Credit Default Swap Spreads and Systemic Financial Risk

Transcription

1 Credit Default Swap Spreads and Systemic Financial Risk Stefano Giglio Harvard University May 3, 2011

2 Introduction What is the joint probability of default of large financial institutions? Systemic default risk: Pr{at least r LFIs default} Banks are interconnected and exposed to common shocks Defaults not independent even at short horizons Severe consequences of multiple defaults

3 Introduction: this paper Difficult to measure Rare events Prices that reflect individual defaults (bonds,cdss) but not multiple defaults In this paper 1. Exploit counterparty risk to learn about P(A i A j ): enrich information set 2. Derive tightest bounds on multiple default risk

4 CDS and counterparty risk Credit Default Swap is an OTC contract designed to transfer the credit risk of the reference entity Counterparty risk in CDSs: if seller defaults, contract terminates Double default relevant for pricing: discount relative to the corresponding bond

5 CDS and counterparty risk Collateral static: very costly -> dynamic Not widely used with dealers (66% of contracts in 2008) When margin set to current exposure, subject to jumps Collateral can be less than current exposure (Goldman) Buyers aware of counterparty risk (CDS against seller)

6 Theory Assume we observe p i : P(A i ) z ji : P(A i ),P(A i A j ) Look for P r : P{at least r default} (information of order N - systemic) P 1 = P(A 1 A 2 A 3 ) P 2 = P((A 1 A 2 ) (A 2 A 3 ) (A 1 A 3 )) P 3 = P(A 1 A 2 A 3 )

7 Theory Becomes max Pr{at least r default} P(A i ) = a i P(A i A j ) = a ij max p c rp Ap = b Consistent probability system p 0 i p = 1

8 Implementation Assume a simple, discretized pricing model for bonds and CDSs Constant hazard rates Assume recovery rates R=30%, S=30% Impose a lower bound for the liquidity process γ of bonds Nonnegative Calibrated to 2004 Calibrated to nonfinancial firms

9 Implementation Observe average CDS spreads: zji linear function of P(A i ) and P(A i A j ) 1 zi linear function of P(A i ) and N 1 j i P(A i A j ) One constraint for each i

10 Systemic risk 0.05 Pr [ at least 1 ] Pr [ at least 4 ]

11 Systemic risk - γ i t 0 Pr [ at least 1 ] Only bonds Only CDS Pr [ at least 4 ]

12 Systemic risk - γ i t 0 Pr [ at least 1 ] Only bonds Only CDS Bonds and CDS Pr [ at least 4 ]

13 Systemic risk measures: assumptions on liquidity Nonneg liq Only bonds Only CDS Bonds and CDS Liq Liq nonfinancials

14 Picture of network 8/4/2008 MS Citi 0.27% 0.19% 0.08% 0.02% 0.14% 0.13% 0.05% 0.17% 0.08% 0.18% JPM 0.13% Merrill 0.33% 0.30% 0.10% 0.38% Lehman

15 Marginal and pairwise probabilities Marginal probability 8 x Merrill Lehman Citi 0 Jan 2007 Apr 2007 Jul 2007 Oct 2007 Jan 2008 Apr 2008 Jul 2008 Lehman Joint probability 4 x Merrill & Citi Merrill & Lehman Citi & Lehman 0 Jan 2007 Apr 2007 Jul 2007 Oct 2007 Jan 2008 Apr 2008 Jul 2008 Lehman

16 Contribution: Pr{at least 4 j} Pr [ bank involved in systemic default ] 8 x Citi Lehman Merrill BofA Others

17 Explore: Contrib vs. MES 0.01 Citigroup Contrib MES

18 Explore: Contrib vs. MES 5 x 10 3 Lehman Contrib MES

19 Explore: Contrib vs. MES 5 x 10 3 Lehman Contrib CDS

20 Conclusion We learn that systemic risk really started to increase in late 2008: If systemic risk was so high in January-March 2008, why did the average CDS spread go up so much? Why were people so keen to buy insurance from unreliable counterparties? Things to explore Correlation of contribution to systemic risk with other measures (MES, CoVar, stress test) Pairwise default risk and correlation of equity returns Systemic risk and puts

21 Extra Slides Additional details Simple Example of Bounds Linear Programming Algorithm Implementation Pricing Formulas Data Symmetry

22 Extra Slides Robustness to assumptions on recovery rates: Robustness to R and S Stochastic and Time Varying Recovery Rates All other robustness tests: Robustness Results Derivations References: Arora et al.

23 Theory: simple example Bank 1 Bank 1 Bank 2 Bank 2 Bank 3 Bank 3

24 Theory: simple example Suppose we observe, from bonds: P(A 1 ) = P(A 2 ) = P(A 3 ) = 0.2 From CDSs: P(A 1 A 2 ) = P(A 2 A 3 ) = 0.07 Tightest bounds P(A 1 A 3 ) = P P P

25 Theory: simple example Heterogeneity. Suppose still P(A 1 ) = P(A 2 ) = P(A 3 ) = 0.2 but now we only know Then P(A 1 A 2 ) + P(A 2 A 3 ) + P(A 1 A 3 ) 3 = 0.05 Full information Only average 0.45 P P P P P P

26 Linear Programming Algorithm Start from problem: max P r s.t. P(A i ) = a i... P(A i A j ) = a ij

27 Linear Programming Algorithm Obtain: s.t. max p c r p p 0 i p = 1 Ap = b

28 Linear Programming Algorithm Start with matrix B (2 N,N) Rows are binary representation of N B = Each row is event of type A 1 A 2... A N where A j = A j if element j of the row is 1, and A j = A j otherwise

29 Linear Programming Algorithm p contains probabilities of these events p 0 P(A i ): or P(A i ) = p i = 1 p j j:b(j,i)=1 P(A i ) = a i p a i j = B(j,i)

30 Linear Programming Algorithm P(A i A k ): or: P(A i A k ) = p j j:b(j,i)=1and B(j,k)=1 P(A i A k ) = b ik p for a vector b ik of size (2 N,1) s.t.: b ik j = B(j,i)B(j,k)

31 Linear Programming Algorithm P r : or P r = j:( h=1:n B(j,h)) r P r = c r p for a vector c r of size (2 N,1) s.t.: [ ] cj r = I B(j,h) r h=1:n p j

32 Implementation: 1 - Pricing Contracts span long horizons Contracts priced again every time t looking forward, assuming Constant hazard rates h i t Constant bond liquidity premium γ i t Discretize by month Joint default in a month <=> double default Seller default and reference survives until next month -> small change in reference risk Recovery R=30%, S=30%

33 Implementation: 1 - Pricing (bonds) ( T ij B ij (t,t ij ) = c ij δ(t,s)(1 ht) i s t (1 γt) )+ i s t s=t+1 +R +δ(t,t ij )(1 h i t) T ij t (1 γ i t) T ij t ( ) T ij δ(t,s)(1 ht) i s t 1 (1 γt) i s t 1 ht i s=t+1

34 Implementation: 1 - Pricing (bonds) Bond liquidity: constant conveniency yield γ i t Interpretation. Garleanu and Pedersen (2010): E t [R ij t+1 Rf t+1] = Cov t(m t+1,r ij t+1 Rf t+1 ) + m i E t [M t+1 ] tx t ψ t m i t margin for senior unsecured bonds of firm i xt proportion of agents constrained ψt shadow cost of capital γ i t m i tx t ψ t = α i λ t

35 Implementation: 1 - Pricing (CDSs) T 1 δ(t,s)(1 P(A i A j )) s t z ji = s=t = T δ(t,s)(1 P(A i A j )) s t 1 s=t+1 {[ P(Ai ) P(A i A j ) ] (1 R) + S [ P(A i A j ) ] (1 R) }

36 Implementation: 2 - Calibration of liquidity γ i t Bond liquidity: constant conveniency yield γ i t = α i λ t λt : common variations in margins, cost of capital, constrained agents Calibrating γ i t γ i t, obtain P(A i ) h i (γ i t ) 1. γ i t 0 2. γ i t α i : liquidity at least as of 2004

37 Implementation: 2 - Calibration of liquidity γ i t 3. For a group K of A-rated (or better) nonfinancial firms double default risk is low calibrate matching the bond-cds basis γ k t = α k λ t and assume that for financials γ i t α i λ t

38 Implementation: 3 - Availability of CDS spreads Observe average CDS spreads: zji linear function of P(A i ) and P(A i A j ) 1 zi linear function of P(A i ) and N 1 j i P(A i A j ) One constraint for each i Do not observe contributors of Markit quotes Pick 15 dealers covering 90% of CDS market

39 Pricing formulas: bonds Bonds: ( T ij B ij (t,t ij ) = c ij δ(t,s)(1 ht) i s t (1 γt) )+ i s t s=t+1 +δ(t,t ij )(1 ht) i T ij t (1 γt) i T ij t ( T ij ) +R s=t+1 δ(t,s)(1 ht) i s t 1 (1 γt) i s t 1 ht i

40 Pricing formulas: CDSs T 1 δ(t,s)(1 P(A i A j )) s t z ji = s=t = T s=t+1 δ(t,s)(1 P(A i A j )) s t 1 {[ P(Ai ) P(A i A j ) ] (1 R) + S [ P(A i A j ) ] (1 R) }

41 Pricing formulas: CDSs Linearize to use as a constraint: [ T z ji,t = (P(A i ) (1 S)P(A i A j )) s=t+1 δ(t,s) ] (1 R) [ ] T s=t 1 δ(t,s)

42 Data Bonds Look on Bloomberg and Markit for all bonds that are issued by institution i Restrict to senior unsecured fixed or zero coupon: no callable, putable, sinkable, structured TRACE-eligible bonds: use TRACE closing price Other bonds: generic closing price

43 Data Risk-free rate: zero-coupon government bonds CDS: Markit Period: 2004 to June 2010

44 Data Table 1 Avg valid bonds Abn Amro Bank of America Barclays Bear Stearns Bnp Paribas Citigroup Credit Suisse Deutsche Bank Goldman Sachs JP Morgan Lehman Brothers Merrill Lynch Morgan Stanley UBS Wachovia Note: first column reports average number of bonds for each institution that are used for the estimation of marginal default probabilities. Columns 2-8 break this number down by year.

45 Data Table 2 Avg CDS spread Std CDS spread Min spread Max spread Abn Amro Bank of America Barclays Bear Stearns Bnp Paribas Citigroup Credit Suisse Deutsche Bank Goldman Sachs JP Morgan Lehman Brothers Merrill Lynch Morgan Stanley UBS Wachovia

46 Data Table 2 Avg basis Std basis Min basis Max basis Abn Amro Bank of America Barclays Bear Stearns Bnp Paribas Citigroup Credit Suisse Deutsche Bank Goldman Sachs JP Morgan Lehman Brothers Merrill Lynch Morgan Stanley UBS Wachovia

47 Symmetry p is symmetric if it does not depend on the ordering of A i s. A LP problem max c p s.t.ap b is symmetric if c p and all constraints do not depend on the ordering of A i s. Example: union of all events, average of probabilities Proposition 3: problem is symmetric => symmetric solution Corollary: symmetric systems have the widest bounds given average probabilities

48 Symmetry Average bonds and CDS All bonds and CDS Nonneg liq Liq nonfinancials

49 Robustness: R and S Dependence on R Two-period case: p i = 1 (1 R)P(A i ) z ji = (1 R)P(A i ) (1 S)(1 R)P(A i A j ) Higher R lower yield bond-implied probability scales up Higher R lower CDS spread cds-implied probabilities scale up Bounds scale up

50 Robustness: R and S Dependence on S Depends on whether the basis can be all explained by counterparty risk Remember the constraints: P(A i ) a i (γ i ) P(A i ) (1 S) j i P(A i A j ) N 1 = p i (z i ) S=1 For each i, P(Ai ) = p i Decrease S P(Ai ) > p i : counterparty risk But if S large, P(Ai ) p i requires high counterparty risk

51 Robustness: R and S P(A i ) a i (γ i ) P(A i ) (1 S) i j P(A i A j ) N 1 = p i (z i ) S decreases more P(A i A j ) can fill a larger gap systemic risk increases This ignores constraints from bonds Once P(A i ) hits the upper bound a i (γ i ), P(A i A j ) has to decrease

52 Robustness: R and S Example: June 25, Bank of America, Citigroup, GS. Probabilities are average monthly risk-neutral probabilities in bp. Bank a i (0) p i

53 Robustness: R and S Only CDS CDS + bonds Only bonds r= r= S

54 Robustness: R and S Bank a i (0) p i 1 15(not 25) x Bonds and CDSs Only bonds Only CDS

55 Robustness: R and S Model R S 2007 Jan 2008 to Bear Bear to Lehman Max P1 Month after Lehman Oct 2008 to April 2009 After April

56 Robustness: R and S Model R S 2007 Jan 2008 to Bear Bear to Lehman Max P4 Month after Lehman Oct 2008 to April 2009 After April

57 Robustness: R and S Model R S 2007 Jan 2008 to Bear Bear to Lehman Min P1 Month after Lehman Oct 2008 to April 2009 After April

58 Time-varying recovery rates R could be lower in bad times Adjusting R would imply bounds S could be lower in bad times lower S -> joint default risk has greater effect on basis in peak episodes, basis is small -> joint default risk even smaller

59 Stochastic Recovery Rates Bonds and CDSs price in stochastic recovery rate Recovery rate depends on number of defaults Simple case: R H if 1 bank defaults, R L if more banks default Call B(R H,R L ) the price of a bond, z(r H,R L ) the price of a CDS

60 Stochastic Recovery Rates Show that: And: = T s=1 B(R H,R L ) = B(R L,R L ) + Y bond (R H, R L ) T δ(0,s 1)(1 P(A i A j )) s 1 z ji (R H,R L ) = s=1 δ(0,s 1)(1 P(A i A j )) s 1 z ji (R L,R L ) Y cds (R H, R L ) with Y bond Y cds

61 Stochastic Recovery Rates Yields and CDS spreads are Rescaled as if R = RL Shifted by a constant Adding Y to both bonds and CDSs does not change the basis The relevant rate is R L

62 Other robustness tests Model Baseline Using swap rates US banks US banks, larger trans Reweight top 5 banks Reweight, decreasing Alternative bond model 2007 Jan 2008 to Bear Bear to Lehman Max P1 Month after Lehman Oct 2008 to April 2009 After April

63 Other robustness tests Model Baseline Using swap rates US banks US banks, larger trans Reweight top 5 banks Reweight, decreasing Alternative bond model 2007 Jan 2008 to Bear Bear to Lehman Max P4 Month after Lehman Oct 2008 to April 2009 After April

64 Other robustness tests Model Baseline Using swap rates US banks US banks, larger trans Reweight top 5 banks Reweight, decreasing Alternative bond model 2007 Jan 2008 to Bear Bear to Lehman Min P1 Month after Lehman Oct 2008 to April 2009 After April

65 Other robustness tests - derivations Alternative Pricing Model Using Swap Rates Different weighting in CDS Different currencies TRACE, larger transactions

66 Robustness: Pricing model Hazard rate deterministic but not constant: h t+s = (1 ρ t )h t + ρ t h t+s 1 CDS: assume joint default risk inherits ρ t and h t /h t from reference entity Approximate around h t = 0

67 Robustness: Swap rates Interest Rate Swaps contain counterparty risk are not indexed to a risk-free short rate Swap rates are higher than Treasuries -> lower systemic risk However, partly offset by calibrated liquidity process

68 Robustness: Weighting scheme If not all dealers post quotes every day, observed average will overrepresent more active banks Assume CDS spread is: [ ( )] [ T z i = P(A i ) (1 S) w j P(A i A j ) s=t+1 δ(t,s) ] (1 R) [ i j T 1 s=t δ(t,s) ] with w j 1 N 1

69 Robustness: Weighting scheme Obtain list of top 5 counterparties by trade count Two schemes (call w the weight of banks 6-15): 1. 5w,5w,5w,5w,5w 2. 10w,8w,6w,4w,2w

70 Robustness: Currencies Bonds and CDSs denominated in different currencies What assumptions do we need to mix them? Two bonds, same firms, different currencies s = 0 or i, default state e exchange rate m se SDF Joint distribution of s and e f (s,e) = π s f s (e)

71 Robustness: Currencies p $ i = π 0 E[m se s = 0] + Rπ i E[m se s = i] = E[m se ] (1 R)π i E[m se s = i] p E i e 0 = E[e m se ] (1 R)π i E[e m se s = i] t $ = E[m se ] t E e 0 = E[e m se ]

72 Robustness: Currencies From Euro bonds, we obtain P(A i ) = π i E[m se s = i] E[m se ] π i E[m se s = i] E[m se ] So we can mix if: E[e m se s = i] E[m se s = i] = E[e m se] E[m se ]

73 Robustness: Currencies Now take Euro-denominated CDS for i. Counterparty j American. s {i,j,ij,0} z ji e 0 = (1 R)π i E [e m se s = i] + (1 R)Sπ ij E [e m se s = ij] ( ) E [e m se s = i] E [e m se s = ij] = E[e m se ] (1 R)π i + (1 R)Sπ ij E[e m se ] E[e m se ] So: condition is for every s E[e m se s] E[m se s] = E[e m se] E[m se ]

74 Robustness: TRACE, larger transactions Quoted data might have lags and matrix prices Small trades might be less reflective of credit risk Results using only US banks TRACE trades >= $100,000

75 Discussion: Arora et al. Arora, Gandhi and Longstaff (2010) run the regression for bond k: z jk,t = a k,t + bz j,t 1 + e jk,t Find that b is negative but small Default probability of the counterparty little cross-sectional effect on price. First point: The starting point of my paper is the difference between P(A j ) and P(A i A j ) Only P(Ai A j ) is priced in the CDS, not P(A j ) z j,t. Second point: ak,t removes all average counterparty risk This paper is based only on the pricing of average counterparty risk Even if for some reason there is compression of quotes around an average, the estimation goes through as long

76 Discussion: Arora et al. Third point: cross-sectional difference in S (collateralization) might induce lower dispersion of quotes If Sj is different by j the average quote reflectes a weighted average of P(A i A j ) z 1i + z 2i 2 = P(A i ) (1 S 1) 2 P(A i A 1 ) (1 S 2) P(A i A 2 ) 2 = P(A i ) (1 S)[ (1 S 1) (1 S)2 P(A i A 1 )+ (1 S 2) (1 S)2 P(A i A 2 )] where S = S 1+S 2 2 Lower collateral requirement -> lower S -> higher weight Robustness: biggest dealers (Goldman, DB, JPM) safer -> less collateral Smaller dealers (Lehman, Merrill) -> more collateral

77 Explore: ETF binary puts Loss of 10% 2 1 binary put P4 P Loss of 15% Loss of 20%