Synheic CDO s and Baske Defaul Swaps in a Fixed Income Credi Porfolio Louis Sco June 2005
Credi Derivaive Producs CDO Noes Cash & Synheic CDO s, various ranches Invesmen Grade Corporae names, High Yield Corporaes, Emerging Marke Credis, Corporae Loans Credi Derivaives Single name defaul swaps Baske Loss Proecion Firs o Defaul, Second o Defaul, Porfolio Managemen Synheic CDO provide leverage and enhanced spreads Model Correlaion for Relaive Value Trading Elemens of gap risk and convexiy risk Sress Tess: wha happens o ranches when one or several names go ino financial disress?
Correlaion of Defaul and Correlaion Beween Defaul and Recovery Moody's Defaul & Recovery Raes Over Time 4.5% 70% 4.0% 60% 3.5% 3.0% 50% 2.5% 40% 2.0% 30% 1.5% 20% 1.0% 0.5% 10% 0.0% 0% 1970 1975 1980 1985 1990 1995 2000 2005 Defaul Raes (LHS Recovery Raes (RHS
Typical Synheic CDO Srucure AAA High- Qualiy Collaeral Cash Ineres & Principal Originaing Bank Defaul Swap Swap Premium Credi Proecion Based on Reference Porfolio Trus or Eniy o Manage Porfolio Ineres & Principal Funding Senior Noes Mezzanine Noes Equiy
Example of a Synheic CDO Srucure Managed Synheic CDO, Parially Funded, $250MM Collaeral (Reference Porfolio, $1,000MM Tranche Losses From To Cap. Srucure Spread Raing Supersenior swap 250MM 1,000MM 25%-100% 0.03% AAAA Mezzanine I 136MM 250MM 13.6%-25% 0.65% AAA Mezzanine II 96MM 136MM 9.6%-13.6% 1.00% AA Mezzanine III 71MM 96MM 7.1% -9.6% 1.75% A Mezzanine IV 32.5MM 71MM 3.25%-7.1% 2.85% BBB Firs Loss 0 32.5MM 0%-3.25% Equiy
Quoes for Tranched Dow-Jones Trac-X 5 Year March 2004 Tranche Raing Bid Offer Correlaion (Old Trac-X N.A. 5y II A- o BBB+ 69 70-0%-3% Equiy * 44.4% 48.4% 20% 3%-7% BBB- 424 464 3% 7%-10% AA+ 138 158 16% 10%-15% AAA 60 70 21% 15%-30% AAA+ 12 15 26% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched CDX 5 Year (U.S. December 2004 Tranche Raing Bid Offer Correlaion (Base CDX N.A. 5y A- o BBB+ 46 bp 0%-3% Equiy * 30.3% 31.3% 19% 3%-7% BBB- 179 188 30% 7%-10% AA+ 67 72 35% 10%-15% AAA 22 26 44% 15%-30% AAA+ 8 9 66% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched CDX 5 Year (U.S. June 2005 Tranche Raing Bid Offer Correlaion (Base CDX N.A. 5y A- o BBB+ 56 bp 0%-3% Equiy * 47.0% 47.3% 10% 3%-7% BBB- 157 161 28% 7%-10% AA+ 47 48 37% 10%-15% AAA 19 21 49% 15%-30% AAA+ 10 11 73% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched CDX 10 Year (U.S. December 2004 Tranche Raing Bid Offer Correlaion (Base CDX N.A. 10y A- o BBB+ 68 0%-3% Equiy * 52.5% 56.0% 18% 3%-7% BBB- 500 535 26% 7%-10% AA+ 185 210 32% 10%-15% AAA 85 98 42% 15%-30% AAA+ 30 35 63% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched CDX 10 Year (U.S. June 2005 Tranche Raing Bid Offer Correlaion (Base CDX N.A. 10y A- o BBB+ 80 0%-3% Equiy * 64.0% 65.0% 8% 3%-7% BBB- 745 755 14% 7%-10% AA+ 173 180 25% 10%-15% AAA 49 53 42% 15%-30% AAA+ 27 33 74% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched DJ Europe itraxx-x, 5 Year December 2004 Tranche Raing Bid Offer Correlaion (Base DJ Europe itraxx 5y A- o BBB+ 35.25 35.75 0%-3% Equiy * 22.9% 23.4% 19.5% 3%-6% 130 132 29.3% 6%-9% 42.5 44.5 37% 9%-12% 26 27.5 42.6% 12%-22% 13.5 14.5 56.5% * Equiy premium is an upfron premium, plus 500 bp running.
Quoes for Tranched DJ Europe itraxx-x, 10 Year December 2004 Tranche Raing Bid Offer Correlaion (Base DJ Europe itraxx 5y A- o BBB+ 52 53 0%-3% Equiy * 45.5% 46.75% 17.7% 3%-6% 362 377 26% 6%-9% 148 155 33% 9%-12% 83 88 39% 12%-22% 44 47 52.4% * Equiy premium is an upfron premium, plus 500 bp running.
Simulaed Porfolio Reurns Over 5 Years Example from March 2004 Porfolio of 100 Bonds 7-10% CDO Tranche of Porfolio Hisogram for Bond Porfolio Hisogram for 7-10% CDO Tranche 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 < -1.0% 0.0% 0.2% 0.4% 0.6% 0.8% 1% 1.2% 1.4% 1.6% Average Spread Over 5 Years 0.0 < -1.0% 0.0% 0.2% 0.4% 0.6% 0.8% 1% 1.2% 1.4% 1.6% Average Spread Over 5 Years Average Spread 0.59% Sandard Deviaion 0.24% Average Spread 1.45% Sandard Deviaion 0.68%
Focus on Lef Tail of Disribuion Porfolio of 100 Bonds 7-10% CDO Tranche of Porfolio Disribuion for Bond Porfolio Disribuion for 7-10% CDO Tranche 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% < -1.0% 0.0% 0.2% 0.4% 0.6% 0.8% 1% 1.2% 1.4% 1.6% Average Spread over 5 years 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% < -1.0% 0.0% 0.2% 0.4% 0.6% 0.8% 1% 1.2% 1.4% 1.6% Average Spread over 5 years
Valuaion Models for Baske Trades CDO Noes Valuaion is he value of he collaeral (a par plus he value of selling baske loss proecion. Valuaion of Baske Loss Proecion Gaussian copula model has become he indusry sandard. Calibrae risk neural survival probabiliy curves for individual names o quoes for single name defaul swaps. Se correlaion of defaul across names in he porfolio and simulae mulivariae normal random variables. For each simulaion pah, compue correlaed uniform (0,1 simulaions by applying he cumulaive normal disribuion funcion o he simulaed normal random variables. Use he correlaed uniform random numbers and he survival probabiliy funcions o compue simulaed ime o defaul for each name in he porfolio. Compue relevan cashflows and discouned presen value for each ranche. Simulaed model is consisen wih he pricing of single name defaul swaps.
Deficiencies of he Copula Model Wha is he correlaion of defaul? The model can handle differen correlaions across he names in he porfolio, bu how should hese correlaions be se? Mos users se he correlaions o be equal across all names. A single correlaion is no consisen wih observed marke quoes for baske ranches. Typically, each ranche has is own marke implied correlaion. The model can handle random recovery, bu i canno handle correlaion beween defaul and recovery. Curren procedures for exracing risk neural survival probabiliy funcions do no work if defaul and recovery are correlaed. New models would be necessary for valuing single name defaul swaps.
Why does he marke pu a pricing premium on defaul correlaion? Alernaive view. Finance heory ells us ha markes pu risk premiums ino prices as compensaion for risk. In a ime-sae preference approach o asse pricing (Arrow- Debreu, he marke increases he subjecive probabiliies for hose saes in which marginal uiliy of wealh is high, relaive o hose saes associaed wih lower marginal uiliy of wealh. The marginal uiliy of wealh for risk averse invesors is high when wealh is low, or when values of marke porfolios are low. Defaul raes increase during recessions when marke porfolios are falling in value. Invesors impue higher risk neural probabiliies for corporae defauls. Defaul and recovery raes are correlaed so ha when massive defauls occur, recovery raes will be lower. These rare saes when defaul raes are high and recovery raes are low ge higher riskneuralized probabiliies. The ne resul is higher value for defaul proecion agains hese rare evens. Copula models capure his risk premium by increasing he correlaion of defaul, o effecively increase he probabiliy of simulaing massive defauls. Bu he model misses he correlaion beween defaul and recovery.
Time permiing, skech a model wih correlaed defaul and recovery Model for pricing single name defaul swaps. Le h be he defaul inensiy, so ha h is he defaul probabiliy over a small ime inerval. Survival probabiliies are Pr(, T = Eˆ e h( s ds Defaul swap raes are se so ha he ne presen value of he defaul swap is zero. D(,T is he discouning funcion, Prem is he defaul swap premium, and Rec is he recovery rae. Swap premiums are over N ime inervals, and defauls are assumed o occur a a se of M discree ime poins. T. PV N = Prem( j D(, j Pr(, j j= 1 M k = 1 [ 1 Eˆ (Rec] D(, [ Pr(, Pr(, ] = 0 k k1 k To derive his model, one mus assume ha defaul inensiies, ineres raes, and recovery raes are independen.
The More General Pricing Model for Single Name Defaul Swaps If ineres raes are independen, we can use he discouning funcion from he ineres rae marke. 0 Loss ( ˆ ˆ Prem ( 1 ( ( ( 1 j ( 1 = = = = + M k ds s h ds s h ds s r k N ds h r j k k k j s s e e e E e E PV 0 Loss( ˆ, ( ˆ, ( Prem( 1 ( ( 1 j ( 1 = = = = M k ds s h ds s h k k N ds s h j j k k j e e E D e E D PV
Skech of a Model wih Correlaed Defaul and Recovery Need o define sochasic processes for he facors ha deermine boh defaul inensiies (and spreads and loss raes. Models ha generae closed form soluions for single name defaul swaps will be easier o calibrae. Need o model correlaions across spreads. Calibrae model o single name defaul swap marke and quoes for baske loss proecion. Look for relaive value opporuniies across differen ranches and across differen porfolios.