Electronic copy available at: http://ssrn.com/abstract=2400324

KVA: Capital Valuation Adjustment∗

Andrew Green†, Chris Kenyon‡and Chris Dennis§

February 20, 2014

Version 1.0

Abstract

Credit (CVA), Debit (DVA) and Funding Valuation Adjustments (FVA)are now familiar valuation adjustments made to the value of a portfolio ofderivatives to account for credit risks and funding costs. However, recentchanges in the regulatory regime and the increases in regulatory capital re-quirements has led many banks to include the cost of capital in derivativepricing. This paper formalises the addition of cost of capital by extendingthe Burgard-Kjaer (2013) semi-replication approach to CVA and FVA toinclude an addition capital term, Capital Valuation Adjustment (KVA).1

Two approaches are considered, one where the (regulatory) capital is re-leased back to shareholders upon counterparty default and one where thecapital can be used to offset losses in the event of counterparty default.The use of the semi-replication approach means that the flexibility aroundthe treatment of self-default is carried over into this analysis. The paperfurther considers the practical calculation of KVA with reference to theBasel II (BCBS-128 2006) and Basel III (BCBS-189 2011) Capital regimesand its implementation via CRD IV (EU 2013b; EU 2013a). The paperassesses how KVA may be hedged, given that any hedging transactionsthemselves would lead to regulatory capital requirements and hence KVA.To conclude, a number of numerical examples are presented to gauge thecost impact of KVA on vanilla derivative products.

1 Introduction

Capital is a legal requirement for financial institutions holding derivatives, andrequirements have increased over the past few years (Dodd and Frank 2010;Department of the Treasury 2013; EU 2013b; EU 2013a). Hence it is surprisingthat few papers include capital in derivatives pricing, (Kenyon and Green 2013a;Kenyon and Green 2013b). Here we extend the hedging framework of (Burgardand Kjaer 2011; Kenyon and Kenyon 2013; Burgard and Kjaer 2013) to pricecapital requirements of derivatives trades by replicating its costs, together with

∗The views expressed are those of the authors only, no other representationshould be attributed.†Contact: andrew.green2@lloydsbanking.com‡Contact: chris.kenyon@lloydsbanking.com§Contact: chris.dennis2@lloydsbanking.com1i.e. Kapital Valuation Adjustment to distinguish from CVA.

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Electronic copy available at: http://ssrn.com/abstract=2400324

the costs from credit and funding. Thus we present a Capital Valuation Adjust-ment (KVA) alongside the existing adjustments for credit and funding.

Capital pricing appears challenging for several reasons:

1. Diversity and length of regulations (see below). In Basel III there areseveral major categories, each with hundreds of pages.

2. Lifetime capital costs are needed, not just the spot costs.

3. Calculations must be done at several different levels of granularity andcombined. For example, for counterparty credit risk and CVA capitalnetting sets are important, while bank-level portfolio can also be neededsuch as for determination of the stressed period for Market Risk for SVARcalculation. Under the standardized approach CVA capital is calculatedacross all counterparties.2

4. The date when new regulation comes into force, and the exact contentof them is often uncertain, for example a series of new regulations arecurrently in a consultation phase, fundamental review of the trading book(BCBS-219 2012; BCBS-265 2013), NIMM (BCBS-254 2013), margin re-quirements for non-centrally cleared derivatives (BCBS-261 2013) and pru-dent valuation (EBA 2013).

We present a brief list below of typical capital regulations here based on BaselIII, and the type of calculation they require in Table 1.

Whilst capital calculation may appear challenging, these calculations do notat first seem to introduce anything fundamentally different from CVA or FVAcalculation. The truly new element is handling the capital itself, that is, decidingwhat happens to the capital on counterparty default. There are two approacheshere that we call Capital Allocation and Capital Insurance.

• Capital Allocation = the desk has been allocated the capital it pays for,but the desk takes no ownership: on counterparty default the capital isunaffected.

• Capital Insurance = the desk has bought insurance against unexpectedrisks by paying for capital: on counterparty default the first losses areborn by the capital.

Since bank losses hit capital the second alternative appears more realistic, al-though in practice it will depend on the policy applied by individual banks.Overall, there are also new complexities when we consider the capital require-ments on the positions that we use to hedge the capital costs and this is exploredin more detail in section 3 below. We deal with other, systematic, theoreticalconsequences elsewhere (Kenyon and Green 2013b).

The main contribution of this paper is to complete the pricing picture byincluding the costs of capital, the Capital Valuation Adjustment (KVA), inderivatives pricing by replication. Given the increased regulatory focus on cap-ital post-crisis, continuing regulatory developments, and its cost, this is longoverdue.

2In practice for large numbers of counterparties it is well approximated by a summationover terms against individual counterparties as is described in section 4.3.1.

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Classification Alternatives Calculation TypeCounterparty EAD CalculationCredit Risk CEM Function of Netting set Value

Standardized Function of Netting set ValueInternal Model Method Exposure profileWeight CalculationStandardized External RatingsFIRB Internal & External RatingsAIRB Internal & External Ratings, Inter-

nal LGDsCVA Capital Standardized Function of EAD

Advanced VAR / SVAR on Regulatory CVA orCS01

Market Risk Standardized Deterministic formulaeInternal Model Method VAR + SVAR

Table 1: Typical categories of capital regulations, their diversity (alternatives),and the type of calculations they require.

1.1 Diversity in Regulatory Capital Requirements

Different quantities of capital are required for the same portfolio dependingon the institution’s regulatory status, and its interpretation of the regulations(BCBS-267 2013). The capital requirements also change depending on the inten-tion of the institution, hold-to-maturity positions (Banking Book) have differentcapital requirements to available-for-sale (Trading Book) (BCBS-265 2013). Ourreplication pricing is applicable to all these cases. This does however mean thatdifferent institutions will have different replication costs, we go into detail onthe implications of this in (Kenyon and Green 2013b).

In theory capital is a cost to risky businesses because investors require apositive return on risky investments. We assert without proof that derivativesdesks are risky businesses. In practice capital use is charged by the issuingbank’s treasury to derivatives desks. This can be done more, or less, directmethods, for example through budgets and RWA limits, but capital is always acost to the desk.

2 Extending Semi-Replication to Include Capi-tal

To include the cost of (regulatory) capital in pricing alongside Credit and Fund-ing Valuation Adjustments we extend the semi-replication argument of Burgardand Kjaer (2013). We assume that the value of the derivative or portfolio ofderivatives has value V (t, S). We also assume that we have a replicating port-folio, Π, with positions in the underlying stock, S, counterparty bond, PC , twoissuer bonds of different credit ratings, P1 and P2, and cash accounts associatedwith stock position, βS , counterparty bond, βC , collateral, X and capital βK .

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The dynamics of the underlying assets are given by

dS =µsSdt+ σsSdW (1)

dPC =rCPCdt− PCdJC (2)

dPi =riPidt− (1−Ri)PidJB , (3)

where PC is a zero recovery bond and the two bonds Pi have recovery rates, Ri.On default of the issuer, B, and the counterparty, C, the value of the derivativetakes the following values

V (t, S, 1, 0) =gB(MB , X) (4)

V (t, S, 0, 1) =gC(MC , X), (5)

where MB and MC are the respective close out values and X is the collateralassociated with the derivative. The two g functions allow a degree of flexibilityto be included in the model around the value of the derivative after default.The usual assumption is that

gB =(V −X)+ +RB(V −X)−

gC =RC(V −X)+ + (V −X)−. (6)

We assume that the following funding condition holds,

V −X + α1P1 + α2P2 = 0, (7)

where α1 and α2 are the holdings of the two bonds. The changes in the cashaccount positions are given by,

dβS =δ(γS − qS)Sdt (8)

dβC =− αCqCPCdt (9)

dX =− rXXdt, (10)

where δ is the stock position, γS is the stock dividend yield, qS is the stockrepo rate, αC is the counterparty bond holding, qC is the repo rate on thecounterparty bond and rX is the yield on the collateral position.

In the portfolio, Π, we have to account for two different sources of regulatorycapital requirements, the derivative and the replicating portfolio. Positions inthe stock and counterparty bond will themselves attract a capital requirement.Hence we write that

K ≡ KV (t, V, “market risk”, X,C) +KΠ(δ, αC) (11)

reflecting the fact that the requlatory capital associated with the derivative isa function of the derivative portfolio value, its sensitivites through market riskcapital, the collateral account value and the rating of the counterparty and thatof the hedge portfolio a function of the position in stock and bond. In practice wecollapse these two terms into one overall capital requirement K. Nevertheless,this effect reflects the fact that regulatory capital applies at the level of thewhole derivative portfolio and not individual trades or counterparties. As willbe discussed below, market risk capital is calculated on the net position of allderivatives, while CVA capital under the standardized approach is calculated

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across all counterparties. Some elements of the regulatory capital framework doneed capital to be attributed to portfolios from an overall net position.

The change in the cash account associated with the capital position is givenby

dβK = −γK(t)Kdt+ ∆KdJC , (12)

where γK(t) is the cost of capital and ∆K reflects potential use of the capitalposition on default of the counterparty.3 This approach reflects the treatmentof capital as a borrowing action, where capital is borrowed from shareholders tosupport derivative trading activities. The cost of capital is thus the cost of thereturn expected by shareholders for putting their capital at risk. In essence thederivatives business borrows the capital and pays cash profits to the shareholdersat a given rate. The capital is not cash but it is an asset with value K thatcould be used to reduce losses in the event of counterparty default. It shouldalso be noted that there is no term in dJB and no impact on the default ofthe issuer. This reflects that any capital available to compensate the creditorsof the issuer on default is already incorporated in the recovery rate RB . Thefinal point to state is that we have implicitly assumed that the rating of thecounterparty remains constant although the model could be extended to takeaccount of rating transitions.

Using Ito’s lemma the change in the value of the derivative portfolio is givenby

dV =∂V

∂tdt+

1

2σ2S2 ∂

2V

∂S2dt+

∂V

∂SdS + ∆VBdJB + ∆VCdJC , (13)

where ∆VB and ∆VC are the changes associated with the default of B and Crespectively.The change in the hedging portfolio is given by

dΠ =δdS + δ(γS − qS)Sdt+ α1dP1 + α2dP2 + αCdPC

− αCqCPCdt− rXXdt− γKKdt+ ∆KdJC .(14)

Adding the derivative and replicating portfolio together we obtain

dV + dΠ =

[∂V

∂t+

1

2σ2S2 ∂

2V

∂S2+ δ(γS − qS)S

+ α1r1P1 + α2r2P2 + αCrCPC − αCqCPC − rXX − γKK

]dt (15)

+ εhdJB (16)

+

[δ +

∂V

∂S

]dS (17)

+[gC + ∆K − V − αCPC

]dJC , (18)

whereεh =

[∆VB − (P − PD)

](19)

3The assets comprising the capital may themselves have a dividend yield and this can beincorporated into γK(t) by reducing the cost of capital.

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is the hedging error on the default of the issuer. We make the condition thatthe portfolio be self-financing except at the default of the issuer,

dV + dΠ = 0, (20)

and so make the usual assumptions to eliminate the remaining sources of riskso that

δ =− ∂V

∂S(21)

αCPC =gC + ∆K − V , (22)

and this leads to the PDE

0 =∂V

∂t+

1

2σ2S2 ∂

2V

∂S2− (γS − qS)S

∂V

∂S− (r + λB + λC)V

+ (gC + ∆K)λC + gBλB − εhλB − sXX − γKKV (T, S) = H(S). (23)

where the bond funding equation (7) has been used.4

Writing the derivative portfolio value, V , as the sum of the underlying deriva-tive portfolio value, V and a valuation adjustment U and recognising that Vsatisfies the Black-Scholes PDE,

∂V

∂t+

1

2σ2S2 ∂

2V

∂S2− (γS − qS)S

∂V

∂S− rV =0

V (T, S) =0, (24)

allows a PDE to be formed for the valuation adjustment,

∂U

∂t+

1

2σ2S2 ∂

2U

∂S2− (γS − qS)S

∂U

∂S− (r + λB + λC)U =

(gC + ∆K)λC + gBλB − εhλB − sXX − γKKU(T, S) = 0 (25)

Hence formally applying the Feynman-Kac theorem gives (using the terminologyof Burgard and Kjaer),

U = CVA + DVA + FCA + COLVA + KVA, (26)

4α1r1P1 + α2r2P2 = rX − (r + λB)V − λB(εh − gB)

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where

CVA =−∫ T

t

λC(u)e−∫ ut

(r(s)+λB(s)+λC(s))ds

× Et[V (u)− gC(V (u), X(u))−∆K(u, V (u),

∂V

∂S(u), X(u))

]du

(27)

DVA =−∫ T

t

λB(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [V (u)− gB(V (u), X(u))] du

(28)

FCA =−∫ T

t

λB(u)e−∫ ut

(r(s)+λB(s)+λC(s))duEt [εh(u)] du (29)

COLVA =−∫ T

t

sX(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [X(u)] du (30)

KVA =−∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [K(u)] du. (31)

Only the CVA and KVA terms differ from Burgard and Kjaer (2013) so theresults of that paper apply to the other terms and will not be discussed furtherhere. Hence the modified CVA and the new KVA term will be the focus of theremainder of this paper.

As noted earlier there are two cases to consider:

Case 1: Capital Allocation ∆K = 0

The capital is returned to the shareholders untouched on the default of thecounterparty. This is the simplest case as the CVA term is unchanged in thiscase, leaving only the KVA term to be calculated and this is described in section4. Capital is held against the possibility of loss and so this may seem unrealistic.However, historically capital has had no bearing on the derivative modelling in-cluding the calculation of CVA or counterparty credit risk in general. Secondly,in practice a bank may have no mechanism for the capital to be used to offsetlosses explicitly. In such a case capital is indirectly impacted by loss through thebalance sheet. Hence to assume that there is no offset is not an unreasonableassumption. Capital then simply becomes a cost to the derivatives trading desk.

Case 2: Capital Insurance ∆K = −KThe capital is used in full to offset against the loss on the default of the coun-terparty explicitly. In practice, as we shall see later, the regulatory capital isplaced against both market risk and counterparty risk so assuming that all ofthe capital allocated to a particular portfolio is available in the event of counter-party default is a limiting case. However, making this assumption is instructive.In this case the CVA term becomes,

CVA =−∫ T

t

λC(u)e−∫ ut

(r(s)+λB(s)+λC(s))ds

× Et [V (u)− gC(V (u), X(u))−K(u)] du. (32)

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Setting gC = RC(V −X)+ + (V −X)− +X gives

CVA =−∫ T

t

λC(s)e−∫ ut

(r(s)+λB(s)+λC(s))ds

× Et[(1−RC)(V −X)+ −K(u)

]du, (33)

that is the loss is offset by the Capital.We can alternatively group the ∆K term into the capital formula leaving the

CVA term unchanged. This would give a revised KVA:

KVA′ = −∫ T

t

(γK(u)− λC(u))e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [K(u)] du. (34)

Here we can interpret this as adjusting the cost of capital for the probability ofdefault of the counterparty.

Both case 1 and case 2 resolve to calculating integrals over the capital profileEt[K(u)] which is a strictly positive quantity. The generation of this profile isthe subject of section 4.

3 Capital at Portfolio Level

Regulatory capital is a portfolio level rquirement. The above model describes thecalculation of KVA for an individual counterparty, while what we are actuallyinterested in understanding is the total KVA for the whole portfolio, that is

KVATOT =

all ctpy’s and hedge securities∑i

KVAi, (35)

although in practice this may not be a simple sum. This is no great surprise asCVA and FVA desks, for example, in general manage the total CVA and FVA.Counterparty credit sensitivities may be hedged individually in some cases, par-ticularly if the are appropriate single name CDS contracts available. However,the interest rate and other market risk of the CVA portfolio will be hedgedacross all counterparties.

When pricing derivatives it is no longer sufficient to look at the impact of justthe new trade, the impact of the trade and all hedging transactions should beconsidered. The hedge trades will themselves create additional capital require-ments, although they may also mitigate other capital requirements. Considera ten year interest rate swap traded with a corporate client on an unsecuredbasis. This trade has market risk, counterparty credit risk and CVA capitalrequirements associated with it. To hedge the market risk the trading desk en-ters another ten year swap with a market counterparty on a collateralised basis.This hedge trade generates a small amount of counterparty credit risk and CVAcapital but drastically reduces the market risk capital on the whole book.

KVA itself, like CVA and FVA, has market risk sensitivities. The CCRterm, for example, is clearly driven by the EAD and hence by the exposure tothe counterparty. Capital requirements go up as exposures rise irrespective toany impact on credit quality. KVA could be hedged and KVA hedging could beviewed as using trades to generate retained profits to offset additional capital re-quirements arising from market moves. However, KVA hedges will again gener-ate capital requirements. Collateral will mitigate the additional CCR and CVA

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capital positions but the Market Risk capital will be affected as the hedge tradewill look like a naked market risk position under the current capital regime.5

The implication of the introduction of KVA is that just like CVA and FVA,KVA should be managed and hedged. KVA can be aligned with the counterpartyand clearly has contingency on the survival of the counterparty and issuer. Themost appropriate approach would be to manage KVA alongside CVA and FVA atportfolio level. KVA and capital management become part of the responsibilityof a central resource management desk.

4 Calculating KVA for Regulatory Capital

In this section we consider the KVA associated with a derivative portfolio witha single counterparty. The aggregated KVA position will be obtained fromequation (35) and as noted earlier this will require some capital attributiondown to portfolio level.

Here we will only consider the three main capital requirements that mostderivative trades are subject to, Market Risk Capital, Counterparty Credit RiskCapital and Credit Valuation Adjustment Capital. Hence we can divide K(u)up into three separate terms,

K = KMR(u,∂V

∂S) +KCCR(u, V, C,X) +KCVA(u, V, C,X). (36)

Here the Market Risk is written as a function of the sensitivity of the unadjustedvalue V to reflect the fact that it is driven by Market Risk, while the other termsare written as functions of the value, the collateral and of the properties of thecounterparty.

4.1 Market Risk Capital

Market Risk Capital is a capital requirement held to offset against the riskof losses due to market risk on traded products and forms part of the BaselII framework (BCBS-128 2006). As currently implemented market risk capitalcan be calculated in two ways, Standardised Method and Internal Models Method(IMM) for those institutions with appropriate regulatory approvals. Changesto the market risk capital framework are included in the Fundamental Reviewof the Trading Book (BCBS-265 2013) but these changes will not be consideredfurther here as the implementation date is unknown and the final proposals arenot yet available.6

It should be noted that in the case of both standardised and IMM approachesthe market risk capital is calculated on a net basis across the portfolio. Thisis problematic from a calculation perspective as it implies that the market riskcapital associated with a given portfolio will need to be attributed from theoverall net requirement. For the purposes of the examples given in section 5this will not be considered and the full standardised market risk capital for theportfolio given assuming no hedging.

5Note that a similar situation has been avoided in the context of CVA capital for CDSspread hedges. Qualifying CDS positions that are designated as CVA hedges are exempt fromfurther capital requirements.

6The proposed changes would change certain aspects of the calculation under both Stan-dardised and IMM approaches.

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One argument that could be made is that the market risk is hedged infull on a back-to-back basis and that in such circumstances the market riskcapital is zero. If trades are hedged back-to-back then they can be taken outof the market-risk capital regime entirely. However, the existence of valuationadjustments means that in this case the overall delta will not be zero, even ifthe capital requirement is. A true hedge of portfolio delta, adjusting for thedelta on valuation adjustments would give zero delta but still have a capitalrequirement as the valuation adjustments do not feature in the current marketrisk capital regime.7

4.1.1 Standardized Method

The standardised measurement method for Market Risk resolves to a formulabased approach to generating the capital requirement with different approachesfor interest rate, equity, foreign exchange and commodities risk. In each casethere are a number of different optional approaches to the calculation availableto the bank. Options are treated separately, again with multiple ways of quan-tifying the capital requirement. It is not the purpose of this paper to describeall of these approaches in detail and the reader is referred to the Basel II docu-mentation (BCBS-128 2006) for a detailed description. However, the numericalexamples in section 5 will be based on interest rate swaps and so the selectedapproach is summarised here.

An interest rate swap is treated as two positions in government securities,that is a notional position in a floating rate instrument with a maturity equalto the period until the next interest rate fixing and an opposite position ina fixed-rate instrument with a maturity equal to the residual maturity of theswap. Consider a GBP interest rate swap with a maturity of 10 years, wherethe bank pays a fixed rate of 2.7% on a notional of GBP 100m, and receives3 month LIBOR. The swap has an annual payment frequency and we assumethat the first coupon has exactly three months to the next fixing.

Assuming the use of the maturity method, at the start of the simulationthe floating leg will give a risk falling into the 3 to 6 month time-band andhence a risk weight of 0.40%. The fixed leg will fall into the 9.3 to 10.6 yearstime-band as the coupon is less than 3%, giving a risk weight of 5.25%. Fora portfolio the short and long positions are summed in each band to give aweighted long and weighted short position. A vertical disallowance equal to 10%of the smaller of the weighted long and weighted short in each band would thenbe calculated. Banks are then conduct two levels of horizontal offsetting withinthree wider time zones spanning, 0-12 months, 1-5 years and 5+ years. Withineach band there is a horizontal disallowance and a second one between bands.The disallowance between zones 1 and 3 is 100%. This then yields an overallmarket risk capital figure. A good description of the practical implementationof the standardized method is given in BIPRU (FCA 2014).

It should be clear from the above discussion that the market risk capitalunder standardized method for market risk is simply a function of trade prop-erties such as residual maturity, coupon and notional. It is not a function of the

7That is CVA capital is treated separately in the regime and is not part of the core frame-work as a valuation adjustment.

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current mark to market or risk. Hence the KVA formula reduces to

KVAstdMR =−

∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt[Kstd

MR(u)]du.

=−∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsKstdMR(u,Mi, Si, Ni)du (37)

where Mi(u), Si(u) and Ni(u) are the residual maturity, coupon and notionalrespectively for trade i. The inner expectation has dropped out and a MonteCarlo simulation is not required for this calculation. In practice this would becalculated using a simple numerical integral and this is done in the examplesbelow.

4.1.2 IMM

The exact methodology used for internal model method market risk depends onan internal choice made by the bank in question and agreed with the appropriateregulatory body. The general approach is the same in all cases, however, usingValue-at-Risk at the 99th percentile with price shocks generated from 10 daymovements in prices.8 The time-series of data must be at least a year. A numberof different can be used including variance-coveriance, historical simulation andMonte Carlo. VAR models can also use full revaluation or delta-gamma-vegaapproximation (that is, a Taylor series).

All these IMM approaches to market risk capital will be expensive to com-pute KMR as they will typically involve Monte Carlo within Monte Carlo. So forexample, a historical simulation full re-valuation model would require a histor-ical simulation and full revaluation at each point inside the outer Monte Carlothat captures the capital exposure. This paper will not address the use of IMMfor market risk, rather the reader is referred to Green and Kenyon (2014) fordetails of a suitable computational technique to accelerate this calculation.9.

4.2 Counterparty Credit Risk Capital

Counterparty Credit Risk Capital (CCR) is calculated for OTC derivatives using

RWA = w × 12.5× EAD (38)

where w is the weight and EAD is the (regulatory) Exposure at Default of thecounterparty. The calculation methodology is divided into two separate partsto estimate the weight and the EAD. The weight can be calculated using threedifferent approaches in order of increasing sophistication and regulatory ap-proval, Standardized Approach, Foundation Internal Rating-Based (FIRB) andAdvanced Internal Rating-Based (AIRB). The EAD can be calculated usingthree different approaches, two simplified approaches based on trade mark-to-markets Current Exposure Method (CEM) and Standardized and the InternalModel Method (IMM) using the banks own internal expected exposure engine.10

8Under the current proposals contained in the Fundamental Review of the Trading Book,expected shortfall (CVAR) will replace VAR and the price shocks will be in most cases takenover periods longer than 10 days (BCBS-265 2013).

9Green and Kenyon (2014) examines the calculation of the cost of VAR-based initial mar-gin, but this approach can be directly translated to the calculation of market risk capital

10CEM and Standardized will be replaced by NIMM at some point in the future.

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4.2.1 Weight Calculation

Standardized Method In the Standardized Approach the weight is simplygiven by the external rating of the counterparty and the sector in which itoperates. For unrated counterparties the weight is set at 100%. Tables of theweights can be found in (BCBS-128 2006).

Internal Ratings-Based Approach In the Internal Ratings-Based (IRB)approach banks estimate key risk components themselves: the probability ofdefault (PD) and the loss given default (LGD). In the Foundation IRB approachbanks provide PD estimates but use supervisory estimates for the LGD. In theAdvanced-IRB approach banks are also allowed to estimate the LGD. In bothcases the weight is calculated according to the following formula,

ρ =0.121− e−50×PD

1− e−50+ 0.24

1− (1− e−50×PD)

1− e−50(39)

b =(0.11852− 0.05478 log(PD))2 (40)

w =LGD

(Φ

(Φ−1(PD)√

1− ρ+ Φ−1(0.999)

√ρ

1− ρ

)− PD

)(41)

× 1 + (M − 2.5)b

1− 1.5b(42)

where Φ is the cumulative Normal distribution, and Φ−1 its inverse.The PD is the greater of 0.03% and the bank’s internal estimate for proba-

bility of default over one year. Under FIRB the LGD = 45% for corporates. Mis the effective maturity of the netting set and this is given by

M = min

(5.0,max

(1.0,

∑Ntrades

i=1 miNi∑Ntrades

i=1 Ni

)), (43)

where mi is the residual trade maturity and Ni is the trade notional.

EAD using CEM In the CEM banks must get replacement costs by markingcontracts to market, and then add a factor (the add-on) to capture exposureover the remainder of the contract life. Hence the EAD is given by

EAD = V +A(mi, Ni, assetclass). (44)

The add-on reflects the asset class (Interest Rates, FX and Gold, Equities,Other Precious Metals, Other Commodities) and the remaining maturity (lessthan one year, one to five, longer). Add-ons are deterministic percentages ofthe contract notional.

Some legally-supported bilateral netting is permitted with the net add-onANet calculated as:

ANet = 0.4AGross + NGRAGross

where NGR is the ratio of net to gross replacement costs and AGross is the grossadd-on amount. The net to gross ratio is given by

NGR =(∑Ntrades

i=1 Vi)+∑Ntrades

i=1 (Vi)+. (45)

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For a single uncollateralized 10Y IR swap, the add-on for EAD is 1.5%of notional. The CEM approach to calculating EAD will be adopted in thenumerical examples.

EAD using Standardized Approach For the Standardized Approach theEAD is calculated as:

EAD =β ×max

(∑i

Vtransaction i −∑l

Vcollateral l,

∑j

∣∣∣∑i

Rtransaction ij −∑l

Rcollateral lj

∣∣∣× CCFj

(46)

Where: Vtransaction value of transactions; Vcollateral value of collateral; Rtransaction

risk from transactions; Rcollateral risk from collateral; CCF supervisory creditconversion factor for the hedging set. Indices: i for transactions; j for supervisory-designated hedging sets, these correspond to risk factors; l collaterals.

Risk positions for linear debt instruments are mapped to suitably-definedinterest rate swap positions. The size of the risk position is the effective no-tional value of the outstanding payments in domestic currency, multiplied bythe modified duration.

The CCF for debt positions are: 0.6% when there is high specific risk; 0.3%for a reference debt instrument beneath a CDS and has low specific risk; 0.2%otherwise. β is set to 1.4.

Calculating KVA for formula-based Approaches In both standardizedmethod and CEM the EAD is a function of the value of the trades in the nettingset Vi. Hence we calculate the KVA as follows

KVAformulaCCR = −

∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))ds12.5cEt [wEAD(Vi, u)] du,

(47)where c is the capital multiplier that is currently set at 8%. The inner expec-tation is broadly similar to the expected exposure calculation in CVA and FVAterms as the EAD at any point is simply a function of the portfolio value.

EAD under Internal Model Method EAD is calculated according to thefollowing formulae:

EAD =α× Effective EPE

Effective EEti = max(Effective EEti−1 ,EEti)

Effective EPE =

min(1year,maturity)∑k=1

Effective EEtk ×∆tk

∆tk =tk − tk−1

α =1.4

where EE is the expected exposure (always greater than or equal to zero bedefinition of exposure).

13

The formula for KVA has the same form as equation (47) with the innerexpectation now given by

EQwαmin(1year,maturity)∑

k=1

max(Effective EE(u)ti−1 ,EE(u)ti)∆tk

∣∣∣∣∣Ft , (48)

where11

EEti = EQ [max(V (ti), 0)|Fu] . (49)

The filtrations have been specified to aid clarity on exactly what expectationsare being calculated. It is clear that we need to estimate future expected ex-posures inside the expectation used to give the EAD profile under IMM. Thisis problematic as it points to the need to use Monte Carlo within Monte Carloto solve. American Monte Carlo (for example the Longstaff-Schwartz (2001)approach) techniques offer one possible solution but this is a subject for furtherresearch.

4.3 CVA Capital

CVA Capital was introduced in Basel III (BCBS-189 2011) in response to thelarge CVA losses some financial institutions faced during the 2007-2009 finan-cial crisis. CRD-IV, the European implementation of Basel III, removes therequirement to calculate CVA Capital for corporate counterparties that are EUdomiciled but it must still be calculated for other counterparties. Two meth-ods of calculation aren offered, standardized and advanced for those banks withIMM approval for both exposure and VAR calculation.

4.3.1 Standardised

The standardized CVA risk capital charge in (BCBS-189 2011), paragraph 104,gives the formula to generate CVA capital:

KCVA =2.33√h

(∑

i

0.5ωi

(MiEADtotal

i −Mhedgei Bi

)−∑ind

ωindMindBind

)2

+∑i

0.75ω2i

(MiEADtotal

i −Mhedgei Bi

)2}1/2

(50)

Where:

• h one year risk horizon in units of years, i.e. h=1;

• ωi risk weight of ith counterparty based on external rating (or equivalent);

• EADi exposure at default of counterparty i, discounted using 1−e−0.05Mi

0.05Mi

(as we are using the non-IMM point of view);

11Note that in equation (49) the measure is specified as the Q measure. In practice the IMMapproved model may in fact be set in the P measure. This would add further complexity asthe implied dynamics used to physically calculate the IMM exposures would be different fromthose used to estimate the KVA term which are risk-neutral. IMM is not restricted to use aP -measure exposure engine and the Q-measure is acceptable as long as the model fulfills theback-testing requirements.

14

• Bi notional of purchased single name CDS hedges, discounted as above;

• Bind notional of purchased index CDS hedges, discounted as above;

• ωind risk weight of index hedge using one of seven weights using the averageindex spread;

• Mi effective maturity of transactions with counterparty i, for non-IMMthis is notional weighted average, and is not capped at five years;

• Mhedgei maturity of hedge instrument with notional Bi;

• Mind maturity of index hedge ind.

The standardized CVA charge is calculated across all counterparties. Miti-gation is given for CDS that are used to hedge counterparty credit risk. In theabsence of hedging then formula reduced to

KCVA = 2.33√h

(∑

i

0.5ωiMiEADtotali

)2

+∑i

0.75ω2i

(MiEADtotal

i

)2

1/2

(51)However, in the limit of a large number of counterparties is is well approximatedas a sum over terms for individual counterparties,

KiCVA ≈

2.33

2

√hωiMiEADtotal

i . (52)

This is a simple expression in terms of the EAD so KVA for CVA capital underthe standardized approach is given by

KVAstdCVA = −

∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt[

2.33

2

√hωMEAD(Vi, u)

]du.

(53)

4.3.2 IMM

If the bank has Specific Interest Rate Risk VaR model approval and IMM ap-proval for EAD calculation then it must use the Advanced CVA risk capitalcharge. The model uses VAR on the credit spread sensitivity of a unilateralCVA formula where the expected exposure is generated from a stressed calibra-tion. Where the bank uses a VAR model with full revaluation then the followingCVA formula must be used directly,

CVA =LGDMKT

T∑i=1

max

(0, exp

(− si−1ti−1

LGDMKT

)− exp

(− siti

LGDMKT

))×(

EEi−1Di−1 + EEiDi

2

)(54)

where D are discount factors, and si are market-observed CDS spreads. If thebank uses a VAR model based on credit spread sensitivities then the creditspread sensitivity is given by

Regulatory CS01i = 0.0001ti exp

(− siti

LGD

)(EEi−1Di−1 + EEi+1Di+1

2

)(55)

15

The CVA capital under IMM is therefore given by the same approach asMarket Risk Capital under IMM. To proceed we need to generate the forwardexpected exposures as was the case with CCR capital under IMM. As in thatcase, American Monte Carlo could be used to generate these. This would thenneed to be coupled with a lifetime VAR technique as discussed in the contextof IMM Market Risk Capital in section 4.1.2.

5 Numerical Examples

Here we provide a number of example results to allow the impact of KVA to beassessed and compared to the existing valuation adjustments. In all cases thevaluation adjustments have been calculated using numeric integration of equa-tions (27) through (31). To simplify the presentation and to allow assessment ofthe work case KVA, we assume that the trade is unsecured and so X and henceCOLVA is zero. We also choose to calculate the case of semi-replication with noshortfall at own default, “strategy 1” in Burgard and Kjaer (2013). This choicegives the following formulae for CVA, DVA and FCA:

CVA =−∫ T

t

λC(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt[(V (u))+

]du (56)

DVA =−∫ T

t

λB(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt[(V (u))−

]du (57)

FCA =−∫ T

t

λB(u)e−∫ ut

(r(s)+λB(s)+λC(s))duEt[(V (u))+

]du (58)

(59)

we illustrate KVA in both cases given in section 2 of Capital Allocation andCapital Insurance, giving

KVAalloc =−∫ T

t

γK(u)e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [K(u)] du (60)

KVAinsur =−∫ T

t

(γK(u)− λC(u))e−∫ ut

(r(s)+λB(s)+λC(s))dsEt [K(u)] du. (61)

The examples have been calculated under the assumption the issuer calcu-lates Market Risk under the standardized approach, uses the current exposuremethod to estimate the EAD and applies the standardized approach with ex-ternal ratings for CCR and the standardized approach for CVA using the ap-proximation for large numbers of counterparties give in equation (52). The useof the standardized approaches avoids the complexity and bespoke nature ofinternal model methods. We assume that the issuer holds the minimum cap-ital ratio requirement of 10.5% (including minimum capital and capital bufferrequirements) and that the issuer cost of capital is 10%.

The examples are calculated using a single 10 year GBP interest rate swapwith semi-annual payment schedules. The fixed rate on the swap is 2.7% ensur-ing the unadjusted value is zero at trade inception. We consider both the casewhere the issuer pays the fixed rate and the case where the issuer receives thefixed rate. The issuer spread information is is assumed to be flat 100bp accrossall maturities and the issuer recovery rate is assumed to be 40%.

16

We calculate all valuation adjustments for 4 different counterparty ratingsand spread combinations, AAA, A, BB and CCC. The spreads assumed ineach case are given in table 2 alongside the risk-weight that is applied in thestandardized CCR calculation. The counterparty recovery rate is assumed tobe 40%.

Counterparty Rating bp Standardized Risk WeightAAA 30 20%A 75 50%BB 250 100%CCC 750 150%

Table 2: Counterparty spread data used in the examples.

The results of the example calculations are given in table 3. Setting asidethe Market Risk component of the capital we see that KVA from CCR and CVAterms gives an adjustment of similar magnitude to the existing CVA, DVA andFCA terms, demonstrating that KVA is a significant contributor to the price ofthe derivative.

The market risk is assumed to be unhedged and so this KVA component isrelatively large compared to the CCR and CVA terms. Under the standardizedapproach to market risk the capital requirement on a ten year transaction ofthis type is scaled according to a 60 bp move in rates. Practical applicationswould calculate the market risk capital requirement over all trades in a portfolioand then attribute these to trade level.

Comparing the Capital Allocation and Capital Insurances cases shows a re-duction in capital costs if capital is used to offset losses on counterparty default.This reduction becomes increasingly significant with increased probability ofcounterparty default.

6 Conclusions

We have presented a unified model for valuation adjustments that includes theimpact of Capital and in so doing have introduced a new “XVA” term, KVA.The impact of capital upon the default of the counterparty has been exploredgiving two cases, Capital Allocation and Capital Insurance. We have describedhow KVA can be calculated in the case of the formula based approached toregulatory capital calculation and sketched how this may be calculated in thecase of internal model approaches. Practical examples of KVA on an interestrate swap have demonstrated how significant capital costs are and that KVAis broadly similar in size to the other components of XVA. The assumption ofCapital Insurance results in significant reductions in KVA for counterpartieswith a high probability of default.

References

BCBS-128 (2006, June). International Convergence of Capital Measurementand Capital Standards. Basel Committee for Bank Supervision.

17

BCBS-189 (2011). Basel III: A global regulatory framework for more resilientbanks and banking systems. Basel Committee for Bank Supervision.

BCBS-219 (2012). Fundamental review of the trading book — consultativedocument. Basel Committee for Bank Supervision.

BCBS-254 (2013). The non-internal model method for capitalising counter-party credit risk exposures - consultative document. Basel Committee forBank Supervision.

BCBS-261 (2013). Margin requirements for non-centrally cleared derivatives.Basel Committee for Bank Supervision.

BCBS-265 (2013). Fundamental review of the trading book - second consul-tative document. Basel Committee for Bank Supervision.

BCBS-267 (2013). Second report on the regulatory consistency of risk-weighted assets in the trading book issued by the Basel Committee. BaselCommittee for Bank Supervision.

Burgard, C. and M. Kjaer (2011). Partial differential equation representa-tions of derivatives with bilateral counterparty risk and funding costs.The Journal of Credit Risk 7, 75–93.

Burgard, C. and M. Kjaer (2013). Funding Strategies, Funding Costs.Risk 26 (12).

Department of the Treasury (2013). 12 CFR Parts 208, 217, and 225. Regula-tory Capital Rules: Regulatory Capital Rules: Regulatory Capital, Imple-mentation of Basel III, Capital Adequacy, Transition Provisions, PromptCorrective Action, Standardized Approach for Risk-weighted Assets, Mar-ket Discipline and Disclosure Requirements, Advanced Approaches Risk-Based Capital Rule, and Market Risk Capital Rule; Final Rule. FederalRegister, Vol. 78(198), pp62017-62291. Department of the Treasury.

Dodd, C. and B. Frank (2010). Dodd-Frank Wall Street Reform and Con-sumer Protection Act. H.R. 4173, http://www.sec.gov/about/laws/

wallstreetreform-cpa.pdf.

EBA (2013). On prudent valuation under Article 105(14) of Regulation (EU)575/2013. Technical report, European Banking Authority. EBA-CP-2013-28.

EU (2013a). Directive 2013/36/EU of the European Parliament and of theCouncil of 26 June 2013 on access to the activity of credit institutionsand the prudential supervision of credit institutions and investment firms,amending Directive 2002/87/EC and repealing Directives 2006/48/ECand 2006/49/EC Text with EEA relevance. European Commission.

EU (2013b). Regulation (EU) No 575/2013 of the European Parliament andof the Council of 26 June 2013 on prudential requirements for credit insti-tutions and investment firms and amending Regulation (EU) No 648/2012Text with EEA relevance. European Commission.

FCA (2014). Prudential Sourcebook for Banks, Building Societies andInvestment Firms (BIPRU). Online; accessed 17 Feb 2014; http://

fshandbook.info/FS/html/handbook/BIPRU.

Green, A. and C. Kenyon (2014). Calculating the Funding Valuation Adjust-ment (FVA) of Value-at-Risk (VAR) based Initial Margin. Forthcoming.

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Kenyon, C. and A. Green (2013a). Pricing CDSs’ capital relief. Risk 26 (10).

Kenyon, C. and A. Green (2013b). Regulatory-Compliant Derivatives Pricingis Not Risk-Neutral. SSRN . http://ssrn.com/abstract=2349103.

Kenyon, C. and R. Kenyon (2013). DVA for Assets. Risk 26 (2).

Longstaff, F. and E. Schwartz (2001). Valuing american options by simulation:A simple least-squares approach. The Review of Financial Studies 14 (1),113–147.

19

KVA

Allocation

KVA

Insu

rance

Swap

Rating

CVA

DVA

FCA

MR

CCR

CVA

MR

CCR

CVA

Pay

AA

A-4

38

-14

-260

-4-2

-248

-4-2

Pay

A-1

037

-14

-253

-9-5

-225

-8-5

Pay

BB

-31

33

-12

-230

-17

-10

151

-11

-6P

ayC

CC

-69

23

-9-1

79

-19

-12

-25

-3-1

Rec

AA

A-1

214

-38

-260

-9-5

-248

-8-4

Rec

A-2

814

-37

-253

-21

-12

-225

-19

-10

Rec

BB

-82

12

-33

-230

-37

-21

-151

-25

-14

Rec

CC

C-1

799

-23

-179

-40

-24

-25

-9-3

Tab

le3:

XV

Ava

lues

for

aG

BP

10ye

arp

ayer

san

d10

year

rece

iver

sin

tere

stra

tesw

ap

.R

esu

lts

are

qu

ote

din

bp

of

the

trad

en

oti

on

al.

20