Probabilities of Default and the Market Price of Risk in a Distressed Economy

Transcription

1 WP//75 Probabiliies o Deaul and he Marke Price o Risk in a Disressed Economy Raphael Espinoza and Miguel Segoviano

2 20 Inernaional Moneary Fund WP// IMF Working Paper Sraegy, Policy and Review Deparmen Probabiliies o Deaul and he Marke Price o Risk in a Disressed Economy Prepared by Raphael Espinoza and Miguel Segoviano Auhorized or disribuion by Caherine Paillo April 20 This Working Paper should no be repored as represening he views o he IMF. The views expressed in his Working Paper are hose o he auhor(s) and do no necessarily represen hose o he IMF or IMF policy. Working Papers describe research in progress by he auhor(s) and are published o elici commens and o urher debae. Absrac We propose an original mehod o esimae he marke price o risk under sress, which is needed o correc or risk aversion he CDS-implied probabiliies o disress. The mehod is based, or simpliciy, on a one-acor asse pricing model. The marke price o risk under sress (he expecaion o he marke price o risk, condiional on i exceeding a cerain hreshold) is compued rom he price o risk (which is he variance o he marke price o risk) and he discoun acor (which is he inverse o he expeced marke price o risk). The hreshold is endogenously deermined so ha he probabiliy o he price o risk exceeding i is also he probabiliy o disress o he asse. The price o risk can be esimaed via dieren mehods, or insance derived rom he VIX or rom he acors in a Fama-MacBeh regression. JEL Classiicaion Numbers: G3; G2 Keywords: Price o risk; credi deaul swap; CDS; risk-neural probabiliy Auhor s Address: respinoza@im.org; msegoviano@cnbv.gob.mx Miguel Segoviano is Direcor General o Risk Analysis and Quaniaive Mehodologies a he Mexican Financial Auhoriy (Comisión Nacional Bancaria y de Valores). The views expressed in his Working Paper are hose o he auhors and do no necessarily represen hose o he IMF or o he Mexican Financial Auhoriy. We are graeul o Thanawan Chaiwaana or ousanding research assisance. This paper beneied grealy rom discussions wih Tobias Adrian, Carlos Caceres, Vincenzo Guizzo, Huyn Shyn, Dimirios Tsomocos, and rom paricipans a he IMF and a he LSE/CCBS Bank o England conerence on Measuring Sysemic Risk and Issues or Macroprudenial Regulaion. Any errors are solely he auhors responsibiliy.

3 2 Conens Page I. Inroducion...3 II. Theoreical Background...4 A. Risk-neural Probabiliies o Deaul and he Marke Price o Risk...4 B. Condiional Expecaion o he Marke Price o Risk...6 C. Esimaing he Price o Risk...7 III. Endogenous Threshold...9 IV. Applicaion o US Banks during he Crisis... V. Conclusion...2 Reerences...4 Figures Figure. Raio o CDS-implied Probabiliy o Disress o Moody s KMV EDF...4 Figure 2. Uniqueness o he Threshold...0 Figure 3. Adjusmen Facor and Marke Price o Risk... Figure 4. Esimaed Probabiliies o Deaul...3

4 3 I. INTRODUCTION During he credi crisis o , he esimaion o deaul probabiliies o banks has been a ocal poin o ineres. Deaul probabiliies can be esimaed using markes assessmen, as capured by CDS spreads, or using models o he value o he irm derived rom he Black-Scholes-Meron model (Black and Scholes, 973; Meron, 974). The laer mehod has been popularized by he Moody s KMV Esimaed Deaul Frequency (EDF) daase, which provides deaul probabiliies or many o he larges companies in he world, and mos U.S. banks. CDS spreads are also widely used, alhough hey are only proxies or deaul probabiliies as hey are inluenced by he marke price o risk (he cos o insurance) as well as he probabiliies o disress o he irms. The wo measures o probabiliies o sress have diverged markedly during he pos-lehman imes. We show in Figure he raio o he CDS-implied probabiliy o deaul over he EDF probabiliy o deaul or dieren U.S. banks. This raio, which could be inerpreed as he marke price o insurance i he EDF and he CDS spread were represening perecly he probabiliy o deaul and he risk-neural probabiliy o deaul, has also varied dramaically across banks. I has been shown o vary across secors (Bernd e al., 2005), and o correlae negaively wih real aciviy (as one would expec rom a pricing kernel see secion II) and posiively wih real ineres raes (Amao, 2005, Amao and Luisi, 2006). The objecive o his paper is o provide a mehod or compuing he marke price o risk under disress and hereore he probabiliies o deaul implied by he CDS spreads. The objecive is o provide an alernaive measure o he EDFs, which are based on slowlymoving balance shee inormaion and have been lagging marke inormaion during he crisis. The probabiliies o disress in he recen abnormal imes canno be esimaed using pas daa, so he mehod proposed by Jackwerh (2000) o compue risk aversion and acual probabiliies is also ill-suied. In our paper, he calculaions assume a one acor model (or insance a Consumpion CAPM, alhough his is no needed or he resuls) and show how inormaion on he mean and he variance o he acor can be used o derive he marke price o risk under a siuaion o disress. This marke price o risk is he condiional expecaion o he price o risk, condiioning on i exceeding a hreshold (secion II). The acor we use is a ransormaion o he VIX (which has been shown o correlae srongly wih he irs principal componen o asse reurns). The approach is ree o any assumpion on he shape o he uiliy uncion (e.g., he coeicien o risk aversion) and o assumpions on saionariy o reurns, bu noneheless i allows us o srip he eec o risk aversion on asse prices. However, he wo undamenal assumpions are ha pricing is based on a oneacor model, and ha his acor is normally disribued.

5 4 Figure. Raio o CDS-implied Probabiliy o Disress o Moody s KMV EDF AIG AXP BAC GS JPM MS WFC 0 2-Jan-06 2-Mar-06 2-May-06 2-Jul-06 2-Sep-06 2-Nov-06 2-Jan-07 2-Mar-07 2-May-07 2-Jul-07 2-Sep-07 2-Nov-07 2-Jan-08 2-Mar-08 2-May-08 2-Jul-08 2-Sep-08 2-Nov-08 2-Jan-09 2-Mar-09 2-May-09 Source: Moody s KMV and Bloomberg We inally show ha he marke price o risk under disress has o be calculaed joinly wih he probabiliy o disress o ensure he condiioning hreshold is compaible wih he probabiliy o deaul (secion III). We apply our mehod o he major U.S. banks during he crisis (secion IV) and show ha CDS-implied probabiliies o deaul overesimaed credi risk by roughly 50 percen. Secion V concludes. II. THEORETICAL BACKGROUND A. Risk-neural Probabiliies o Deaul and he Marke Price o Risk In his secion we review he basic asse pricing ramework linking he price o risk o he CDS-implies probabiliies o deaul. A one acor model can be derived rom a consumpion Euler equaion: u( c ) P E x () u( c ) where P is asse price a ime and x is he one period-ahead payo. The sochasic discoun acor is deined as u( c m ) (2) u( c ) and he linear pricing ormula or he one-acor model is hereore: P E [ m x ]. (3)

6 5 Wriing he gross reurn R = x P, he pricing ormula is equivalen o E [ m R ]. (4) Applying he previous equaion o a risk-ree asse: E[ m R ] or R E [ m ] (5) I he saes o naure a ime + are indexed by s, and he probabiliy o sae s is () s P () s m()() s x s E[ m x] (6) The risk-neural probabiliy ˆ is such ha 2 s ˆ( s) R m() s () s (7) This probabiliy measure does no correspond o any acual probabiliy o naure-in paricular i does no imply ha he asse pricing model is based on risk-neural invesors. Noneheless, he erm risk-neural probabiliy is used in he lieraure because his is he probabiliy measure ha a risk-neural invesor would need o believe in o agree on he asse prices given by he markes. Indeed, he price o an asse can be wrien as P Ex ˆ( ) ˆ( s)() x s (8) R s R where E ˆ( x ) is he expeced value o x+ associaed wih he risk-neural probabiliy measure ˆ. For a risk-neural invesor ha believes ha ˆ( s) are he acual probabiliies o naure, he price P given by he markes is coheren wih he payo x +. The risk-neural probabiliy ˆ( s) is wha is obained rom CDS spreads. Indeed, ˆ( s) /R + is he price o an asse ha pays x(s) = dollar in he sae o disress s (rom equaion 8). A S N common approximaion is ˆ where S N is he CDS spread o bank N and K he ( K) recovery rae (assumed o be a 60 percen), and R can be proxied by he OIS rae. The missing elemen is hereore he marke price o risk m(s). 2 ˆ is a probabiliy measure since all ˆ (s) are posiive (rom he undamenal heorem o inance, m(s) is posiive in absence o arbirage opporuniies). Furhermore, ˆ d() s R m() s d() s E[ R m ] s s

7 6 B. Condiional Expecaion o he Marke Price o Risk Since our ocus is on he sae o naure where he banks are under disress, we need no esimae m(s) or all saes o naure. We group he saes o naure under wo headings: disress, and no disress and rewrie he linear pricing ormula: P E [ m y ] d E[ m y disress] ( d ) E[ m y no disress] The relaionship beween he risk-neural probabiliy under sress and he real probabiliy o sress is given by equaion (7) ˆ d d R E[ m disress] The marke price o risk under sress E[ m disress] is no observable and our objecive is o esimae i rom marke prices. Calibraions based on he shape o he uiliy uncions, derived rom equaion (2), could also be used, bu he link beween asse prices and uiliy uncions is subjec o many diiculies, as evidenced by he numerous puzzles spurred by he consumpion-capm lieraure. Our mehod relies on he esimaion o E[ m disress] based on he mean and he variance o m +, which can boh be measured. Indeed, E [ m ] is deduced rom he Overnigh Indexed Swap (OIS) rae, ollowing equaion (4). Furhermore, he volailiy o m + ( Var [ m ] / E [ m ] ) is he price o risk, which has been he subjec o much aenion in he asse pricing lieraure. Finally disress is deined as he siuaion where he marke price o risk exceeds a cerain hreshold (see below). Under he addiional assumpion ha m + is normally disribued, E [ m disress ] can be deermined using he runcaed normal disribuion ormula: E where [ m disress] E[ m m hreshold] ( ) ( hreshold ) / ( T ) / E[ m ] var [ m ] ( ) ( ) /[ ( )] and (.) is he sandard normal cumulaive disribuion uncion. λ (α) is called he inverse Mills raio.

8 7 We need o deermine a hreshold T above which banks are considered o be under sress (his hreshold should ideally correspond o he saes o naure where he CDS payos are high). We irs assume, somewha arbirarily, ha he hreshold is one hisorical sandard deviaion away rom he average sochasic discoun acor E ] bu we show in [ m secion III how o exen he model so ha he hreshold is in line wih he esimaed probabiliy o deaul. C. Esimaing he Price o Risk The remaining variable o esimae is he variance o m +, which is linked o he price o risk λ m hanks o he equaion: Var [ m ] m E [ m ] λ m is called he price o risk as he CAPM predics his acor o be he main driver o excess reurns, ogeher wih he quaniy o risk. Indeed, wriing equaion (4) as and re-arranging i, he CAPM equaion is: ER ( ) R E [ mr ] = E( m ) E( R ) cov( m, R ) i, m m where cov( m, R ) Var( m ) i, m is he quaniy o risk and m is he price o risk. Var( m ) E( m ) In he empirical lieraure (see Adrian and Moench, 2009 and Adrian and Shin, 200, or recen applicaions) he price o risk has been relaed o he VIX index, or alernaively o he Principal Componens (PC) o marke yields. The VIX index and he irs principal componen are srongly correlaed (see Couder and Dex, 2008) so he choice in he lieraure o one variable or he oher does no seem o be crucial. Mehod. VIX and he Maximum Sharp Raio In our second mehod, we use a propery o he price o risk o normalize he VIX (we could do he same wih he PC). For a given asse reurn, he pricing equaion implies i i E( m) E( R ) ivar( R ) Var( m) m, R i Var( m) i E( R ) R i Var( R ) m, R E( m)

9 8 Since he correlaion coeicien i m,r canno be greaer han, we deduce i E( R ) R i Var( R ) Var( m) E( m) Theoreically, he price o risk is hereore he maximum Sharpe Raio aainable or an eicien porolio. Hisorically, Sharpe raios higher han 3 would be considered very high: we hereore decide o normalize he VIX series (by a acor o 4) in order o scale he VIX index o a series consisen wih his propery o he price o risk. Mehod 2. Principal Componens and he Price o Risk We could also use he PC mehod proposed by Adrian and Moench (2008) o esimae he price o risk in he U.S. sock markes. Adrian and Moench (2008) applied heir mehod o bond reurns and Adrian, Eula and Shin (200) applied i o currencies, whils our ocus is on S&P sock reurns. The mehod relies on a CAPM equaion: R ˆ i, R i( m, m, ) where ˆm, is he expeced componen o he marke price o risk and m, is he innovaion in he marke price o risk. The expeced componen o he marke price o risk is assumed o be an aine uncion o he expeced PC o sock marke reurns: ˆm, 0 PC where 0, are he coeiciens we need o normalize he PC o make conver i ino he esimaed price o risk. I one esimaes, or each asse he equaion, R R a bpc cpc i, i i i Adrian and Moench (2008) show ha a i = β i λ 0 and ha β i λ = c i, wo equaions ha ideniy he normalizaion coeiciens λ 0 and λ. They apply he model o he Principal Componen o bond yields.

10 9 III. ENDOGENOUS THRESHOLD The acual probabiliy o deaul is deduced rom he risk-neural probabiliy using he ˆ ˆ ormula: ( r ) E [ m m T ] ( r )( ( )) T where ( ) ( ) /[ ( )] is he inverse Mills raio and. The hreshold T was chosen once and or all, such ha P [ m T ] 0. 84, i.e., he price o risk under disress was one uncondiional sandard deviaion rom he uncondiional mean /(+ r ): T E[ m] [0.84]* var( m) r where [0.84] and is he inverse o he cumulaive normal disribuion. The esimaion o he marke price under risk would gain in coherence i one could choose he hreshold T such ha he deiniion o he sress scenario is in line wih he probabiliy o disress ha is inally esimaed. This would also ensure ha he hreshold is bank-speciic and consisen wih he probabiliy o deaul. Hence, we wan o se T E[ m] [ ]* var( m) [ ] r where π is he probabiliy o deaul we are looking or. (Noe ha a very high hreshold means we are looking a unlikely evens, i.e., a lower π, which is why - π is he argumen in.) The probabiliy o deaul hereore solves he ollowing non-linear equaion: ( r ) ˆ r r [ ] (9) or ). (

11 0 Proposiion There is a unique probabiliy o deaul ha solves equaion 9. Proo and λ are sricly increasing so ( ) lim ( x) and lim ( x) 0 x0 x The limi o ( ) is 0 since lim ( x) x is an increasing uncion o. Furhermore ( ) ˆ implies and i is well known ha 0 ( x) x. This implies ( ) 0 x We can also show ha he derivaive o ( ) is ininie in 0: Since ( x) x x, we can rewrie, and b ) a b [ ] ( '( ) ( a b [ ] [ ]) 2 Since [ ], we ind [ ] '( ) b 2 [ ] ( a b [ ]) 0 X ( a bx ) 0 we proved ( ) 2 X Finally, since. Thereore, ( ) is a coninuous, sricly increasing, uncion over ]0, ˆ ], wih ˆ < < and wih a derivaive in 0 greaer han. This shows ha he equaion = ( ) (equaion 8) admis exacly one soluion wihin ]0,]. We plo and ( ) in Figure 2. Figure 2. Uniqueness o he Threshold π (π)

12 IV. APPLICATION TO US BANKS DURING THE CRISIS We show in Figure 3 he adjusmen acor (beween risk-neural and real probabiliies o disress) obained wih he irs mehod described above. In crisis imes, he real probabiliies o disress end o be much lower han he risk-neural probabiliies since he insurance coss increase as invesors seek higher compensaion or risk. We compare our calculaed subjecive deaul probabiliy wih he Moody s KMV Esimaed Deaul Frequency. The EDF measure is based on he modeling o equiy as a call opion on asses (since liabiliies have o be paid beore shareholders in case o deaul). The asse value is derived orm he marke capializaion o he irm, and an esimae o asse volailiy is used o model he probabiliy disribuion o asse value. When he value o deb, he value asses and he volailiy o asses are esimaed, he value o equiy (which is a call opion on he asses) yields he probabiliy o deaul. Figure 4 shows he risk-neural probabiliy o deaul and he EDF o several US inancial insiuions, ogeher wih our esimae o he deaul probabiliy under mehod 2. The riskneural probabiliy and associaed subjecive probabiliy respond o he crisis very quickly while EDF seems o lag. The value, however, o he subjecive probabiliy and EDF are more in line han he risk-neural deaul probabiliy. Indeed, we esimaed ha he marke price o risk (or saes o naure in which banks would be under sress) increased by 30 percen in he recen urmoil (see Figure 3) Figure 3. Adjusmen Facor and Marke Price o Risk Adjusmen Facor 3.4 E(M / M>T) /4/200 6/4/2002 2/4/2002 6/4/2003 2/4/2003 6/4/2004 2/4/2004 6/4/2005 2/4/2005 6/4/2006 2/4/2006 6/4/2007 2/4/2007 6/4/2008 2/4/2008 6/4/2009 2/4/200 6/4/2002 2/4/2002 6/4/2003 2/4/2003 6/4/2004 2/4/2004 6/4/2005 2/4/2005 6/4/2006 2/4/2006 6/4/2007 2/4/2007 6/4/2008 2/4/2008 6/4/2009 Source: auhors calculaions Noe: he adjusmen acor is ( r ) E [ m m T ]

13 2 V. CONCLUSION Compuing he marke price o risk under siuaions o disress is a crucial elemen when one needs o use asse prices o assess probabiliies o disress. We oered in his paper a simple bu heoreically consisen approach o such a calculaion. The approach is also ree o model assumpions on he shape o he uiliy uncion, because i is based solely on an empirical one-acor model. I does no require daa on pas deauls and hereore is appropriae or esimaing probabiliies o exreme evens alhough i sill depends on CDS marke assessmen o risk. We applied our mehod o US banks during he subprime crisis, bu he mehod is general enough o be applied o oher CDS markes. For insance, using his mehod, Caceres, Guizzo and Segoviano (200) analyzed he conribuion o risk aversion o he evoluion o sovereign spreads in he eurozone. 3 The model is based on a one-acor model bu a muliple acor model would be beer a iing he daa and a capuring he deerminan o asse prices. The diiculy lies in maching he risk ha we wan o measure (he marke o price o risk under a siuaion o disress) wih he momens o he pricing acor we can observe. When several acors are assumed o deermine pricing, ideniying which acor o iler ou requires addiional undersanding o he meaning o he acors and o he meaning o he risk one wans o iler ou. In a one acor model, his issue does no arise. A second assumpion in our mehod is ha he price o risk is normally disribued. Relaxing his assumpion may also require esimaing more acors in he asse pricing model because wih a one acor model, only one momen (i.e., he variance) o he price o risk can be ideniied. Furher research is needed o generalize he mehod o hese more complex seings. 3 Caceres, Guizzo and Segoviano (200) showed ha in he aermah o he subprime crisis, global risk aversion and conagion acors were he main drivers behind increases in euro CDS spreads, bu he urning poin was in he second hal o 2009, when counry-speciic undamenals drove CDS in Greece, Ireland and Porugal. Anoher applicaion o his mehod or Asia is available in Caceres and Filiz Unsal (20).

14 3 Figure 4. Esimaed Probabiliies o Deaul 0.7 AIG 0.2 AXP M A M J J A S O N D J F M A M M A M J J A S O N D J F M A M 2008 Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF BAC BBT M A M J J A S O N D J F M A M Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF BK 0.75 COF M A M J J A S O N D J F M A M M A M J J A S O N D J F M A M J 2008 Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF Risk-neural Probabiliy Esimaed Acual Probabiliy Moody`s KMV EDF Source: Moody s KMV, Bloomberg and auhors calculaions

15 4 REFERENCES Adrian, T., and E. Moench, 2008, Pricing he Term Srucure wih Linear Regressions, Federal Reserve Bank o New York Sa Repor No. 340., E. Eula, and H. Shin, 200, Risk Appeie and Exchange Raes, Fed Reserve Bank o New York Sa Repor No. 36., and H. Shin, 200, Liquidiy and Leverage, Journal o Financial Inermediaion, Vol. 9, pp Amao, J., 2005, Risk Aversion and Risk Premia in he CDS Marke, BIS Quarerly Review, December, pp , and M. Luisi, 2006, Macro Facors in he Term Srucure o Credi Spreads, BIS Working Paper No. 203 (Basel: Bank or Inernaional Selemens). Bernd, A., D, Rohan, D. Duie, M. Ferguson, and D. Schranzk, 2005, Measuring Deaul Risk Premia rom Deaul Swap Raes and EDFs, BIS Working Paper No. 73 (Basel: Bank or Inernaional Selemens). Black, F., and M. Scholes, 973, The Pricing o Opions and Corporae Liabiliies, Journal o Poliical Economy, Vol. 7, pp Caceres, C., and D., Filiz Unsal, 20, Sovereign Spreads and Conagion Risks in Asia, IMF Working paper, orhcoming., V. Guzzo, and M. Segoviano, 200, Sovereign Spreads: Global Risk Aversion, Conagion or Fundamenals? IMF Working Paper 0/20 (Washingon: Inernaional Moneary Fund). Cochrane, J., 2005, Asse Pricing, Princeon NJ: Princeon Universiy Press. Couder V., and M. Gex, 2008, Does Risk Aversion Drive Financial Crises? Tesing he Predicive Power o Empirical Indicaors, Journal o Empirical Finance, Vol. 5, pp Jackwerh, J., 2000, Recovering Risk Aversion rom Opion Prices and Realized Reurns, Review o Financial Sudies, Vol. 3, pp Meron R., 974, On he Pricing o Corporae Deb: The Risk Srucure o Ineres Raes, Journal o Finance, Vol. 29, pp , 974.