Models of Defaul Risk Models of Defaul Risk 1/29
Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed by Black and Scholes (1973) and Meron (1974) and values a rm s defaul-risky deb as an explici funcion of he rm s capial srucure and he value and risk of a rm s asses. The reduced form approach simply assumes ha defaul is a Poisson process wih a ime-varying defaul inensiy and defaul recovery rae wihou explicily modeling a rm s asses and capial srucure. Examples of he reduced form approach include Jarrow, Lando, and Turnbull (1997), Madan and Unal (1998), and Du e and Singleon (1999). Models of Defaul Risk 2/29
Srucural Approach Assumpions Consider a model similar o Meron (1974) where a rm owns risky asses wih dae marke value A () and dynamics da=a = ( ) d + dz (1) where and are he expecaion and he sandard deviaion of he rae of reurn on asses and is he rae a which asses are paid ou as dividends o he rm s shareholders. Along wih shareholders equiy, he rm has issued a zero-coupon deb ha promises o pay he amoun B a dae T >, where T. The dae marke values of shareholders equiy and he deb are E () and D (; T ), respecively, so ha A () = E () + D (; T ). Models of Defaul Risk 3/29
Assumpions (coninued) A dae T, he rm pays B o he debholders if here is su cien rm asse value. Else, bankrupcy occurs and he debholders ake ownership of he rm s asses. Thus, he payo o debholders is D (T ; T ) = min [B; A (T )] (2) = B max [0; B A (T )] Le P (; T ) be he curren dae price of a defaul-free, zero-coupon bond ha pays $1 a dae T and assume ha he Vasicek (1977) model holds for he defaul-free erm srucure speci ed earlier in (9.41) o (9.43). Models of Defaul Risk 4/29
Marke Value of Deb Recognizing ha deb s payo in (2) equals he defaul-free value B less he value of a pu opion wrien on he rm s asses wih srike B, i is valued using opion pricing resuls in Chapers 9 and 10: D (; T ) = P (; T ) B P (; T ) BN ( h 2 ) + e AN ( h 1 ) = P (; T ) BN (h 2 ) + e AN ( h 1 ) (3) where h 1 = ln e A= (P (; T ) B) + 1 2 v 2 =v, h 2 = h 1 v, and v () is given in (9.61). Noe ha he deb s promised yield-o-mauriy is R (; T ) 1 ln [B=D (; T )] and is credi spread is 1 R (; T ) ln [1=P (; T )]. Models of Defaul Risk 5/29
Marke Value of Shareholders Equiy Given (3), shareholders equiy equals E () = A () D (; T ) (4) = A P (; T ) BN (h 2 ) e AN ( h 1 ) h i = A 1 e N ( h 1 ) P (; T ) BN (h 2 ) Equiy is similar o a call opion on he rm s asses since is payo is max [A (T ) B; 0]. However, i di ers if he rm pays dividends o equiyholders prior o he deb s mauriy, as re eced in he rs erm in he las line of (4). Models of Defaul Risk 6/29
Discussion of Srucural Models Meron (1974) analyzes he properies of deb and equiy formulas similar o equaions (3) and (4). Noe ha an equiy formula such as (4) is useful when rms have publicly raded equiy, since observaion of he marke value of equiy and is volailiy can be used o esimae A () and, which can hen be used o value D (; T ). There is now a vas lieraure on srucural models ha modify and exend he original Meron (1974) framework. Examples include Black and Cox (1976), Leland (1994), and Collin-Dufresne and Goldsein (2001). Models of Defaul Risk 7/29
The Reduced-Form Approach As before, le D (; T ) be he dae value of a defaul-risky, zero-coupon bond ha promises o pay B a is mauriy dae of T. Le () d be he insananeous probabiliy of defaul occurring during he inerval (; + d), so ha () is he physical defaul inensiy, or hazard rae. Then he bond s (physical) survival probabiliy from daes o is R E e (u)du (5) Models of Defaul Risk 8/29
A Zero-Recovery Bond Firs, consider a bond ha, if i defauls, has zero recovery value, so ha D (T ; T ) = B if here is no defaul or D (T ; T ) = 0 if defaul occurs over he inerval from daes o T. Applying risk-neural pricing, his bond s value, D Z (; T ), is D Z (; T ) = E b R T i he r (u)du D (T ; T ) (6) where r () is he dae insananeous defaul-free ineres rae, and b E [] is he dae risk-neural expecaions operaor. Models of Defaul Risk 9/29
Sae Variables and Pricing Kernel Suppose he defaul-free erm srucure and () depend on an n 1 vecor of sae variables, x, ha follows he process dx = a (; x) d + b (; x) dz (7) where x = (x 1 :::x n ) 0, a (; x) is an n 1 vecor, b (; x) is an n n marix, and dz = (dz 1 :::dz n ) 0 is an n 1 vecor of independen Brownian moions so ha dz i dz j = 0 for i 6= j. Assuming complee markes, he sochasic discoun facor for pricing he rm s defaul-risky bond is dm=m = r (; x) d (; x) 0 dz (; x) [dq (; x) d] (8) where (; x) is an n 1 vecor of he marke prices of risk associaed wih dz and (; x) is he marke price of risk associaed wih he acual defaul even. Models of Defaul Risk 10/29
Defaul as a Poisson Process The defaul even is recorded by dq, which if defaul occurs q () jumps from 0 (he no-defaul sae) o 1 (he absorbing defaul sae) a which ime dq = 1. The risk-neural defaul inensiy, b (; x), is hen given by b (; x) = [1 (; x)] (; x). Defaul is a doubly sochasic process, also referred o as a Cox process, because i depends on he Brownian moion vecor dz ha drives x and deermines how he likelihood of defaul, b (; x), changes over ime, bu i also depends on he Poisson process dq ha deermines he arrival of defaul. Hence, defaul risk re ecs wo ypes of risk premia, (; x) and (; x). Models of Defaul Risk 11/29
Value of he Zero-Recovery Bond Based on (5), we can solve for D Z (; T ): D Z (; T ) = b E e R T r (u)du e R T b(u)du B (9) = b E he R i T [r (u)+ (u)]du b B Noe ha (9) is similar o valuing a defaul-free bond excep ha he discoun rae r (u) + b (u), raher han jus r (u), is used. Given speci c funcional forms for r (; x), b (; x), x and (; x), (9) can be compued. Models of Defaul Risk 12/29
Specifying Recovery Values Suppose ha if he bond defauls a dae, where < T, bondholders recover an amoun w (; x) a dae. Then he risk-neural probabiliy densiy of defauling a is R e (u)dub b () (10) In (10), b h R () is discouned by exp b i (u) du because defaul a dae is condiioned on no having defauled previously. Models of Defaul Risk 13/29
Valuing a Bond wih Recovery Value Thus, he presen value of recovery, D R (; T ), is: Z T D R (; T ) = E b e Z T = E b e R r (u)du w () e R b (u)dub () d R [r (u)+ b (u)]dub () w () d Puing his ogeher wih (9) gives he bond s oal value: (11) D (; T ) = D Z (; T ) + D R (; T ) (12) = b E he + Z T R T e [r (s)+ b (s)]ds B R [r (s)+ b (s)]ds b () w () d Models of Defaul Risk 14/29
Recovery Proporional o Par Value Le us consider paricular speci caions for w (; x). Le he defaul dae be and assume w (; x) = (; x) B, where (; x) can be a consan, say,. Then (11) is D R (; T ) = B Z T k (; ) d (13) where k (; ) b E he R [r (u)+ b (u)]dub () i (14) has a closed-form soluion when r (u; x) and b (u; x) are a ne funcions of x and he vecor x in (7) has a risk-neural process ha is also a ne. (13) can be compued by numerical inegraion of k (; ). Models of Defaul Risk 15/29
Recovery Proporional o Par, Payable a Mauriy Assume ha if defaul occurs a dae, bondholders recover (; x) B a dae T, which is equivalen o w (; x) = (; x) P (; T ) B. Then (11) is D R (; T ) Z T = E b e Z T = E b e = b E e R T R [r (u)+ (u)]dub b R T () (; x) e r (u)du Bd R b(u)dub () (; x) e Z T r (u)du e R R T r (u)du Bd b(u)dub () (; x) d B (15) Models of Defaul Risk 16/29
Recovery Proporional o Par, Payable a Mauriy If (; x) =, noe ha R h T R exp b (u) dui b () d is he oal risk-neural h probabiliy of defaul from o T and R i T equal 1 exp b (u) du. Thus, D R (; T ) = E b R T R T e r (u)du 1 e b(u)du B = b E he R T r (u)du e R i T [r (u)+ (u)]du b B = BP (; T ) D Z (; T ) (16) Therefore, he oal value of he bond is D (; T ) = D Z (; T )+D R (; T ) = 1 D Z (; T )+BP (; T ) (17) so only a value for he zero-recovery bond is required. Models of Defaul Risk 17/29
Recovery Proporional o Marke Value Assume ha a defaul, bondholders lose a proporion L (; x) of he bond s value jus prior o defaul: D + ; T = w (; x) = D ; T [1 L (; x)] (18) Treaing he defaulable bond as a coningen claim and applying Iô s lemma: dd (; T ) =D (; T ) = ( D k D ) d + 0 D dz L (; x) dq (19) where D and he n 1 vecor D are given by he usual Iô s lemma expressions, he expeced jump size k D ( ) E [D ( + ; T ) D ( ; T )] =D ( ; T ) = L (; x), so ha he drif erm in (19) is D + (; x) L (; x). Models of Defaul Risk 18/29
Recovery Proporional o Marke Value (coninued) The risk-neural process for D (; T ) replaces D wih r (): dd (; T ) =D (; T ) = r (; x) + b (; x) b L (; x) d(20) + 0 D dbz b L (; x) dq where b L (; x) is he risk-neural loss given defaul. Similar o (11.17), D (; T ) sais es he PDE: 1 2 Trace b (; x) b (; x) 0 D xx +ba (; x) 0 D x R (; x) D +D = 0 (21) where D x is he n 1 vecor of rs derivaives, D xx is he n n marix of second derivaives, ba (; x) = a (; x) b (; x), and R (; x) r (; x) + b (; x) b L (; x). Models of Defaul Risk 19/29
Soluion for he Defaulable Bond Value The PDE (21) is sandard excep ha R (; x) replaces r (; x) in he sandard PDE. Thus, he Feynman-Kac soluion is D (; T ) = b E he R T R(u;x)du i B (22) where R (; x) r (; x) + b (; x) b L (; x) is he defaul-adjused discoun rae. The produc s (; x) b (; x) b L (; x) is he credi spread on an insananeous-mauriy, defaulable bond, and since b (; x) and b L (; x) are no individually ideni ed, a single funcional form can be speci ed for s (; x). Models of Defaul Risk 20/29
Examples Le x = (x 1 x 2 ) 0 be a wo-dimensional vecor, ba (; x) = ( 1 (x 1 x 1 ) 2 (x 2 x 2 )) 0, and b (; x) is a p p diagonal marix wih elemens of 1 x1 and 2 x2. Also assume r (; x) = x 1 () and b (; x) = x 2 (), so ha he defaul-free erm srucure q and b (; x) are independen. De ning r x 1 and 1 2 1 + 22 1, he CIR bond price is P (; T ) = A 1 () e B 1()r (), where (23) " # 2 1 e ( 1+ 1 ) 21 r = 2 1 2 A 1 () ( 1 + 1 ) (e 1 1) + 2 1 (24) 2 e 1 1 B 1 () ( 1 + 1 ) (e 1 1) + 2 1 (25) Models of Defaul Risk 21/29
Examples (coninued) Also de ne x 2, hen based on (9) we have D Z (; T ) = b E he R i T [r (s)+ (s)]ds b B where = b E he R T r (s)ds i b E he R T b(s)ds i B = P (; T ) V (; T ) B (26) V (; T ) = A 2 () e B 2() b () (27) and where A 2 () is he same as A 1 () in (24), and B 2 () is he same as B 1 () in (25) excep ha q 2 replaces 1, 2 replaces 1, replaces r, and 2 2 2 + 22 2 replaces 1. Models of Defaul Risk 22/29
Example: Recovery Proporional o Par, Payable a Mauriy If recovery is a xed proporion,, of par, payable a mauriy, hen from (17): D (; T ) = 1 D Z (; T ) + BP (; T ) = + 1 V (; T ) P (; T ) B (28) In (27), V (; T ) is analogous o a bond price in he sandard Cox, Ingersoll, and Ross erm srucure model and is inversely relaed o b () and sricly less han 1 whenever b () is sricly posiive, which can be ensured when 2 2 2 2. Thus, (28) con rms ha he defaulable bond s value declines as is risk-neural defaul inensiy rises. Models of Defaul Risk 23/29
Example: Recovery Proporional o Marke Value Assume recovery is proporional o marke value and s (; x) b (; x) b L (; x) = x 2 and de ne s x 2. Then from (22): D (; T ) = b E he R i T [r (u)+s(u)]du B where = b E he R T r (u)du i b E he R T s(u)du i B = P (; T ) S (; T ) B (29) S (; T ) = A 2 () e B 2()s() and where A 2 () is he same as A 1 () in (24) and B 2 () is he same as B 1 () in (25) excep ha q 2 replaces 1, 2 replaces 1, s replaces r, and 2 2 2 + 22 2 replaces 1. (30) D (; T ) is like P (; T ) bu R () = r () + s () replaces r (). Models of Defaul Risk 24/29
Coupon Bonds Suppose a defaulable coupon bond promises n cash ows, wih he i h promised cash ow being equal o c i and being paid a dae T i >. Then he value of his coupon bond in erms of our zero-coupon bond formulas is np i=1 D (; T i ) c i B (31) Models of Defaul Risk 25/29
Credi Defaul Swaps A credi defaul swap is a conrac in which one pary, he proecion buyer, makes periodic paymens unil he conrac s mauriy as long as a paricular issuer does no defaul. The oher pary, he proecion seller, receives hese paymens in reurn for paying he di erence beween he issuer s deb s par value and is recovery value if defaul occurs prior o he swap s mauriy. Le he conrac specify equal periodic paymens of c a fuure daes +, + 2,..., + n. Since hese paymens are coningen on defaul no occurring, heir value is c np D Z (; + i) (32) B i=1 where D Z (; T ) is given in (9). Models of Defaul Risk 26/29
Credi Defaul Swaps (coninued) Le w (; x) be he recovery value of he defaulable deb underlying he swap conrac. Then similar o (11), he value of he swap proecion is Z +n R be e [r (u)+ (u)]dub b () [B w ()] d (33) Suppose he proecion seller s paymen in he even of defaul is B w () = B B = B 1. Then (33) is B 1 Z +n k (; ) d (34) where k (; ) is de ned in (14). Given funcional forms for r (; x), b (; x), w (; x), and x, he value of he swap paymens, c, ha equaes (32) o (33) can be deermined. Models of Defaul Risk 27/29
Implemening a Reduced-Form Approach A general issue when implemening he reduced-form approach is deermining he proper curren values b (), s (), or w () ha may no be direcly observable. One or more of hese defaul variables migh be inferred by seing he acual marke prices of one or more of an issuer s bonds o heir heoreical formulas. Then, based on he implied values of b (), s (), or w (), one can deermine wheher a given bond of he same issuer is over- or underpriced relaive o oher bonds. Alernaively, hese implied defaul variables can be used o se he price of a new bond of he same issuer or a credi derivaive (such as a defaul swap) wrien on he issuer s deb. Models of Defaul Risk 28/29
Summary There are wo main branches of modeling defaulable xed-income securiies. The srucural approach models defaul based on he ineracion beween a rm s asses and is liabiliies. Poenially, i can improve our undersanding beween capial srucure and corporae bond and loan prices. In conras, he reduced-form approach absracs from speci c characerisics of a rm s nancial srucure, bu i permis a more exible modeling of defaul probabiliies and may beer describe acual he prices of an issuer s deb. Models of Defaul Risk 29/29