V Time Varying Covariance and Correlaion DEFINITION OF CONDITIONAL CORRELATIONS. ARE THEY TIME VARYING? WHY DO WE NEED THEM? FACTOR MODELS DYNAMIC CONDITIONAL CORRELATIONS Are correlaions/covariances ime varying? Yes Correlaions are ime varying Derivaive prices of correlaion sensiive producs imply changes. Derivaives on correlaion now are raded. Time series esimaes change. There are many varieies. 1
How do we define ime varying correlaions? Recall ha for a single asse, we defined he condiional variance as he variance of he unpredicable par. Tha is, if y hen he condiional 2 variance is given by h E F 1. Hence if here is predicabiliy in he mean, we don include ha variaion in he condiional variance, we remove i firs. The same is rue for he condiional variance covariance marix. We define he condiional variance covariance marix for he par of y ha is no predicable. The condiional variance covariance marix is given by Eεε F 1 where ε y μ If here are no dynamics in he mean hen y 2
If mean reurns are no predicable hen y = and a wo dimensional covariance marix looks like: 2 r 1, 1, 1,2, Eyy F 1 E r1, r2, F 1 2 r 2, 1,2, 2, Condiional Covariance: E r r 1, 2, F 1 Condiional Variances CONDITIONAL CORRELATIONS Condiional correlaions are hen given by 1,2, 1,2, 1, 2, E 1 r1, r2, 2 2 1, 2, E r E r 1 1 3
ESTIMATION HISTORICAL CORRELATIONS i, j, s r r i, m j, m m1 s s 2 2 ri, mrj, m m1 m1 Exponenial smooher: q r r 1 q i, j, i, 1 j, 1 i, j, 1 Then i, j, q q i, j, q ii,, j, j, Use he same value for for each series 4
.9 100 day hisorical correlaions beween AXP and GE.8.7.6.5.4.3.2.1.0 94 95 96 97 98 99 00 01 02 03 04 C100_AXP_GE MULTIVARIATE MODELS 5
SOME MODELS ONE FACTOR MODEL ONE FACTOR MODEL WITH GARCH IDEOSYNCRATIC ERRORS MANY FACTOR MODEL MULTIVARIATE GARCH DYNAMIC CONDITIONAL CORRELATION ONE FACTOR ARCH One facor model such as CAPM There is one marke facor wih fixed beas and consan variance idiosyncraic errors independen of he facor. The marke has 2 some ype of ARCH wih variance m,. r r e i, i m, i, 2 2 ii,, i m, i If he marke has asymmeric volailiy, hen individual socks will oo. 6
CORRELATIONS Beween sock i and sock j assuming idiosyncracies are uncorrelaed. 2 i, j, i j m, i, j, 2 i j m, 2 2 2 2 i im, j jm, Assuming beas are boh posiive, correlaions range from zero o one and increase wih marke volailiy. HOW TO ESTIMATE A ONE FACTOR MODEL FIT THE VOLATILITY OF THE MARKET ESTIMATE THE BETAS OF THE STOCKS AND THE VARIANCE OF THE IDIOSYNCRACIES 7
MARKET VOLATILITY.030.025.020.015.010.005.000 94 95 96 97 98 99 00 01 02 03 04 Condiional Sandard Deviaion CALCULATE DYNAMIC CORRELATIONS h 1 2 2 2 2 2 1h 1 2h 2 8
AXP AND GE AGAIN.9.8.7.6.5.4.3.2.1 94 95 96 97 98 99 00 01 02 03 04 C4_AXP_GE The mulivariae facor model Consider he case of n asses and le is he vecor of beas for he n socks. Then ' hm, Vn Where V n is he diagonal marix wih he idiosyncraic variances (v i s) on he diagonal. 9
The correlaions are given by: 1 ' 1 2 2 R diag diag FACTOR DOUBLE ARCH Idiosyncraic errors also follow a GARCH models 2 ' hm, V, V ~ diagonal garch sd where h 0 V h 0 h n 1 2 2 where i i iri1 hi 1 10
Mulifacor models Iniially, suppress he ime subscrip and consider our K facor model for asse i K ri i i 1F1i2F2... ikfk ei i ijfj ei j1 0 if i j Where Eee i j 2 I is convenien o wrie in marix noaion: r ' i i Bi F ei 1xK Kx1 vi if i=j If he facors are mean zero and he idiosyncraic errors are uncorrelaed wih he facors we ge ha he variance of asse i is: 2 Eri i EBF i eibf i ei EBF e F' B e i i i i E BFF B E BFe E e 2 ' 2 i i i i i BB v i i i EFFis he covariance marix of he facors. 0 11
The ideosyncraic errors are uncorrelaed across asses so ha he covariance of asse i wih asse j is E ri i rj i E BF i ei BjF e j E BF e F B e ' i i j j i ' j i j i j i j E BFF B B EFe E ef B E ee BB i j 0 0 0 Now le r denoe a vecor of n asse reurns. r α B F e nx1 nx1 nxk Kx1 nx1 E E F F E F F EB FFB ee BEFFB EeeBB V rαrα B eb e B e Be Where I is a nxn ideniy marix and V is a nxn marix wih diagonal elemen i equal o v i. B is a marix wih i h row given by B i 12
Now le s exend his o he ime varying case: E EFF E r α r α B B ee B B V Can be made ime varying as in: E 1 E 1FF E 1 r α r α B B ee B B V where E FF 1 If he facors are observable (like porfolios), we can apply a model for ime varying (co)variances o he facors. E FF 1 13
Principal Componens Can we idenify facors from he reurns daa raher han specifying hem? Recall ha he variance covariance marix of he K facor model reurns are given by: E ' r α r α B B V where E FF Wih he facors unknown, B is idenified only up o an orhogonal ransformaion. To see his, le G be ANY KxK marix such ha GG I hen B * =B G and F * =G F also yields he same variance covariance srucure since he marix *' * *' * E F F V GG GG B Β B' ' ' BV B' ΒV 14
This lack of idenificaion can be solved by imposing some condiion (if we don have names on he facors we don care which of he equivalen models we choose)! Typically, we impose ha he facors are uncorrelaed. This yields: B' DBV where D is a diagonal marix We can find a represenaion of he facors by finding he linear combinaion of he daa ha has he highes variance. In pracice, we solve Max xˆ x subjec o xx 1 x 1 1 1 1 1 where ˆ is he esimaed variance covariance marix of he reurns. 15
The soluion is ha x 1 will be he eigenvecor associaed wih he larges eigenvalue of ˆ. Nex we simply normalize x 1 o sum o one so ha i is a valid vecor of porfolio weighs. The second principal componen solves he problem Max xˆx subjec o xx 1 and xˆx 0 x 2 2 2 2 2 1 2 The new se of porfolio weighs correspond o he porfolio ha has he highes variance bu is uncorrelaed wih he firs porfolio. The soluion is ha x 2 will be he eigenvecor associaed wih he second larges eigenvalue of ˆ. Again, he weighs x 2 can be normalized o sum o 1. 16
This process can be repeaed K imes unil we have weighs w 1, w 2,, w K which are he normalized eigenvecors of K larges eigenvecors of he variance covariance marix of reurns. Once he weighs are known, we can consruc he porfolios and rea he facors as if hey are observed. How many facors? The larges eigenvalues should be associaed wih he common componens. The smaller eigenvalues will be associaed wih he ideosynraic variances saring wih he larges and hen becoming smaller. When n ges large, he ideosyncraic variances become an arbirarily small fracion of he overall variance. The facors do no go away. 17
Hence in large samples we can idenify he K facors up o an orhogonal ransformaion. Ofen we look a he scree plo of he eigenvalues from larges o smalles. Number of facors is deermined by he drop off. Looks like 4 facors here: Eigen value 18
Dynamic Condiional Correlaion o DCC is a new ype of mulivariae GARCH model ha is paricularly convenien for big sysems. See Engle(2002) or Engle(2005). o Jus las monh Engle s new ex came ou. DYNAMIC CONDITIONAL CORRELATION OR DCC 1. Esimae volailiies for each asse and compue he sandardized residuals or volailiy adjused reurns. 2. Esimae he ime varying covariances beween hese using a maximum likelihood crierion and one of several models for he correlaions. 3. Form he correlaion marix and covariance marix. They are guaraneed o be posiive definie. 19
HOW IT WORKS When wo asses move in he same direcion, he correlaion is increased slighly. This effec may be sronger in down markes (asymmery in correlaions). When hey move in he opposie direcion i is decreased. The correlaions ofen are assumed o only emporarily deviae from a long run mean CORRELATIONS UPDATE LIKE GARCH Approximaely, z z 1, 1 2, 1 1 1 1 z z 1, 1 2, 1 1 20
An Asymmeric model allows correlaions o increase more when boh prices move down ogeher (like our asymmeric GARCH models. z z z z ( I )( I ) 1, 1 2, 1 1, 1 2, 1 z 0 z 0 1 1 2.9 DCC Correlaions.8.7.6.5.4.3.2.1.0 95 96 97 98 99 00 01 02 03 04 C9_AXP_GE 21
.9.8.7.6.5.4.3.2.1.0 95 96 97 98 99 00 01 02 03 04 C100_AXP_GE C4_AXP_GE C9_AXP_GE A simple approach o esimaing and forecasing large covariance marices Engle(2002), (2006) H DRD, D is diagonal marix of condiional sandard deviaions R is correlaion marix Based on sandardized reurns z 1 D r, V z R z 1 22
The general, mulivariae version of he DCC model looks like: ' 1 1 2 2 R diag Q Q diag Q 1 1 1 1 Q S z z Q where and are scalars. Hence he dynamics are he same for all series. S is he uncondiional correlaion marix. Esimaion T 1 1 L log H r ' H r 2T 1 1 1 1 2T 2 T T T 2 1 2log D r ' D r log R z ' R z z 1 2T 1 T 1 The log likelihood can be wrien as he sum of a par ha depends on he variances and a par ha depends on he correlaions. I is addiive and can be separaed ino wo esimaion problems. ' z 23
Bivariae Esimaion of correlaion parameers 1 z z 2* z z L 2 1 qi, j, 1 ri, jzi, 1 zj, 1 qi, j, 1 qi, j, i, j, q q T 2 2 2 i, j, i, j, i, j, log(2 ) log 1 i, j, 2 1 i, j, ii,, ii,, o Engle proposes esimaing a model for all univariae pairs and hen averaging parameer esimaes (easy, bu no saisically sound). o An alernaive approach is o rea he pairs as a panel daase and simulaneously esimae DCC parameers (inference works here). Asymmeric Dynamic Correlaions of Global Equiy and Bond Reurns Lorenzo Capiello, Rober Engle and Kevin Sheppard Journal of Financial Economerics (2006) 24
Daa Weekly $ reurns Jan 1987 o Feb 2002 (785 observaions) 21 Counry Equiy Series from FTSE All-World Index 13 Daasream Benchmark Bond Indices wih 5 years average mauriy Europe AUSTRIA* BELGIUM* DENMARK* FRANCE* GERMANY* IRELAND* ITALY THE NETHERLANDS* SPAIN SWEDEN* SWITZERLAND* NORWAY UNITED KINGDOM* Ausralasia AUSTRALIA HONG KONG JAPAN* NEW ZEALAND SINGAPORE Americas CANADA* MEXICO UNITED STATES* *wih bond reurns 25
GARCH Models (asymmeric in orange) GARCH AVGARCH NGARCH EGARCH ZGARCH GJR-GARCH APARCH AGARCH NAGARCH 3EQ,8BOND 0 1BOND 6EQ,1BOND 8EQ,1BOND 3EQ,1BOND 0 1EQ,1BOND 0 26
AVERAGE EMU COUNTRY BOND RETURN CORRELATION 27
RESULTS Asymmeric Correlaions correlaions rise afer negaive reurns Shif in level of correlaions wih formaion of Euro Correlaions are rising no jus wihin EMU EMU Bond correlaions are especially high 28