## In the Balance

### Version 1.0, 13 Mar 2011

### Christoph Burgard

Quantitative Analytics Barclays Capital 5 The North Colonnade

Canary Wharf London E14 4BB

United Kingdom

christoph.burgard@barclayscapital.com +44 (0)2077731462

### Mats Kjaer

Quantitative Analytics Barclays Capital 5 The North Colonnade

Canary Wharf London E14 4BB

United Kingdom

mats.kjaer@barclayscapital.com +44(0)2031341191

Abstract

### 1

### Introduction

Funding and bilateral counterparty credit risk of derivatives positions are intimately related to each other. In Burgard and Kjaer [1], we have developed a unified framework that combines both effects. On one hand, the framework specifies, how a positive cash account related to the hedging strategy of an uncollateralised derivative can be used to fund the repurchase of the issuer’s own bonds in order to hedge out its own credit risk, justifying the inclusion of a debt value adjustment (DVA) in a bilateral counterparty credit value setting. On the other hand, the framework also includes the funding costs associated with a negative cash account, thus yielding an additional funding cost adjustment to the derivatives price. The size of the funding cost adjustment depends on the specific way the funding is achieved.

It is a widespread practise among banks with major un-collateralised derivatives positions to operate an internal counterparty credit trading (CCT) desk, which, against payment of a credit value adjustment (CVA), insures the derivatives desks against losses on the derivatives positions caused by counterparty defaults. With the framework developed in Burgard and Kjaer [1] we show here how such an internal setup can be extended to include the DVA and funding costs in a consistent way. We outline, how a central funding unit (CFU) can be established that provides and receives funding to and from the derivatives desk at the risk free rate in exchange for a funding value adjustment.

Finally we establish a link between the funding cost adjustment and the balance sheet of the issuer and demonstrate it in a simple balance sheet model. We demonstrate that balance sheet effects of derivatives assets can decrease the effective funding costs. We also show that if the balance sheet can actively be managed such that the impact of the funding of the derivative position is neutralised, the funding cost adjustment term can be reduced to zero.

This paper is organised as follows: Section 2 reviews the unified framework for bilateral counterparty risk and funding costs presented in Burgard and Kjaer [1]. This is followed in Section 3 by a description of how a funding desk could be setup and operate. Section 4 present a strategy for reducing the funding cost adjustment to zero by trading own bonds of different seniority. Finally we conclude in Section 5.

### 2

### Review of the unified framework for bilateral counterparty

### risk and funding adjustments

In Burgard and Kjaer [1], we model the hedging strategy for derivative contracts including the risk of defaults of the issuerB (”own” credit) and the counterparty C. The derivative is assumed to be a derivative on an underlying asset S with risk free value V(S, t), where the sign ofV is defined as seen from the counterparty. Here, risk-free means without credit risks to the counterparty or the issuer and without potential funding costs. V does, however, include the market risk of the underlying asset S. The risky value ˆV, in contrast, is defined to include the credit risks and funding adjustments. 1 The underlying processes for the asset processS and the zero-recovery bondsPBandPC of the issuer and counterparty, respectively,

1_{As such, value and sign of ˆ}_{V} _{corresponds, at the start of the trade, to the premium that the counterparty}

are assumed to follow log-normal process for S and independent jump-to-default processes

dJB and dJC for the issuer and countparty, respectively:

dS

The jump-to-default processesdJB and dJC jump from 0 to 1 upon default of the issuer and

counterparty, respectively. For simplicity, the framework assumes deterministic rates and default intensities. As such, it does not include any convexity effects between funding rates and market factors, as discussed in Piterbarg [3], but could be extended to do so.

The bonds PB and PC are zero-recovery bonds for simplicity. Zero-recovery bonds together

with risk free assets can be used as building blocks to replicate more complicated structures, e.g. bonds with recovery. In Section 4, we will consider a case where we include bonds of different seniority (and therefore recovery) in order to neutralise the balance sheet impact of the derivatives.

The boundary condition of the risky value of the derivative ˆV(S, t, JB, JC) upon default of

the issuer or counterparty are defined as

ˆ

V(t, S,1,0) = M−(t, S) +RBM+(t, S) B defaults first

ˆ

V(t, S,0,1) = RCM−(t, S) +M+(t, S) C defaults first

(4)

Contractual details will determine, which values for the close-outsM should be used upon default of the issuer or counterparty.

In Burgard and Kjaer [1] we model a replication strategy of holding δ amounts of the un-derlying asset S, αC amounts of the zero-recovery bonds PC of the counterparty, and β(t)

amounts of cash. It is shown, that this cash, if positive, can be used to fund the (re-)purchase of the required numberαB of the issuer’s own zero-recovery bondsPB needed to hedge out

the issuer’s credit risk on the derivative position. Any remaining cashβF = ˆV−αBPB, if

pos-itive, is being used to purchase risk-free assets (yielding rater), so as not to spoil this hedge position, and, if negative, has to be financed as usual from an external funding provider at a funding raterF. The cost of this funding is included in the cost of the hedging strategy and

therefore the risky price ˆV. It shall be noted, that while the risk of the issuer’s default on the derivative contract is hedged out by the strategy, that the risk of the funding provider to the issuer’s credit, is not, and the funding provider correspondingly earns the spreadsF =rF−r

to be compensated for this risk. The hedge ratios found in Burgard and Kjaer [1] are:

These hedge ratios then result in the following PDE for the risky value of the derivative:

andqS is the net share position financing cost and γS is the dividend income.

Two cases for the close-out amountM are considered in Burgard and Kjaer [1]: one whereM

corresponds to the risky value, i.e. M(t, S) = ˆV(t, S,0,0), and one where it corresponds to the risk-less valueM(t, S) =V(t, S,0,0). The M =V(t, S,0,0) case more closely describes contracts that follow the ISDA master agreements of 1992 and 2002, albeit even with these agreements legal uncertainties remain with respect to the inclusion of funding costs and own credit adjustment (alias DVA) of the non-defaulting party in the calculation of the close-out amount, as well as the potential inclusion of set-offs against other obligations between the counterparties. For the following discussion we will focus on theM(t, S) =V(t, S,0,0) case. For the M = V case, splitting the risky value into the riskless value and adjustment, ˆV =

V +U, Burgard and Kjaer [1] obtain the following integral representation forU:

U(t, S) = CV A+DV A+F CA (14)

The first term is often referred to as the (modified) unilateral CVA, the second term as the DVA and the third term is a funding cost adjustment, which we shall call FCA. It is clear from this representation that while both, the DVA as well as the FCA are related to the credit position of the issuer, they do not double count the issuer’s credit but capture exposures of the mark-to-market value of the derivative of opposite sign and as such are opposite sides of the same coin. The DVA term itself can be seen as a funding benefit term, in the sense that it arises from the issuer using a positive cash account to buy back his own bonds, thus hedging out his own credit risk on the derivative position while earning its spread on it.

### 3

### Practical Setup

Equation (14) deserves some discussions.

First, it should be noted, that the hedging strategy leading to this equation involves the issuer repurchasing its own bonds. It does not involve any dealings in its own CDS. The term λB

is the spread of the yield of a zero-recovery bond over the risk free rate. It is not the hazard rate derived from the CDS market. Thus, if there is a basis between bonds and CDS for the issuer B, it is the bond market that counts for determining λB used in the DVA and FCA

terms in equations (16) and (17).

Second, it is instructive to discuss the hedge ratios obtained for theM =V case, for which equations (5) read

δ = ∂SV +∂SU, (18)

αB =

U + (1−RB)V+ PB

, (19)

αC =

U + (1−RC)V− PC

, (20)

βF = V−+RBV+ (21)

For positive V, the combination of zero-recovery zero-coupon bonds of value (1−RB)V+ and RBV+ amount of cash invested in risk free assets, replicates an investment of value V+ in a

zero-coupon bond with recovery RB 2. Thus, for the issuer to hedge its own credit risk on

the risk free partV+ of the option value, it could as well invest the full amountV+ in bonds with recoveryRB. The contributions of U to the hedge ratios, on the other hand, come from

the fact that upon default the close-out amount of V differs from the (risky) value of the derivative just prior to default by the amount U, because the credit and funding adjustments for the trade disappear upon default of the counterparty or the issuer. Thus, the credit risk on the full amount of U needs to be hedged out, and this is achieved by means of taking positions in zero-recovery bonds ofB and C of value U.

Third, the spreadsF in the F CAterm (17) is the spread paid for funding negative balances

on the cash account. In Burgard and Kjaer [1] we mention two cases, which we want to discuss here in more detail.

3.1 Case where derivative can be used as collateral

In principle the derivative itself could be used as collateral for the funding required for negative balances on the cash account. The cash account is negative when the valueV is negative (to the counterparty), thus positive (−V) to the issuer. It is an asset for the issuer, which in principle, could be ring-fenced and used as security for the required funding. To be precise, the negative cash balance βF that needs funding is V−, i.e. the risk-free value of the derivative,

if V is negative. While the value of the derivative to the issuer is the risky value (−Vˆ), its close-out value is (−V), so it could indeed be used to secure the required amount of funding of−V if no haircut were applied. Now in practise, there are a number of technical difficulties to be taken into account when considering whether and how the derivative could be used as

collateral. One of them is that the value of the derivative (and the amount of cash requiring funding) changes constantly. Another one is, that, in general, one should expect a healthy haircut to be applied between the value of the derivative collateral and the secured funding amount. In the extreme case, though, where the derivative can be used as collateral with zero haircut and the secured funding rate is the risk free rate, the spreadsF and, correspondingly,

the FCA of equation (17) disappears and we are left with the first two terms for the adjustment U in equation (14), representing the usual bilateral CVA as described in many papers (e.g. see Gregory [2]), but its use being well justified in our framework through the hedging strategy of repurchasing the issuer’s own bonds from its positive cash account. At the same time, since the derivative has been pledged as collateral when the cash accounts is negative, it does not appear as an asset on the balance sheet of the issuer.

3.2 Case where derivative cannot be used as collateral

If the derivative cannot be used as collateral, we consider in Burgard and Kjaer [1] the case where the funding for a negative balance in the cash account is done unsecured through the issuers usual funding channels. In this case, sF does not vanish. This may be the most

realistic case in practice. As discussed in [1], if sF is the normal unsecured senior funding

rate then it can be linked to the spreadλB via the relation sF = (1−RB)λB. The DVA and

FCA can then be combined into a single term, which we may refer to as the funding value adjustment (FVA):

This leads to the following interpretation of the equation and natural split between derivatives desk, CCT and CFU units as depicted in figure 1: when the derivatives desk enters into the contract with the counterparty and receives premium ˆV it pays the modified unilateral CV A

as premium to the CCT desk in exchange for getting protection on counterparty risk, i.e. CCT agrees to pay the derivatives desk −(1−RC)V− if the counterparty defaults first. The CV A in equation (24) is the expectation value of this protection payout, discounted by the risky discounting factor because it is conditional to issuer and counterparty surviving up to the point of default. At the same time, the derivatives desk enters into a funding agreement with the CFU desk. The agreement is that the CFU desk will provide/receive funding at the risk free rate r of the risk-free mark-to-market value V to/from the derivative desk as long as neither the counterparty nor the issuer has defaulted. So the cashflow each day will be

Figure 1: Cashflows between derivatives, Counterparty Risk Trading (CCT ) and Economic Funding Unit (CFU ) desks.

desk. For this funding arrangement, the derivatives desk will pay the CFU desk the premium FVA in equation (25) at the start of the trade. This FVA term is the expectation value of the present value of the cost that CFU encounters for providing/receiving the funding on V at the risk free rate rather than the funding rate rF through the life of the trade,

i.e. it is the funding spread times the expected mark-to-market value integrated over time, discounted by the risky discount factor, because, again, funding is conditional on both issuer and counterparty surviving up to that point.

The advantages of the setup in Figure 1 are that the hedging of the funding risk and counter-party credit risk are done centrally by the CFU and CCT units rather than by each derivatives desk, and can take portfolio and netting effects into account. Moreover, as soon as the deriva-tives desk has paid the CVA and FVA charges, it does not have to care about funding costs or counterparty risk. On a practical level, only the risk-systems of the CCT and CFU desks need to be aware of funding curves and credit risk, whereas the risk-system of the derivatives desk can be spared such complexities.

Table 1 summarizes the positions the derivatives, CCT and CFU desks take.

It is worthwhile discussing the case where ˆV is negative, i.e. out-of-the-money for the coun-terparty but in-the-money for the issuer, and the issuer defaults. In this case, just prior to default, the cash account is negative and CFU is providing funding of−V− at risk free rater

to the derivatives desk. CFU itself gets the funding on this amount from the external funding provider, payingrF. If the issuer then defaults, the derivatives desk will settle the derivative

Desk Under- Own bonds Cpty Bonds Unsecured Risk

Table 1: Hedge positions for the derivatives, CCT and CFU desks and for the issuing insti-tution as a whole.

here. The external funding provider, on the other hand, being a senior creditor, will only receiveRB(−V−) on his funding, so loosing (1−RB)(−V−). The external funding provider

is compensated for this potential loss by receiving the spread sF while the issuer was alive.

The position of the issuer upon its default, i.e. getting −V− for the derivative and paying

RB(−V−) to the funding provider, is a positive (1−RB)(−V−), which goes towards the

recovery of the bondholders of the issuer. So the following picture arises: the FCA term of equation (17) is the premium for the ”windfall” to the bondholders of the issuer in the case where the issuer defaults and the derivative has positive value for the issuer. This premium is a cost that arises while the issuer is alive. This cost is included in the derivatives price ˆV

charged to the counterparty by means of the funding term in equation (23).

There are two discussion points arising from this interpretation of the funding term and the potential windfall to the bondholders: first, derivatives are potential assets and can contribute to the asset pool upon the issuer’s default, as in the case just discussed. Therefore, they impact the balance sheet and should ultimately help to lower the funding costs of the issuer. This link is hard to make in practise, in particular to quantify the impact of this balance sheet effect on the funding costs, but in a purely theoretical setting, the impact on the funding costs via the balance sheet is such, that it compensates the funding term in equation (23). This can be seen in a simple balance sheet and funding model, which we will discuss next.

3.2.1 Simple model for impact of derivatives asset on balance sheet and funding

Assume that, as in a reduced form credit model, default of the issuer is driven by an instanta-neous default process with intensityλand that, prior to entering into the derivative contract, the expected recovery upon default isR0. In particular, we assume that the expected assets

upon default areA0 and the expected liabilities areL0. Within this simple setup, the funding

spreadsF of the issuer over the risk free rate that compensates for the expected loss upon its

default is sF = (1−R0)λ. Thus, the instantaneous funding costs f0 per unit of time of the

issuer over timedtfor his total liability L0 is

f0 = (r+ (1−R0)λ)L0 (27)

Let us now add a derivative with positive value das an asset to the issuer, so get new total assets ofA1=A0+d3 and fund the corresponding negative cash by adding a corresponding

3_{The positive value}_{d}_{to the issuer corresponds to}_{−}_{V}−

liability, giving new total liability ofL1=L0+d. Thus, the expected recovery changes to

R1 = A1 L1

= A0+d

L0+d

(28)

Adding the derivatives asset, assuming the issuer has hedged the market and counterparty risk, does not change the default intensity of the issuer. Thus, the instantaneous funding costs after adding the derivatives is

f1dt = (r+ (1−R1)λ)L1dt (29)

= r(L0+d)dt+ (L1−A1)λdt (30)

= rL0dt+r·d·dt+ (L0−A0)λdt (31)

= r·d·dt+r·L0·dt+ (1−R0)λL0dt (32)

= r·d·dt+f0dt (33)

It follows, that while the new liability ddraws the new funding spread (1−R1)λthe change

on the recovery and its effect on the funding of the total liabilies results in an effective funding rate for d that is the risk free rate. Thus, within this balance sheet model the spread sF is

zero. While this balance sheet model is somewhat simplistic, it shows, that accounting for the effects of the derivative asset on the balance sheet and the overall funding position does have the potential to lower the FCA term - in the extreme case down to 0.

The second discussion point arising from the interpretation of the funding cost adjustment term is that the reason that the derivative asset and its funding impact the balance sheet and cause a potential ”windfall” to the bondholders is the constraint of using a single means of funding for negative values of the cash account of the derivative hedging strategy. This constraint means that there are not enough degrees of freedom available to monetise the ”windfall” and corresponding impact on the balance sheet prior to default. One way of neutralising the impact on the balance sheet is by obtaining funding using the derivative as collateral, if possible, as discussed in section 3.1, where indeed the funding cost term vanishes. Another way of doing so is to extend the range of funding instruments available and thus manage the balance sheet such, that impact of the derivatives asset and its funding are neutralised, as will be discussed in the following section.

### 4

### Balance sheet management to mitigate funding costs

In this section we show how that if the issuer have two bonds own P1 and P2 of different

seniority with recovery rates R1 and R2, and is free to trade these bonds to manage the

funding needed for the negative cash positions in such a way that the total impact if its own default on the funding position vanishes, then the issuer can achieve in effect risk-free financing of the cash account associated with the derivatives hedging strategy and the funding cost adjustment disappears.

issuingP1 and/or P2 bonds. The assets follow the dynamics R2 need to equal the derivative recovery rate RB.

As before we setup a replicating portfolio Π given by

Π =α1P1+α2P2+αCPC+δS+βS+βC, (35)

whereβS=−δSis the funding account for the share position andβC =−αCPC is the funding

position for thePC−bonds. The fact that Π is replicating implies that Π = ˆV anddΠ =dVˆ

so repeating the delta hedging arguments of Burgard and Kjaer [1] and definings1 =r1−r

ands2 =r2−r yield

Furthermore, in order to ensure that all positive or negative cash is invested or raised through

P1 andP2 we impose the constraint

α1P1+α2P2= ˆV .

From these equations we can determineα1 and α2 to be

α1 =

which implies the following pricing PDE

∂tVˆ +AtVˆ −rVˆ =s1

which by Section 3.1 implies financing the negative cash account at the risk-free rate r. In particular the strategy involves issuing the senior P2-bonds and use the proceeds to

re-purchase the junior and hence higher yieldingP1−bonds. The excess return generated by

this strategy exactly offsets the additional funding costs, so the net-financing rate becomesr. At the same time, there is no windfall to the bondholders in the case of default of the issuer while V (and the cash account) is negative. Table 2 summarises the total bond position at the issuers own default, which matches perfectly the value of the derivative at that point as defined in equation (4).

P1−position R1R2 ˆ

V−R1M−−R1RBM+

R2−R1 P2−position R2M

−_{+}_{R2R}

BM+−R1R2Vˆ

R2−R1

Total position M−+RBM+

Table 2: Value of the issuer’s P1 and P2 bond positions post own default.

So if the issuer is able to offset the impact of the derivative and its funding on the balance sheet with a combination of going long and short its own bonds of different seniority, then it can reduce the funding cost term to zero.

### 5

### Conclusion

First we showed how valuing derivatives taking funding costs and counterparty risks into account leads to a derivative valuation formula ˆV = V +CV A+F V A. We then propose that the derivative issuer could setup a separate funding desk which task is to manage the FVA in a similar way that counterparty risk trading desks currently manage the CVA. This is followed by a short description on how such a funding desk would operate, including the hedges it would use.

Second we proved that the issuer is able to reduce the FCA to zero by going long and short its own bonds of different seniority.

Acknowledgements

The authors would like to thank Tom Hulme and Vladimir Piterbarg for useful comments and suggestions.

### References

[1] C. Burgard, M. Kjaer. PDE representations of options with bilateral counterparty risk and funding costs. Submitted. http://ssrn.com/abstract=1605307.

[2] J. Gregory. Being Two-faced over Counterpartyrisk.Risk, February 2009.

[3] V. Piterbarg. Funding beyond discounting: Collateral agreements and derivatives pricing.