How Best To Incorporate The Leverage Ratio, LCR and NSFR into XVA?

Transcription

1 How Best To Incorporate The Leverage Ratio, LCR and NSFR into XVA? Risk Minds 2015, Amsterdam Andrew Green

2 Contents 1 Introduction 2 Leverage Ratio 3 LCR 4 5 Conclusion 6 Bibliography

3 Disclaimer Joint work with Chris Kenyon. The views expressed in this presentation are the personal views of the speaker and do not necessarily reflect the views or policies of current or previous employers. Chatham House Rules apply to the reporting on this presentation and the comments of the speaker A. Green / 51

4 Introduction Theoretical Economic Context I Pre-Crisis, 2005 Complete markets Perfect execution Zero profit and loss under all states of the world No funding costs No capital costs Single-curve pricing Spot Risk analysis Post-Crisis, 2015 Funding costs including IM (Burgard and Kjaer 2013; Green and Kenyon 2015) Multi-curve pricing CSA-based pricing Regulatory costs Capital costs present since 2008, only formalized in 2014 (Green, Kenyon, and Dennis 2014) A. Green / 51

5 Introduction Theoretical Economic Context II Lifetime costs largely accepted, but not yet systematically applied Incomplete markets - recognized but limited work on implications (Kenyon and Green 2014a; Kenyon and Green 2014b; Kenyon and Green 2015) A. Green / 51

6 Introduction Existing and Expected Regulations Pillar 1 Capital Liquidity RWA LCR (60%)-17 NSFR +271, 302 by Jan 2018 PruVal +EBA 90%, final RTS Market Risk +265, 305 FRTB Credit Risk Concentration, WWR Leverage Ratio +270, IM/IA +317, Sep Exit Funding Market Price, term funding Default (CCR) &279 SA-CCR, Jan 2017 Large Exposures +283, Jan 2019 Concentration Close-out CVA Changes Admin Model Risk CCP +282, Jan 2017 Early Termination Accounting CVA DVA -214, Jan Operational Risk A. Green / 51

7 Introduction Economics: Lifetime Credit, Funding, Capital, and Tax Break-even price level Market Risk (Basel 2) Trade Price = B-S-M + XVA KVA desk costs CVA/FVA desk costs Institutional Costs Capital Funding Credit CCR RWA (Basel 2) CVA RWA (Basel III) CCP (BCBS 227,253) AVA (Prudent Valuation) FVA / Collateral cost IM/IA (BCBS 261, PruVal) CVA DVA Institutional Costs TVA Tax Tax Tax Tax Black-Scholes-Merton A. Green / 51

8 Introduction Trade pricing operates at different granularities Break-even price level Market Risk (Basel 2) Trade Price = B-S-M + XVA KVA desk costs CVA/FVA desk costs Institutional Costs Capital Funding Credit CCR RWA (Basel 2) CVA RWA (Basel III) CCP (BCBS 227,253) AVA (Prudent Valuation) FVA / Collateral cost IM/IA (BCBS 261, PruVal) CVA DVA Institutional Costs TVA Tax Tax Tax Tax Black-Scholes-Merton A. Green / 51

9 Introduction Capital and Funding at Portfolio Level Regulatory capital contains portfolio level requirements - Leverage Ratio Funding has portfolio level requirements - Liquidity Coverage Ratio Need a portfolio-level XVA model! A. Green / 51

10 Leverage Ratio Leverage Ratio I The Basel III leverage ratio is given by LR = Tier 1 Capital Exposure Measure (1) Tier 1 Capital is as defined in Basel III Exposure Measure On-balance sheet assets including collateral Derivatives Exposure Measure = max(rc Net, 0)+(0.4 A Gross )+(0.6 NGR A Gross ) (2) where RC Net is the net replacement cost, A Gross is the gross add-on and NGR is the net to gross ratio i.e. Exposure as calculated by CEM... A. Green / 51

11 Leverage Ratio Leverage Ratio II The leverage ratio is constrained to be greater than 3%, otherwise restrictions are placed on payments of dividends, bonuses etc. Note that the actual US and EU implementations differ from Basel III. A. Green / 51

12 LCR Liquidity Coverage Ratio I (BCBS ) sets out requirements for the Liquidity Coverage Ratio Key aim is to: Ensure banks have adequate stock of unencumbered high quality liquid assets (HQLA) that can be converted easily into cash in order to meet liquidity needs during a 30 calendar day liquidity stress scenario. Rules phased in over period 1st Jan 2015 to 1st Jan 2019 LCR is defined by LCR = HQLA Net cash outflows over next 30 calendar days (3) By 1st Jan 2019 LCR 100%. The calculation of net cash outflow spans the entire bank: loans and derivatives A. Green / 51

13 LCR Liquidity Coverage Ratio II retail, commercial and wholesale Rules applying to derivatives are specified in (BCBS ) paragraphs , 158, 159 A. Green / 51

14 LCR Liquidity Coverage Ratio III Derivative Net Outflow Net Deriv = derivative cash outflow over next 30 days Associated collateral inflow (if rehypothecable) derivative cash inflow over next 30 days + Associated collateral inflow (if rehypothecable) + Collateral would be posted in 3-notch downgrade + 20% value of non-level 1 posted collateral + 100% non-segregated received collateral above requirement + 100% Contractually due collateral that has not been called + 100% HQLA coll. that can be substituted with non-hqla coll. + Largest absolute net 30-day collateral flow in preceding 24 months (4) A. Green / 51

15 LCR Liquidity Coverage Ratio IV Banks must maintain a buffer of HQLA Cannot be used for any other purpose - unencumbered Must be funded unsecured FVA A. Green / 51

16 Extending Burgard-Kjaer semi-replication I Extend the Burgard-Kjaer (Burgard and Kjaer 2013; Green, Kenyon, and Dennis 2014) semi-replication PDE model to M counterparties j and N assets S i. Include the capital valuation adjustment (KVA) and impact of LCR The dynamics of the underlying assets are given by (for all i and j) ds i =µ Si S i dt + σ Si SdW i (5) dp Cj =r Cj P Cj dt P Cj dj Cj (6) dp l =r l P l dt (1 R l )P l dj B. (7) A. Green / 51

17 Extending Burgard-Kjaer semi-replication II On the default of the issuer, B, and counterparty C j, the value of the derivatives book takes the following values, ˆV (t, S, 1, 0) =g B (M B (V 1,..., V M ), X 1,..., X M ) (8) ˆV (t, S, 0, J) =g CJ (M CJ, X ) + ˆV J (t, S). (9) where ˆV J (t, S) is the value of the derivative portfolio after the default of counterparty j, excluding the positions with counterparty J. As usual the g functions allow different close-out conditions to be considered and the usual close-out assumption is that, M g B = (V j X j ) + + R B (V j X j ) + X j (10) g CJ =R CJ (V J X J ) + + (V J X J ) + X J, (11) i=1 A. Green / 51

18 Extending Burgard-Kjaer semi-replication III The generalized funding condition becomes, ˆV X j + L B + α 1 P 1 + α 2 P 2 φk = 0, (12) i=1 where, φk represents the use of capital to offset derivative funding and L B (t) is the liquidity buffer generated by the LCR Here the funding condition spans the whole derivative book. The cash accounts grow at the following rates, prior to rebalancing, with one cash account per counterparty bond, one per stock and one per collateral account (for all i and j), d β Si =δ i (γ Si q Si )S i dt (13) d β Cj = α Cj q Cj P Cj dt (14) d X j = r Xj X j dt. (15) A. Green / 51

19 Extending Burgard-Kjaer semi-replication IV K is the capital requirement for the replicating portfolio and the derivative portfolio. The change in the cash account associated with the capital position is, d β K = γ K (K, t)kdt (16) As is discussed above, the capital is effectively borrowed from shareholders to fulfill the regulatory requirement, for which the shareholders are paid a yield, γ K. There is no term in dj B as the capital is assumed to be part of the recovery rate R B. A. Green / 51

20 Extending Burgard-Kjaer semi-replication V Using multi-dimensional Itô s lemma, the change in the value of the derivative portfolio is given by, d ˆV = ˆV t dt + 1 N N 2 ˆV σ a σ b S a S b dt (17) 2 S a S b + N a=1 a=1 b=1 ˆV S a ds a + ˆV B dj B + ˆV Cj dj Cj. We assume that the portfolio is self-financing so the change in value of Π is given by N N dπ = δ a ds a + δ a (γ Sa q Sa )S a dt + α 1 dp 1 + α 2 dp 2 + α Cj dp Cj a=1 a=1 (18) A. Green / 51 α Cj q Cj P Cj dt r Xj X j dt γ K Kdt.

21 Extending Burgard-Kjaer semi-replication VI Hence the combined portfolio of derivative positions and replicating portfolio is given by, [ d ˆV + dπ = ˆV t + 1 N N 2 σ aσ b S ˆV N as b + δ a(γ Sa q Sa )S a 2 S a S b a=1 b=1 a=1 + α 1 r 1 P 1 + α 2 r 2 P 2 α Cj q Cj P Cj + α Cj r Cj P Cj ] r Xj X j γ K K dt ] + [ ˆV B α 1 (1 R 1 )P 1 α 2 (1 R 2 )P 2 + [ ˆV ] Cj α Cj P Cj dj Cj ( ) N + δ a + ˆV ds a. (19) S a=1 a A. Green / 51

22 Extending Burgard-Kjaer semi-replication VII Assuming replication of the derivative by the hedging portfolio, except at the default of the issuer gives, d ˆV + dπ = 0. (20) To eliminate the remaining sources of risk, the δ a and α Cj using, are set δ a = ˆV S a (21) α Cj P Cj =g Cj + ˆV j ˆV. (22) A. Green / 51

23 Extending Burgard-Kjaer semi-replication VIII Applying these expressions gives the PDE for ˆV, ˆV t +1 2 N a=1 b=1 N 2 ˆV σ a σ b S a S b S a S b + α 1 r 1 P 1 + α 2 r 2 P 2 + N a=1 ˆV S a (γ Sa q Sa )S a [g Cj + ˆV j ˆV ](r cj q Cj ) r Xj X j γ K K = 0 ˆV (T, S) = H(S). (23) A. Green / 51

24 Extending Burgard-Kjaer semi-replication IX The derivative funding equation (12) can be used to give, α 1 r 1 P 1 + α 2 r 2 P 2 = r X j r ˆV + ɛ h λ B + λ B [g B ˆV ] + rφk rl B, i=1 (24) where ɛ h = ˆV B α 1 (1 R 1 )P 1 α 2 (1 R 2 )P 2, (25) is the hedging error on issuer default. A. Green / 51

25 Extending Burgard-Kjaer semi-replication X Hence the PDE becomes, ˆV t +1 2 N a=1 b=1 N 2 ˆV σ a σ b S a S b S a S b r ˆV + ɛ h λ B + λ B [g B ˆV ] + N a=1 ˆV S a (γ Sa q Sa )S a (26) [g Cj + ˆV j ˆV ]λ Cj s Xj X j γ K K + rφk rl B = 0 ˆV (T, S) = H(S), (27) where λ Cj = r Cj q Cj. A. Green / 51

26 Extending Burgard-Kjaer semi-replication XI To proceed we now introduce the usual ansatz, ˆV = V + U = V j + U, (28) where we know that all the V j s satisfy the multi-asset Black-Scholes PDE, V j t a=1 b=1 N N 2 V j N σ a σ b S a S b S a S b a=1 V j S a (γ Sa q Sa )S a rv j = 0. (29) A. Green / 51

27 Extending Burgard-Kjaer semi-replication XII Hence we can write a PDE for U, U t +1 2 N a=1 b=1 N 2 U σ a σ b S a S b S a S b = ɛ h λ B λ B [g B V ] + N a=1 U S a (γ Sa q Sa )S a (r + λ B )U (30) [g Cj + Kj + ˆV j V U]λ Cj s Xj X j + γ K K rφk + rl B U(T, S) = 0, (31) A. Green / 51

28 Extending Burgard-Kjaer semi-replication XIII This PDE spans all counterparties j so to solve it we would like to be able to separate U into individual contributions from counterparties, for ease of computation, that is we would like to write and hence U j t +1 2 N a=1 b=1 U = U j, (32) N 2 U j σ a σ b S a S b S a S b (r + λ B + λ Cj )U j N a=1 = ɛ hj λ B λ B [g Bj V j ] [g Cj V j ]λ Cj U j S a (γ Sa q Sa )S a + s Xj X j + γ K K j rφk j + rl Bj U j (T, S) = 0. (33) A. Green / 51

29 Extending Burgard-Kjaer semi-replication XIV In order to be able to do this, all the terms on the right hand side of the PDE must also be able to be expressed at counterparty level, either by construction or through a valid allocation process. A. Green / 51

30 Separating the Issuer Terms I To separate the issuer terms we need to separate the hedging error per counterparty, that is we define ɛ hj, that satisfies For this to hold, ɛ h = ɛ hj. (34) ɛ hj = ˆV Bj α 1j (1 R 1 )P 1 α 2j (1 R 2 )P 2 (35) =g Bj V j U j α 1j (1 R 1 )P 1 α 2j (1 R 2 )P 2. A. Green / 51

31 Separating the Issuer Terms II The issuer bond positions can easily be attributed to counterparty level so that α 1 = α 2 = α 1j (36) α 2j (37) and from equation (10) that under standard closeout conditions g B = g Bj. (38) In general this must be true as g B is simply the total close-out claim made against the issuer on their default. A. Green / 51

32 Separating the Issuer Terms III The relationship between ɛ h and the counterparty level ɛ hj is now clear, ɛ h =g B ˆV α 1 (1 R 1 )P 1 α 2 (1 R 2 )P 2 (39) = g Bj V j U j (α 1j (1 R 1 )P 1 + α 2j (1 R 2 )P 2 ) = ɛ hj. which relies on the fact that U j itself is separable, which will only be true if the capital terms are also separable. A. Green / 51

33 Separating the Capital Terms The key question then is whether the capital term on the right hand side of the PDE can be separated by counterparty. Writing ν (γ K rφ)k, (40) we see that γ K and K will determine if this separation is possible. A. Green / 51

34 Capital and Capital Attribution I Ideally we would like to be able to write K = K j, (41) that is we would like to allocate all the capital to individual counterparty positions. Capital allocation approaches are widely represented in the literature with considerable emphasis on Euler Allocation (Tasche 2008) although a number of other approaches have also been proposed (Balog 2010). In general a capital allocation will always be possible, although Euler Allocation does not work for all elements of regulatory capital. A. Green / 51

35 Capital and Capital Attribution II Earlier we considered three components of capital, market risk, counterparty credit risk and CVA in the counterparty level KVA model. Here we add the impact of the leverage ratio which is a capital measure spanning most bank positions. A. Green / 51

36 Euler Allocation A function is a homogenious function or order 1 if we can write f (tα) = tf (α). (42) According to Euler s homogeneous function theorem a homogeneous function f of order m can be written mf (α) = n i=1 α i f α i. (43) Hence a homogeneous function of order 1 can be allocated. A. Green / 51

37 Market Risk I The standardized method is a formula based approach as can be seen from the discussion above on the standardised approach for interest rates. It is clear that the interest rate methodology involves linear operations on netted position information. Hence, given the operations are linear then Euler Allocation can be applied. The same is true for the remaining risk categories with a small number of exceptions (although even though these are still homogeneous functions of order one). Hence in general Euler Allocation can be used with the standardized method. The Internal Model Method uses a Value-at-Risk (VAR) approach to estimate the regulatory capital requirement and under the fundamental review of the trading book, expected shortfall will be used. A. Green / 51

38 Market Risk II Euler allocation can be used with both methods (Tasche 2008). A. Green / 51

39 Counterparty Credit Risk (CCR) Counterparty Credit Risk capital is calculated at netting set level under the current Basel III regulatory framework (BCBS ) for those institutions with IMM approval and for those using the Current Exposure Method and Standardized method. Under the proposed revised standardized approach (BCBS ) this remains the case. Capital as calculated under CCR is already defined on a per counterparty basis. A. Green / 51

40 CVA Capital I IMM All calculations are performed on a per counterparty basis. The VAR model is used estimate the capital requirement either by full revaluation of using the regulatory CS01 formula. Standardized As noted earlier, nn the absence of CVA hedging and for large numbers of conterparties this formula is well approximated by a sum over counterparties, KCVA i 2.33 hωi M i EAD total i. (44) 2 In the general case with hedging K CVA is cannot be easily expressed as a sum of counterparty level terms. Furthermore the obvious choice of Euler attribution methodology cannot be applied as equation as the standardized CVA formula is not a homogenous function of order one. A. Green / 51

41 CVA Capital II However, it is possible to define a suitable attribution with the desired properties, ( M i EAD total i K CVAj = which clearly satisfies M (M i EAD total i ) M hedge i B i M hedge i B i ) K CVA, (45) K CVA = K CVAj. (46) A. Green / 51

42 Allocating the Leverage Ratio Ignoring other contributors, the leverage ratio for derivatives is given by M j K j + K LR = M E. (47) j where K is any additional capital required to satisfy the Leverage Ratio and E j is the exposure calculated for counterparty j. Rearranging and imposing the constraint shows that K = max 0.03 E j K j, 0. (48) Hence it is possible to attribute any leverage ratio capital using Euler Attribution However, the size of any leverage ratio capital will be state dependent. j A. Green / 51 j

43 Allocating the LCR Liquidity Buffer Recall equation (4) Only the last item, Largest absolute net 30-day collateral flow in preceding 24 months, is not defined at counterparty level. If this can be allocated to counterparty level then we can write L B = L Bj (49) One possible approach would to allocate the 30-day collateral outflow on a pro-rata basis with the remaining counterparty aligned LCR contributions. A. Green / 51

44 Solving for XVA I RHS of equation (30) can be written as a sum over terms at counterparty level. Hence U can be written as a sum of terms U j that satisfy equations (33). To solve for the U j we apply the Feynman-Kac theorem in the usual way to obtain: U j = CVA j + DVA j + FCA j + COLVA j + KVA j, (50) A. Green / 51

45 Solving for XVA II where T CVA j = t λ Cj (u)e u t (r(s)+λ B (s)+λ Cj (s))ds E t [ V j (u) g Cj (V j (u), X j (u)), V j ] S (u), X j (u)) du (51) T DVA j = λ B (u)e u t (r(s)+λ B (s)+λ Cj (s))ds t ] E t [V j (u) g Bj (V j (u), X j (u)) du (52) T FCA j = λ B (u)e u ] t (r(s)+λ B (s)+λ Cj (s))ds Et [ɛ hj (u) du t T t T COLVA j = t T KVA j = t r(u)e u t (r(s)+λ B (s)+λ Cj (s))ds Et [ LBj (u) ] du (53) s Xj (u)e u t (r(s)+λ B (s)+λ Cj (s))ds Et [ Xj (u) ] du (54) (γ K (K, u) r(u)φ j )e u t (r(s)+λ B (s)+λ Cj (s))ds Et [ Kj (u) ] du. (55) A. Green / 51

46 LCR FVA The LCR has generated an additional funding cost term T r(u)e u t (r(s)+λ B (s)+λ Cj (s))ds E t [L Bj (u)] du (56) t To evaluate we need to calculate the expected size of the liquidity buffer in every Monte Carlo state L Bj will in some cases have terms associated with rating downgrade Could ignore the probability of downgrade and just calculate the buffer based on the outflow associated with a downgrade of 3 notches from the initial rating However, to capture the full effect a rating transition model is required A. Green / 51

47 Conclusion Conclusion Introduced the Leverage Ratio and Liquidity Coverage Ratio Presented a unified model for valuation adjustments including Capital and LCR at Portfolio Level Provided an allocation mechanism that allows the total XVA to be calculated at netting set level A. Green / 51

48 Bibliography Balog, D. (2010). Risk based capital allocation. In Proceedings of FIKUSZ 10 Symposium for Young Researchers, pp Keleti Faculty of Business and Management, buda University. BCBS-189 (2011). Basel III: A global regulatory framework for more resilient banks and banking systems. Basel Committee for Bank Supervision. BCBS-238 (2013). Basel III: The Liquidity Coverage Ratio and liquidity risk monitoring tools. Basel Committee for Bank Supervision. BCBS-279 (2014). The standardised approach for measuring counterparty credit risk exposures. Basel Committee for Bank Supervision. Burgard, C. and M. Kjaer (2013). Funding Strategies, Funding Costs. Risk 26(12). Green, A. and C. Kenyon (2015). MVA: Initial Margin Valuation Adjustment by Replication and Regression. Risk 28(5). Green, A., C. Kenyon, and C. R. Dennis (2014). KVA: Capital Valuation Adjustment by Replication. Risk 27(12). Kenyon, C. and A. Green (2014a, September). Regulatory costs break risk neutrality. Risk 27. Kenyon, C. and A. Green (2014b, October). Regulatory costs remain. Risk 27. A. Green / 51

49 Bibliography Kenyon, C. and A. Green (2015, February). Warehousing credit risk: pricing, capital and tax. Risk 28. Tasche, D. (2008). Capital allocation to business units and sub-portfolios: the euler principle. In In: Pillar II in the New Basel Accord: The Challenge of Economic Capital, pp available at: A. Green / 51

50 Bibliography Thanks for your attention questions? A. Green / 51

51 LR, LCR, NSFR, XVA Bibliography Available Now: A. Green / 51