John H. J. Einmahl Tilburg University, NL. Juan Juan Cai Tilburg University, NL

Transcription

1 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Estimatio of the margial exected shortfall Jua Jua Cai Tilburg iversity, NL Laures de Haa Erasmus iversity Rotterdam, NL iversity of Lisbo, PT Joh H. J. Eimahl Tilburg iversity, NL Che Zhou De Nederladsche Ba Erasmus iversity Rotterdam, NL

2 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Exected shortfall of a asset robability level is X at ( ( )) E X X F where F ( x) : = P{ X x} ad X X F X the iverse fuctio of F. X 2

3 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 A ba holds a ortfolio i i R = i yr Exected shortfall at robability level ( ) E R R< VaR Ca be decomosed as i ( VaR ) ye R R< i i The sesitivity to the i-th asset is ( VaR ) E R R< (is margial exected shortfall i this case) i 3

4 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 More geerally: Cosider a radom vector ( X, Y ) Margial exected shortfall (MES) of X at level is ( > ( )) E X Y F (these are losses hece Y big is bad). All these are ris measures i.e. characteristics that are idicative of the ris a ba occurs uder stress coditios. Y 4

5 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 We are iterested i MES uder excetioal stress coditios of the id that have occurred very rarely or eve ot at all. This is the id of situatio where extreme value ca hel. ( > ) E X Y F We wat to estimate ( ) for small o the basis of i.i.d. observatios X, Y, X, Y,, X, Y ( ) ( ) ( ) 2 2 ad we wat to rove that the estimator has good roerties. Y 5

6 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Whe we say that we wat to study a situatio that has hardly ever occurred, this meas that we eed to cosider the case i.e., whe a o-arametric estimator is imossible, sice we eed to extraolate. O the other had we wat to obtai a limit result, as (the umber of observatios) goes to ifiity. Sice the iequality to assume = ad = O( ) as. is essetial, we the have 6

7 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Note that a arametric model i this situatio is also ot realistic: The model is geerally chose to fit well i the cetral art of the distributio but we are iterested i the (far) tail where the model may ot be valid. Hece it is better to let the tail sea for itself. This is the semi-arametric aroach of extremevalue theory. 7

8 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Notatio: (t big ad small, t = ) () t : = F X t () t : = F 2 Y t θ : = E X Y > 2 MES 8

9 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 θ P X x, Y dx > > 2 0 E X Y 2 P Y > 2 = > = = P X > x, Y > dx 2 0 = P X x, Y dx 2 > > 0 i.e., θ = P X x, Y dx 2 > > 0 9

10 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202. We cosider the limit of this as 0 Coditios (): First ote (tae x = ustairs) P X >, Y > 2 = P F X <, F Y < { ( ) ( ) } 2 where F ad F are the distributio fuctios of 2 X ad Y. This is a coula. 0

11 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 : We imose coditios o the coula as 0 Suose there exists a ositive fuctio R( x, y ) (the deedece fuctio i the tail) such that for all 0 xy,, x y > 0, x y< x y lim P X >, Y R x, y 0 2 > = ( ) lim P F X <, F Y < R x, y 0 2 = x y ( ) ( ) ( ). i.e.,

12 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 This coditio idicates ad secifies deedece secifically i the tail. (usual coditio i extreme value theory) (2): Comare: i the defiitio of θ we have P X > x, Y > 2 ad i the coditio we have (for y = ) x P X >, Y > 2. 2

13 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 I order to coect the two we imose a secod coditio, o the tail of X : for x > 0 P X > tx lim t P X > t { } { } Where γ is a ositive arameter. = x γ. This secod coditio imlies a similar coditio for the quatile fuctio () t = F t amely lim t tx t ( ) () x > 0. γ = x ( ) 3

14 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 P X t We say that { > } is regularly varyig at ifiity with idex γ (. ) regularly varyig, with idex γ. RV ad γ (usual coditio is extreme value theory) is also These two coditios are the basic coditios of oe-dimesioal extreme value theory. 4

15 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Examles: Studet distributio, Cauchy distributio. It is quite geerally acceted that most fiacial data satisfy this coditio. Sufficiet coditio: F ( t) ct γ = + lower order owers. der these coditios we get the first result: 5

16 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 > θ = = E X Y 2 R x dx ( ) γ lim lim, Hece θ goes to ifiity as 0 at the same rate as, the value-at-ris for X. Now we go to statistics ad loo at how to θ. estimate 6

17 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 We do that i stages: First we estimate θ where = ( ), ( ) 0 as. Clearly we ca estimate θ o-arametrically (it is just iside the samle). The secod stage will be the extraolatio from θ to θ with. For the time beig we suose that X is a ositive radom variable. 7

18 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 θ = E X Y > Recall 2 by First ste: relace quatile ( ) 2 corresodig samle quatile, statistic from above). The obvious estimator of θ is the θ X { Y > Y } i, i= : = = X { } > PY 2 i i= Y ( th order. { > } i Y Y i, 8

19 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 First result: der some stregtheig of our coditios (relatig to R ad to the sequece ( ) ) θ θ d Θ, a ormal radom variable that we describe ow. 9

20 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Bacgroud of limit result is our assumtio lim P F X <, F Y < R x, y 0 2 = x y Now defie ( ) V ( ) ( ) ( ). : = F X W : = F Y. 2 ( ) V ad W have a uiform distributio, their joit distributio is a coula. 20

21 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Now cosider the i.i.d. r.v. s,, ( VW) = F( X) F( Y) ( i ) i i i 2 i i=. Emirical distributio fuctio: { V x, W y} i i We cosider the lower tail of ( VW ) i.e., the higher X, Y. tail for ( ) i i That is why we relace ( x, y ) by, xy>, 0 defie the tail versio T ( x, y) : = i= Vi, Wi x y i, i x y ad for 2

22 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202, Now T ( X Y ) is close to its mea which is P F ( X), F ( Y) 2 x y ad this is close to R( x, y. ) P T x, y R x, y Hece ( ) ( ) ad eve better T x, y R x, y ( ( ) ( )) coverges i distributio to a mea zero Gaussia rocess W ( x, y ) (i D- sace). R 22

23 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 This stochastic rocess W ( x, y ) has ideedet icremets that is, EW x, y W x, y = R x x, y y R R ( ) ( ) ( ) R ad i articular Var W x, y = R x, y. R ( ) ( ) 23

24 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Formulated i a differet way: Idex the rocess by itervals: ( ) ( ) R W 0, x 0, y : = W x, y ( ) ( ) R The for two itervals I ad I 2 (abuse of otatio) ( ) ( ) ( ) R R EW I W I = R I I

25 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Hece W is the direct aalogue of R Browia motio i 2-dimesioal sace. 25

26 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 How do we use this covergece for θ? T x, dx = dx { } Xi>, Yi> 2 ( ) γ 0 i= 0 x R. V. i= 0 i= 0 = γ { } γ Xi> x, Yi > 2 { } Xi> x, Yi > 2 dx dx γ 26

27 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 = i= 0 = { } i Yi > = 2 0 = i= { } { } Xi> x Yi > 2 ( ) Xi/ i X i X dx { } Yi > 2 dx ˆ θ 2 R. V. i = i= ( ) { Y } ( ) i> Y, sice: X P, R.V.. ( ) 27

28 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Hece ad we get θ θ ( ) ( ( ) ( )) γ θ θ T x, R x, dx 0 d γ W, R s, ds + W s, ds 0 0 ( ) ( ) ( ) γ ( ) R R γ a mea zero ormally distributed radom variable. 28

29 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Last ste: extraolatio from to θ (iside the samle) θ (outside the samle). Agai we use the reasoig tyical for extreme value theory. 29

30 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Cosider our first (o-statistical) result agai: lim, 0 E X Y > 2 R( = x ) γ dx 0 I articular this holds for = i.e. E X Y 2 R( = x ) γ dx 0 lim, >. 30

31 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Combie the two: θ = E X Y > 2 E X Y θ > = 2. 3

32 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 This leads to a estimate for θ θ = θ : Here θ is the estimator we discussed before ad = X, as before. 32

33 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 It remais to defie ad to study = as. Now with is a oe-dimesioal object (oly coected with X, ot Y ). Such quatile is beyod the scoe of the samle. Recall our coditio: P{ X > t} R.V. which imlies 33

34 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 lim ( ) t t tx () γ =. x Hece for large t ad (say) x > tx t x γ ( ) ( ) We use this relatio with t relaced by tx relaced by 34

35 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 The x= ( ). We get.. γ This suggests the estimator for : γ :. X, = = where γ is a estimator for γ. γ 35

36 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Sice γ > 0 we use the well-ow Hill estimator: γ = X : log log X. i,, i= 0 Proerty of Hill s estimator: ( d γ γ γ N ( ( ) N stadard ormal) may differ from but satisfies similar coditios) 36

37 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 X : Proerty of, X, d N 0 (stadard ormal) ( N ad N are ideedet). 0 Combie the two relatios: 37

38 X, = Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 R. V. γ γ γ γ γ X X,, = log N ( ) 0 + ex γ γ. 38

39 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Now assume that log 0 ( ) (this meas that ca ot be too small). The (exasio of fuctio ex ) ad hece log N ( ) γ γ 39

40 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, log d N γ (i.e. asymtotically ormal). Fial result:

41 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Coditios Suose γ ( 0, 2) ad X > 0. Assume : = ad d log d lim = 0. Deote r : = lim [ 0, ]. The as, log d Θ+ rγ N,ifr ; log d θ Θ + γ N,if r r >. d θ mi, Corer cases are r = 0 ad r = +. 4

42 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 So far we assumed X > 0. For geeral X we eed some extra coditios: mi X,0. Thier left tail: ( ) γ E <. 2. A further boud o =. The the left tail ca be igored. 42

43 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Estimator i case X : θ γ : = X i=. { X > 0, > } i Y Y i i, Has same behaviour as i case X > 0. 43

44 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Simulatio setu: Trasformed Cauchy distributio o ( 0, ) 2 : Tae ( Z, Z ) stadard Cauchy o 2 ad defie 2 ( X, Y) : = ( Z, Z ) 2 Studet t distributio o ( 0, ) 2. 3 With ( Z, Z ) as before 2 ( ) ( ) 2 5 (( ) 2 ( ) ( ) ( ) ) XY, = max 0, Z + mi 0, Z,max 0, Z + mi 0, Z 44

45 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Table : Stadardized mea ad stadard deviatio of θ log θ = 2,000 = 2,000 = 5,000 = 5,000 Trasformed Cauchy distributio () 0.52 (.027) 0.07 (.054) Studet-t 3 distributio (0.929) 0.48 (0.964) Trasformed Cauchy distributio (2) (.002) (.002) The umbers are the stadardized mea of log θ θ ad betwee bracets, the ratio of the samle stadard deviatio ad the real stadard deviatio based o 500 estimates with = 2,000 or 5,000 ad =. 45

46 Estimatio of the margial exected shortfall Laures de Haa, Poitiers,

47 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Alicatio Three ivestmets bas: Goldma Sachs (GS), Morga Staley (MS), ad T. Rowe Price (TROW). Data (X): mius log returs betwee 2000 ad 200. Data (Y): same for maret idex NYSE + AMES + Nasdaq. 47

48 Estimatio of the margial exected shortfall Laures de Haa, Poitiers,

49 Estimatio of the margial exected shortfall Laures de Haa, Poitiers,

50 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Table 2 : MES of the three ivestmet bas Ba Goldma Sachs (GS) Morga Staley (MS) T. Rowe Price (TROW) γ θ Here γ is comuted by taig the average of the Hill estimates for [ 70,90]. θ is give as before, with = 253, = 50 ad = =

51 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Iterretatio table 2: θ = 0.30 (Goldma Sachs) Hece i a oce-er-decade maret crisis the exected loss i log retur terms is 30% (erhas about 26% i equity rices) 5

52 Estimatio of the margial exected shortfall Laures de Haa, Poitiers, 202 Refereces V.V. Acharya, L.H. Pederso, T. Philio ad M. Richardso. Measurig systemic ris. Prerit, 200. C. Browlees ad R. Egle. Volatility, correlatio ad tails for systemic ris measuremet. Prerit, 20 52

53 Estimatio of the margial exected shortfall Laures de Haa, Poitiers,

54 Estimatio of the margial exected shortfall Laures de Haa, Poitiers,