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  • 1. Category - Based Tail Comovement Christophe Villa AUDENCIA - Nantes and CREST February, 25 2010 Christophe Villa () EM Lyon February, 25 2010 1 / 22
  • 2. Co-authors Arthur Charpentier University of Rennes and Ecole Polytechnique Emilios Galariotis Audencia Nantes School of Management Christophe Villa () EM Lyon February, 25 2010 2 / 22
  • 3. Comovement there are numerous patterns of comovement in the data Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 4. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 5. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 6. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry small stocks Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 7. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry small stocks value stocks Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 8. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry small stocks value stocks closed-end funds Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 9. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry small stocks value stocks closed-end funds what is the source of this comovement ? Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 10. Comovement there are numerous patterns of comovement in the data common factors in the returns of certain groups of assets stocks within the same industry small stocks value stocks closed-end funds what is the source of this comovement ? why do certain assets comove while others do not ? Christophe Villa () EM Lyon February, 25 2010 3 / 22
  • 11. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 12. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 13. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 14. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 15. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 16. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows correlated changes in their discount rates Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 17. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows correlated changes in their discount rates changes in interest rates Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 18. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows correlated changes in their discount rates changes in interest rates changes in risk aversion Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 19. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows correlated changes in their discount rates changes in interest rates changes in risk aversion correlated changes in rational perception of risk Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 20. Traditional view : Fundamentals - based comovement derived from economies without frictions and with rational investors assets comove because their "fundamental values" comove fundamental value = rational forecast of future cash ‡ows discounted at rate appropriate for risk in this view, assets comove because of : correlated news about their cash ‡ows correlated changes in their discount rates changes in interest rates changes in risk aversion correlated changes in rational perception of risk useful framework for understanding many types of comovement Christophe Villa () EM Lyon February, 25 2010 4 / 22
  • 21. Evidence on comovement Twin stocks (Froot/Dabora [1999]) Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 22. Evidence on comovement Twin stocks (Froot/Dabora [1999]) claims to same cash-‡ow stream, but traded in di¤erent locations (eg Royal Dutch / Shell) Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 23. Evidence on comovement Twin stocks (Froot/Dabora [1999]) claims to same cash-‡ow stream, but traded in di¤erent locations (eg Royal Dutch / Shell) Royal Dutch, traded primarily in New York, is a claim to 60% of the cash ‡ow Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 24. Evidence on comovement Twin stocks (Froot/Dabora [1999]) claims to same cash-‡ow stream, but traded in di¤erent locations (eg Royal Dutch / Shell) Royal Dutch, traded primarily in New York, is a claim to 60% of the cash ‡ow Shell, traded primarily in London, is a claim to the remaining 40% Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 25. Evidence on comovement Twin stocks (Froot/Dabora [1999]) claims to same cash-‡ow stream, but traded in di¤erent locations (eg Royal Dutch / Shell) Royal Dutch, traded primarily in New York, is a claim to 60% of the cash ‡ow Shell, traded primarily in London, is a claim to the remaining 40% under traditional view of comovement, expect them to move in lockstep Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 26. Evidence on comovement Twin stocks (Froot/Dabora [1999]) claims to same cash-‡ow stream, but traded in di¤erent locations (eg Royal Dutch / Shell) Royal Dutch, traded primarily in New York, is a claim to 60% of the cash ‡ow Shell, traded primarily in London, is a claim to the remaining 40% under traditional view of comovement, expect them to move in lockstep in fact, Royal Dutch comoves more with the U.S. stock market, Shell with the U.K. market rRD,t rSH,t = α + 0.207 rS&P,t 0.428 rFTSE ,t + εt Christophe Villa () EM Lyon February, 25 2010 5 / 22
  • 27. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 28. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) funds traded in one location, fund assets in another Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 29. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) funds traded in one location, fund assets in another under traditional view of comovement, expect fund returns and NAV returns to move together closely Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 30. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) funds traded in one location, fund assets in another under traditional view of comovement, expect fund returns and NAV returns to move together closely in fact, fund returns comove as much with market where fund is traded as with market where assets are traded Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 31. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) funds traded in one location, fund assets in another under traditional view of comovement, expect fund returns and NAV returns to move together closely in fact, fund returns comove as much with market where fund is traded as with market where assets are traded Domestic closed-end funds (Lee et al. [1991]) Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 32. Evidence on comovement ctd. Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al. [1995]) funds traded in one location, fund assets in another under traditional view of comovement, expect fund returns and NAV returns to move together closely in fact, fund returns comove as much with market where fund is traded as with market where assets are traded Domestic closed-end funds (Lee et al. [1991]) closed-end funds invested in large stocks often comove with small stocks Christophe Villa () EM Lyon February, 25 2010 6 / 22
  • 33. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 34. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 35. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Fama/French [1995] test whether these factors are due to cash-‡ow news Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 36. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Fama/French [1995] test whether these factors are due to cash-‡ow news do …nd cash-‡ow factors but they line up poorly with the return factors Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 37. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Fama/French [1995] test whether these factors are due to cash-‡ow news do …nd cash-‡ow factors but they line up poorly with the return factors Commodities (Pindyck/Rotemberg [1990]) Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 38. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Fama/French [1995] test whether these factors are due to cash-‡ow news do …nd cash-‡ow factors but they line up poorly with the return factors Commodities (Pindyck/Rotemberg [1990]) …nd strong comovement in price changes of seven commodities Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 39. Evidence on comovement ctd. Small stocks and value stocks (Fama/French [1995]) there is a strong common factor in returns of small stocks and in returns of value stocks (Fama/French [1993]) Fama/French [1995] test whether these factors are due to cash-‡ow news do …nd cash-‡ow factors but they line up poorly with the return factors Commodities (Pindyck/Rotemberg [1990]) …nd strong comovement in price changes of seven commodities hard to explain comovement through news about aggregate demand Christophe Villa () EM Lyon February, 25 2010 7 / 22
  • 40. Category-based comovement Barberis/Shleifer [2003] Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 41. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 42. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 43. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem makes it easier to evaluate money managers Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 44. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem makes it easier to evaluate money managers categories are based on a salient common characteristic Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 45. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem makes it easier to evaluate money managers categories are based on a salient common characteristic impressive past performance often spurs category formation Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 46. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem makes it easier to evaluate money managers categories are based on a salient common characteristic impressive past performance often spurs category formation categories can be identi…ed by looking at labels on money managers’ products Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 47. Category-based comovement Barberis/Shleifer [2003] many investors allocate funds at the level of an asset category simpli…es the portfolio problem makes it easier to evaluate money managers categories are based on a salient common characteristic impressive past performance often spurs category formation categories can be identi…ed by looking at labels on money managers’ products e.g. small-cap, large-cap, growth, index Christophe Villa () EM Lyon February, 25 2010 8 / 22
  • 48. Category-based comovement ctd. suppose categories are adopted by noise traders with correlated sentiment Christophe Villa () EM Lyon February, 25 2010 9 / 22
  • 49. Category-based comovement ctd. suppose categories are adopted by noise traders with correlated sentiment if the noise traders a¤ect prices ) assets will comove simply because they are classi…ed into the same category Christophe Villa () EM Lyon February, 25 2010 9 / 22
  • 50. Category-based comovement ctd. suppose categories are adopted by noise traders with correlated sentiment if the noise traders a¤ect prices ) assets will comove simply because they are classi…ed into the same category even if fundamentals are uncorrelated Christophe Villa () EM Lyon February, 25 2010 9 / 22
  • 51. Category-based comovement ctd. Consider an simple economy with a riskless asset and 2n risky assets Christophe Villa () EM Lyon February, 25 2010 10 / 22
  • 52. Category-based comovement ctd. Consider an simple economy with a riskless asset and 2n risky assets risky asset i is a claim to a single liquidating dividend Di,T to be paid at some later time T: Di,T = Di,0 + ei,1 + ... + ei,T where et = (e1,t , ..., e2n,t )0 ~ N (0, ΣD ) , i.i.d. over time Christophe Villa () EM Lyon February, 25 2010 10 / 22
  • 53. Category-based comovement ctd. Consider an simple economy with a riskless asset and 2n risky assets risky asset i is a claim to a single liquidating dividend Di,T to be paid at some later time T: Di,T = Di,0 + ei,1 + ... + ei,T where et = (e1,t , ..., e2n,t )0 ~ N (0, ΣD ) , i.i.d. over time Denote the price of asset i at time t as Pi,t and the return as ∆Pi,t = Pi,t Pi,t 1 Christophe Villa () EM Lyon February, 25 2010 10 / 22
  • 54. Category-based comovement ctd. Consider an simple economy with a riskless asset and 2n risky assets risky asset i is a claim to a single liquidating dividend Di,T to be paid at some later time T: Di,T = Di,0 + ei,1 + ... + ei,T where et = (e1,t , ..., e2n,t )0 ~ N (0, ΣD ) , i.i.d. over time Denote the price of asset i at time t as Pi,t and the return as ∆Pi,t = Pi,t Pi,t 1 Some investors group the risky assets into two categories, X and Y Christophe Villa () EM Lyon February, 25 2010 10 / 22
  • 55. Category-based comovement ctd. Consider an simple economy with a riskless asset and 2n risky assets risky asset i is a claim to a single liquidating dividend Di,T to be paid at some later time T: Di,T = Di,0 + ei,1 + ... + ei,T where et = (e1,t , ..., e2n,t )0 ~ N (0, ΣD ) , i.i.d. over time Denote the price of asset i at time t as Pi,t and the return as ∆Pi,t = Pi,t Pi,t 1 Some investors group the risky assets into two categories, X and Y suppose assets 1 through n in category X, assets n + 1 through 2n in Y Christophe Villa () EM Lyon February, 25 2010 10 / 22
  • 56. Category-based comovement ctd. assume the following cash-‡ow structure: Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 57. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 58. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 59. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t fM,t is the market-wide shock Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 60. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t fM,t is the market-wide shock fX ,t and fY ,t are the group-speci…c shocks Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 61. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t fM,t is the market-wide shock fX ,t and fY ,t are the group-speci…c shocks εi,t and εj,t are idiosyncratic shocks Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 62. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t fM,t is the market-wide shock fX ,t and fY ,t are the group-speci…c shocks εi,t and εj,t are idiosyncratic shocks ψM and ψS are constants that control the relative importance of the three components Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 63. Category-based comovement ctd. assume the following cash-‡ow structure: for an asset i 2 X ei,t = ψM fM,t + ψS fX ,t + q 1 ψ2 S ψ2 M εi,t for an asset j 2 Y ej,t = ψM fM,t + ψS fY ,t + q 1 ψ2 S ψ2 M εj,t fM,t is the market-wide shock fX ,t and fY ,t are the group-speci…c shocks εi,t and εj,t are idiosyncratic shocks ψM and ψS are constants that control the relative importance of the three components Each shock has unit variance and is orthogonal to the other shocks Christophe Villa () EM Lyon February, 25 2010 11 / 22
  • 64. Category-based comovement ctd. Suppose that Christophe Villa () EM Lyon February, 25 2010 12 / 22
  • 65. Category-based comovement ctd. Suppose that category-based investors are noise traders, who channel funds in and out of the categories depending on their sentiment Christophe Villa () EM Lyon February, 25 2010 12 / 22
  • 66. Category-based comovement ctd. Suppose that category-based investors are noise traders, who channel funds in and out of the categories depending on their sentiment the economy also contains "fundamental traders" who invest at the level of individual assets and act as arbitrageurs Christophe Villa () EM Lyon February, 25 2010 12 / 22
  • 67. Category-based comovement ctd. Suppose that category-based investors are noise traders, who channel funds in and out of the categories depending on their sentiment the economy also contains "fundamental traders" who invest at the level of individual assets and act as arbitrageurs A simple representation for asset returns is then ∆Pi,t = ei,t + ∆uX ,t , for i = 1, ..., n ∆Pi,t = ei,t + ∆uY ,t , for i = n + 1, ..., 2n where uX ,t uY ,t N 0 0 , σ2 u 1 ρu ρu 1 i.i.d. over time Christophe Villa () EM Lyon February, 25 2010 12 / 22
  • 68. Category-based comovement ctd. Suppose that category-based investors are noise traders, who channel funds in and out of the categories depending on their sentiment the economy also contains "fundamental traders" who invest at the level of individual assets and act as arbitrageurs A simple representation for asset returns is then ∆Pi,t = ei,t + ∆uX ,t , for i = 1, ..., n ∆Pi,t = ei,t + ∆uY ,t , for i = n + 1, ..., 2n where uX ,t uY ,t N 0 0 , σ2 u 1 ρu ρu 1 i.i.d. over time uX ,t (uY ,t ) can be thought of as time t noise trader sentiment about the securities in category X (Y ) Christophe Villa () EM Lyon February, 25 2010 12 / 22
  • 69. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 70. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock assets in the same category comove too much Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 71. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock assets in the same category comove too much assets in di¤erent categories comove too little Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 72. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock assets in the same category comove too much assets in di¤erent categories comove too little and reclassifying an asset into a new style raises its correlation with that category Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 73. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock assets in the same category comove too much assets in di¤erent categories comove too little and reclassifying an asset into a new style raises its correlation with that category category-based view suggests that there is excess comovement in stock prices, with stocks in the same category moving together more than their intrinsic values say that they should. Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 74. Category-based comovement ctd. so long as arbitrage is limited in some way, two assets in same category comove not only because of correlated cash-‡ow news, but because of a correlated sentiment shock assets in the same category comove too much assets in di¤erent categories comove too little and reclassifying an asset into a new style raises its correlation with that category category-based view suggests that there is excess comovement in stock prices, with stocks in the same category moving together more than their intrinsic values say that they should. comovement = correlation Christophe Villa () EM Lyon February, 25 2010 13 / 22
  • 75. Evidence on extreme comovement Mr. A. Greenspan: "work that characterizes the distribution of extreme events would be useful as well." Christophe Villa () EM Lyon February, 25 2010 14 / 22
  • 76. Evidence on extreme comovement Mr. A. Greenspan: "work that characterizes the distribution of extreme events would be useful as well." During the unfolding …nancial crisis stocks have experienced both extreme movement and extreme comovement in returns. ) Category-based tail comovement Christophe Villa () EM Lyon February, 25 2010 14 / 22
  • 77. Evidence on extreme comovement No simple analogue of the correlation coe¢ cient for extreme dependencies ) various statistical measures. Christophe Villa () EM Lyon February, 25 2010 15 / 22
  • 78. Evidence on extreme comovement No simple analogue of the correlation coe¢ cient for extreme dependencies ) various statistical measures. Poon, Rockinger and Tawn (2004) derive a general multivariate framework with two types of extreme dependence structures that allow for both asymptotic dependence and independence. Christophe Villa () EM Lyon February, 25 2010 15 / 22
  • 79. Evidence on extreme comovement No simple analogue of the correlation coe¢ cient for extreme dependencies ) various statistical measures. Poon, Rockinger and Tawn (2004) derive a general multivariate framework with two types of extreme dependence structures that allow for both asymptotic dependence and independence. four types of dependence: perfect dependent, independent, asymptotically dependent and asymptotically independent Christophe Villa () EM Lyon February, 25 2010 15 / 22
  • 80. Quantile dependence Quantile dependence is a measure of the dependence in the tails of the distribution and describes the limiting proportion that Christophe Villa () EM Lyon February, 25 2010 16 / 22
  • 81. Quantile dependence Quantile dependence is a measure of the dependence in the tails of the distribution and describes the limiting proportion that one margin exceeds a certain threshold Christophe Villa () EM Lyon February, 25 2010 16 / 22
  • 82. Quantile dependence Quantile dependence is a measure of the dependence in the tails of the distribution and describes the limiting proportion that one margin exceeds a certain threshold given that the other margin has already exceeded that threshold. Christophe Villa () EM Lyon February, 25 2010 16 / 22
  • 83. Quantile dependence Quantile dependence is a measure of the dependence in the tails of the distribution and describes the limiting proportion that one margin exceeds a certain threshold given that the other margin has already exceeded that threshold. If Z1 and Z2 are random variables with distribution functions F1 and F2, ) there is quantile dependence in the lower tail at threshold α, whenever P Z2 F 1 2 (α) j Z1 F 1 1 (α) 6= 0 ) there is quantile dependence in the upper tail at threshold α, whenever P Z2 F 1 2 (α) j Z1 F 1 1 (α) 6= 0 Christophe Villa () EM Lyon February, 25 2010 16 / 22
  • 84. Asymptotic dependence Upper tail dependence index between Z1 and Z2 is de…ned as λU = lim α!1 P(Z1 F 1 1 (α) j Z2 F 1 2 (α)) = lim α!1 P(Z2 F 1 2 (α) j Z1 F 1 1 (α)), Christophe Villa () EM Lyon February, 25 2010 17 / 22
  • 85. Asymptotic dependence Upper tail dependence index between Z1 and Z2 is de…ned as λU = lim α!1 P(Z1 F 1 1 (α) j Z2 F 1 2 (α)) = lim α!1 P(Z2 F 1 2 (α) j Z1 F 1 1 (α)), Lower tail dependence index is λL = lim α!0 P(Z1 F 1 1 (α) j Z2 F 1 2 (α)) = lim α!0 P(Z2 F 1 2 (α) j Z1 F 1 1 (α)). Christophe Villa () EM Lyon February, 25 2010 17 / 22
  • 86. Asymptotic dependence Upper tail dependence index between Z1 and Z2 is de…ned as λU = lim α!1 P(Z1 F 1 1 (α) j Z2 F 1 2 (α)) = lim α!1 P(Z2 F 1 2 (α) j Z1 F 1 1 (α)), Lower tail dependence index is λL = lim α!0 P(Z1 F 1 1 (α) j Z2 F 1 2 (α)) = lim α!0 P(Z2 F 1 2 (α) j Z1 F 1 1 (α)). 0 λ 1 Christophe Villa () EM Lyon February, 25 2010 17 / 22
  • 87. Asymptotic dependence asymptotically dependent if λ > 0 ) using multivariate EVT implicitly assume asymptotic dependence Christophe Villa () EM Lyon February, 25 2010 18 / 22
  • 88. Asymptotic dependence asymptotically dependent if λ > 0 ) using multivariate EVT implicitly assume asymptotic dependence asymptotically independent if λ = 0 ) a range of extremal dependence models describe dependence but have λ = 0 Christophe Villa () EM Lyon February, 25 2010 18 / 22
  • 89. Asymptotic dependence asymptotically dependent if λ > 0 ) using multivariate EVT implicitly assume asymptotic dependence asymptotically independent if λ = 0 ) a range of extremal dependence models describe dependence but have λ = 0 perfectly dependent if λ = 1 Christophe Villa () EM Lyon February, 25 2010 18 / 22
  • 90. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 91. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. To this end, PRT introduce another (weak) tail dependence index Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 92. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. To this end, PRT introduce another (weak) tail dependence index upper tail dependence index between Z1 and Z2 is de…ned as ηU = lim α!1 log(1 α) log P(Z1 > F 1 1 (α), Z2 > F 1 2 (α)) , Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 93. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. To this end, PRT introduce another (weak) tail dependence index upper tail dependence index between Z1 and Z2 is de…ned as ηU = lim α!1 log(1 α) log P(Z1 > F 1 1 (α), Z2 > F 1 2 (α)) , lower tail dependence index is ηL = lim α!0 log(α) log P(Z1 F 1 1 (α), Z2 F 1 2 (α)) . Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 94. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. To this end, PRT introduce another (weak) tail dependence index upper tail dependence index between Z1 and Z2 is de…ned as ηU = lim α!1 log(1 α) log P(Z1 > F 1 1 (α), Z2 > F 1 2 (α)) , lower tail dependence index is ηL = lim α!0 log(α) log P(Z1 F 1 1 (α), Z2 F 1 2 (α)) . η > 0, η = 1 2 and η < 1 loosely correspond respectively to when Z1 and Z2 are positively associated in the extremes, exactly independent, and negatively associated. Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 95. Asymptotic independence Although the random variables are asymptotically independent, di¤erent degrees of dependence are attainable at …nite levels of α. To this end, PRT introduce another (weak) tail dependence index upper tail dependence index between Z1 and Z2 is de…ned as ηU = lim α!1 log(1 α) log P(Z1 > F 1 1 (α), Z2 > F 1 2 (α)) , lower tail dependence index is ηL = lim α!0 log(α) log P(Z1 F 1 1 (α), Z2 F 1 2 (α)) . η > 0, η = 1 2 and η < 1 loosely correspond respectively to when Z1 and Z2 are positively associated in the extremes, exactly independent, and negatively associated. λ = 0 and η 2 (0, 1) signi…es asymptotic independence, in which case the value of η determines the strength of dependence within this class (also known as dependence in independence). Christophe Villa () EM Lyon February, 25 2010 19 / 22
  • 96. Results The Category-based comovement is also a Category-based (weak) tail comovement Christophe Villa () EM Lyon February, 25 2010 20 / 22
  • 97. Results The Category-based comovement is also a Category-based (weak) tail comovement Under the category-based comovement model, for two assets i and j we have λ ∆Pi , ∆Pj = λ ei , ej = 0 Christophe Villa () EM Lyon February, 25 2010 20 / 22
  • 98. Results The Category-based comovement is also a Category-based (weak) tail comovement Under the category-based comovement model, for two assets i and j we have λ ∆Pi , ∆Pj = λ ei , ej = 0 if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y , then η ∆Pi,t , ∆Pj,t > η ei,t , ej,t Christophe Villa () EM Lyon February, 25 2010 20 / 22
  • 99. Results The Category-based comovement is also a Category-based (weak) tail comovement Under the category-based comovement model, for two assets i and j we have λ ∆Pi , ∆Pj = λ ei , ej = 0 if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y , then η ∆Pi,t , ∆Pj,t > η ei,t , ej,t suppose that asset j, previously a member of category Y , is reclassi…ed as belonging to X. Then η ∆Pj,t , ∆PX ,t increases after j is added to category X where ∆PX ,t = 1 n ∑ l2X ∆Pl,t . Christophe Villa () EM Lyon February, 25 2010 20 / 22
  • 100. Results ctd. we assume that uX has heavy tails Christophe Villa () EM Lyon February, 25 2010 21 / 22
  • 101. Results ctd. we assume that uX has heavy tails if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y , then Christophe Villa () EM Lyon February, 25 2010 21 / 22
  • 102. Results ctd. we assume that uX has heavy tails if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y , then if uX has right-heavy tails 1 = λU ∆Pi,t , ∆Pj,t > λU ei,t , ej,t = 0 Christophe Villa () EM Lyon February, 25 2010 21 / 22
  • 103. Results ctd. we assume that uX has heavy tails if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y , then if uX has right-heavy tails 1 = λU ∆Pi,t , ∆Pj,t > λU ei,t , ej,t = 0 if uX has left-heavy tails 1 = λL ∆Pi,t , ∆Pj,t > λL ei,t , ej,t = 0 We thus observe that even if cash-‡ow shocks are asympotically independent, ∆Pi and ∆Pj are perfectly dependent. Christophe Villa () EM Lyon February, 25 2010 21 / 22
  • 104. Results ctd. we assume that uX has heavy tails Christophe Villa () EM Lyon February, 25 2010 22 / 22
  • 105. Results ctd. we assume that uX has heavy tails Suppose that asset j, previously a member of category Y , is reclassi…ed as belonging to X. Then Christophe Villa () EM Lyon February, 25 2010 22 / 22
  • 106. Results ctd. we assume that uX has heavy tails Suppose that asset j, previously a member of category Y , is reclassi…ed as belonging to X. Then if uX has right-heavy tails, λU ∆Pj,t , ∆PX ,t increases after j is added to category X where ∆PX ,t = 1 n ∑ l2X ∆Pl,t . Christophe Villa () EM Lyon February, 25 2010 22 / 22
  • 107. Results ctd. we assume that uX has heavy tails Suppose that asset j, previously a member of category Y , is reclassi…ed as belonging to X. Then if uX has right-heavy tails, λU ∆Pj,t , ∆PX ,t increases after j is added to category X where ∆PX ,t = 1 n ∑ l2X ∆Pl,t . if uX has left-heavy tails, λL ∆Pj,t , ∆PX ,t increases after j is added to category X where ∆PX ,t = 1 n ∑ l2X ∆Pl,t . Christophe Villa () EM Lyon February, 25 2010 22 / 22