how best to incorporate the leverage ratio, lcr and nsfr...

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

Risk Minds 2015, Amsterdam

Andrew Green

Contents

1 Introduction

2 Leverage Ratio

3 LCR

4 Extending XVA to include Leverage Ratio & LCR

5 Conclusion

6 Bibliography

LR, LCR, NSFR, XVA

Disclaimer

Joint work with Chris Kenyon.

The views expressed in this presentation are the personal views ofthe speaker and do not necessarily reflect the views or policies ofcurrent or previous employers.

Chatham House Rules apply to the reporting on this presentationand the comments of the speaker

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Introduction

Theoretical Economic Context I

Pre-Crisis, 2005

Complete marketsPerfect executionZero profit and loss under all states of the worldNo funding costsNo capital costsSingle-curve pricingSpot Risk analysis

Post-Crisis, 2015

Funding costs including IM (Burgard and Kjaer 2013; Green andKenyon 2015)Multi-curve pricingCSA-based pricingRegulatory costs

Capital costs present since 2008, only formalized in 2014 (Green,Kenyon, and Dennis 2014)

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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 andGreen 2015)

A. Green 10.12.2015 5 / 51

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Introduction

Existing and Expected Regulations

Pillar 1 Capital

RWA

PruVal +EBA 90%, final RTS

Market Price, term funding

Close-out

Model Risk

Accounting CVA

Exit Funding

Concentration

Admin

Early Termination

Operational Risk

Market Risk +265, 305 FRTB

Credit Risk

Default (CCR) &279 SA-CCR,

Jan 2017

CVA Changes

CCP +282, Jan 2017

DVA -214, Jan 2014-2018

Concentration, WWR

Large Exposures +283, Jan 2019

Leverage Ratio +270, 2017-18

Liquidity

LCR +272 2015(60%)-17

NSFR +271, 302 by Jan 2018

IM/IA +317, Sep 2016-20

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Introduction

Economics: Lifetime Credit, Funding, Capital, and Tax

Capital

CCR RWA (Basel 2)

CVA RWA (Basel III)

CCP (BCBS 227,253)

Market Risk (Basel 2)

AVA (Prudent Valuation)

Institutional Costs

KVA desk costs

Credit CVA

DVA

Funding FVA / Collateral cost

IM/IA (BCBS 261, PruVal) CVA/FVA desk costs

Break-even price level

Institutional Costs

Black-Scholes-Merton

Trade

Price

= B-S-M

+ XV

A

Tax

Tax

Tax

Tax

TVA

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Introduction

Trade pricing operates at different granularities

Capital

CCR RWA (Basel 2)

CVA RWA (Basel III)

CCP (BCBS 227,253)

Market Risk (Basel 2)

AVA (Prudent Valuation)

Institutional Costs

KVA desk costs

Credit CVA

DVA

Funding FVA / Collateral cost

IM/IA (BCBS 261, PruVal) CVA/FVA desk costs

Break-even price level

Institutional Costs

Black-Scholes-Merton

Trade

Price

= B-S-M

+ XV

A

Tax

Tax

Tax

Tax

TVA

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Introduction

Capital and Funding at Portfolio Level

Regulatory capital contains portfolio level requirements - LeverageRatio

Funding has portfolio level requirements - Liquidity Coverage Ratio

⇒ Need a portfolio-level XVA model!

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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 collateralDerivatives

Exposure Measure = max(RCNet, 0)+(0.4×AGross)+(0.6×NGR×AGross)(2)

where RCNet is the net replacement cost, AGross is the gross add-onand NGR is the net to gross ratioi.e. Exposure as calculated by CEM. . .

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Leverage Ratio

Leverage Ratio II

The leverage ratio is constrained to be greater than 3%, otherwiserestrictions are placed on payments of dividends, bonuses etc.

Note that the actual US and EU implementations differ from BaselIII.

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LCR

Liquidity Coverage Ratio I

(BCBS-238 2013) sets out requirements for the Liquidity CoverageRatio

Key aim is to:

Ensure banks have adequate stock of unencumbered high qualityliquid assets (HQLA) that can be converted easily into cash in orderto meet liquidity needs during a 30 calendar day liquidity stressscenario.

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

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LCR

Liquidity Coverage Ratio II

retail, commercial and wholesale

Rules applying to derivatives are specified in (BCBS-238 2013)paragraphs 116 - 123, 158, 159

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LCR

Liquidity Coverage Ratio III

Derivative Net Outflow

NetDeriv =∑

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)

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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

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Extending XVA to include Leverage Ratio & LCR

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 jand N assets Si .

Include the capital valuation adjustment (KVA) and impact of LCR

The dynamics of the underlying assets are given by (for all i and j)

dSi =µSi Sidt + σSi SdWi (5)

dPCj =rCj PCj dt − PCj dJCj (6)

dPl =rlPldt − (1− Rl)PldJB . (7)

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Extending Burgard-Kjaer semi-replication II

On the default of the issuer, B, and counterparty Cj , the value ofthe derivatives book takes the following values,

V (t,S , 1, 0) =gB(MB(V1, . . . ,VM),X1, . . . ,XM) (8)

V (t,S , 0, J) =gCJ(MCJ

,X ) + V−J(t,S). (9)

where V−J(t,S) is the value of the derivative portfolio after thedefault of counterparty j , excluding the positions with counterpartyJ.

As usual the g functions allow different close-out conditions to beconsidered and the usual close-out assumption is that,

gB =M∑j=1

(Vj − Xj)+ + RB

M∑j=1

(Vj − Xj)− +

M∑i=1

Xj (10)

gCJ=RCJ

(VJ − XJ)+ + (VJ − XJ)− + XJ , (11)

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Extending Burgard-Kjaer semi-replication III

The generalized funding condition becomes,

V −M∑i=1

Xj + LB + α1P1 + α2P2 − φK = 0, (12)

where, φK represents the use of capital to offset derivative fundingand LB(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 oneper collateral account (for all i and j),

d βSi =δi (γSi − qSi )Sidt (13)

d βCj =− αCj qCj PCj dt (14)

dXj =− rXj Xjdt. (15)

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Extending Burgard-Kjaer semi-replication IV

K is the capital requirement for the replicating portfolio and thederivative portfolio.

The change in the cash account associated with the capital positionis,

d βK = −γK (K , t)Kdt (16)

As is discussed above, the capital is effectively borrowed fromshareholders to fulfill the regulatory requirement, for which theshareholders are paid a yield, γK .

There is no term in dJB as the capital is assumed to be part of therecovery rate RB .

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Extending Burgard-Kjaer semi-replication V

Using multi-dimensional Ito’s lemma, the change in the value of thederivative portfolio is given by,

dV =∂V

∂tdt +

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2V

∂Sa∂Sbdt (17)

+N∑

a=1

∂V

∂SadSa + ∆VBdJB +

M∑j=1

∆VCj dJCj .

We assume that the portfolio is self-financing so the change in valueof Π is given by

dΠ =N∑

a=1

δadSa +N∑

a=1

δa(γSa − qSa)Sadt + α1dP1 + α2dP2 +M∑j=1

αCj dPCj

−M∑j=1

αCj qCj PCj dt −M∑j=1

rXj Xjdt − γKKdt.

(18)A. Green 10.12.2015 20 / 51

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Extending Burgard-Kjaer semi-replication VI

Hence the combined portfolio of derivative positions and replicatingportfolio is given by,

dV + dΠ =

[∂V

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2V

∂Sa∂Sb+

N∑a=1

δa(γSa − qSa )Sa

+ α1r1P1 + α2r2P2 −M∑j=1

αCjqCj

PCj+

M∑j=1

αCjrCj

PCj

−M∑j=1

rXjXj − γKK

]dt

+[∆VB − α1(1 − R1)P1 − α2(1 − R2)P2

]+

M∑j=1

[∆VCj

− αCjPCj

]dJCj

+N∑

a=1

(δa +

∂V

∂Sa

)dSa. (19)

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Extending Burgard-Kjaer semi-replication VII

Assuming replication of the derivative by the hedging portfolio,except at the default of the issuer gives,

dV + dΠ = 0. (20)

To eliminate the remaining sources of risk, the δa and αCj are setusing,

δa =− ∂V

∂Sa(21)

αCj PCj =gCj + V−j − V . (22)

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Extending Burgard-Kjaer semi-replication VIII

Applying these expressions gives the PDE for V ,

∂V

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2V

∂Sa∂Sb−

N∑a=1

∂V

∂Sa(γSa − qSa)Sa

+ α1r1P1 + α2r2P2 +M∑j=1

[gCj + V−j − V ](rcj − qCj )

−M∑j=1

rXj Xj − γKK = 0

V (T ,S) = H(S). (23)

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Extending Burgard-Kjaer semi-replication IX

The derivative funding equation (12) can be used to give,

α1r1P1 +α2r2P2 = rM∑i=1

Xj − r V + εhλB +λB [gB − V ] + rφK − rLB ,

(24)where

εh = ∆VB − α1(1− R1)P1 − α2(1− R2)P2, (25)

is the hedging error on issuer default.

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Extending Burgard-Kjaer semi-replication X

Hence the PDE becomes,

∂V

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2V

∂Sa∂Sb−

N∑a=1

∂V

∂Sa(γSa − qSa)Sa (26)

− r V + εhλB + λB [gB − V ] +M∑j=1

[gCj + V−j − V ]λCj

−M∑j=1

sXj Xj − γKK + rφK − rLB = 0

V (T ,S) = H(S), (27)

where λCj = rCj − qCj .

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Extending Burgard-Kjaer semi-replication XI

To proceed we now introduce the usual ansatz,

V = V + U =M∑j=1

Vj + U, (28)

where we know that all the Vj ’s satisfy the multi-asset Black-ScholesPDE,

∂Vj

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2Vj

∂Sa∂Sb−

N∑a=1

∂Vj

∂Sa(γSa −qSa)Sa− rVj = 0.

(29)

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Extending Burgard-Kjaer semi-replication XII

Hence we can write a PDE for U,

∂U

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2U

∂Sa∂Sb−

N∑a=1

∂U

∂Sa(γSa − qSa)Sa − (r + λB)U

(30)

= −εhλB − λB [gB − V ]−M∑j=1

[gCj + ∆Kj + V−j − V − U]λCj

+M∑j=1

sXj Xj + γKK − rφK + rLB

U(T ,S) = 0, (31)

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Extending Burgard-Kjaer semi-replication XIII

This PDE spans all counterparties j so to solve it we would like to beable to separate U into individual contributions from counterparties,for ease of computation, that is we would like to write

U =M∑j=1

Uj , (32)

and hence

∂Uj

∂t+

1

2

N∑a=1

N∑b=1

σaσbSaSb∂2Uj

∂Sa∂Sb−

N∑a=1

∂Uj

∂Sa(γSa − qSa)Sa

− (r + λB + λCj )Uj

= −εhjλB − λB [gBj − Vj ]− [gCj − Vj ]λCj

+ sXj Xj + γKKj − rφKj + rLBj

Uj(T ,S) = 0. (33)

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Extending Burgard-Kjaer semi-replication XIV

In order to be able to do this, all the terms on the right hand side ofthe PDE must also be able to be expressed at counterparty level,either by construction or through a valid allocation process.

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Separating the Issuer Terms I

To separate the issuer terms we need to separate the hedging errorper counterparty, that is we define εhj , that satisfies

εh =M∑j=1

εhj . (34)

For this to hold,

εhj =∆VBj − α1j(1− R1)P1 − α2j(1− R2)P2 (35)

=gBj − Vj − Uj − α1j(1− R1)P1 − α2j(1− R2)P2.

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Separating the Issuer Terms II

The issuer bond positions can easily be attributed to counterpartylevel so that

α1 =M∑j=1

α1j (36)

α2 =M∑j=1

α2j (37)

and from equation (10) that under standard closeout conditions

gB =M∑j=1

gBj . (38)

In general this must be true as gB is simply the total close-out claimmade against the issuer on their default.

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Separating the Issuer Terms III

The relationship between εh and the counterparty level εhj is nowclear,

εh =gB − V − α1(1− R1)P1 − α2(1− R2)P2 (39)

=M∑j=1

gBj −M∑j=1

Vj −M∑j=1

Uj −M∑j=1

(α1j(1− R1)P1 + α2j(1− R2)P2)

=M∑j=1

εhj .

which relies on the fact that Uj itself is separable, which will only betrue if the capital terms are also separable.

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Separating the Capital Terms

The key question then is whether the capital term on the right handside of the PDE can be separated by counterparty.

Writingν ≡ (γK − rφ)K , (40)

we see that γK and K will determine if this separation is possible.

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Capital and Capital Attribution I

Ideally we would like to be able to write

K =M∑j=1

Kj , (41)

that is we would like to allocate all the capital to individualcounterparty positions.

Capital allocation approaches are widely represented in the literaturewith 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 EulerAllocation does not work for all elements of regulatory capital.

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Capital and Capital Attribution II

Earlier we considered three components of capital, market risk,counterparty credit risk and CVA in the counterparty level KVAmodel.

Here we add the impact of the leverage ratio which is a capitalmeasure spanning most bank positions.

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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 homogeneousfunction f of order m can be written

mf (α) =n∑

i=1

αi∂f

∂αi. (43)

Hence a homogeneous function of order 1 can be allocated.

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Market Risk I

The standardized method is a formula based approach as can beseen from the discussion above on the standardised approach forinterest rates. It is clear that the interest rate methodology involveslinear operations on netted position information.

Hence, given the operations are linear then Euler Allocation can beapplied.

The same is true for the remaining risk categories with a smallnumber of exceptions (although even though these are stillhomogeneous functions of order one).

Hence in general Euler Allocation can be used with the standardizedmethod.

The Internal Model Method uses a Value-at-Risk (VAR) approach toestimate the regulatory capital requirement and under thefundamental review of the trading book, expected shortfall will beused.

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Market Risk II

Euler allocation can be used with both methods (Tasche 2008).

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Counterparty Credit Risk (CCR)

Counterparty Credit Risk capital is calculated at netting set levelunder the current Basel III regulatory framework (BCBS-189 2011)for those institutions with IMM approval and for those using theCurrent Exposure Method and Standardized method.

Under the proposed revised standardized approach (BCBS-279 2014)this remains the case.

Capital as calculated under CCR is already defined on a percounterparty basis.

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CVA Capital I

IMM

All calculations are performed on a per counterparty basis.The VAR model is used estimate the capital requirement either byfull revaluation of using the regulatory CS01 formula.

Standardized

As noted earlier, nn the absence of CVA hedging and for largenumbers of conterparties this formula is well approximated by a sumover counterparties,

K iCVA ≈

2.33

2

√hωiMiEADtotal

i . (44)

In the general case with hedging KCVA is cannot be easily expressedas a sum of counterparty level terms.Furthermore the obvious choice of Euler attribution methodologycannot be applied as equation as the standardized CVA formula isnot a homogenous function of order one.

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CVA Capital II

However, it is possible to define a suitable attribution with thedesired properties,

KCVAj =

(MiEADtotal

i −Mhedgei Bi

)∑M

j=1

(MiEADtotal

i −Mhedgei Bi

)KCVA, (45)

which clearly satisfies

KCVA =M∑j=1

KCVAj . (46)

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Allocating the Leverage Ratio

Ignoring other contributors, the leverage ratio for derivatives is givenby

LR =

∑Mj Kj + K ′∑M

j=1 Ej

. (47)

where K ′ is any additional capital required to satisfy the LeverageRatio and Ej is the exposure calculated for counterparty j .

Rearranging and imposing the constraint shows that

K ′ = max

0.03×M∑j

Ej −M∑j

Kj , 0

. (48)

Hence it is possible to attribute any leverage ratio capital usingEuler Attribution

However, the size of any leverage ratio capital will be statedependent.

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Allocating the LCR Liquidity Buffer

Recall equation (4)

Only the last item, Largest absolute net 30-day collateral flow inpreceding 24 months, is not defined at counterparty level.

If this can be allocated to counterparty level then we can write

LB =M∑j=1

LBj (49)

One possible approach would to allocate the 30-day collateraloutflow on a pro-rata basis with the remaining counterparty alignedLCR contributions.

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Solving for XVA I

RHS of equation (30) can be written as a sum over terms atcounterparty level.

Hence U can be written as a sum of terms Uj that satisfy equations(33).

To solve for the Uj we apply the Feynman-Kac theorem in the usualway to obtain:

Uj = CVAj + DVAj + FCAj + COLVAj + KVAj , (50)

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Solving for XVA II

where

CVAj = −∫ T

tλCj

(u)e−

∫ ut (r(s)+λB (s)+λCj

(s))ds

× Et

[Vj (u) − gCj

(Vj (u),Xj (u)),∂Vj

∂S(u),Xj (u))

]du (51)

DVAj = −∫ T

tλB(u)e

−∫ ut (r(s)+λB (s)+λCj

(s))ds

× Et

[Vj (u) − gBj

(Vj (u),Xj (u))]du (52)

FCAj = −∫ T

tλB(u)e


(s))dsEt

[εhj (u)

]du

−∫ T

tr(u)e


(s))dsEt[LBj (u)

]du (53)

COLVAj = −∫ T

tsXj

(u)e−

∫ ut (r(s)+λB (s)+λCj

(s))dsEt[Xj (u)

]du (54)

KVAj = −∫ T

t(γK (K , u) − r(u)φj )e


(s))dsEt[Kj (u)

]du.

(55)

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LCR FVA

The LCR has generated an additional funding cost term

−∫ T

t

r(u)e−∫ ut

(r(s)+λB (s)+λCj(s))dsEt [LBj(u)] du (56)

To evaluate we need to calculate the expected size of the liquiditybuffer in every Monte Carlo state

LBj will in some cases have terms associated with rating downgrade

Could ignore the probability of downgrade and just calculate thebuffer based on the outflow associated with a downgrade of 3notches from the initial rating

However, to capture the full effect a rating transition model isrequired

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Conclusion

Conclusion

Introduced the Leverage Ratio and Liquidity Coverage Ratio

Presented a unified model for valuation adjustments includingCapital and LCR at Portfolio Level

Provided an allocation mechanism that allows the total XVA to becalculated at netting set level

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Bibliography

Balog, D. (2010). Risk based capital allocation. In Proceedings of FIKUSZ 10 Symposium forYoung Researchers, pp. 17–26. Keleti Faculty of Business and Management, budaUniversity.

BCBS-189 (2011). Basel III: A global regulatory framework for more resilient banks and bankingsystems. 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 andRegression. Risk 28(5).

Green, A., C. Kenyon, and C. R. Dennis (2014). KVA: Capital Valuation Adjustment byReplication. 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 10.12.2015 48 / 51

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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. InIn: Pillar II in the New Basel Accord: The Challenge of Economic Capital, pp. 423–453.available at: http://arxiv.org/pdf/0708.2542.pdf.

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Bibliography

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A. Green 10.12.2015 50 / 51

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