Collateral flows, funding costs, and counterparty-risk-neutral swap rates

Transcription

1 1/32 Collateral flows, funding costs, and counterparty-risk-neutral swap rates Enrico Biffis Imperial College London BASED ON JOINT WORKS WITH Damiano Brigo (King s College) Lorenzo Pitotti (Imperial & Algorithmics) AND David Blake (Cass Business School) Ariel Sun (Imperial & RMS) HIPERFIT Workshop, Copenhagen, December 1, 2011

2 2/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

3 3/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

4 /32 Overview Global financial crisis Counterparty risk and counterparty risk mitigation tools matter collateral rules and funding costs integral part of the transaction implications for pricing, hedging, market-to-market accounting Multicurve modelling LIBOR, EURIBOR, EONIA, EUREPO? New regulation (Dodd-Frank, EMIR) clearing, netting, collateralization collateral quality, segregation, re-hypothecation replacement cost, close-out conventions Valuation challenges (e.g., Brigo s Counterparty Risk FAQ, Nov 2011) Credit/Debit Valuation Adjustment (CVA/DVA)

5 /32 Questions Consistent valuation with counterparty risk and liquidity risk Swap rates endogenize collateral flows and funding/opportunity costs Root finding, stochastic approximation algorithms. Impact of different collateral rules / conventions Partial vs. full collateralization Symmetric vs. asymmetric collateral rules Segregation vs. rehypothecation Quantifying the cost of collateralization Benchmark: interest-rate swaps (IRS) market Case study: bespoke longevity swaps

6 6/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

7 7/32 Common pitfalls Interest-rate swaps (IRS) almost every IRS bilaterally collateralized cash collateral in over 90% of the cases Party A (fixed payer) p Party B (fixed receiver) L T

8 /32 Common pitfalls Interest-rate swaps (IRS) almost every IRS bilaterally collateralized cash collateral in over 90% of the cases Duffie/Singleton valuation formula: unitary notional, single payment LIBOR default spread, λ ( V 0 = E [exp Q T 0 (r t +λ t )dt ) ] (L T p) Exceptions: He (2001) and Collin-Dufresne/Solnik (2001) set λ = 0.

9 /32 Bilateral default risk Allow for credit quality of counterparties (Duffie/Huang, 1997) party A pays fixed, party B pays floating default intensities λ A t, λ B t, and recovery rates R A,R B fixed payer s viewpoint V 0 = E Q [exp Λ t := ( T 0 (r t +Λ t )dt { (1 R A )λ A t if V t < 0 (1 R B )λ B t if V t 0 ) (LT p d)]

10 /32 Bilateral default risk Allow for credit quality of counterparties (Duffie/Huang, 1997) party A pays fixed, party B pays floating default intensities λ A t, λ B t, and recovery rates R A,R B fixed payer s viewpoint V 0 = E Q [exp Λ t := ( T 0 (r t +Λ t )dt { (1 R A )λ A t if V t < 0 (1 R B )λ B t if V t 0 ) (LT p d)] full collateralization, R A = R B = 1 ( V 0 = E [exp Q T 0 ) (LT r t dt p d)]...default-free, risk-neutral valuation formula...

11 /32 Collateralization Collateral fractions (c h t) t 0 (ITM), (c p t) t 0 (OTM) [hedger s viewpoint] c h tv t 1 {Vt >0} cash held, c p tv t 1 {Vt <0} cash posted funding cost / opportunity cost / capital relief δt h net gain from holding collateral (r rebated) δ p t net cost of posting collateral (r rebated)

12 9/32 Collateralization Collateral fractions (c h t) t 0 (ITM), (c p t) t 0 (OTM) [hedger s viewpoint] c h tv t 1 {Vt >0} cash held, c p tv t 1 {Vt <0} cash posted funding cost / opportunity cost / capital relief δ h t net gain from holding collateral (r rebated) δ p t net cost of posting collateral (r rebated) swap s market value V 0 = E Q [exp Γ t := ( T 0 (r t +Γ t )dt ) { (1 c p t) + λ A t δtc p p t if V t < 0 (1 c h t) + λ B t δt h c h t if V t 0 (S T p c ) ]

13 9/32 Collateralization Collateral fractions (c h t) t 0 (ITM), (c p t) t 0 (OTM) [hedger s viewpoint] c h tv t 1 {Vt >0} cash held, c p tv t 1 {Vt <0} cash posted funding cost / opportunity cost / capital relief δ h t net gain from holding collateral (r rebated) δ p t net cost of posting collateral (r rebated) swap s market value V 0 = E Q [exp Γ t := ( T 0 (r t +Γ t )dt ) { (1 c p t) + λ A t δtc p p t if V t < 0 (1 c h t) + λ B t δt h c h t if V t 0 (S T p c ) ] Full collateralization (c p,h = 1), symmetric costs/spreads (δ,λ): r t +Γ t = r t δ t

14 10/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

15 11/32 Swap rates Swap rate p c from V 0 = 0 p c = E Q [L T ]+ Cov Q( ( exp ) ) T 0 (r t +Γ t )dt,l T ( E [exp Q )] T 0 (r t +Γ t )dt

16 1/32 Swap rates Swap rate p c from V 0 = 0 p c = E Q [L T ]+ Cov Q( ( exp ) ) T 0 (r t +Γ t )dt,l T ( E [exp Q )] T 0 (r t +Γ t )dt Root finding (V 0 (p c ) = 0) and stochastic approximations Robbins-Monro, Polyak-Ruppert averaging Main issue is unbiased estimator of V 0 (p) when using American MC

17 11/32 Swap rates Swap rate p c from V 0 = 0 Cov Q( ( exp ) ) T p c = E Q 0 (r t +Γ t (c h,p ))dt,l T [L T ]+ ( E [exp Q )] T 0 (r t +Γ t (c h,p ))dt Root finding (V 0 (p c ) = 0) and stochastic approximations Robbins-Monro, Polyak-Ruppert averaging Main issue is unbiased estimator of V 0 (p) when using American MC Collateral rule examples collateral thresholds based on credit ratings, CDS spreads, etc. c p t = c h t = 1 (full collateralization) c p t = α, c h t = β, with α,β [0,1] c p t = 1 {Lt β(t)}, c h t = 1 {Lt α(t)} {λ B t γ}, with α( ) > β( ), γ 0

18 11/32 Swap rates Swap rate p c from V 0 = 0 Cov Q( ( exp ) ) T p c = E Q 0 (r t +Γ t (V t (p c )))dt,l T [L T ]+ ( E [exp Q )] T 0 (r t +Γ t (V t (p c )))dt Root finding (V 0 (p c ) = 0) and stochastic approximations Robbins-Monro, Polyak-Ruppert averaging Main issue is unbiased estimator of V 0 (p) when using American MC Collateral rule examples collateral thresholds based on credit ratings, CDS spreads, etc. c p t = c h t = 1 (full collateralization) c p t = α, c h t = β, with α,β [0,1] c p t = 1 {Lt β(t)}, c h t = 1 {Lt α(t)} {λ B t γ}, with α( ) > β( ), γ 0 c p t = 1 {Vt(p c ) v} and c h t = 1 {Vt(p c ) v}, with v < v

19 12/32 Symmetric collateralization with re-hypothecation 0.05 Symmetric case with re hypothecation Swap Rate Collateralization fraction (c h =c p )

20 13/32 (A)symmetric collateralization with segregation Asymmetric collateralization and segregation: "A B" Spread Swap Rate Collateralization B Collateralization A

21 14/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

22 15/32 Longevity swaps Date Hedger Size Term (yrs) Type Interm./supplier Jan 08 Lucida Not disclosed 10 indexed JPM ILS funds Jul 2008 Canada Life GBP 500m 40 bespoke JPM ILS funds Feb 2009 Abbey Life GBP 1.5bn run-off bespoke DB ILS funds Partner Re Mar 2009 Aviva GBP 475m 10 bespoke RBS Jun 2009 Babcock GBP 750m 50 bespoke Credit Suisse International Pacific Life Re Jul 2009 RSA GBP 1.9bn run-off bespoke GS (Rothesay Life) Dec 2009 Berkshire GBP 750m run-off bespoke Swiss Re Council Feb 2010 BMW UK GBP 3bn run-off bespoke DB Paternoster Dec 2010 Swiss Re USD 50m 8 indexed ILS funds (Kortis bond) Feb 2011 Pall GBP 70m 10 indexed JPM Pension Fund

23 16/32 Bespoke longevity swaps Stylized example: single payment at time T > 0 notional n > 0, fixed payment p (0,1) variable payment S T (realized survival rate) n p Party A (hedger) Party B (hedge supplier) n S T

24 16/32 Bespoke longevity swaps Stylized example: single payment at time T > 0 notional n > 0, fixed payment p (0,1) variable payment S T (realized survival rate) n p Party A (hedger) Party B (hedge supplier) n S T Swap value (hedger s viewpoint) ( V 0 = ne [exp Q T 0 ) ] r t dt (S T p)

25 16/32 Bespoke longevity swaps Stylized example: single payment at time T > 0 notional n > 0, fixed payment p (0,1) variable payment S T (realized survival rate) n p Party A (hedger) Party B (hedge supplier) Longevity swap rate p = E Q [S T ]+ n S T Cov Q( ( exp ) ) T 0 r tdt,s T ( E [exp Q )] T 0 r tdt

26 16/32 Bespoke longevity swaps Stylized example: single payment at time T > 0 notional n > 0, fixed payment p (0,1) variable payment S T (realized survival rate) n p Party A (hedger) Party B (hedge supplier) n S T Longevity swap rate (r, S uncorrelated) p = E Q [S T ]+0

27 16/32 Bespoke longevity swaps Stylized example: single payment at time T > 0 notional n > 0, fixed payment p (0,1) variable payment S T (realized survival rate) n p Party A (hedger) Party B (hedge supplier) n S T Longevity swap rate (r, S uncorrelated) p = E Q [S T ]+0 Useful baseline case p = E P [S T ] (best estimate).

28 17/32 Cashflows and marking-to-market MTM baseline MTM 50 bps MTM 100 bps Swap cashflow

29 18/32 Longevity swap rates LC forecast year

30 19/32 Hedge supplier s credit deterioration 0 5% 10% 15% 20% 25% 30% 35 % 50 bps: MTM (%) 100 bps: MTM (%) 50 bps: MTM (% of P&L) 100 bps: MTM (% of P&L) 40 %

31 0/32 Fully fledged calibration Building blocks two-factor short rate model TED spread for λ B λ A = λ B +, > 0 net cost of collateral in IRS market (calibration of Johannes/Sundaresan, 2007) Lee-Carter mortality model Two approaches to collateral costs δ h,δ p funding costs associated with collateral flows simulate Solvency II capital charges (1-year 99.5% VaR) accruing from representative longevity-linked liability; opportunity cost of (say) 6% + LIBOR incurred on capital charges

32 21/32 Fully fledged calibration Q-dynamics of state variable process X dxt 1 = ( k 1(Xt 2 Xt) η 1 1) dt+σ 1dWt 1 dxt 2 = ( k 2(θ 2 Xt) η 2 2) dt+σ 2dWt 2 dxt 3 = ( κ 3(θ 3 Xt)+κ 3 3,1(Xt 1 θ 2)+κ 3,4(Xt 4 ) θ 4) η 3 dt+σ3dwt 3 dxt 4 = ( κ 4(θ 4 Xt)+κ 4 4,1(Xt 1 θ 2)+κ 4,2(Xt 2 ) θ 2) η 4 dt+σ4dwt 4 dxt 5 = ( κ 5(θ 5 Xt)+κ 5 5,1(Xt 1 θ 2)+κ 5,2(Xt 2 θ 2)+κ 5,3(Xt 3 θ 3) +κ 5,4(Xt 4 θ 4)+κ 5,6(Xt 6 E 0[Xt]) η 6 ) 5 dt+σ5dwt 5 dxt 6 = ( A(t)+B(t)(Xt 6 a(t)) ) dt+σ 6(t)dWt 6 r = X 1, mean reverting to random target X 2 λ B = X 3, TED spread X 4, net cost of collateral in IRS markets (Johannes/Sundaresan, 2007) X 5 net cost of collateral for longevity risk exposures X 6 continuous time version of Lee-Carter model

33 22/32 Parameter estimates Parameter estimates Treasury/IRS market: Johannes/Sundaresan (2007) Mortality: US/UK HMD data Net cost of collateral: i) δ h = δ p = λ A = X (3) +, {0,0.01,0.02} ii) δ h = δ p = X (5) κ η σ UK κ η σ δ K κ η σ σ K κ η σ US κ κ 5, σ δ K κ 3, κ 5, θ σ K κ 4, κ 5, θ κ 3, κ 5, θ κ 4, κ 5, θ ρ 1,

34 23/32 Longevity swap spreads Underlying: 10,000 US males aged 65 at beginning of Term: 25 years. swap spreads (basis points), p c T E P [S T]: Maturity c A = 0 c A = 0 c A = 1 c A = 1 payment c B = 0 c B = 1 c B = 0 c B = 1 (yrs) (bps) (bps) (bps) (bps) λ A,B = λ, δ A,B = δ, δ = λ λ A = λ B +, δ i = λ i, = 100 bps λ A = λ B +, δ i = λ i, = 200 bps

35 Longevity swap margins swap margin and percentiles (%) payment date 4/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles for = 0 (dashed), = 100 bps (solid): no collateral (squares), full collateralization (circles).

36 Longevity swap margins swap margin and percentiles (%) λ A =λ B payment date 4/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles for = 0 (dashed), = 100 bps (solid): no collateral (squares), full collateralization (circles).

37 Longevity swap margins λ A =λ B + no collateral swap margin and percentiles (%) λ A =λ B payment date 4/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles for = 0 (dashed), = 100 bps (solid): no collateral (squares), full collateralization (circles).

38 Longevity swap margins λ A =λ B + no collateral swap margin and percentiles (%) λ A =λ B λ A =λ B + full collateral payment date 4/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles for = 0 (dashed), = 100 bps (solid): no collateral (squares), full collateralization (circles).

39 One-sided vs. two-sided collateralization swap margin and percentiles (%) no collateral full collateral payment date 5/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles. = 100 bps. No collateral (squares) vs. full collateralization: two-sided (circles), one-sided A (stars), one-sided B (diamonds).

40 One-sided vs. two-sided collateralization full (hedge supplier) 0.6 swap margin and percentiles (%) no collateral full collateral payment date 5/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles. = 100 bps. No collateral (squares) vs. full collateralization: two-sided (circles), one-sided A (stars), one-sided B (diamonds).

41 One-sided vs. two-sided collateralization full (hedge supplier) 0.6 swap margin and percentiles (%) no collateral full collateral full (hedger) payment date 5/32 p Funding costs case. Swap margins c E P 1 against Lee-Carter mortality [S T ] improvements quantiles. = 100 bps. No collateral (squares) vs. full collateralization: two-sided (circles), one-sided A (stars), one-sided B (diamonds).

42 6/32 Capital charges approach swap margin and percentiles (%) payment date Opportunity cost case. Swap margins p c T i /p Ti 1 against Lee-Carter mortality improvements quantiles for = 0: no collateral (squares), two-sided full collateralization (circles), one-sided A (stars), one-sided B (diamonds).

43 27/32 Understanding longevity swap rates Two effects at play here longevity risk interest rate risk p c = E Q [S T ]+ Cov Q( ( exp ) ) T 0 (r t +Γ t )dt,s T ( E [exp Q )] T 0 (r t +Γ t )dt

44 27/32 Understanding longevity swap rates Two effects at play here longevity risk interest rate risk Intuition p c = E Q [S T ]+ Cov Q( ( exp ) ) T 0 (r t +Γ t )dt,s T ( E [exp Q )] T 0 (r t +Γ t )dt A receives collateral when S T is high, liability more capital intensive A posts collateral when S T is low, liability less capital intensive

45 7/32 Understanding longevity swap rates Two effects at play here longevity risk interest rate risk Intuition p c = E Q [S T ]+ Cov Q( ( exp ) ) T 0 (r t +Γ t )dt,s T ( E [exp Q )] T 0 (r t +Γ t )dt If A is ITM, collateral higher in low interest rate environments If A is OTM, collateral lower in higher interest rate environments

46 28/32 Comparison with IRS market IRS spreads: difference betweeen futures price (δ = r) and swap rate of collateralized IRS of corresponding maturity IRS longevity Maturity c A = 0 c A = 1 c A = 1 c A = 0 c A = 1 c A = 1 payment c B = 1 c B = 0 c B = 1 c B = 1 c B = 0 c B = 1 (yrs) (bps) (bps) (bps) (bps) (bps) (bps) λ A,B = λ, δ A,B = δ, δ = λ λ A = λ B +, δ i = λ i, = 100 bps

47 29/32 Agenda 1 Overview 2 Consistent valuation of swaps 3 Equilibrium swap rates 4 Cost of collateralization: case study 5 Conclusion

48 0/32 Conclusion Swap valuation with counterparty risk and liquidity risk Swap rates endogenize collateral flows generated by MTM procedure and associated funding/opportunity costs Root finding and stochastic approximation algorithms Even standard collateral rules may pose significant challenges Impact of collateral rules / conventions Partial vs. full collateralization Symmetric vs. asymmetric collateral rules Segregation vs. rehypothecation Funding costs vs. opportunity costs Quantifying the cost of collateralization The case of IRSs and bespoke longevity swaps Sign and magnitude of costs are far from obvious

49 31/32 End THANK YOU

50 2/32 Some references S. Assefa, T.R. Bielecki, S. Crépey and M. Jeanblanc (2010), CVA computation for counterparty risk assessment in credit portfolios. In T. Bielecki, D. Brigo and F. Patras (eds.), Recent Advancements in the Theory and Practice of Credit Derivatives, Bloomberg Press. E. Biffis, D. Brigo, L. Pitotti (2011), Collateral flows, funding costs, and counterparty-risk-neutral swap rates. E. Biffis, D. Blake, L. Pitotti, A. Sun (2011), The cost of counterparty risk and collateralization in longevity swaps, WP on SSRN. D. Brigo (2011), Brigo s Counterparty Risk FAQ, WP on ArXiv. D. Brigo and A. Capponi (2009). Bilateral counterparty risk valuation with stochastic dynamical models and application to CDSs. WP, Kings College London. D. Brigo, A. Pallavicini and V. Papatheodorou (2011). Collateral margining in arbitragefree counterparty valuation adjustment including re-hypothecation and netting. WP, Kings College London. P. Collin-Dufresne and B. Solnik (2001), On the term structure of default premia in the swap and LIBOR markets, Journal of Finance. D. Duffie and M. Huang (1997), Swap rates and credit quality, Journal of Finance. D. Duffie and K. Singleton (1997), An econometric model of the term structure of interest rate swap yields, Journal of Finance. H. He (2001), Modeling term structures of swap spreads, WP, Yale. H.J. Kushner and G.G. Yin (2003), Stochastic Approximation and Recursive Algorithms and Applications, Springer. M. Johannes and S. Sundaresan (2007), The impact of collateralization on swap rates, Journal of Finance.