A New Model for the Pricing of Defaultable Bonds

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Reynard Neil Clark
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1 A Nw Mol fo h Picing of Dflbl Bon Pof. D. Ri Zg Mnich Univiy of chnology Mi 6. Dzmb 004 HVB-Ini fo Mhmicl Finnc

2 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion - Yil Cv Bhvio US y Sip - Ci Sp Bhvio US Inil A - Ci Sp Bhvio US Inil BBB - conomic Bhvio US Go Domic Poc (GDP) h Mol fo h Picing of Dflbl Bon Mol Compion Fh Rch HVB-Ini fo Mhmicl Finnc

3 y n Sp D US Mk Amicn y Sip A- n BBB- Amicn Inil Mii bwn 3 monh n 30 y im i of wkly bon pic fom Okob o Jn 00 (in mpl) im i of wkly bon pic fom Jn 8 00 o Ag (o of mpl) All pic in US Doll i.. no cncy ik in ci p Pm imion ing Klmn fil HVB-Ini fo Mhmicl Finnc 3

4 - o 0-Y US y Sip in % im Pio: HVB-Ini fo Mhmicl Finnc Y 3 Y 5 Y 7 Y 0 Y Soc: Bloombg

5 - o 0-Y Ci Sp of US Inil A in % im Pio: Soc: Bloombg HVB-Ini fo Mhmicl Finnc Y 3 Y 5 Y 7 Y 0 Y

6 - o 0-Y Ci Sp of US Inil BBB in % im Pio: Soc: Bloombg HVB-Ini fo Mhmicl Finnc Y 3 Y 5 Y 7 Y 0 Y

7 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion h Mol fo h Picing of Dflbl Bon - Schmi n Zg [000] (Sp pn on fim-pcific fco) - Bkhi Mn Zhng [00] (Sp pn on y & fim-pcific fco) - Gnliz Schmi n Zg [004] (Sp pn on mcoconomic & fim-pcific fco) Mol Compion Fh Rch HVB-Ini fo Mhmicl Finnc 7

8 h Mol of Schmi n Zg [000] Moling of h Sochic Poc Un h Mingl M Q Dynmic of h yil cv (non-flbl ho ) () [ θ ( ) ( ) ] + σ W ( ) [0 ] > 0 σ > 0 Dynmic of h nobvbl conpy qliy (nciny inx)* () [ θ ( ) ] + σ ( ) W ( ) [0 ] θ 0 > 0 σ > 0 Dynmic of h yil p (ho- ci p) ( ) [ b ( ) ( ) ] + σ ( ) W ( ) [0 ] b > 0 > 0 σ > 0 h Win poc W W n W ncol * Lvg n book-o-mk cn b cib by mn-ving ochic poc (.g. Bkhi Mn Zhng [00] ) HVB-Ini fo Mhmicl Finnc 8

9 9 HVB-Ini fo Mhmicl Finnc hom (Hll n Whi [990] Hll[997]). h im vl P() of non-flbl zo-copon bon wih miy i givn by wih n ( ) ( ) ( ) B A P ( ) ( ) ( ) B ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). ln 4 0 ln P P P B A σ h Mol of Schmi n Zg [000] Picing of Non-Dflbl Bon

10 0 HVB-Ini fo Mhmicl Finnc hom (Schmi n Zg [000]). h im vl P () of flbl zo-copon bon wih miy i givn by wh ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D C A D C B A P P ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) v v A A A A v v v D k K K K C k k ln wih complic wih n wih + + σ θ σ δ σ δ δ δ h Mol of Schmi n Zg [000] Picing of Dflbl Bon ymic fim pcific

11 h Mol of Schmi n Zg [000] Pm imion Pm imion Pm imion imion y A BBB ( l ) Sip (0.44) ( l ) σ (%) (.9995) (.34760) σ (%) l-wol mn vion lvl fo (%) Avg obv 3-Monh R: y: 5. % Sp A: 0.59 % Sp BBB: 0.76 % l-wol mn vion lvl fo (%) θ (%) ( l ) σ (%) l-wol mn vion lvl fo (%) (0.9990) ( ) HVB-Ini fo Mhmicl Finnc

12 h Mol of Schmi n Zg [000] Lin Rgion of Mol v. mpiicl y Sip* In mpl: O of mpl: qáãéíçj~íìêáíó b R² im o Miy b R² Y * Y * (.86057*0-6 ) (0.8693) (0.7500) (-7.807*0-5 ) ( ) (0.7060) 3 Y * Y * ( *0-6 ) (.03008) (0.9560) (-.9950*0-5 ) (0.9967) (0.994) 5 Y * Y.03988* (-.684*0-6 ) (.08467) (0.996) (.96055*0-5 ) (.0860) (0.99) 7 Y * Y * (.03906*0-6 ) (.08400) (0.9606) ( *0-5 ) (.09057) (0.979) 0 Y 6.58* Y 6.778* ( *0-6 ) (.07958) (0.9404) ( *0-5 ) (.0770) (0.948) * Vl in bck wiho.5% of mx. oli HVB-Ini fo Mhmicl Finnc

13 h Mol of Schmi n Zg [000] Lin Rgion of A - Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R² im o Miy b R² Y * Y.6433* ( *0-5 ) (.38773) (0.6690) ( *0-6 ) (.3570) (0.64) 3 Y * Y.7865* ( 5.93*0-6 ) (.36455) (0.46) ( *0-6 ) (.46988) (0.3304) 5 Y * Y.3477* (-.78*0-6 ) (.537) (0.386) (.86*0-5 ) (.8500) (0.403) 7 Y -6.85* Y -.76* (-3.608*0-6 ) (.55750) (0.745) (.838*0-5 ) (0.694) (0.89) 0 Y.3400* Y -.88* ( *0-6 ) (.596) (0.680) (-5.6*0-6 ) (0.6547) (0.0406) * Vl in bck wiho.5% of mx. oli HVB-Ini fo Mhmicl Finnc 3

14 h Mol of Schmi n Zg [000] Lin Rgion of BBB - Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R² im o Miy b R² Y * Y * ( *0-6 ) (.449) (0.8503) ( *0-6 ) (.3738) (0.973) 3 Y * Y.93* ( *0-6 ) (.749) (0.5643) (.44*0-5 ) (.4689) (0.750) 5 Y * Y ( *0-6 ) ( ) (0.968) (.36*0-5 ) (0.5877) (0.9) 7 Y * Y ( *0-6 ) (0.9787) (0.3489) ( *0-5 ) ( ) (0.48) 0 Y * Y * ( *0-5 ) (.0975) (0.4397) ( *0-5 ) (0.9395) (0.453) * Vl in bck wiho.5% of mx. oli HVB-Ini fo Mhmicl Finnc 4

15 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion h Mol fo h Picing of Dflbl Bon - Schmi n Zg [000] (Sp pn on fim-pcific fco) - Bkhi Mn Zhng [00] (Sp pn on y & fim-pcific fco) - Gnliz Schmi n Zg [004] (Sp pn on mcoconomic & fim-pcific fco) Mol Compion Fh Rch HVB-Ini fo Mhmicl Finnc 5

16 h Mol of Bkhi Mn Zhng [00] Moling of h Sochic Poc Un h Mingl M Q Dynmic of h yil cv (non-flbl ho ) () () () () [ ] + σ ρ W + σ ρ W ( ) [0 ] > 0 σ > 0 ρ < Dynmic of h nobvbl if iving poc ( ) ( ) + ( ) [0 ] 0 > 0 > 0 θ σ W θ σ Dynmic of h nobvbl nciny inx ρ ρ () θ () + σ W () + σ W () ρ ρ θ 0 > 0 σ > 0 ρ < ρ h Win poc W W n W ncol HVB-Ini fo Mhmicl Finnc 6

17 h Mol of Bkhi Mn Zhng [00] Moling of h Sochic Poc Un h Mingl M Q h ho p i givn by ( ) ( ) ( ) ( ) Λ + Λ +Λ [0 ] Λ Λ Λ IR i ( ) θ ( ) + ( ) ( ) + σ ( ) θ θ ( ) b b W wih : Λ +Λ b : Λ b : Λ 0 ρ ρ n σ W() : ( Λ) σ ρ +Λ σ W () () ( ) ( ) +Λσ W + Λ σρ W ρ ρ HVB-Ini fo Mhmicl Finnc 7

18 8 HVB-Ini fo Mhmicl Finnc hom 5. h im vl P() of non-flbl zo-copon bon wih miy i givn by wih n ( ) ( ) ( ) ( ) B A P ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B + ( ) ( ) ( ) ( ) ( ) ( ). + + B B A θ σ σ ρ σ σ h Mol of Bkhi Mn Zhng [00] Picing of Non-Dflbl Bon

19 9 HVB-Ini fo Mhmicl Finnc hom 6. h im vl P () of flbl zo-copon bon wih miy i givn by wh ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Λ Λ D B A D B A P P ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Λ Λ + Λ + Λ + Λ + + Λ Λ Λ Λ D B D B D B A A A A D B B θ θ ρ σ σ ρ σ σ σ σ σ 0 h Mol of Bkhi Mn Zhng [00] Picing of Non-Dflbl Bon ymic fim pcific ymic

20 h Mol of Bkhi Mn Zhng [00] Pm imion Pm ( l ) ρ w σ (%) l-wol mn vion fo (%) θ w (%) w ( wl ) σ w (%) l-wol mn vion fo w (%) imion y (0.49) -0.36* (mp. 5.) (0.87) HVB-Ini fo Mhmicl Finnc Pm θ (%) ( l ) ρ σ (%) l-wol mn vion fo (%) Λ 0 Λ l-wol mn vion fo ho p (%) imion A *0-6 ( ) (mp. 0.59) imion BBB *0-6 (0.839) (mp. 0.76) * Ngiv colion bwn IR movmn n long-m mn (.g. Chn Sco [93] Dfn Solnik [99] Di Singlon [00] Bkhi Mn Zhng [0] 0

21 h Mol of Bkhi Mn Zhng [00] Lin Rgion of Mol v. mpiicl y Sip* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ) (0.8855) (0.853) Y ( ) (0.8049) (0.83) 3 Y ( ).069 (.03) (0.978) 3 Y ( ) (.044) (0.993) 5 Y ( ).0485 (.044) (0.984) 5 Y ( ).0886 (.068) (0.9663) 7 Y ( ).049 (.0355) (0.98) 7 Y ( ) (.0364) (0.9786) 0 Y ( ).000 (.007) (0.655) 0 Y ( ) (0.9936) (0.986) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli

22 h Mol of Bkhi Mn Zhng [00] Lin Rgion of A-Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ).0747 (.79) (0.88) Y ( ) (0.693) (0.0787) 3 Y ( ).589 (.558) (0.693) 3 Y ( ).0793 (.9753) (0.6493) 5 Y ( ).7 (.386) (0.434) 5 Y ( ).07 (0.90) (0.368) 7 Y ( ).8 (.97) (0.459) 7 Y ( ) (0.65) 0.36 (0.36) 0 Y ( ) (.0874) (0.3953) 0 Y ( ) (0.875) 0.5 (0.083) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli

23 h Mol of Bkhi Mn Zhng [00] Lin Rgion of BBB-Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ).989 (.335) (0.769) Y ( ).7096 (.000) (0.3598) 3 Y 5 Y ( ) ( ) (.4778) (0.739) (0.745) (0.48) 3 Y 5 Y ( ) ( ) (.790) (0.590) (0.793) (0.574) 7 Y ( ) (0.8576) 0.38 (0.68) 7 Y ( ) (0.846) 0.3 (0.305) 0 Y ( ) (0.985) (0.355) 0 Y ( ) (0.9598) (0.4069) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli 3

24 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion h Mol fo h Picing of Dflbl Bon - Schmi n Zg [000] (Sp pn on fim-pcific fco) - Bkhi Mn Zhng [00] (Sp pn on y & fim-pcific fco) - Gnliz Schmi n Zg [004] (Sp pn on mcoconomic & fim-pcific fco) Mol Compion Fh Rch HVB-Ini fo Mhmicl Finnc 4

25 US y Sip n Inil v. US Go Domic Poc im Pio: y y (lh) 3 monh US GDP gowh (h) Soc: Bloombg Lin Rgion: R( ) R(+) +.03 w() + ε Colion (993-00): ρ(r(+)w()) 008 HVB-Ini fo Mhmicl Finnc 5

26 US Ci Sp v. US Go Domic Poc im Pio: y BBB p o i 3 monh US GDP gowh y A p o i Soc: Bloombg Lin Rgion: HVB-Ini fo Mhmicl Finnc S A ( ) S A (+) w() + ε S BBB ( ) S BBB (+) w() + ε Colion (993-00): ρ(s A (+)w()) -0 ρ(s BBB (+)w()) -05 Colion (993-00): ρ(w()fl pobbiliy BBB)

27 h Gnliz Mol of Schmi n Zg [004] Moling of h Sochic Poc Un h Mingl M Q Dynmic of h yil cv (non-flbl ho ) ( ) [ θ ( ) + b ( ) ( ) ] + σ W () [0 ] > 0 σ > 0 Dynmic of h conomy inx Dynmic of h nciny inx ( ) [ θ ( ) ] + σ W ( ) [0 ] θ 0 > 0 σ > 0 Dynmic of h yil p (ho- ci p) ( ) [ θ ( ) ] + σ W ( ) [0 ] θ 0 > 0 σ > 0 () [ θ + b () b () () ] + σ W () h Win poc W W W n W ncol [0 ] θ 0 b > 0 b > 0 σ > 0 > 0 HVB-Ini fo Mhmicl Finnc 7

28 8 HVB-Ini fo Mhmicl Finnc hom 7. h im vl P() of non-flbl zo-copon bon wih miy i givn by wih n ( ) ( ) ( ) ( ) B A P ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b B + ( ) ( ) ( ) ( ) ( ) ( ). + B B A θ θ σ σ h Gnliz Mol of Schmi n Zg [004] Picing of Non-Dflbl Bon

29 9 HVB-Ini fo Mhmicl Finnc hom 8. h im vl P () of flbl zo-copon bon wih miy i givn by wh ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + D C A D C B A P P ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b b D C D C A A A A D C θ θ θ σ σ σ σ wih wih h Gnliz Mol of Schmi n Zg [004] Picing of Non-Dflbl Bon ymic fim pcific ymic

30 h Gnliz Mol of Schmi n Zg [004] Pm imion Pm imion Pm imion imion y A BBB ( l ) (0.076) ( l ) (.49603).0934 (.66) b w θ (%) σ (%).9036 σ (%) l-wol mn vion lvl fo (%) θ w (%) w ( wl ) σ w (%) l-wol mn vion lvl fo w (%) 5. (mp. 5.) (.4388) (mp..37) b w l-wol mn vion lvl fo (%) θ (%) ( l ) σ (%) l-wol mn vion lvl fo (%) (mp. 0.59) (0.7408) (mp. 0.76) (0.6993) HVB-Ini fo Mhmicl Finnc 30

31 h Gnliz Mol of Schmi n Zg [004] Lin Rgion of Mol v. mpiicl y Sip* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ) (0.758) (0.7605) Y ( ) (0.709) 0.67 (0.706) 3 Y ( ) (0.9577) (0.965) 3 Y ( ) (0.8997) (0.956) 5 Y ( ) (.0078) (0.9999) 5 Y ( ).004 (.0087) (0.9999) 7 Y ( ) (.009) (0.9556) 7 Y ( ) (.0077) (0.9666) 0 Y ( ).09 (.0085) (0.934) 0 Y ( ) (.006) (0.9349) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli 3

32 h Gnliz Mol of Schmi n Zg [003] Lin Rgion of A-Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ).3997 (.6) (0.8987) Y ( ).3765 (.36) 0.9 (0.9006) 3 Y ( ).097 (.0683) (0.488) 3 Y (0.0000).367 (.30) (0.378) 5 Y ( ).3654 (.359) (0.5806) 5 Y ( ).4094 (.97) (0.5358) 7 Y ( ).4064 (.360) (0.5494) 7 Y (0.0000).080 (.0088) (0.45) 0 Y (0.0000).073 (.50) (0.4044) 0 Y ( ) 0.93 (0.9345) 0.44 (0.975) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli 3

33 h Gnliz Mol of Schmi n Zg [004] Lin Rgion of BBB-Mol v. mpiicl Ci Sp* In mpl: O of mpl: im o Miy b R im o Miy b R Y ( ).488 (.4005) (0.87) Y ( ).545 (.457) (0.9057) 3 Y ( ).69 (.59) (0.59) 3 Y ( ).5953 (.484) (0.546) 5 Y ( ) (0.76) 0.47 (0.87) 5 Y ( ) (0.6675) 0.59 (0.444) 7 Y ( ) (0.9573) (0.854) 7 Y ( ).63 (.65) 0.97 (0.3998) 0 Y ( ).8693 (.37) (0.4549) 0 Y ( ).83 (.087) (0.3560) HVB-Ini fo Mhmicl Finnc * Vl in bck wiho.5% of mx. oli 33

34 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion h Mol fo h Picing of Dflbl Bon Mol Compion - Scl Infomion - Siicl - Dfl Pobbilii Fh Rch HVB-Ini fo Mhmicl Finnc 34

35 Mol Compion Sniiviy of Ci Sp wih Rpc o Fim-pcific Fco 070% 060% % Bkhi Mn Zhng [00] D(0)/*m% 050% 040% 030% 00% 00% long-m inflnc by D(0)/*0.08% % % conn inflnc ov im US Inil BBB im o Miy US Inil A C(0)/*m % 000% % 060% 050% 040% 030% 00% US Inil A im o Miy ho-m inflnc by US Inil BBB Schmi Zg [000] 00% 000% im o Miy HVB-Ini fo Mhmicl Finnc US Inil BBB US Inil A 35

36 Mol Compion Sniiviy of Ci Sp wih Rpc o Fim-pcific Fco 05% 00% % Bkhi Mn Zhng [00] D(0)/*m% 05% 00% 005% long-m inflnc by D(0)/*0.08% % 000% im o Miy US Inil BBB US Inil A conn inflnc ov im 08% 07% Schmi Zg [004] % im o Miy US Inil BBB US Inil A C(0)/*m% 06% 05% 04% 03% ho-m inflnc by 0% 0% 00% im o Miy US Inil BBB US Inil A HVB-Ini fo Mhmicl Finnc 36

37 Mol Compion Avg Abol Dviion of y Sip Ci Sp n R Avg bol Dviion of yil / p (%) Schmi Zg [000] Bkhi Mn Zhng [00] Schmi Zg [004] Avg R Schmi Zg [000] Bkhi Mn Zhng [00] Schmi Zg [004] y Sip (0.8467) 0.45 (0.9366) (0.965) y Sip ( ) (0.84) ( ) Inil A (0.033) (0.6887) 0.06 ( ) Inil A (0.9966) (0.6440) (0.443) Inil BBB (0.496) 0.08 (0.354) (0.643) Inil BBB (0.3936) (0.3646) (0.4639) * Vl wiho bck in mpl vl in bck o of mpl HVB-Ini fo Mhmicl Finnc 37

38 Mol Compion Do h Inocion of Poiiv Sp o GDP Py? Avg bol viion of yil / p (%) Poiiv p no mco vibl (SZ 000) Noml p no mco vibl (b w b w 0) Noml p mco vibl (SZ 004) Avg R Poiiv p no mco vibl (SZ 000) Noml p no mco vibl (b w b w 0) Noml p mco vibl (SZ 004) y Sip (0.8467) (0.8467) (0.965) y Sip ( ) ( ) ( ) Inil A (0.033) (0.6375) 0.06 ( ) Inil A (0.9966) (0.3854) (0.443) Inil BBB (0.496) ( ) (0.643) Inil BBB (0.3936) (0.303) (0.4639) * Vl wiho bck in mpl vl in bck o of mpl HVB-Ini fo Mhmicl Finnc 38

39 Mol Compion Unciny Inx v. Dfl Pobbiliy fo Ring A Unciny Inx/ Dfl R Schmi Zg 000 Schmi Zg 004 Bkhi Mn Zhng 00 Dfl R D HVB-Ini fo Mhmicl Finnc 39

40 Mol Compion Unciny Inx v. Dfl Pobbiliy fo Ring BBB Schmi Zg 000 Schmi Zg 004 Bkh Mn Zhng 00 Dfl R Unciny Inx / Dfl R D HVB-Ini fo Mhmicl Finnc 40

41 A Nw Mol fo h Picing of Dflbl Bon Ovviw Mk Infomion h Mol fo h Picing of Dflbl Bon Mol Compion Fh Rch - Inclion of Oh Mol (Dff [996]) - Ingion of Oh Mcoconomic Fco (Inflion) - Picing of Ci Diviv (Ci Dfl Swp) - Foc n Simlion of niion (Dfl) Pobbilii - Ing Mk n Ci Pofolio Mngmn HVB-Ini fo Mhmicl Finnc 4

42 HVB-Ini fo Mhmicl Finnc ïïïkã~íüñáå~ååékã~kíìãkçé HVB-Ini fo Mhmicl Finnc 4

Bayesian Credibility for Excess of Loss Reinsurance Rating. By Mark Cockroft 1 Lane Clark & Peacock LLP

Bayesian Credibility for Excess of Loss Reinsurance Rating. By Mark Cockroft 1 Lane Clark & Peacock LLP By Cly o c o Lo Rc Rg By M Coco L Cl & Pcoc LLP GIRO coc 4 Ac Th pp c how o v cly wgh w po- pc-v o c o lo c. Th po co o Poo-Po ol ch wh po G o. Kywo c o lo c g By cly Poo Po G po Acowlg cl I wol l o h