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7/31/2019 Economic Catastrophy Bonds Covan Jurek Stafford
1/41
628
Aerican Econoic Review 2009, 99:3, 628666tt://www.aeaweb.org/articles.?oi=10.1257/aer.99.3.628
T -
. P
q w , W
S . A y
z w () y , y . T -
z w w . Hw, yy ,
y .
T K J. Aw (1964) G D (1959) y y
. S y w
w w , y w -
. Cqy,
y .
Iy, w
y , w -
. C y y y . Hw, y
y w , y .
N, , y y y
Economic Catastrophe Bonds
By Joshua D. Coval, Jakub W. Jurek, and Erik Stafford*
Te central insigt o asset ricing is tat a secritys vale eens bot onits istribtion o ayos across econoic states an on state rices. In fxeincoe arkets, any investors ocs exclsively on estiates o execte
ayos, sc as creit ratings, witot consiering te state o te econoyin wic ealt occrs. Sc investors are likely to be attracte to secri-ties wose ayos reseble econoic catastroe bonsbons tat ealtonly ner severe econoic conitions. We sow tat any strctre fnance
instrents can be caracterize as econoic catastroe bons, bt oer arless coensation tan alternatives wit coarable ayo rofles. (JEL G11,G12)
* C: H B S, S F R, B, MA 02459 (-: j@.); J:B C F, P Uy, 26 P A, P, NJ 08540 (-: jj@-.); S: H B S, S F R, B, MA 02459 (-: @..)W J C, M C, J C, P C-D, F L, B M,A P, C S, Jy S , B C, B Uy, G S Uy, H M Cy, H Uy,O S Uy, P Uy, MIT, NYU, Uy C, Uy Py, VT, Y Uy, 2007 NBER A P S I, 2007 UNC/D A P C, 2008 U W F C, 15 M L Sy F M, 2008W F A C. W y C .
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VOL. 99 NO. 3 629COVAL ET AL.: ECONOmIC CATASTROphE BONdS
, w -
y w y . I , -
w y . T w
y , w - - .
T y , w -
w. I W F. S (1964) J L(1965) CAPM, w z .T w q D
B R Lz (1978). I y w y w w . W y -
, y y z
w, Aw-D z-
. T - y, w y R C. M
(1974) , w y .T w y y z
. S w , -
y y y y y
. Fy, , w -
y y - . A -
w y
, -y . T w
w- , w -
y .
O w- w y y w w
. B w y z
z , w q y conitional
, w
. T y . U -
y , w y
y y w (CDX). T . W w
y y y,
y . Sy,
y 30 wy CDX, w y .
Ry, , w CDX
y - . W w -
AAA- CDX y y
. T y y
, y, y y
US . I , y -
w y -y 60
. O 2004 2007, CDX y
qy , w , y
y .
7/31/2019 Economic Catastrophy Bonds Covan Jurek Stafford
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JuNE 2009630 ThE AmERICAN ECONOmIC REVIEW
I. The Impact o Pooling and Tranching on Asset Prices
S w . I , -
(, , w, .) oole . I
, w , trance, - . T y w
, , w w ( ) y y . I ,
.
T y w
y y w qy. A z
, y j .
M, z , -
, w y w . T y
y y (Ew J. E .2001; J D 2005; J H, M P, A W 2005).
T y y
. W w
w y -
w , y y y
- w . Sy,
y .
I , -
Aw (1964) D (1959).T S II ,
w .
A. Te Econoic Setting
I y ,
, w -y . S
- w w y y w -
y , , s, -
y. I w w, w s y
y , (-) (). I y Aw (1964) D (1959) w y -y.
S y y y , w w qy, y
y y , y
econoic assets, y y y .1 T
, w y, , w y , w y,d(s), y z. T y y w
, s. I , w
1 Ey, w w qy S, , E F. F K R. F (1993).
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VOL. 99 NO. 3 631COVAL ET AL.: ECONOmIC CATASTROphE BONdS
s, y (d(s)/s < 0).Fy, w y s y (s).
I , y
qy y - . T (..,
My, S&P, F) - .2 T y w
. Cqy, y w yy
y y
.
B. Caracterizing Trance prices
T w - y
, w y w
$1, N qy w y . T , , y y
1 X, z w. T X , y w
y.3 I -
z y z y .
I y z
, s, y
w , w d(s) N. I , y q -
y , w w y w ,Nd(s), ,Nd(s)(1 d(s)).
4 T
w N ,
w y . T
w , w y
qy . U
, - y ,dX(s), y
y y ,X,
(1) dX(s) = 1
Q__N
(X d(s))_______________
_____________d(s)(1 d(s)) R ,w () . F 1 - y, y 1 d
X(s), w X - y y w y w
2 I , y z, y .
3 I -w , w w , y w (X) (Y) , y X, z Y, y y w X Y.
4 T y w , s, y w d(s) d(s)(1-d(s))/N. A yw , w y wN.
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JuNE 2009632 ThE AmERICAN ECONOmIC REVIEW
w y. Hw, N, y w
y, .
T y
(2) dX=
s
dX(s) (s)s.
F , y y -z, X. L w y N. I ,
s
d1(X), N. T
y y,d
X, w
y w ,
w . Iy,
z y X
w y y w, ,
N y ,d
X.
M q w z ,Ny , w
y *, .., w .
Recession Boom
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
N (small)
N (medium)
N (large)
Economic state (s)
Tranche
payoff
Figure 1. The Effect of Collateral Pool Diversification on the State-Contingent Payoffs
of a CDO Tranche
Notes: T y - y CDO y y , X, y z w ( ). T y N , wN 10, 100, 1,000.
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VOL. 99 NO. 3 633COVAL ET AL.: ECONOmIC CATASTROphE BONdS
U - Aw (1964) D (1959), w s y q(s) Aw-D s y y - y y
. F , - y y
(3) B= s
(1 d(s)) q(s)s.A , , y
q . I , w
X(*,N) w ,d
X q
*, w y N, w y
(4) BX(*,N) =
s
Q__N (X(*,N) d(s))_____________________________d(s) (1 d(s)) Rq(s)s.By y , -
y q - -
y,d
X(*,N)(s); . By y
,X, q , y N, w
y *. T yX(*,N).
W w q w y.
PROPOSITION 1:holing te nconitional ealt robability constant at *, te vale o a
igital trance, BX(*, N), eclines as te nber o secrities in te nerlying bon ortolio,
N, increases.
PROOF:
S A A.
T y P 1 N, w
, y w y () ww y (). Cqy, , - q y . I
y , -
, .
C. Ceaest to Sly
I ,
, y, y,*. I
, w w -
, w w w y - , w y . U -
w, w w w
7/31/2019 Economic Catastrophy Bonds Covan Jurek Stafford
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JuNE 2009634 ThE AmERICAN ECONOmIC REVIEW
y 1 *, w w (.., y ),5 w w . W y
ceaest to sly.
PROPOSITION 2: Te ceaest secrity to sly wit an nconitional execte ayoo1 * ays zero on a set wit easre * containing te worst econoic states, an oneelsewere.
PROOF:
S A A.
F , w w , s, y z
, y w -
, w *- - . Ay, y z
* z, w.A F 1 , y y -
N . T , -
, - y y ,
1 d(s). I , w - X y z - w s, wd(s) =X. T, w *, y y
y y w y *.
PROPOSITION 3: For any esire robability o ealt *, te ceaest asset to sly wit
tat robability o ealt can be constrcte by issing a trance wit a loss attacent ointo X(*,) against an asytotically iversife collateral ool (N). I te clativeistribtion o econoic states is given by (s), te attacent oint o te liiting trance isgiven by, d(
1(*)).
PROOF:
S A A.
Fy, w -
y y . I ,
y , w w .
Hw, N , y, y y y . T
y F 2, w y
,
y. A w , y , y
() w y (). Cqy, y -
, , y
. I w y
5 Y , , z y y y y y y - w .
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VOL. 99 NO. 3 635COVAL ET AL.: ECONOmIC CATASTROphE BONdS
y y y y
, w y .
II. Pricing CDO Tranches
O wy z ,
CDO. A CDO w z
y . I y CDO, y y ( CDO) w (y CDO), w .6 T
, w, - y
w. Cqy, w w, w -
CDO, qy w w y CDO, w
j y.
T - y y
-w CDO . T q () y , () y z
6 A w (CDS) y , y .
Recession Boom
0.0
0.2
0.4
0.6
0.8
1.0
Economic state (s)
Payoff
Bond portfolioCheapest to supply
Figure 2. State-Contingent Payoffs of a Bond Portfolio and a CDO Tranche with the Same Expected Loss
Note: T y - y CDO ( ) - CDO y .
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JuNE 2009636 ThE AmERICAN ECONOmIC REVIEW
. I S (1964) L (1965) CAPM, w z y z qy . T
w y
q B Lz (1978).
T z y , w - y y y . W -
y , CDO y y y
, y y
y y . I z -
y, w y - y,
d(s), - y , w w y z. T , w y M (1974) y - .7 Sy, w y
CAPM-y , w w -
y z . By w -
, w - , w y y
y .8
Iy, w y y y z
, w -
. T - (O V1987, 1991; P S 2000; H, P, W 2006), w - -
.9 F, y - (..,) , y w y -
, w y yz - .Fy, CDO y, w y qy
. By - , w w j. F,
w w y w- , w
y y y
(G, -, ).10 A, , w y , w y $1 .
A.Integrating mertons Creit moel wit te CApm
I M (1974) ,
,Ai,T, w , di.11 Fy, y y y CDO, w w w
w N, w . I
, w y
7 S Y H E, J Hw, J-Z H (2004), , y .
8 C Z (2001) y .
9 T V , w y , G , q CDO , w ilie correlation.
10 S z , - .
11 F B J C. C (1976) , w .
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VOL. 99 NO. 3 637 COVAL ET AL.: ECONOmIC CATASTROphE BONdS
w y N (oogenos ortolio asstion). Cqy, z y-
. I , M , w -
z y : , w w y
, a, -- ,d/At, y , a.I y z , - , w y M y
. Sy, w y -
, Z, y , Zi,. Iy, w
y y y, w
, w y, -G
. Oy conitional z w
G (.., Zi, y ). M, w ww , w , -
, y y, T. I w y , y, T t, () y
(5) Ai,T Ai,t = Qr+a a2
___
2 R + a_ Z+ _ Zi,,
(6) mT mt = Qr + 2
___
2 R + _ Z,
w r , a , a CAPM , y y, y, qy -
, y. Fy, y, a, y, _________a22 + 2, CAPM a a2/2 = a( 2/2).S w y z
, w w y w w
A i,T. I w y, ,
, mT, t ,mt ((r )), w , A i,T, y q ,
y
(7) A i,T( ) = (Ai,t) (r + a + _
Zi,).
T, y i z ,
w y y A i,T( )
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JuNE 2009638 ThE AmERICAN ECONOmIC REVIEW
U , w y ( = 0), econoic assets CAPM ( > 0). Cqy, y y (i
d()/< 0).12
T , - y, w
y y y . I M (1974) - , y y y , A i,T. Hw, w w w y y
y y . Cqy,
, , y (Hy L 1994). F, , Mj C, D, P M (2008) y y q 50 .13 W
y , - y qy w N
y y
(10) p( )= 1__Ni=1
N
Q1 (1 1A i,T( ) d) + (1 )
A i,T()_____________d 1A i,T( ) dR ,w 1A i,T( ) d
z
i , z w.
T , w - y -
qy . S w
, y w --
y . W i ,
w - y
(11) Et[p( )]= 1 Q1 (1 )Et[AT( )| AT( )
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VOL. 99 NO. 3 639COVAL ET AL.: ECONOmIC CATASTROphE BONdS
Fy, y , w y Aw-D -
q - y. S w -
z , y qy
. I , B Lz (1978) w
E y w y . By
y , q( ), z , w () y , V:
(13) V=
Et[p( )]q( ) .
S y , - y-
z y , p( ). F , w w X Y. T y q z y 1 Y; y 1 X; j y w z y w 1 Y 1 X. T y y y w 1/(Y X) y 1 X, 1/(YX) 1 Y:
(14) p[X,Y]( )= 1 1_____
YXQ A(1 X) p( ), 0B A(1 Y) p( ), 0BR .
Cy, , V[X,Y],
wy y , y -
y , (13). Hw, , - y yy
y p( ) y y , (14).Cqy, y y
. I , -
y w , y y
y y .
III. Data Description
O y w . T y
CDO w w DJ CDX N A I G I. T
, w F L A Rj (2008), qy w 125 q -y w (CDS) US w .15 A CDS y ,
w w w y. T
y y , w ,
,
15 D D K J. S (2003) CDS CDO. Ay y D N G (2001), .
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JuNE 2009640 ThE AmERICAN ECONOmIC REVIEW
y . T
CDX -
qy w y . I w
CDS, w w j y y
. W y y , CDX w y q
y CDS .
O S 2004 S 2007 y
CDX w 03, 37, 710, 1015, 1530 CDX .16 T CDX
w y y CDX,
. F , $1 710
y $1 CDX 7 ; $0
CDX 10 ; y y j CDX w
7 10 . A w CDS, q
y w y y. 17 I
, y y y . Cqy,
, y y .
I , CDX y M S
q . I ,
w CDX, y ,
. S jy y -- , w
S 3 S 8 CDX NA IG -
-- -y , L Rj (2008). F , w CDX y
AA, A, BBB, BB, B- . T q -y CDS y .
Fy, y q ---y
. T CDX y, w
q y y. Uy, ,
w y y z . I w
y y -- q -y S&P 500 C.
T q 13 w z y 0.70
(30 ---y) 1.30 (30 --y) 5 ,w y - y.18
16 A , J 2007, w S2006. B A 2007, w y . T y w w S 2007 . T w ://w..//5725..
17 Fy, q 03 q - y y , 500. I 03 , w - y q .
18 A oneyness K/Ft,, w K Ft, -y (= 5) S&P 500 .
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VOL. 99 NO. 3 641COVAL ET AL.: ECONOmIC CATASTROphE BONdS
A. Sary Statistics
T 1 y CDX w
y. P A
. A ,
qy . T CDX y
BBB , w CDS
CDX BBB, y A K By M (2004).P B wy P C
. C - y y w , y y -
.19 T w - y
CDX w .
IV. Calibrating the Bond Pricing Model
A CDX y -
y y y. I
y , q CDX
19 J P K J. S (2006) w w y.
Table 1Summary Statistics for US Credit Market (9/20049/2007)
5y AA BBB CDX [03] [37] [710] [1015] [1530]
panel A. Tie series eans an stanar eviations o aily seriesM 18.20% 15.60 47.90 45.90 34.00% 138.00 39.10 18.20 8.60
S 1.40% 5.00 7.10 10.10 8.50% 61.40 25.10 11.00 5.30
panel B. Correlations between weekly series
5y 1.00AA 0.59 1.00BBB 0.61 0.77 1.00CDX 0.20 0.50 0.51 1.00
[03] 0.19 0.47 0.49 0.93 1.00[37] 0.03 0.26 0.32 0.80 0.66 1.00[710] 0.18 0.18 0.23 0.70 0.55 0.95 1.00[1015] 0.00 0.33 0.37 0.87 0.73 0.97 0.94 1.00[1530] 0.17 0.47 0.49 0.95 0.86 0.81 0.73 0.89 1.00
panel C. Correlations between canges in weekly series
5y 1.00AA 0.22 1.00BBB 0.13 0.67 1.00CDX 0.45 0.28 0.37 1.00
[03] 0.38 0.14 0.25 0.69 1.00[37] 0.37 0.10 0.16 0.67 0.87 1.00[710] 0.32 0.18 0.21 0.69 0.77 0.91 1.00[1015] 0.37 0.23 0.27 0.76 0.76 0.88 0.94 1.00[1530] 0.37 0.26 0.26 0.75 0.73 0.77 0.79 0.86 1.00
Notes: T y . R -y w y w . T CDX Dw J CDX N A I G -y w. S CDX y w,X, , Y, , [XY]. A , [03] , (1 = 0.01 ). F-y --y
y S&P 500 5y .
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y y
y ( ) y q y ( ). Cqy, w CDX , w
, w w q . T ,w, y y w w y
y CDX, V, w S II.
I y y y w
CDX, w - . I , w
S II - y y
, (11), w w -y , (13). I w, w j y y , Aw-D
y . T q
w y .
T y , w y y y CDX , .., y N-
. Cqy, ,
d/At, y y, , , a , w y , w z
. O y, w , w q
w : () - CDX y ; () - qy q
__
; , () - w qy q __
.A , w y ,
CDX -j , y qy . I, L, Sjy M, E N
(2005) y , w, w
qy y .
T w - y -
z y (d/At,,a) - . I y y ,
w y CDX y w
, CDX . T y
, w y . F, w
w y y. S, w w y . T, w w
w R2 CDX y
q, y y
. T - y
CDX . W w w
, w
.
A.Extracting State prices
I - y S
II, w z - ,
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VOL. 99 NO. 3 643COVAL ET AL.: ECONOmIC CATASTROphE BONdS
q( ). A , w y y () y, w ,mt+, - t,
(15) = mt+____
Ft,=
mt+_______________(m
t)
((r
)
)
.
T w Aw-D
. I , B Lz (1978) w E w - y K,C(K,) Aw-D y q w . Cqy,
(16) q( )=2C(K,)_______
K2 |K=xFt,,
wx = ( ) y . T Aw-D y w y w - .
A w w , w, y , w
---y w y
- . T , w B
Lz (1978) y . Sy, w , C(K,), B-S w y , CBS(K,(K,),), w
(17) q( )=2CBS_____K2 + ___K Q2 Q2CBS_____KR+2CBS_____2 Q___KRR+ 2___K2 CBS____ |K=x Ft,.
F y w- y , (K,), Aw-D . C
w , y ,
w - . W
y w w y w -
y .
S y w- y Aw-D
, y w -y . O , w y
w y 0.7 1.3, w
y . Ty
y (J R RF. E 2001) (R B N Pz 2004).20 I, w , w y z y ,
w y , ,
20 Y A-S Aw W. L(1998
) ,
- y, q y qy. F w - y , JC. Jw (1999) B B R H (2003).
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w ( A B ). O y
(18) (x,)=a+b (c x) (a>b> 0),
wx y . T
y , w Aw-D ,
13 E w w , z q () . T w -
, w y ,
Aw-D (17 ) . T y y w ,
y q 0.29 .
F 3 y -y y
CDX . T -y --y y 20 10 y y . A y z,
y y 30 . T y
0 0.5 1.0 1.5 2.0 2.5 3.00
0.001
0.002
0.003
0.004
0.005
Moneyness
Stateprice[$]
Five-year optionimplied state prices
Series 4
Series 6
Series 8
0 0.5 1.0 1.5 2.0 2.5 3.0
10
20
30
Moneyness
Impliedvo
latility[%]
Fitted five-year optionimplied volatility function
Series 4
Series 6
Series 8
Figure 3. Calibrated Five-Year Implied Volatility Functions and State Prices as of
Selected CDX Initiation Dates
Notes: T w y y -yS&P 500 . T y . T y 13 w z y -y S&P 500 (y), 0.7 1.3, 0.05 . T y - q B Lz (1978), j y w.
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VOL. 99 NO. 3 645COVAL ET AL.: ECONOmIC CATASTROphE BONdS
w y 0 0.5,
.
T y Aw-D z ,
w w , y. T y
y y - w , w w q
CDX . T y, w
y y q -
y w w . T y
y w w y 0 0.7
y 0.7. T y w
w ,21 w -
--y . W
y
y .
B.Ilying te Conitional payo
Eq (13) w y y CDX y y - y y , -
, . By - qy
, w y y CDX y
w . T w y C,
D, M (2008), w w w - .
G y , CDX , y, w -
(d/At,a,) CDX y. T q , w w . I
, w q qy w
. W w qy ,__
, - CDX 2003 2007 0.2 qy ,
__
, .T CDX - y
US .
O y y , (11), M y , w yy. O y-
y y , w w - q . T , w w
y . Sy, w y
M y - y ,__
R , 40
:22
(19) Et[p()]= 1 (1 __
R)d( ).
21 T y q 3.20 .
22 T CDS- w 40 y , w wA (2006). W qy w 50 y .
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W , y CDX 0.40
z, w w M y , CDX y
z . A , , , ww- -
y y .
W y , (d/At,a,), - w y w y
. T y
y y . Sy
T 2. A -
, y , .
T y
y ---y
CDX . M, , q .
C.Evalating te Bon pricing moel
S ,
CDX y . T,
y y z ,
w y -
Aw-D . I y y-
, y - y CDX . Cqy, yz j
y CDX, w
Table 2CDX Calibration Results (9/20049/2007)
M 5-y MD-- Iy y
M A y y
M y (1) 0.7359 0.3435 0.2688 0.0410 0.4103(0.0196) (0.0249) (0.0140) (0.0029) (0.0042)
S- 0.7317 0.3494 0.2672 0.0423 0.4000
y (2) (0.0186) (0.0236) (0.0136) (0.0032) (0.0000)
M y w 0.7150 0.3725 0.2612 0.0417 0.4110 y (3)
(0.0197) (0.0251) (0.0139) (0.0030) (0.0043)
S- y 0.7133 0.3750 0.2605 0.0423 0.4000w y (4)
(0.0187) (0.0237) (0.0135) (0.0032) (0.0000)
Notes: T y y , - y ,
CDX y. T CDX Dw J CDX N A I G -y w. A y y . M 1 3 M (1974) y . M 2 4 - y 40 . M 3 4 y y q y y -y . T (a), -- (d/A), y y (). T - -y y y . T wy .
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VOL. 99 NO. 3 647 COVAL ET AL.: ECONOmIC CATASTROphE BONdS
wy CDX y , w
y .
T 3 . W
t t+ 1 w y t+ 1, t, y t. T y -
y w R2 . I -
(M 1), t- 8.70 R2 0.34.W w - ,
S&P 500 , -y --y y, -y -
y w. T y CDX
, y w . T
y w, w y .
T ,
y y CDX. A,
y y
y CDX ,
y qy . Fy, y w
y CDS (P C-D, R S. G, J. S M 2001; Bj Z, H Z, H Z ).
Table 3Regressions Explaining the Time Series of Changes in CDX Yield Spreads (9/20049/2007)
I E[y1] E[y2] E[y3] E[y4] r S&P 5y Sw R2[N]
0.0000 0.0687 0.0096 0.0188 0.0089 0.4138(1.47) (3.16) (5.16) (3.50) (3.00) [146]
0.0000 0.3787 0.3402(0.78) (8.70) [146]
0.0000 0.6100 0.0088 0.0022 0.0152 0.0169 0.4423(1.37) (2.87) (0.30) (0.49) (1.17) (4.21) [146]
0.0000 0.4050 0.3497
(0.85) (8.89) [146]
0.0000 0.7217 0.0002 0.0043 0.0179 0.0172 0.4461(1.37) (3.04) (0.01) (0.87) (1.36) (4.33) [146]
0.0000 0.4002 0.3400
(0.96) (8.70) [146]
0.0000 0.5609 0.0107 0.0021 0.0055 0.0165 0.4336(1.45) (2.44) (0.33) (0.41) (0.49) (3.86) [146]
0.0000 0.4200 0.3479(1.01) (8.85) [146]
0.0000 0.6381 0.0046 0.0036 0.0071 0.0166 0.4360(1.45) (2.56) (0.14) (0.66) (0.62) (3.97) [146]
Notes: T wy CDX . T CDX Dw J CDX NA I G -y w. T CDX ,E[yi], w y t+ 1, t, y t, . A y y. M 1 3 M (1974) y . M 2 4 - y 40 . M 3 4 y y q y y -y . T -yw r . S&P S&P 500 . F-y --y y
S&P 500 5y . Sw w y -y30 ---y y -y --y . Wy y . T jR-q R2, t- , q .
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V. Pricing Credit Derivatives
T M (1974) w , CAPM, - y . T y-
y z , w w . I w, w
w qy . T
y w w y, -
w y . W w q w
w .
T w . H
( , , y y) - CDX, w - y CDX (10).23S (.., ) y y CDX(.., y y), y y y y . I , y y CDX y. Fy, , y -
y y Aw-D .
T - y w. F, w y
y (.., z ), 0 10 0.005. S, y , , w
q (10) CDX (N= 125), w . T y. W
10,000 - y , w
y - .24
T - y CDX 710 , - , y
F 4.
T 4 y w
y CDX . I , , w
w wy y
y. T -, qy, (03) , y .25
O y
qy . A , y y
710 1015 . T 710 y
y . F 1015 . F 5
. A ,
23 M w y - q - . I , w w 60 (1-y ) 25 -, A (2006).
24 A y y execte- CDX y, , -CDX y. A J y y , y y .
25 T - 03 q CDX , w . T 03 T 4 y y .
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qy . O , wy
w y . T
y , y
y .
T Aw-D . I -, w
z . T
y . A y -
y - y w -
y--
(.., w y w w ) w y . T w y . F, y w
. S, y w
, .T w
. W wy ,
Figure 4. Simulated State-Contingent Payoffs for the CDX and the 7 percent10 percent CDX Tranche
Notes: T y CDX 7 10 CDX y , w M (1974) y . T - y () (). T y y w y -, w w 0.4 0.25.
VOL. 99 NO. 3 649COVAL ET AL.: ECONOmIC CATASTROphE BONdS
0 0.5 1 1.5 2 2.5 30.0
0.2
0.4
0.6
0.8
1.0
Moneyness
Bondportfolio
(CDX)payoff
Merton recovery
0 0.5 1 1.5 2 2.5 30.0
0.2
0.4
0.6
0.8
1.0
Moneyness
Bondportfolio
(CDX)payoff
State-independent recovery
0 0.5 1 1.5 2 2.5 30.0
0.2
0.4
0.6
0.8
1.0
Moneyness
7%10%t
ranche
payoff
Merton recovery
0 0.5 1 1.5 2 2.5 30.0
0.2
0.4
0.6
0.8
1.0
Moneyness
7%10%t
ranche
payoff
State-independent recovery
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w y y execteCDX
y. F , w
w X, Y, w (11) q 1 X, y , . Cy, y- y ---y (.., ). T -- y (.., ), w CDX y q 1 Y.
T 5 -- .
T - y
.26 A, y y -
qy CDO . M, w wy
q . T ,
CDX
26 I y w y q.
Table 4 Comparison of Actual and Model Tranche Spreads (9/20049/2007)
C C T M M
() [] [] () ()
merton recovery30100 1 NA NA NA1530 28 9 0.94 0.681015 87 18 0.85 0.66710 150 39 0.67 0.5837 267 138 0.80 0.5803 914 1,508 0.92 0.75
State-ineenent recovery30100 1 NA NA NA1530 25 9 0.95 0.701015 85 18 0.85 0.67710 150 39 0.67 0.5737 273 138 0.80 0.5803 953 1,517 0.93 0.75
merton recovery wit constraine ilie volatility nction30100
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VOL. 99 NO. 3 651COVAL ET AL.: ECONOmIC CATASTROphE BONdS
w y , w w
.
VI. Discussion
O w CDO
- ,
w . T y y
y y
4 5 6 7 80
20
40
60
80
100
Series
Spread
[bps]
CDX
Model Actual
4 5 6 7 8500
1,000
1,500
2,000
2,500
3,000
3,500
Series
Spread
[bps]
0%3% Tranche
Model Actual
4 5 6 7 80
200
400
600
800
Series
Spread
[bps]
3%7% Tranche
Model Actual
4 5 6 7 80
100
200
300
400
Series
Spread
[bps]
7%10% Tranche
Model Actual
4 5 6 7 80
50
100
150
200
250
Series
Spread
[bps]
10%15% Tranche
Model Actual
4 5 6 7 80
20
40
60
80
100
Series
Spread
[bps]
15%30% Tranche
Model Actual
Figure 5. Time Series Comparison of Model and Actual Tranche Spreads (9/20049/2007)
Notes: O y, CDX , y y - CDX y y- . Dy y - y - . T CDX Dw J CDX N A I G -y w.T y w .
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- , y . I
, CDO w , w
y .T y y j y (F
1970). T, CDX , w
.
A.moel Robstness
T y y : () ; ()
; () CDX y y y . T w .
Table 5Comparison of Actual and Put Spread Approximated Tranche Spreads (9/20049/2007)
T()
M
[]
M
[]
C ()
C ()
merton recovery30100 2 NA NA NA1530 29 9 0.59 0.201015 82 18 0.45 0.19710 139 39 0.30 0.1237 263 138 0.42 0.0903 837 1,508 0.55 0.34
State-ineenent recovery30100 1 NA NA NA1530 26 9 0.59 0.201015 80 18 0.45 0.19710 139 39 0.30 0.1237 269 138 0.42 0.09
03 872 1,517 0.54 0.35
merton recovery wit constraine ilie volatility nction30100
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VOL. 99 NO. 3 653COVAL ET AL.: ECONOmIC CATASTROphE BONdS
Single-Factor.A w w
w y q w - -
. T , y y 10
qy , w
- -w. T y w y
w y y . O ,
wy y -
. F , (.., ) w y , - y y
q, - .
F 6 y y
w w -y -
. T w ( = 0.60) w w 1987. T , ,
w . T - , w -y
y - w
. L w
. O, -w
- qy -
w US 19 y.
Ilie Volatility Extraolation.T y -
-y --
. I , y y 0.7 1.3. My y w y
w w 0.7. T, y
-y y y w 0.7, y
OTC y w , w -
- S&P 500 (SPX). Sy, w y SPX
. T w y y wy
w y w 0.7, w 0.4. T -
y w y -y -
y. M y, -
y w y w 1.0. I, , w w y 1.0 - w y ww
y y.
W , y w
w y 0.7 y w
y y 0.7. A S IV, y -
wy. A, y, , w
y (x,)=a+b (cx), wx y. B y y
y ---y , .
I , 2007 y w y ---y y w CDX
, .
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hoogeneity.A y
125 CDX y y . F,
w CDX w R2 w , y
w .
S, w - , y
y w-y -
. T -,
qy zz , -
zz . O , y
- w , y , . A, CDX
w , y y -
7 . A , w w-y
y w wy. Ry,
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Downgraderate(percent)
40
30
20
10
0
10
20
30
40
50
Laggedone-yearmarketreturn(percen
t)
Downgrade rate Lagged return
Figure 6. US Bond Downgrade Rate and Lagged Market Returns (12/19874/2007)
Notes: Dw y- US w S&P- w . T -y S&P500 12 . U , w:
P (Dw i,t= 1)= (X)/(1 + (X)), wX=b0+b1R,t+b2(Ri,tR,t)
T y w, w :
b0 b1 b2 N
3.90 3.27 3.01 129,330
(0.03) (0.13) (0.06)
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VOL. 99 NO. 3 655COVAL ET AL.: ECONOmIC CATASTROphE BONdS
y- My 2005 w GM
F, w y y- w. T
y zz ww
y . Eqy w y
zz , y y- w qy
zz .
4 5 6 7 80
20
40
60
80
100
Series
Spread
[bps]
CDX
Model Actual
4 5 6 7 8500
1,000
1,500
2,000
2,500
3,000
3,500
Series
Spread
[bps]
0%3% Tranche
Model Actual
4 5 6 7 80
200
400
600
800
Series
Spread
[bps]
3%7% Tranche
Model Actual
4 5 6 7 80
100
200
300
400
Series
Spread
[bps]
7%10% Tranche
Model Actual
4 5 6 7 80
50
100
150
200
250
Series
Spread
[bps]
10%15% Tranche
Model Actual
4 5 6 7 80
20
40
60
80
100
Series
Spread
[bps]
15%30% Tranche
Model Actual
Figure 7. Robustness of Model Results to Ruling Out Extreme Market States (9/20049/2007)
Notes: T w w y w 0.4 z, w y . O y, CDX , y y - CDX y y- . Dy y - y - . T CDX Dw J CDX N A I G -y w. T y w .
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B.Rling Ot Extree States
T , CDX
- w w -
, . O , w y y , . T
y , y , -
(.., ---y ) w .S w y
, y y . R
y ---y w w , w
q z. T -
y , -
y , y
CDX . I , w w y 0.40 z,
w y .27 T - y y w
0.40 ,
, y ---y . Ny, -
y
w j z . W
y CDX CDX ( F 7).Iy, y j ,
qy , w 37, 710, 1015
. T
y .
C. Wic market Is misrice?
O y w -
: S&P , ,
CDX . A w
CDO y , q w w -
y CDX y w .
T q q y
.28 I
CDO , y w CDO y y CDO
w 2008,
w.
A y y-
. T
j y . W
27 T j y y y. T 0.40 y w y -qy .
28 O B (2003) D B (2006), , -, w M B, M C, M J (2007) w .
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VOL. 99 NO. 3 657 COVAL ET AL.: ECONOmIC CATASTROphE BONdS
y w . A
w . F , y
L Rj (2008), S Azz Ky G (2008), A E(2008) CDX CDX w - ,
y y .I , w
, qy y, jy CDX CDX
. T y, w -
z y y y . T y w
y y. W
y , w jy
CDX CDX . S q CDX
, CDX. B ,
w y w : CDO
bot .
D. Creit Risk preia
A w w S I, y w
, -
. I , (.., ) z , , y w
y. W y yy
(N),
, y . I y y , y ECB y
. T , y,
, w , =Yiel Srea/Loss Rate.29 W y
w , z -
y w .30
W w ,
- .
I , w , w
y . T, w y , w
, , q 5 y y, , y --y -y- y. T w
. T T 6.T
, y w
y . F , y, q
29 W y z,
- -
y - .30 T y,Loss Rate=(1/T) E[Py] q y w y-
w .
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.
Cy, y y w
y . C w ,
y - .31 T
w y w
w w y y . Hy, -
CDX BBB A. F , K
M (2004) CDX BBB+ J2004. O w . T
y CDX 31 , -y y 1.54 ,w w A- (0.50 ) BBB- (2.08 ) , R C . (2005). T CDX 2.9, y , 2.3 3.3.
R y w y
( ), z y , . A , , y
- . I
31 E . (2001) Aj B . (2004) y - ; H, P, W (2005) w A- BBB- w 1996 2004.
Table 6Credit Risk Premia (9/20049/2007)
MIy A M 5-y M
D-- y ySy A y [] [] []
Five-year creit ealt swa inicesAA 0.85 0.19 0.31 16 16 36 0.41 4.3A 0.81 0.25 0.29 27 27 73 0.41 3.5BBB 0.73 0.35 0.27 48 48 162 0.41 2.8BB 0.44 0.75 0.16 145 145 724 0.43 1.9B 0.12 1.14 0.05 250 249 1,616 0.47 1.6
dow Jones CdX Nort Aerica IG inexCDX 0.74 0.34 0.27 46 46 153 0.41 2.9
CdX trances(ercentage)30100 NA 1 0.03 0.96 6,106.61530 9 28 4 0.81 364.51015 18 87 22 0.56 69.6710 39 150 77 0.45 25.237 138 267 599 0.61 7.203 1,508 914 5,943 0.75 1.7
Notes: T y - . R -y w y w . T CDX Dw J CDX NA I G -y w. T CDX y w . T (a), -- (d/A), y y (). T . T , , y . L z y . T - , -y y y j , w y S&P 500 .
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VOL. 99 NO. 3 659COVAL ET AL.: ECONOmIC CATASTROphE BONdS
y --
- , w w
y . I , w
w y y y
. T y T 6, w CDX -CDS w y, y -
, y y . T y , y -
y q y- .
F 8 y - - , y .
T y , w y
, y . I
qy CDX y w
- . I , y - , y
. F 8 y y
5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.00
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Model[03]
Model[37]
Model[710]
Model[1015]
Model[1530]
CDXAAA
BBB
BBB
Actual[03]Actual
[37]
Actual[710]
Actual
[1015]
Actual[1530]
Annualized loss rate (log10
)
(log10
)
Cheapest to supply
Figure 8. Credit Risk Premia for Single-Name Credits and CDX Tranches (9/20049/2007)
Notes: T y 10 , , 10 y . M () y () y . L z y . - - y . T CDX y q CDX y .
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VOL. 99 NO. 3 661COVAL ET AL.: ECONOmIC CATASTROphE BONdS
w CDO , w q -
, y y w ---
y . W y w
w y CDO w,
, y q .
Appendix A: Proofs of Propositions
PROOF OF PROPOSITION 1:
L d
X(*,N)(s) - y, w
y *, y N. S
w w y
y w w
(A1)
ss
d
X(*,N)______N (s) (s) s+
s>s
d
X(*,N)______N (s) (s) s= 0,
w s () w() s. N - y y 1
dX(
*,N)(s),w q , y -
,
(A2)BX____N |
dX
=*=
ss
d
X(*,N)______
N (s) Qq(s)____(s)R(s) s
s>s
d
X(*,N)______
N (s) Qq(s)____(s)R(s) s.
Hw, s w ,
q(s)/(s).., s y w y s,
. U y , w w w w __
_, w__ > _, w q
(A3) BX____
N
|
dX
=*=
__
ss
d
X(*,N)______
N (s) (s) s _
s>s
d
X(*,N)______
N (s) (s) s.
Fy, (A1), w w w q , w . I w , , w y
(A4) BX____
N |
dX=*= (
__ _ ) < 0.
Cqy, y ,N, w j- X(*,N), y , y.
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JuNE 2009662 ThE AmERICAN ECONOmIC REVIEW
PROOF OF PROPOSITION 2:
C w - , d(s), y *. T y -
, y z y , q 1 *. I w
y y , y r,
(A5) B(*)= (r) s
d(s) (s) (s) s
= (r) S1 *s
d(s)(s)___________s d(s) (s)s
((r)(s)) s T
= (r)[(1 *)__ (*,d(s))],
w w
- , (r), (s) q y, q(s)/(s). A , ,(s) w y s. Fy,
__ (*,d(s)),
(s) w , w w w (.., y ) y w * w .
PROOF OF PROPOSITION 3:
C q , w X(*,N), -
y ,N, - y *. AN, - y y, w:
(A6) N
a__N (X(*,N) d(s))__________________________d(s)(1 d(s)) b = {01 s
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VOL. 99 NO. 3 663COVAL ET AL.: ECONOmIC CATASTROphE BONdS
wx y,
- . T y w y . F, y y w w
--y, yz - . S, -
w, w y w w . T
y
(D S 1993; J M. C . 1997), y . T y G Bw K B. T
(1999) . W q (x,) - --y, (1,), x. Fy, , w q Aw-D y y y
y .
F y , y
w y y
(B3)(x,)______x = Qbc___x R 2(c x),
(B4)2(x,)_______
x2 =Qbc___
x2R 3(c x)( (c x)+ 2c (c x)).
T y
(B5) (x,)______x =bc(cx),(B6)
2(x,)_______x2
=bc2 (cx).
T Aw-D
y B-S w
. D - ty Ft,, w
(B7) 2CBS_____K2
|K=x Ft,= er
(2(x))_____________xFt, (x, )_ ,(B8) C
BS____
|K=x Ft,= erxFt,(2(x))
_
,
(B9) 2CBS_____2
|K=x Ft,= erx Ft,(2(x))
_
1(x)2(x)________(x,)
(B10) 2CBS_____
K|K=x Ft,= er 1(x) (2(x))__________(x,) ,
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JuNE 2009664 ThE AmERICAN ECONOmIC REVIEW
w
(B11) 1(x)=x________
(x,)_
+ 1__
2(x,)
_
,
(B12) 2(x)=1 (x,)_
.
S Aw-D , y -
y z y
.
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