficien (minimum bias for EW) and convergen esimaor for specific volailiy. [ ] EW E CSV = σ ε () 1 σ () N ε N Var 2 i N 1 2 N [ ] EW 4 2 CSV = σ 0 ε N 25

26 Relaxing he Homogeneous Residual Variance Assumpion If we relax he homogeneous residual variance assumpion, we have ha: N N ( w ) E CSV = w σ w [ ] ( ) ( ) i= 1 i ε i iσ ε i i= 1 Hence, he CSV appears as a (biased) esimaor for he average idiosyncraic variance. If we focus on he EW scheme, we obain: N 1 = N i [ ] EW 2 σ ( ) N ε i E CSV = 26

27 Relaxing he Homogeneous Bea Assumpion If we relax he homogeneous bea assumpion, we have ha: where: N N = i ε i σ i ε i i= 1 i= 1 2 N N β = wi βi wi βi i= 1 i= 1 [ ( ) ] w β ( ) ( ) [ ] E CSV w σ w + E F CSV CSV When he homogenous bea assumpion does no hold, he CSV is no an unbiased esimaor for average specific variance, even for an increasingly large number of socks. Empirically, however, his bias is found o be small (less han 1% on average). 27

28 Comparison wih Compeing Measures Using CRSP daa from January 1964 o December 2006, we esimae he CW and EW CSV. We also esimae he CW and EW daily average idiosyncraic variance relaive o he FF model (using overlapping monhly rolling windows). The relaionship is srong for boh weighing schemes: 82.63% for EW and 73.95% for CW. Much higher values are obained a monhly frequency (wih no arificial smoohing for FF): 99.75% for EW and 98.53% for CW. 28

29 Volailiy: Definiions and Uses Benefis and Limis of Exising Volailiy Indices A New Form of Volailiy Index: he CVIX Approach Properies of he CVIX Index 29

30 VIX versus CVIX While represening a priori wo differen underlying risk measures, sysemaic and average specific volailiy indicaors should be highly correlaed, since hey boh reflec he aggregae uncerainy faced by invesors a a given poin in ime regarding economic fundamenals. In wha follows, we confirm his inuiion and find high correlaion level beween he VIX index, an (imperfec) measure of sysemaic risk based on opion prices, and he CVIX index, a model-free efficien and unbiased proxy for average specific risk. 30

31 CVIX as a Measure of Economic Uncerainy Time evoluion of he CVIX index and recession periods 31 Daily ime series for he CVIX index. The shaded areas are he NBER recessions. The sample period is January 1990 o November 2010.

32 Comparison wih VIX Time evoluion of he CVIX index and he VIX index Quanile-based CVIX index performance, compared wih he VIX index, based on daily sock reurn daa for he S&P500 universe on he sample period ranging from January 1990 o November

33 Increasing Correlaion Time evoluion of he CVIX index and he VIX index Quanile-based CVIX index correlaion wih he VIX index, based on 10-year daily sock reurn rolling window daa on he sample period ranging from January 1990 o November

34 Volailiy Indices and Corresponding CVIX Indices Index Ticker Correlaion CBOE Volailiy Index VIX AEX Volailiy VAEX BEL 20 Volailiy VBEL CAC 40 Volailiy VCAC FTSE 100 Volailiy VFTSE DAX Volailiy VDAX-NEW SMI Volailiy VSMI EURO STOXX 50 Volailiy VSTOXX NIKKEI Volailiy Index VNKY India NSE VIX INVIXN KOSPI 200 Volailiy Index VKOSPI Mexico Volailiy Index VIMEX Sample period

35 CVIX as a Proxy for Toal Risk 35

36 Wrap-Up We inroduce a new form of volailiy index, he cross-secional volailiy index or CVIX, wih daa o be made publically available saring in Q Through formal cenral limi argumens, we show ha he crosssecional dispersion of sock reurns can be regarded as an efficien esimaor for he average idiosyncraic volailiy of socks wihin he universe under consideraion. Amongs he key advanages of he CVIX measure over currenly available measures are is observabiliy bili a any frequency, is model-free naure, and is availabiliy for every region, secor and syle of he world equiy markes, wihou he need o resor o any auxiliary opion marke. 36

37 Wrap-Up Con We also provide some inerpreaion of he CVIX index as a proxy for aggregae economic uncerainy, which suggess ha he CVIX index should be inimaely relaed o opion-based implied volailiy measures. We confirm his inuiion by reporing a high correlaion level beween exising volailiy indices and he corresponding CVIX index in various regions and ime-periods. Overall, hese resuls sugges ha he CVIX index is closely relaed o oher volailiy measures where and when such measures are available, and ha i can be used as a reliable proxy for volailiy when such measures are no available. 37

38 Imporan Informaion CVIX (Cross Vol Index/Cross Secional Volailiy Index) is a regisered rademark. The mehodology described in his documen is proprieary and subjec o a pending paen, USPTO applicaion number