X-Value adjustments: accounting versus economic management perspectives
XX-Value adjustments: accounting versuseconomic management perspectives
Alberto Elices ∗ September 11, 2020
Abstract
This paper provides a mathematical framework based on the prin-ciple of invariance [1] to classify institutions in two paradigms accord-ing to the way in which credit, debit and funding adjustments are cal-culated: accounting and management perspectives. This conceptualclassification helps to answer questions such as: In which paradigmeach institution sits (point of situation)? Where is the market consen-sus and regulation pointing to (target point)? What are the implica-tions, pros and cons of switching perspective to align with future con-sensus (design of a transition)? An improved solution of the principleof invariance equations is presented to calculate these metrics avoidingapproximations and irrespective of the discounting curve used in FrontOffice systems. The perspective is changed by appropriate selectionof inputs always using the same calculation engine. A description ofbalance sheet financing is presented along with the justification of thefunding curves used for both perspectives.
Disclaimer : the views expressed in this article are exclusivelyfrom the author and do not necessarily represent the views of neitherBank Santander nor its affiliates.
Counterparty Credit Risk in financial derivatives and funding cost have in-creasingly become topics of research since the credit crisis in 2008. After the ∗ Head of XVA Model Validation, Bank Santander, Av. Cantabria s/n, 28660 Boadilladel Monte, Spain, [email protected] . a r X i v : . [ q -f i n . P R ] S e p efault of Lehman Brothers the assumption that financial institutions couldnot default was no longer accepted. This hypothesis has been introduced inthe existing pricing framework by using the previous risk free pricing plusa number of adjustments: the CVA (Credit Value Adjustment) to take intoaccount the default of the counterparty and the DVA (Debit Value Adjust-ment) to account for the default of the institution from which the pricingis carried out (from now on “the bank”). To reduce counterparty creditrisk, collateralization was generalized among big institutions in the followingyears.Collateralization came for the over-the-counter market in the form ofvariation margin to cover the liable mark-to-market in case of default. Addi-tionally, regulators have fostered closing operations through central clearingcounterparties which require posting an additional initial margin to cover themarket variation during the margin period of risk (usually two weeks) fromcounterparty default to liquidation. Regulators have also required higheramounts of initial margin posting for over-the-counter markets to foster clos-ing operations with central clearing counterparties. Collateral can be cash orother type of liquid assets (e.g. bonds, Equity, etc). This mechanics requiredfunding this collateral and the need for additional adjustments such as theFVA or Funding Value Adjustment, to account for the funding costs andbenefits of collateral or the MVA, Margin Value Adjustment, to address thecost of funding of initial margin. Finally, the last adjustment which has beenproposed is the Capital Value Adjustment or KVA which accounts for thefunding costs of capital required by regulators. All these adjustments are alldenominated under the acronym XVA.Since 2008, a continuous debate has taken place among institutions andregulators on how these adjustments should be calculated and reported tocomply with accounting standards and to properly foster internal manage-ment. After more than a decade, the CVA has settled down as an acceptedadjustment, DVA and FVA are still under debate and the rest are still farfrom being standardized.Two major paradigms have been discussed around XVA and in particularabout FVA since 2012. The first one, mainly advocated by Hull and White in[2], considers that the value of a firm defined as the shareholder and creditorvalue does not depend on the funding strategy of the institution according toModigliani-Miller theorem [3]. Therefore the FVA should be zero, derivativeprices are symmetric (as so CVA and DVA are) and obey the law of oneprice. This paradigm can well be used for accounting fair value. Burgard2nd Kjaer study the impact of derivatives and their corresponding fundingpositions in the balance sheet upon the issuer default (see [4] and [5]). Theyhave also proposed ways to mitigate this impact so that funding adjustmentscan be dropped.The second paradigm advocates for the inclusion of the FVA in derivativeprices which are charged to final customers in order not to reduce shareholdervalue for the cost of collateral funding of hedges associated with unsecuredoperations. This paradigm implies wealth transfer from shareholders to se-nior creditors as the valuation of the extra funds required from the latterto fund derivatives (monetized when the bank defaults) equals the fundingvalue adjustment charged to customers (see [6]). This paradigm is morealigned with what practitioners and institutions consider how the manage-ment should be carried out. On the other hand, the funding benefits providedby FVA overlap with DVA profit on bank liabilities when funding benefits arevalued at the internal transfer funding rate of the bank which accounts forthe bank credit spread, already included in DVA (see for instance [8]). In or-der to avoid this double counting, some practitioners prefer to drop DVA andconsider only FVA and others to preserve both but only the liquidity compo-nent of the internal funding spread in the FVA. In fact, this latter approachis a point in between both paradigms because avoiding double counting be-tween DVA and FVA allows both adjustments to live together providing asymmetric price between the counterparty and the bank. This price wouldbe compatible with the law of one price and the accounting regulation (theessence of the first paradigm).This paper does not provide additional support for either paradigm butsimply pursues to accommodate both approaches into a common frameworkwhich can be “configured” to align with the first paradigm, the “accountingperspective” or with the second paradigm, the “management perspective”.The choice of perspective will depend on the decision of the senior man-agement of each institution which will progressively accommodate with theevolution of the market consensus (see section 11) and the regulatory require-ments (see section 10).This paper has four major contributions. The first one provides a frame-work, based on the principle of invariance [1], which conceptually allows clas-sifying institutions into these two paradigms in relation with credit, debit andfunding value adjustment calculation: an accounting and an economic man-agement perspective with intermediate transitions among them (see sections10 and 11). This classification allows a conceptual identification of which3erspective each institution sits on, where is the market consensus settlingdown (section 11 summarizes the consensus surveys [9], [10] and [11]), towhich perspective the regulation is pointing to (section 13), what are theimplications, advantages and disadvantages of moving from one perspectiveto the other (see section 12) and the prospective difficulties of institutions toadjust to the tendency of market consensus and regulation to the manage-ment perspective (see section 14). This conceptual mindset provides a routeguide to realize about the current point of situation, the target point, howto design a transition from one to the other and allowing reporting the stageof completion of this transition.The second contribution provides an improved solution of the CVA/DVAequation with funding based on the mathematical framework of the principleof invariance [1] (it is reviewed in sections 5 and 6). This improved solutioneliminates a circular dependence which appears in the original equation (seesections 8 and 9). This avoids some approximations which indeed violatethe condition of the principle of invariance given by equation (14). In addi-tion, this solution calculates the XVA adjustments in terms of the exposuresdiscounted with whatever rate is available in the system as presented in equa-tions (25) to (27) where no conversion or approximation needs to be carriedout.The third contribution is that the mathematical framework is commonfor both perspectives. Each of them is defined by a choice of a set objects ofselection (see section 7). This allows a smooth transition from one perspectiveto another without having to implement changes in the calculation engine.Finally, the fourth contribution justifies how funding rates can be esti-mated under the assumption that the Financial area is not a profit center.For the accounting perspective the internal funding rate is related with theaverage liquidity spread of the bonds issued by the institution. On the otherhand, for the management perspective the internal funding rate is an averageyield of the issued bonds.The paper starts with a tour around the management of balance sheetfinancing from a descriptive point of view. This includes a description of thefunctions of the financial area and the financial management control (section2.1), the bond issuance department (section 2.2), the short term desk (section2.3) and the Securities Financing desk (section 2.4). Section 3 illustrates witha simple example how to estimate the internal FTP (Fund Transfer Pricing)rate based on the bond issuance activity and various hypotheses which will bethereafter aligned for the derivation of the appropriate FTP rate depending4n the perspective adopted (either accounting or management).Section 4 discusses the interaction among credit, debit and funding valueadjustments in terms of collateral and concludes that the best way to managecollateral is to reach a balance between the positions which generate CVA(unsecured assets in favour of the bank) and the positions which generateDVA (unsecured liabilities in favour of the counterparty). This section is abridge from the first descriptive part of the paper and the second part withthe presentation of the mathematical framework, the two perspectives andtheir interpretation and implications.The presentation of the mathematical framework in section 7 is intro-duced by a review and an interpretation of the principle of invariance withand without default in sections 5 and 6. Sections 8 and 9 present the improvedsolution of the invariance equations of section 6 with and without defaultto calculate credit, debit and funding value metrics avoiding approximationsand irrespective of the discounting curve used in Front Office systems.Sections 10 and 11 justify and motivate the accounting perspective (regu-lation) and management perspective (hedging and market consensus). Eachone is defined by making choices for the objects of selection introduced inthe mathematical framework (section 7). Both perspectives are compared insection 12.Section 13 analyzes the FRTB-CVA regulation and justifies why it ispointing to the management perspective. Finally, section 14 analyzes theimplications of the transition from the accounting to the management per-spective and explains why there are institutions which may be reluctant tochange until the market consensus is clearly settled. Section 15 ends up withsome conclusions.The mathematical derivations are not included in the body of the paperbut left for the appendices. Appendix A shows the derivation of the relationbetween two prices discounted with two different curves. Appendix B showsthe derivation of the equations of the principle of invariance with default riskwhere the invariance rate which can be arbitrarily chosen has not yet beenselected. Finally, appendix C shows the derivation of the improved solutionwhich breaks the circular dependence and expresses XVA metrics in termsof the discounting used in the internal Front Office systems rather than theinvariance rate. Developed by Jrme Maetz, Head of XVA Front Office quant team, Santander Bank. Management of balance sheet financing
In order to properly understand how the balance sheet management of a bankis carried out, this section briefly reviews the main areas and departments inwhich this function is located: the financial area which provides structuralfinancing to the bank, the Issuance Department which allows dynamicallyfitting the financial requirements of the bank, the Short Term desk to managethe daily closeout of cash flows and the Securities Financing desk whichoptimally manages collateral. Section 3 provides a worked example whichprovides an approxiamtion of how the internal Funding Transfer Pricing maybe calculated given some hypotheses.
The main purpose of the Financial area of a bank is to provide structuralshort and long term financing to the bank. The structural financing is neededto support the ordinary businesses of the bank and other corporate operationsfor which liquidity is needed. • Ordinary businesses of the bank : examples of this category arecredit lending (e.g. provide credit to retail/institutional customers,mortgages, etc) derivatives, which also involve a part of liquidity lend-ing or borrowing. • Corporate operations : examples of corporate operations can be toacquire or increase participatio in other entities, real state operationsor mergers.The building blocks of the Financial area are the various areas and de-partments which provide the services to allow raising the structural liquiditythat a bank needs in each moment: • Financial management control : this department calculates and dis-tributes the internal financing rates or FTP (Fund Transfer Pricing)rate. As part of the strategic vision or the bank, some businesses maybe fostered providing lower financing rates and others mitigated withhigher financing rates. This is of course a decision of the senior man-agement of the bank and he actual implementation of this strategicviews are carried out by this department.6
Issuance department : this department issues corporate paper toprovide additional liquidity to close the balance of the bank so thatall financing needs are equal to the financial sources of the bank. Ingeneral, the bank is usually organized to lack liquidity. Therefore, theIssuance Departement keeps constantly issuing corporate paper. • Short Term desk : provides liquidity to daily close-out cash paymentsof every desk in every currency. The surplus or extra liquidity needed isfirst given or provided by the Interbank lending/borrowing and there-after by the internal liquidity of the bank for which regulatory ratiosmust be satisfied. • Securities Financing desk : this desk allows short term asset andcollateral optimization to maximize return and collateral reuse whencollateral is rehypothecable.The management of the Financial area is carried out by collecting all theassets and liquidity from every business of the bank. The main ways the bankcan raise money are the deposits or cash coming from the customers, Equityand debt issuance or Central banks. All sources of liquidity are deposited in acommon account joining liquidity provided and consumed by every business. • Equity issuance : Equity issuance is carried out only from time totime to raise internal capital from shareholders. Preparing an Equityissuance is a relevant event for which many market participants needsto be coordinated (that is why Equity issuance is occasional). Thereare examples in which there is a continuous Equity issuance processsuch as the payment of dividends in form of stock or stock dividends.The bank does not pay dividends and in exchange issues shares to payits shareholders. • Businesses of the bank : most of the financing a bank obtains comesfrom customer deposits from their retail business. The retail businessis carried out through the bank agencies and more lately through thedigital customer relationship. In addition, commercial banking (insti-tutional customers) or corporate banking (corporates) may be otherways of raising financing (sometimes they request for liquidity) for thebank through special bank analysts who are specialized in bigger cus-tomers (e.g. private banking). 7
Central banks : In some situations Central Banks or Central Bankagencies may offer packages of cheaper financing than Capital markets.Examples are Targeted Longer-Term Refinancing Operations (TLTRO)from the European Central Bank or excess of liquidity in various cur-rencies lent by agencies such as SAMA (“South Arabian Monetary Au-thority”). • Debt issuance : there is a big range of debt which may be issued bybanks. Bonds are issued for medium and long terms with a wide rangeof seniority depending on the type of credit quality. Commercial paperis usually issued for short term and covered bonds are bonds backed bymortgages. The issuance process is carried out continuously to matchlack of liquidity and renew expired debt.The budget for the finance area is built one year ahead collecting theneeds and sources of liquidity from every area. This budget is calculatedfrom a conceptual basis. However, there are also financing operations closedinternally from the Financial area for a given time horizon with various busi-nesses of the bank as long as they happen. This operations officially transmitliquidity surplus or requirements so that the Financial area may coordinatewith the Issuance department the provisioning of these extra funds. Thereare also situations in which the Financial area requests the desks (e.g. theEquity desk) to coordinate a campaign to raise liquidity if they anticipatethat the Issuance department may not cope with raising the whole amountof liquidity needed.
The Issuance department is the heart of the dynamic liquidity management.This department closes out sources and needs of medium and long termliquidity. The main objectives of the Issuance department are maintainingregulatory ratios within appropriate levels, providing adjustments to the liq-uidity budget (based on internal term operations closed with the Financialarea) and re-finance expiring debt to modulate in terms of current liquidityneeds or strategic views, the total amount of debt outstanding.The regulatory ratios which are under the scrutiny of the Issuance de-partment so that they are kept within appropriate levels (usually higherthan 100%) are the following: 8
LCR (Liquidity Coverage Ratio): liquid assets over payments in thecoming 30 days. This is the most basic short term ratio. When LehmanBrothers collapsed in September 2008, part of the issue was that theywere getting financing from the Interbank market and it got completelydried (no more borrowing or lending). This ratio was introduced toavoid such situations. • NSFR (Net Stable Funding Ratio): ratio of stable funding (e.g. lia-bilities beyond 1 year maturity). This ratio fosters liabilities which donot expire in such a short term as the LCR. • TLAC (Total Loss Absorbing Capacity): this ratio assures enoughEquity and bail-in debt to pass losses to investors to avoid a governmentbailout. This ratio is to avoid governments to have to take money fromtax payers to rescue financial institutions. • MREL (Minimum Requirement for own funds and Eligible Liabilities):this ratio is more related with the degree of resolvability to absorb lossesand restore capital position during and after a crisis. This ratio is apost default orderly management measure.In order to maintain these regulatory ratios beyond 100%, the Issuancedepartment may have to issue various types of debt with varying maturityand credit seniority. The hierarchy of asset quality and the correspondingcapital impacts are the following: • Equity shares, derivative profit-and-loss not including DVA :these are part of the CET1 (Common Equity Tier 1). Equity sharesand traded derivatives are the most liquid type of assets which can beliquidated in the market at once. The reason why DVA and the restof own credit adjustments cannot be considered part of the CommonEquity Tier 1 capital arises from [18] in which the Basel Committeeon Banking Supervision resolves not to recognize this type of asset asCommon Equity for the profit it provides can only be realized in caseof default of the institution. • Commercial paper : this is a liquid debt issued by the bank to obtainshort term liquidity. In many situations, commercial paper is allowedto be recognized as part of CET1 capital. However, there are situationsin which it is not. 9
Preferred participations : this type of debt is part of what is calledthe AT1 (Additional Tier 1 capital). This is the following layer ofCET1. It is not as liquid as CET1 but it is still not considered tier 2capital. • Subordinated debt : this type of debt has the lowest priority withinthe hierarchy of creditors in case of liquidation or bankruptcy. Subor-dinated debt holders get paid just before stock holders. This type ofdebt is part of T2 (Tier 2) capital, the second layer according to Baselregulation. It is also counted as part of TLAC ratio. • Senior non-preferred debt : this type of debt has higher priority inthe creditor hierarchy than subordinated debt. It is counted for thecalculation of the TLAC ratio. • Senior preferred or unsecured debt : this is the debt at the top ofthe hierarchy of creditor priority. • Covered Bonds : these bonds are usually over collateralized by a poolof mortgages and represent the closest debt to risk-free.As a conclusion, the Issuance department allows raising funds to dynam-ically manage deviations in the financing budget of the bank. Budget devi-ations are formally acknowledged through internal term operations betweenvarious areas of the bank and the Financial area. The wide range of debtissued is optimized to maintain in appropriate levels a number of regulatoryratios. The debt issuance is a continuous process which renews expired debtand can be modulated in the long term to shape the structure of the totaldebt outstanding. Depending on the liquidity of the market, the Issuancedepartment may have to change the strategy ahead of time to be able to copewith the liquidity constraints of the market.
The short term desk daily closes out all the cash positions in every cur-rency for the legal entity. The way in which the short term desk operates issummarized in the following points: • When the bank needs to carry out payments and it does not havemoney, the short term desk borrows it on a daily basis from the In-terbank market to accomplish all payments. When there is surplus of10ash, they lend it to the Interbank market. Daily overnight interbanklending/borrowing is carried out at OIS rate to match excess/lack ofliquidity. • When the liquidity of the Interbank market dries out, the short termdesk takes the liquidity from the common account where all sourcesof financing are collected. Daily payments can never be failed as thiswould technically trigger a default. Therefore, the common accountmust be there as a backup to never run out of liquidity. • Shortage in the common account triggers debt issuance through regula-tory ratios. When the short term desk starts to take liquidity from thecommon account, this deteriorates the regulatory capital ratios. There-fore, the Issuance department will need to issue more debt to restorehealthy levels for these ratios. • Liquidity in currencies other than the mayor currency of the bank (e.g.EUR for European, USD for North American or JPY for Japanesebanks) is obtained by borrowing in the major currency at the inter-bank market OIS rate and exchanging it to the required currency bymeans of foreign exchange spot or forward (very short term) operations.Therefore, the effective rate is the OIS rate plus the cross currency basisspread. Penetration in other currencies could make it cheaper to ob-tain liquidity in currencies different from the major one. However, inpractice, most of the liquidity in foreign currencies is obtained throughspot/forward operations.
This desk optimizes a pool of assets and collateral to maximize return andmatch LCR (Liquidity Coverage Ratio) and collateral requirements at thelowest possible cost. The tasks and machinery carried out by this desk arethe following: • Asset rotation : this process consists of leveraging by exchanging highfor low quality assets for a pick-up spread and de-leveraging by gettingback the high quality assets by paying the spread back. The leveragingprocess allows reusing assets received as collateral when they are notneeded to get some extra return and the de-leveraging process allows11o rescue some assets which at a given point in time may be needed topost as collateral or to increase regulatory ratios. Leveraging increasesreturn but reduces liquidity (the liquid assets are temporarily sold orexchanged) and de-leveraging improves liquidity and LCR ratio butreduces return. • Asset lending/borrowing : agreed among institutions for received orpaid pick-up spread from funds or banks. This is part of the machinerydeveloped by the Securities Financing desk to increase support andcoverage from other institutions in case collateral or other assets maybe needed in the present or future. • Internal operations : closed with desks or other areas of the bankto get or request liquidity for specific terms. Sometimes the derivativedesks are fostered to get liquidity when the market situation may not befavorable to obtain it by promoting derivative campaigns to customersthrough the retail network. • Pool of assets : the Securities Financing desk has access to an ex-tensive pool of assets such as bonds, letters, commercial paper, Equityassets, etc. This pool of assets has to be optimized so that the bankcan post appropriate collateral or have assets available for regulatoryratio computation. This desk is more dynamic than the Issuance de-partment and may help it to maintain some of the regulatory ratioswithin appropriate levels on a periodic basis. • Collateral reuse : the asset rotation by the lending/borrowing ma-chinery allows raising the ratio of collateral reuse up to more than80/90%. This is for the collateral which is re-hypothecable and there-fore can be reused. Segregated or non-rehypothecable collateral cannotbe touched and it is usually kept inside a custodian entity. • Typical operations : the range of operations which the Securities Fi-nancing desk usually closes includes repo and reverse repo transactions,total return swaps, security lending, futures and some options.Some institutions reuse extra collateral coming out of liability porfoliosby posting it as initial margin. This is an example of how to reuse collateralin an efficient way. 12
Estimation of Fund Transfer Pricing rate
This section shows a simple and intuitive approximation to determine thefinancing costs of a bank based on the bond issuance process under twodifferent hypotheses. The first one assumes that the bank can default andonly the recovery of the outstanding debt will be paid back to bond holders.The second one will consider that the bank cannot default.Consider a bank B which issues bonds on dates t i expiring on T i withnotional F i,t = F i · { t i
0) for assets and negative bond price (
B <
0) forliabilities of the Financial area. With this criterion, the internal bonds arepositive (the Financial area has to get paid from the desks) and issued bonds13re negative (the Financial area has liabilities with the bond holders). Theterm T = max { T i , i = 1 · · · N B } refers to the longest expiry of issed debtand NB refers to the Number of Bonds issued by the bank. F A t = NB (cid:80) i =1 B i,t − NB (cid:80) i =1 B Bi,t == (cid:82) Tt p rt,s Q Bt,s (cid:40) NB (cid:80) i =1 F i,t (cid:16) r F T Pt i ,s − r Bt i ,s + (cid:16) − R Bs (cid:17) λ Bt,s (cid:17)(cid:41) ds (4)Assuming that the total debt issued is F t = NB (cid:80) i =1 F i,t , the weight of eachbond issuance is w i,t = F i,t /F t and the CDS spread of the bank on date t is CDS
Bt,s = (cid:16) − R Bs (cid:17) λ Bt,s , the balance given by equation (4) turns intoequation (5). For CDS quotation purposes, sometimes the recovery rate isset to a given standard value (e.g. R = 0 . λ Bt,s hasthe information of both the recovery ratio and the default intensity. However,for the purpose of this discussion, R Bs is the varying recovery rate at time s . See that equation (5) adds and subtracts the CDS spread, CDS Bt i , at theissuance time t i : F A t = F t · (cid:90) Tt p rt,s Q Bt,s NB (cid:88) i =1 w i,t (cid:32) r F T Pt i ,s − r Bt i ,s + CDS
Bt,s + CDS Bt i ,s − CDS Bt i ,s (cid:33) ds (5)Now the yield of the issued bond can be decomposed into OIS rate, r OISt i ,s ,plus CDS spread and liquidity spread, l Bt i ,s , according to equation (6): r Bt i ,s = r OISt i ,s + (cid:16) − R Bt i (cid:17) λ Bt i ,s + l Bt i ,s = r OISt i ,s + CDS Bt i ,s + l Bt i ,s (6)If equation (6) is replaced in equation (5), the expression of equation (7)is obtained where the CDS Bt i inside r Bt i ,s cancels with the one from equation(5). F A t = F t · (cid:90) Tt p rt,s Q Bt,s (cid:40) NB (cid:88) i =1 w i,t (cid:32) r F T Pt i ,s − r OISt i ,s − l Bt i ,s + CDS Bt − CDS Bt i (cid:33)(cid:41) ds (7)Finally, for the Financial area not to be a profit center, namely F A t = 0,the following average FTP rate, ¯ r F T Pt , follows, where the rest of averagevalues are provided by equation (9):¯ r F T Pt,s = ¯ r OISt,s + ¯ l Bt,s + CDS Bt − CDS Bt ≈ ¯ r OISt,s + ¯ l Bt,s (8)14 r F T Pt,s = NB (cid:80) i =1 w i,t r F T Pt i ,s ¯ r OISt,s = NB (cid:80) i =1 w i,t r OISt i ,s ¯ l Bt,s = NB (cid:80) i =1 w i,t l Bt i ,s CDS Bt = NB (cid:80) i =1 w i,t CDS Bt i (9)The conclusions which can be extracted from equation (8) are the follow-ing: • Increasing
CDS Bt implies a profit for the Financial area because ithas to pay less recovery to the bond holders. This means that whenthe credit quality of the bank deteriorates, the internal Fund TransferPrice, FTP, gets reduced due to this profit. • Assuming that the CDS spread is within the internal average, the FTPrate so that the financial area is not a profit center essentially dependson the average liquidity spread of already issued bonds, ¯ l Bt,s . This con-clusion is also reached by [7] and [8].
F A t = F t · (cid:90) Tt p rt,s NB (cid:88) i =1 w i,t (cid:16) r F T Pt i ,s − r Bt i ,s (cid:17) ds (10)¯ r Bt,s = NB (cid:88) i =1 w i,t r Bt i ,s (11)Under the hypothesis that the bank cannot default, equation (5) turnsinto equation (10) where the survival probability of the bank, Q Bs is equalto 1 and the CDS spread disappears as it has been assumed that the defaultintensity of the bank is zero: λ Bt,s = 0. For equation (10) to be zero (theFinancial area is not a profit center), it implies that the average FTP rate isequal to the average yield of the issued bonds as given by equation (11). Theconclusion is that considering that the bank can default makes the FTP ratedependent on liquidity and if it is assumed that the bank cannot default, theFTP rate is the average yield of issued bonds.
This section illustrates that credit, debit and funding value adjustments donot behave independently but are closely related to each other. Having into15ccount this inter-relation clarifies issues which will be reviewed later on insections 10 to 12.First of all, positions that generate CVA or DVA are related with their col-lateral requirements. CVA generating positions (those which increase CVA)are positions with positive mark-to-market in favor of the bank (assets) whichare uncollateralized, partially collateralized or unilateral where counterpartydoes not post collateral. Therefore, these positions are assets with positivemark-to-market in favor of the bank which get hedged with collateralizedliabilities. The payments of the assets which represent the CVA generatingpositions cancel the liabilities of the hedges closing out the positions. How-ever, the bank has to post collateral for the hedges as they are collateral-ized liabilities and this collateral cannot be taken from counterparties (thosetransactions are uncollateralized). In conclusion, CVA generating positionsconsume collateral.DVA generating positions on the other hand are positions with negativemark-to-market in favor of the counterparty (liabilities of the bank) whichare uncollateralized, partially collateralized or unilateral where the bank doesnot have to post collateral. Therefore, these positions are liabilities withnegative mark-to-market in favor of the counterparty which get hedged withcollateralized assets. The payments of the hedges will cancel the liabilitiesof the DVA generating positions closing out the positions. However, thebank receives collateral from these hedges (they are collateralized assets)which does not have to post for its liabilities (they are uncollateralized). Inconclusion, DVA generating positions provide collateral. The Equity businessis an example of this situation. This business sells options to uncollateralizedcustomers (e.g. corporates or private banking) and gets the premium. Thispremium is invested in hedges to offset the profit and loss and the hedgesbecome assets for the bank. As they are collateralized, the premiums paidto establish the hedges are given back in form of collateral. Therefore, theoverall effect of the Equity business is to provide liquidity to the bank inform of collateral.If re-hypothecation is assumed, the collateral provided by DVA generat-ing positions can be reused for the CVA generating positions by means of theSecurities Financing desk. If the collateral consumed by the CVA generatingpositions is higher than the collateral provided by the DVA generating posi-tions, this collateral must be raised by the desk by borrowing money from theFinancial area. This situation is graphically presented in figure 1. If, on thecontrary, the amount of collateral provided by DVA generating positions is16igure 1: Balance between CVA and DVA generating positions and the Fi-nancial area.bigger than the collateral consumed by the CVA generating positions, thereis a surplus of collateral which the desk may try to lend to the Financial area(however, this may not always be possible). Therefore, the most optimalmanagement would consist of balancing the amount of collateral providedby DVA generating positions to fulfill the collateral comsumption by CVAgenerating positions. If this balance is achieved, then neither extra collateralis needed nor there is excess of it. Therefore, it is key to balance CVA andDVA generating positions to minimize collateral requirements.An imbalance between CVA and DVA generating positions leads to lackor excess of collateral which has to be raised or remunerated producing afinancing cost/benefit. However, the following considerations have to betaken into account in relation with this lack/excess of collateral: • Generally speaking, funding benefit obtained from surplus of collateralcan only realize the OIS rate (no benefit) unless the collateral excessmay reduce bond issuance (this needs a lot of coordination with theFinancial area which in practice does not happen) or be reused in repos,initial margin or other instruments through the Securities Financingdesk. This increases the return of this collateral but most likely notas much as the internal financing cost which would be achieved byreducing bond issuance. • Funding costs are always realized at the internal bank financing costwhich is generally speaking rather expensive. Therefore, this situationshould be avoided as much as possible.17
CVA/DVA imbalance increases the sensitivity to the financing curveand the need to hedge it. For instance the situation of many banksis an excess of CVA generating positions. These positions consumecollateral which has to be raised by the Financial area at a high cost.Therefore, if the funding curve moves up (increase of funding cost),there is a big loss as funding costs get higher. • Increasing CVA rises counterparty debt which reduces internal recoveryrate (higher CDS spread). If the CVA/DVA imbalance effectively pro-vides liquidity to the counterparties (uncollateralized) at the expenseof the Financial area, the bank increases its counterparty default riskas these counterparties may default. • Increasing DVA rises bank debt leading to potencial systemic risk.From a conceptual point of view, there is nothing wrong by the factthat the bank increases its liabilities if they will be paid back at somepoint in the future. However, if the bank defaults at some time, thoseliabilities may produce systemic risk. Therefore, regulators may not bevery happy about increasing liabilities of a bank.In conclusion, the best way to manage the collateral produced by CVAand DVA generating positions is to balance them so that neither extra col-lateral nor surplus is needed.
This section reviews the formulation of the funding invariance principle forwhich the complete derivation may be found in [1]. Consider first the com-plete set of cash flows ˜ C s as seen from the desk in equation (12). d ˜ C r ∗ s = dC s + (cid:16) r ∗ s − r Cs (cid:17) M s ds + (cid:16) r ∗ s − r Fs (cid:17) ˆ F s ds (12)This equation considers a portfolio of derivatives closed with differentcounterparties which exchange cash flows with the desk. The symbols in thisequation mean the following: • C s : cummulative process of contingent cash flows already received( dC s >
0) from desk or paid ( dC s <
0) by desk to the counterparty.These contingent cash flows represent the payments along the life of18erivatives. This process has a constant evolution until a cash flowis received or paid by the desk to the counterparty. Figure 2 showshow this process looks like. On the left hand side a positive but in-creasing process indicates cash flows which have been received alongthe life of the contract (e.g. a vanilla cap bought by the bank) fromthe counterparty. On the right hand side a decreasing but negativeprocess represents cash flows already paid by the desk (e.g. a vanillacap sold by the bank) to the counterparty. The integral of this processexperiences positive jumps for coupon payments from counterparty todesk and negative ones for payments from desk to counterparty. • r ∗ s : symmetric and arbitrarily chosen money market rate at which thedesk internally borrows/lends money either to counterparties or to theFinancial area. • M s : re-hypothecable collateral deposited in desk ( M s ¿0) or posted bydesk ( M s <
0) to counterparties. It gets remunerated at r Cs . Figure 2shows that for options bought by the bank (left hand side), the deskpays a premium and gets a contract, V t , which is positive (the coun-terparty owes the payoff to the desk) and therefore part of that debt isposted by the counterparty to the desk in form of collateral, M t , whichis also positive. For options sold by the bank (right hand side), the deskreceives a premium and delivers a contract, V t , which is negative (a li-ability of the bank) and therefore part of that liability is posted by thedesk to the counterparty in form of collateral, M t , which is negative. • ˆ F s : funding borrowed ( ˆ F s >
0) or lent ( ˆ F s <
0) by desk from/to Fi-nancial area. It gets charged ( ˆ F s >
0) or remunerated ( ˆ F s <
0) therate r Fs (possibly asymmetric: borrowing at FTP and lending at OIS).The value of ˆ F t > F t < F s = ˆ V s − M s , where ˆ V s is the risk neutral price of the derivativeor netting set under analysis plus the valuation adjustments (collateral,funding, credit and debit).Equation (12) states that the increments of the total cummulative cash19igure 2: Cash flow process for an option bought (left) and sold (right) bythe desk to a counterparty along with sign for derivative price, V t , collateralbalance, M t and funding account, F s .flow process seen by the desk, ˜ C s , is equal to the increments of the cummu-lative cash flow process of the derivative, C s , plus the cash flows to fund thecollateral received, M s > r ∗ s and payingthe collateral remuneration rate r Cs ), plus the cash flows to fund the liquiditygiven to counterparties not covered by collateral, ˆ F s = ˆ V s − M s >
0, whichis remunerated at the fictitious rate, r ∗ s , and pays r Fs , the remuneration rateof the Financial area. V r ∗ t = E t (cid:34)(cid:90) Tt p r ∗ t,s dC s (cid:35) (13)ˆ V t = E t (cid:104)(cid:82) Tt p r ∗ t,s (cid:16) dC s + ( r ∗ s − r Cs ) M s + ( r ∗ s − r Fs ) ˆ F s (cid:17) ds (cid:105) = V r ∗ t + E t (cid:104)(cid:82) Tt p r ∗ t,s (cid:16) ( r ∗ s − r Cs ) M s + ( r ∗ s − r Fs ) ˆ F s (cid:17) ds (cid:105) (14)ˆ F s = ˆ V s − M s (15)The funding invariance principle [1] is obtained integrating equation (12)multiplied by p r ∗ t,s , taking conditional expectation ( E t [ · ]) and replacing equa-tion (13) which relates the risk neutral price discounted with rate r ∗ , with thecummulative process of contingent cash flows. This yields the equation (14)of the invariance principle. This principle states that equation (14) holdsirrespective of the choice of r ∗ s provided that equation (15) holds, namely,the money borrowed/lent from derivatives not covered by collateral must betaken from the Financial area. This means that the net amount of cash flows20oming to the desk from the counterparty, the Financial area and the col-lateral account is zero. Therefore, their remuneration rate, r ∗ s , is irrelevantbecause a zero balance of net cash flows times any rate is still zero. The invariance funding principle including default risk needs to include thecash flows corresponding to either the counterparty or the bank default. How-ever, it is based on the same idea: the financing provided to counterpartiesnot covered by collateral must come from the Financial area: ˆ F s = ˆ V s − M s .Including credit risk yields equations (16) to (19) which are derived in ap-pendix B (see also p. 20 of [7]), where d { τ C ≤ s } is the counterparty defaultindicator at time s further explained in equation (52) of appendix B, ˜ V s isthe exit price on default, R Cs is the recovery of the counterparty, τ C and τ B are respectively the times to default of the counterparty and the bank and τ = min ( τ C , τ B ) is the first to default time. See that as long as ˆ F s = ˆ V s − M s holds, r ∗ s can be arbitrarity chosen.ˆ V t { τ>t } = V r ∗ t { τ>t } + ColV A r ∗ t + F V A r ∗ t + CV A r ∗ t + DV A r ∗ t V r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s dC s (cid:105) (16) ColV A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s M s (cid:16) r ∗ s − r Cs (cid:17) { τ>s } ds (cid:105) F V A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s ˆ F s (cid:16) r ∗ s − r Fs (cid:17) { τ>s } ds (cid:105) (17) CV A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Cs + M s − V r ∗ s (cid:17) { τ B >s } d { τ C ≤ s } (cid:105) DV A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Bs + M s − V r ∗ s (cid:17) { τ C >s } d { τ B ≤ s } (cid:105) (18) H Bs = ( ˜ V s − M s ) + + R Bs ( ˜ V s − M s ) − H Cs = R Cs ( ˜ V s − M s ) + + ( ˜ V s − M s ) − (19)The Collateral Value Adjustment (ColVA) and funding value adjustment(FVA) terms provided by equation (17) come from equation (14). The onlychange is that the calculation is conditioned by the fact that neither thebank nor the counterparty default (see the indicator function { τ>s } insidethe integrands).The Credit Value Adjustment (CVA) and Debit Value Adjustment (DVA)terms provided by equation (18) account for the cash flows when either the21ank or the counterparty default. The terms H Cs and H Bs given by equation(19) and added to the collateral account, M s , represent the default paymentsof the counterparty and the bank at time s . The probability of a joint default(both the counterparty and the bank defaulting simultaneously) is neglected.See that when either the counterparty or the bank defaults, the collateral bal-ance is always part of the default payment (the posted or received collateralis never returned back in case of default).Equation (19) shows that when the counterparty defaults, it only returnsback the recovery fraction of the money it owes to the desk: R Cs ( ˜ V s − M s ) + .See that ˜ V s − M s may be positive for two reasons: either the liquidity providedto the counterparty ( ˜ V s >
0) is not completely covered by the collateralreceived by the desk ( ˜ V s > M s >
0) or the liability of the bank to thecounterparty ( ˜ V s <
0) is overcollateralized ( M s < ˜ V s <
0) and from theexcess of posted collateral, only the recovery gets returned. See that themoney owed by the desk to the counterparty, ( ˜ V s − M s ) − , is always fullyreturned back when counterparty defaults.Equation (19) also shows that when the bank defaults, it only returns backthe recovery fraction of the money it owes to the counterparty: R Bs ( ˜ V s − M s ) − .Again, ˜ V s − M s may be negative for two reasons: either the liability of thedesk to the counterparty, ˜ V s <
0, is not completely covered by the collateralposted by the desk, ˜ V s < M s <
0, or the money owed by the counterpartyto the desk, ˜ V s >
0, is overcollateralized ( M s > ˜ V s >
0) and from the excessof collateral posted to the desk by the counterparty, only the recovery getsreturned when the bank defaults.
CV A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Cs + M s − V r ∗ s (cid:17) E s [ { τ B >s } d { τ C ≤ s } ] (cid:105) DV A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Bs + M s − V r ∗ s (cid:17) E s [ { τ C >s } d { τ B ≤ s } ] (cid:105) F V A r ∗ t = E t (cid:104)(cid:82) ∞ t p r ∗ t,s ˆ F s (cid:16) r ∗ s − r Fs (cid:17) E s [ { τ>s } ] ds (cid:105) (20) E s [ { τ B >s } d { τ C ≤ s } ] ≈ E s [ { τ B >s } ] E s [ d { τ C ≤ s } ] = e − (cid:82) st ( λ Cu + λ Bu ) du λ Cs ds E s [ { τ C >s } d { τ B ≤ s } ] ≈ E s [ { τ C >s } ] E s [ d { τ B ≤ s } ] = e − (cid:82) st ( λ Cu + λ Bu ) du λ Bs ds E s (cid:104) { τ>s } (cid:105) ≈ E s [ { τ C >s } ] E s [ { τ B >s } ] = e − (cid:82) st ( λ Cu + λ Bu ) du (21)Finally, equation (18) shows the CVA and DVA terms which are dis-counted with the arbitrarily chosen rate, r ∗ , multiplying by p r ∗ t,s . These pay-ments are weighted by the indicator functions corresponding to the first to22efault being the counterparty for CVA ( { τ B >s } d { τ C ≤ s } ) and the first todefault being the bank for DVA ( { τ C >s } d { τ B ≤ s } ). An example in which thefirst-to-default effect may be important can be illustrated by the RegionalCommunities of Spain. Their default is highly correlated with a default ofa Spanish bank. If the Regional Community defaults it is likely that thegovernment of Spain has already defaulted and this would also trigger thedefault of the Spanish bank.It is possible to apply the expectation conditioned by time s to the indi-cator functions of equations (17) and (18) applying the Tower law as shownby (20) as the rest of integrand terms are measurable by time s and canget out of the expectation. Equations (21) show the expectations of theseindicator functions in terms of the hazard rates or default intensities , λ Bu and λ Cu , of the bank and the counterparty assuming independence betweenthe counterparty and the bank. The contribution of the indicator functionsin terms of the harzard rates for the Credit, Debit and Funding value adjust-ment formulas have also been derived by [13] in the context of the resolutionof a partial differential equation. The most general case considers that haz-ard rates and the rest of risk factors are jointly simulated to account forthe first-to-default effect and the correlation between market exposure anddefault. The standard CVA/DVA computation usually considers determin-istic hazard rates calibrated at time t . This implies ignoring the correlationbetween market exposure and default (so called wrong/right way risk) andthe first-to-default effect. Some authors have analyzed the impact alone ofthe first-to-default effect by the use of a copula approach (see [14] for moreinformation). This section presents the general framework to consistently calculate credit,debit and funding value adjustments based on the principle of invariance (see[1] and [7]). This framework is partially open to some objects of selection andleads to a family of perspectives which are compatible within the framework.These perspectives focus on how to account and manage these adjustments.The common framework is defined by equations (16) to (19) where dif-ferent perspectives may be achieved by choosing various objects of selection.Although several perspectives may be included, consistency among them is Probability of default at time u provided survival prior to time u . • The way to solve the equation (16): there is a circular relationshipas ˆ V s , which includes the adjustments, depends on ˆ F s which dependsitself on ˆ V s . There are various ways to avoid this problem. Sections 8and 9 address the particular way this circular dependence is avoidedfor this framework. • Curve selection : there are a number of curves which can be selected:the curve associated with the principle of invariance, r ∗ s , the collateralremuneration rate, r Cs , and the funding rate charged by the Financialarea, r Fs . • Exit price : the selection of ˜ V s which is assumed to be the price atwhich the derivatives can be sold to unwind the positions in case ofdefault. This selection may be conditioned by the way the circulardependence in equation (16) is solved. The exit price is sometimescalculated under the same valuation assumption as the variation margin(e.g. OIS or risk-free discounting). However, many netting agreementsassume that liquidation is carried out at the replacement cost as valuedby a third party. This would include the default probability of thesurviving party . • Survival probabilities : equations (17) require bilateral survival ofthe bank and the counterparty (see factor τ>s ) and equations (18)consider the first-to-default. Therefore, it has to be chosen which ofthese hypotheses are included in the formulas to calculate the XVAadjustments. • Collateral optimisation : depending on whether the collateral canbe re-hypothecated and the ability of reusing collateral by the Securi-ties Financing desk (see section 2.4), various options are available to A particular case for which there would be risk for the bank in case of counterpartydefault would be for a position highly in favour of the bank with an unilateral CSA agree-ment where the counteparty does not post collateral. If the exit price is OIS discounted,the bank would expect to receive more as the replacement cost is always lower than therisk free because the default of the surviving party (the bank) would reduce the price. r Fs . These choices may reflect accounting ormanagement perspectives. – Assume that funding rates are symmetric (positive and negativebalance of collateral gets remunerated/charged the same rate) andcommon across counterparties. In this situation, the correct cal-culation considering a single legal entity funding set aggregationlevel including all the deals of every netting set, see equation (30)is equivalent to the sum of the funding adjustments at netting setlevel as expressed in equation (31). The latter approach is themost common among institutions as their systems are not pre-pared to consider a single funding set for the whole institutioninstead of considering separate funding sets for each netting set.Adding metrics at netting set level to obtain metrics at legal en-tity level is always bigger than directly calculating metrics at legalentity level. – When collateral can be re-hypothecated and reused among allcounterparties, it is more reasonable to assume a symmetric fund-ing rate for management purposes under the assumption that thecollateral provided by every CSA aggrement is jointly managedby the Securities Financing desk and the funding rate is set to theaverage rate which considers how excess of collateral is optimizedto reduce funding cost. A symmetric rate may also be consideredwhen the bank decides a rate to mark the funding adjustment foraccounting purposes (e.g. the liquidity curve shown in section 3). – When collateral is not re-hypothecable or rates are not assumedto be symmetric (e.g. there is not a Securities Financing deskoptimizing collateral), then from a management point of view,the aggregation of FVA and ColVA has to be directly aggregatedat legal entity level considering a single funding set as indicatedby equation (31), where positive collateral balance is remuneratedat OIS and negative collateral balance gets charged the internalcost of the bank.These objects of selection share a common framework which is compatiblewith various uses and perspectives. The bank has to decide which perspec-tives are most convenient for best internally manage the bank or most satisfyregulation for accounting purposes. 25
Avoiding circular loop without default This section shows how the circular dependence presented in section 7 hasbeen solved for the invariance principle without default. The case with de-fault is described in section 9.To solve the circular dependence, the rate associated with the principle ofinvariance, r ∗ s , is chosen to be r Fs . This achieves cancelling the term involvingˆ F s in equation (14) and the circular dependence is avoided. However, thisis at the cost of having to discount every derivative in the systems with thesame rate r Fs : V r F t = E t (cid:104)(cid:82) Tt p r F t,s dC s (cid:105) according to equation (22).ˆ V t = V r F t + E t (cid:34)(cid:90) Tt p r F t,s (cid:104)(cid:16) r Fs − r Cs (cid:17) M s (cid:105) ds (cid:35) (22)As deals are discounted in valuation systems (sys) with another rate, r sys ,equation (23) relates r F and r sys discounting (see section 6.3 of [6]). Thisequation has nothing to do with the invariance pricinple. It simply relatestwo prices discounted with different rates. Appendix A shows the derivationof this equation. V r F t = V r sys t + E t (cid:34)(cid:90) Tt p r F t,s V r sys s (cid:16) r syss − r Fs (cid:17) ds (cid:35) (23)If equation (23) is replaced in equation (22) and assuming that F r sys s = V r sys s − M s , the equation (24) is obtained, which avoids the circular de-pendence and also allows discounting with the rate, r syss , used for markingpurposes within the internal pricing systems:ˆ V t = V r sys t + E t (cid:34)(cid:90) Tt p r F t,s (cid:110) M s (cid:16) r syss − r Cs (cid:17) + F r sys s (cid:16) r syss − r Fs (cid:17)(cid:111)(cid:35) (24)This equation calculates the collateral and funding value adjustmentsavoiding the circular dependence and using directly the prices of the deriva-tives which are already available in the pricing systems. This section presents the solution to avoid the circular loop but consideringdefault risk. This result is presented here under the assumption that the26xit price in case of default is assumed to be the price marked in corporatesystems: ˜ V s = V r sys s . Under this hypothesis, the solution is given by equations(25) to (27). Appendix C shows the derivation of these equations from theequations (16) to (19) of the general framework presented in section 7.ˆ V t { τ>t } = V r sys t { τ>t } − ColV A syst − F V A syst − CV A syst − DV A syst (25)
ColV A syst = E t (cid:104)(cid:82) ∞ t p r F t,s M s (cid:16) r Cs − r syss (cid:17) τ>s ds (cid:105) F V A syst = E t (cid:104)(cid:82) ∞ t p r F t,s F r sys s (cid:16) r Fs − r syss (cid:17) τ>s ds (cid:105) (26) CV A syst = E t (cid:20)(cid:82) ∞ t p r F t,s (cid:16) − R Cs (cid:17) (cid:16) V r sys s − M s (cid:17) + { τ B >s } d { τ C ≤ s } (cid:21) DV A syst = E t (cid:20)(cid:82) ∞ t p r F t,s (cid:16) − R Bs (cid:17) (cid:16) V r sys s − M s (cid:17) − { τ C >s } d { τ B ≤ s } (cid:21) (27)See that the adjustments provided by equations (26) and (27) must bediscounted using the rate chosen for the principle of invariance, r ∗ s = r Fs .Now, F r sys s can be easily calculated as it depends on V r sys s which is availablein the systems. Equation (27) shows the well known formulas for CVA andDVA depending on the recovery rates of counterparty and bank, R Cs and R Bs , but with an exit price discounted with the discounting rate used in thecorporate systems. This is the major hypothesis of the particular solutionwhich has been chosen to solve equation (16).Equations (25) to (27) are defined at deal level. However, the aggregationlevels which are considered in practice are CSA, netting set and legal entity.Equation (28) shows the variation margin calculated at CSA level for thedeals corresponding to the perimeter of the CSA and at netting set level forthe CSA contracts included in the netting set. M NS i s = (cid:80) CSA j ∈ NS i M CSA j s M CSA j s = (cid:80) Deal k ∈ CSA j M Deal k s (28)Equation (29) shows the mark-to-future position, V NS i s , at netting setlevel and the funding position, F NS i s , in terms of the positions of mark-to-future and variation margin, M NS i s , at netting set level. See that V NS i s includes the deals of every CSA contract and the deals which are outside theCSA agreements of the netting set considered.27 NS i s = V NS i s − M NS i s V NS i s = (cid:80) Deal k ∈ NS i V Deal k s (29)Equation (30) presents the funding position, F LEt , and the variation mar-gin position, M LEs at legal entity level. They are obtained by summing acrossnetting sets. F LEt = (cid:80) NS i ∈ LE F NS i s M LEs = (cid:80) NS i ∈ LE M NS i s (30)CVA and DVA are defined at netting set level. Their calculation is carriedout using the mark-to-future and variation margin positions at netting setlevel. Namely, replacing V r sys s by V NS i s and M s by M NS i s in equation (27).FVA and ColVA are defined at legal entity level. The calculation is carriedout using the funding and variation margin positions at legal entity level.Namely, replacing F r sys s by F LEs and M s by M LEs in equation (26).In practice FVA and ColVA are calculated at netting set level by usingthe funding and variation margin positions at netting set level. Namely,replacing F r sys s by F NS i s and M s by M NS i s in equations (26). The legal entitylevel is thereafter obtained by summing FVA and ColVA across every nettingset of the legal entity according to equation (31). F V A
LEt = (cid:80) NS i ∈ LE F V A NS i s ColV A
LEs = (cid:80) NS i ∈ LE ColV A NS i s (31)See that the aggregation provided by equation (30) is only true when therates r Cs and r Fs are symmetric (the same irrespective of the sign of F r sys s or M s ). If this is not the case, the correct calculation should be carried outdirectly at legal entity level. This is something that the systems of mostinstitutions are not prepared for and constitutes an important limitationespecially for the calculation of FVA (Funding Value Adjustment) due tothe fact that without a Securities Financing desk the excess of collateral( F r sys s <
0) is very likely to earn just the OIS rate rather than r F .
10 Definition of accounting perspective
This section motivates and illustrates the objects of selection described insection 7 for an accounting perspective. The accounting perspective is mo-tivated by the current regulation which requires including CVA and DVA28nd maximize observable inputs (e.g. cost of own funding) for fair valuecalculation: • Fair value measurement of derivative instruments should include thecredit risk of their counterparty as well as their own credit risk (seeparagraphs 11, 42, 43 and 56 of [15] and section 1.1 of [16]). • Methods for fair value measurement should maximise the use of observ-able inputs and minimise the use of unobservable inputs (paragraph 61of [15] and section 1.1 of [16]).In addition, the current regulation requires to de-recognize DVA fromcalculation of Common Equity Tier 1 capital (CET1): • The Basel Committee on Banking Supervision establishes with regardto derivative liabilities, to de-recognise all accounting valuation adjust-ments arising from the bank’s own credit risk in the calculation ofCommon Equity Tier 1 capital (see [17] and [18]).Given these premises, a proposal for the objects of selection under anaccounting perspective could be the following: • Solution of equation (16): r ∗ s = r Fs and ˜ V s = V r sys s which yieldsequations (25) to (27) as explained in section 6. • Curve selection : r Fs = ¯ r OISt,s + ¯ l Bt,s . The implications of this choice arethe following: – The funding rate from the Financial area is set to the Fund Trans-fer Pricing rate, r Fs = ¯ r F T Pt,s , according to equation (8). This se-lection, as discussed in section 3, is set so that the Financial areapays the recovery of the issued bonds in case of default of the bankand the Financial area is not a profit center. – The average liquidity, ¯ l Bt,s , of equation (8) can be replaced by amarket proxy (e.g. covered bond, bond-CDS basis of a basketof market representative bonds). Therefore, this liquidity curvewould be observable. – The credit default swap (CDS) contribution is also observable andnegligible, assuming that there is stability of the CDS. Namely,current CDS is similar to the past average along the bond issuancehistory. 29
With this selection there is no overlap between DVA (only con-siders pure credit curve) and FVA (only consider a pure liquiditycurve). This non-overlapping idea is also discussed by [8]. • Exit price : it is set to the internal marking in systems: ˜ V s = V r sys s .This choice is conditioned by the solution derived in section 6. • Survival probabilities : unilateral survival for CVA/DVA. This im-plies that CVA only considers the survival of the counterparty (theindicators { τ B >s } d { τ C ≤ s } of equation (27) are replaced by d { τ C ≤ s } )and the DVA only considers the survival of the bank (the indicators { τ C >s } d { τ B ≤ s } of equation (27) are replaced by d { τ B ≤ s } ). Althoughthis selection is not consistent with the framework (both survival ofcounterparty and bank should be considered), it complies with ac-counting regulation (adjustments arising from own credit risk of thebank must be de-recognized from CET1 capital) and it is still bilateralwhere price agreement may still be achieved if the counterparty alsoconsiders unilateral CVA/DVA. • Collateral optimisation : collateral is not optimised. Funding setsare defined at counterparty level with a collateral remuneration ratecommon for the whole legal entity, symmetric and equal to the liquiditycurve given by equation (8). This choice is observable and equivalent toconsidering a single funding set for the legal entity and ColVA and FVAmetrics would be calculated at netting set level and summed together.The rate r Cs for the calculation of ColVA would simply consider theremuneration of collateral of each CSA, which is usually symmetric.The main contribution of this perspective is to report XVA metrics com-parable among institutions. Therefore, they need to be bilateral so thatboth the counterparty and the bank may agree on them and they have touse observable market data.
11 Definition of management perspective
This section addresses a perspective of calculating XVA metrics which allowsthe best practices for the internal management of the bank. These practices30nclude properly fostering portfolio balance between CVA and DVA gener-ating positions (see section 4) and hedging CVA and FVA risk. It is conve-nient and desirable that the management perspective is also aligned acrossinstitutions. Therefore, surveys are useful for aligning best practices. Theconclusions according to three surveys, [9], [10] and [11], published in 2017,2018 and 2019, are the following: • Institutions which use internal FTP rates from Financial area to markFVA similar to equation (11): 70% according to figure 10 of [10] and60% according to page 3, question Q1 of [11]. • DVA is not highly used for profit and loss reporting: 82% as shown inquestion Q6 of [9] and 75% looking at figure 4, P&L, of [10]. • DVA is reported in accounting statements: 60% according to questionQ7 of [9] and 70% as explained in figure. 4, P&L, of [10]. • Institutions which use symmetric FVA: 82% according to question Q6of [9] and 80% as shown in figure 4, P&L, and figure 11 of [10]). •
60% of funding sets are considered at counterparty level and 32% atwhole bank level as shown in question Q8 of [9]) . • Institutions which hedge FVA: 50% according to figure 10 of [10] and64% according to question Q21 of [11]. • Institutions which hedge FVA internally against the Financial area:only 35% according to figure 10 of [10] and 54% according to questionQ21 of [11].According to these surveys, market consensus is clearly trending to theuse of internal FTP rates, not including DVA in P&L formula, markingwith symmetric funding rates (same rate for borrowing and lending) andhedging FVA against the financial area. Institutions still consider fundingsets at counterparty level instead of the legal entity level. Marking withsymmetric rates is most likely a convention rather than a fact (see [12]) asmonetizing funding benefit is only possible either by reducing bond issuance Both approaches are equivalent only if funding rate is symmetric. The sum of FVAfor every counterparty is equal to the FVA of a single funding set whose perimeter of dealsis the union of the perimeters for all counterparties • Solution of equation (16): r ∗ s = r Fs and ˜ V s = V r sys s which yieldsequations (25) to (27) as explained in section 6. This is the samesolution adopted for the accounting perspective. • Curve selection : r Fs = ¯ r Bt,s = NB (cid:80) i =1 w i,t r Bt i ,s . The implications of thischoice are the following: – The funding rate from the Financial area is set to the averageyield of issued bonds, r Fs = ¯ r Bs , according to equation (11). Thisselection, as discussed in section 3, assumes that the bank cannotdefault. In practice r Fs will be set to an average rate which takesinto account the FTP internal rates given by (11) and the col-lateral optimisation carried out by the Securities Financing desk(excess of collateral may get more remuneration than just theovernight OIS rate). – With this selection there is also no overlap between DVA andFVA. DVA does not exist as bank default is not considered andthe FVA includes both the liquidity and credit components as theaverage yield of the issued bonds includes them both. • Exit price : it is set to the internal marking in systems: ˜ V s = V r sys s .This choice is conditioned, similarly to the accounting perspective, bythe solution derived in section 6. • Survival probabilities : as DVA disappears because bank default isnot considered, CVA turns naturally unilateral and FVA is discountedby r Fs = ¯ r Bs , an average bond issuance cost curve.32 Collateral optimisation : as already mentioned before, the Securi-ties Financing desk optimizes re-hypothecable collateral to reuse it upfor instance 85% and determines an average symmetric rate r Fs whichtakes into account the optimization of collateral as a whole. With thiscommon symmetric rate, calculating funding sets for the legal entity orfor each counterparty netting set would be equivalent (the total FVAwould be the sum of FVA across every counterparty). The rate r Cs forthe calculation of ColVA would simply consider the remuneration ofcollateral of each CSA, which is symmetric.The main contribution of this perspective is to foster appropriate manage-ment of XVA by allowing portfolio balance between CVA and DVA generatingpositions and hedging of CVA and FVA market risk as well as counterpartydefault risk from CVA.
12 Comparison among both perspectives
This section compares the advantages and disadvantages of both accountingand management perspectives. They have different uses and therefore theyaddress complementary parts of the XVA managing and reporting. Theadvantages and disadvantages of the accounting perspective are the following: • The XVA metrics and the fair value of derivative prices including ad-justments are comparable among institutions as adjustments are bilat-eral (include information of the both the bank and the counterparty)and they are calculated with market observable data. • Including CVA and DVA in profit and loss allows fostering balancingCVA and DVA generating positions hedging one with the other. • The main disadvantage of this perspective is that hedging counterpartydefault and market funding risk is not possible: – Hedging CVA default imbalances CVA/DVA compensation. There-fore, the use of credit hedging products such as credit defaultswaps, pass through swaps or risk participation agreements willnot allow offsetting risks. 33
The liquidity curve is up to some extent a theoretical constructwhich is good for marking purposes. However, the exposure of themetrics to the movement of this curve can be very material andhedging it may not make a lot of sense. – DVA cannot be hedged in practice.The advantages and disadvantages of the management perspective arethe following: • The main disadvantage of this approach is that fair value and XVAmetrics are not comparable among banks, as metrics are not bilateral(DVA is absent) and depend on an unobservable internal funding cost(the average price of bonds across the issuing history of the bank opti-mized by the Securities Financing desk). • CVA credit and market risk can be hedged as there is no DVA tocompensate CVA risk. • FVA can be hedged with internal term operations between desk andFinancial area.The fact that the management perspective does not provide comparableprices among institutions (law of one price is broken) should not be a majorproblem for the following reasons: • There should not be great price discrepancies for operations betweenbig institutions as they will be fully collateralized and therefore the sizeof the credit, debit and funding value adjustments will be small. • The cases for which bigger discrepancies may appear correspond tounsecured operations of usually non-financial institutions which willaccept the price provided by the other counterparty as they cannoteither effectively challenge the price or have access to an alternativecounterparty.
13 Comparison with FRTB-CVA regulation
The regulation establishes that market risk uncertainty should be covered byan amount of capital currently given by the value-at-risk metric. This frame-work will be replaced in the future by the FRTB or Fundamental Review of34he Trading Book (see [19]) which calculates the expected shortfall instead ofthe value-at-risk. For CVA, a separate value-at-risk calculation is currentlycarried out and will be also replaced in January 2022 by what is called theFRTB-CVA regulation (see [20]). The main objectives of this new regulationare the following: • Capture both credit and market risk factors when calculating regula-tory CVA capital and recognize market risk hedging to reduce it. • Align the regulatory CVA calculation formula with the fair value CVAwhich is incorporated in the profit and loss (P&L) as shown in equation(27). This means that the CVA formula used to calculate capital con-siders the same risk neutral valuation formula as the CVA calculationfor P&L. • Align regulatory CVA with FRTB calculation rules (see [19]). Thisimplies that the expected shortfall of the CVA is calculated through aparametric method based on the sensitivities of the CVA with respectto market and credit risk factors.The analysis of the regulation concludes that the standard approach forCVA calculation (FRTB-CVA) is well aligned with the management perspec-tive for the following reasons: • Regulatory CVA excludes bank own default from regulatory CVA cal-culation (see paragraph 1 of [20]): this implies unilateral CVA withoutDVA. • Regulatory CVA is calculated with the same risk free implied marketcalibration as P&L CVA (see paragraphs 31-34 of [20]). • It recognizes market & credit hedges: reducing P&L risk implies re-ducing CVA capital (see paragraph 37 of [20]).The conclusion is that the regulation is pointing to the management per-spective. In this context, the most sensible approach would be to align ac-counting regulation with the management perspective so that the profit andloss for internal management is the one reported for accounting purposes(this is not currently the case as accounting regulation requires reporting ofCVA and DVA). However, in order to allow regulators to compare figuresamong institutions, reporting of CVA and DVA figures calculated accordingto the accounting perspective may be additionally required.35
According to sections 11 and 13, the market consensus and the regulatorytrends are pointing to the management perspective in the future. The impli-cations of changing from the P&L formula (32) of the accounting perspectiveto the P&L formula (33) of the management perspective are the following: • The DVA profit is eliminated from the P&L (a loss must be realized). • Funding cost increases as the average bank issuance yield ( r F = ¯ r B )given by equation (11) is higher than the average bond liquidity spread( r F = ¯ r OIS + ¯ l B ) given by equation (8). This implies that a bank witha net asset position will have to realize a loss. On the other hand,a bank with a net liability position may realize a profit, if benefit oncollateral excess can be monetized.ˆ V t = V r sys t − ColV A ¯ r OIS +¯ l B t − F V A ¯ r OIS +¯ l B t − CV A ¯ r OIS +¯ l B t − DV A ¯ r OIS +¯ l B t (32)ˆ V t = V r sys t − ColV A ¯ r B t − F V A ¯ r B t − CV A ¯ r B t (33)To implement the transition without negatively affecting shareholders,benefits on new deals should be provisioned and the perspective switchedfrom the accounting formula (32) to the management formula (33) when thisprovision exceeds the loss of the change of perspective. New deals should bemarked to funding curves which match market prices. For institutions withheavy asset portfolios the transition implies a loss and marking positions to amarket transitioning to the management perspective will also reduce benefits(as compared with a marking according to the accounting perspective). Thisimplies that in order to raise the transition loss, the provisioning period willbe longer with the associated business undermining.
15 Conclusions
Two paradigms or perspectives have been defined for consistent credit, debitand funding value adjustment calculation which share the same mathemat-ical framework and implementation but change the input parameters. The Counterparties owe money to the bank and bank must post collateral to hedge.
16 Acknowledgments
The author of this paper wants to acknowledge the important contributionof Jrme Maetz who provided the improved solution which avoids the circulardependence (see sections 8 and 9) and very enriching discussions throughoutthe development process of this paper. Thanks to Carlos Cataln who pro-37ided ideas for the specification of the common framework and how to con-figure it for switching perspectives (section 6). There were key discussionsthat the author wants to thank to Manuel Villa about collateral optimisationand to Robert Smith, Steven Brittan, Gerard Morris and Enrique Rigol fromthe XVA desk. The author found a very collaborative atmosphere whichallowed collecting the descriptive material about balance sheet managementof section 2. Thank you very much to Mercedes Mora and Carmen del Pozo(Financial Management Control), Andrs Castro and Antonio Toro (Financialarea), Carmen Lafont (Short Term desk) and Juan Manuel Bravo, EnriqueVerd and Hctor Ciruelos (Securities Financing desk). Finally, the authorwants to thank Manuel Menndez, my direct supervisor, for supporting thisresearch. Without the contribution of these people, writing this paper wouldhave not been possible. Thank you very much to them all.
References
A Relation between two discounted prices
This section shows the derivation of equation (23) which relates two pricesdiscounted with two different curves (in this case r F and r sys ) through anadjustment. d (cid:16) p rt,s V rs { τ>s } (cid:17) = p rt,s ( dV s − r s V rs ds ) { τ>s } + p rt,s V rs d { τ>s } (34)The left hand side of equation (34) shows the differential of the product ofthe exponential term, p rt,s , defined in (2), the stochastic price, V sr , discountedby a generic curve r and the survival indicator function, { τ>s } , where τ is thetime of the first to default. Applying Ito’s Lemma to this product (secondderivatives are zero) and the fundamental theorem of calculus to the exponentof the exponential yields the right hand side of equation (34). (cid:90) ∞ t d (cid:16) p rt,s V rs { τ>s } (cid:17) = p rt,s V rs { τ>s } (cid:12)(cid:12)(cid:12) s →∞ s → t = − V rt { τ>t } (35) (cid:82) ∞ t d (cid:16) p rt,s V rs { τ>s } (cid:17) = (cid:82) ∞ t p rt,s ( dV s − r s V rs ds ) { τ>s } − p rt,s V rs d { τ ≤ s } = − V rt { τ>t } (36) d { τ>s } = { τ>s + ds } − { τ>s } = { s<τ ≤ s + ds } = − d { τ ≤ s } (37)Integrating the left hand side of equation (34) by applying Barrow’s ruleyields equation (35). Equation (36) arises from (34) and (35) after replacing40 { τ>s } = − d { τ ≤ s } according to equation (37). If conditional expectationsare taken on equation (36), equation (38) is obtained. This equation will beused later in this section and in appendix B to integrate the price conditionedby the survival of the bank and the counterparty. E t (cid:20)(cid:90) ∞ t p rt,s ( dV rs − r s V rs ds ) { τ>s } − p rt,s V rs d { τ ≤ s } (cid:21) = − V rt { τ>t } (38)Replacing equation (13) with r ∗ = r in equation (38), taking the indicatorfunction inside the expectation as it is measurable at time t and reorganizingyields equation (39). See that the indicator function multiplying dC s dependson t and not on s as the other indicator functions of the integral. E t (cid:20)(cid:90) ∞ t p rt,s ( dV rs − r s V rs ds ) { τ>s } − p rt,s V rs d { τ ≤ s } + { τ>t } dC s (cid:21) = 0 (39)To obtain the relation between two prices discounted with different rates,the rate r of equation (39) is set to the two rates, r Fs and r syss , to relate witheach other and equation (39) is divided by p rt,s . This yields equations (40)and (41). See that in the latter equation, r Fs is added and subtracted to theparenthesis of the second term of the integrand with no effect. E t (cid:34)(cid:90) ∞ t ( dV r F s − r Fs V r F s ds ) { τ>s } − V r F s d { τ ≤ s } + { τ>t } dC s (cid:35) = 0 (40) E t (cid:34)(cid:90) ∞ t (cid:16) dV r sys s − ( r syss + r Fs − r Fs ) V r sys s ds (cid:17) { τ>s } − V r sys s d { τ ≤ s } + { τ>t } dC s (cid:35) = 0 (41)Now equation (41) is multiplied by − dC s cancel with each other. E t (cid:90) ∞ t (cid:16) d ( V r F s − V r sys s ) − r Fs ( V r F s − V r sys s ) ds (cid:17) { τ>s } − ( V r F s − V r sys s ) d { τ ≤ s } + ( r syss − r Fs ) V r sys s { τ>s } ds = 0 (42)If equation (42) is multiplied by p r F t,s , the terms multiplying { τ>s } and d { τ ≤ s } can be solved using equation (38) with V rs = V r F s − V r sys s yieldingequation (43). 41 = − (cid:16) V r F t − V r sys t (cid:17) { τ>t } + E t (cid:20)(cid:90) ∞ t p r F t,s ( r syss − r Fs ) V r sys s { τ>s } ds (cid:21) (43)Reorganizing terms yields equation (44). This is the expression to beproved which relates two prices discounted with different curves. V r F t { τ>t } = V r sys t { τ>t } + E t (cid:20)(cid:90) ∞ t p r F t,s ( r syss − r Fs ) V r sys s { τ>s } ds (cid:21) (44) V r F t = V r sys t + E t (cid:20)(cid:90) ∞ t p r F t,s ( r syss − r Fs ) V r sys s ds (cid:21) (45)If it is assumed that there is no default, τ → ∞ , then the indicatorfunctions are always equal to one and equation (44) turns into equation (45). B Derivation of invariance with default risk
This section shows the derivation of equations (16) to (18) of the mathemat-ical framework common to both perspectives (see sections 10 and 11). Thestarting point is the financing condition, ξ s , of equation (46) which must besatisfied at any time for the principle of invariance to hold. See section 5 forthe definition of these symbols. ξ s = ˆ V s − M s − ˆ F s = 0 (46)The evolution of the financing condition from time s to s + ds may considerthree excluding scenarios: ξ Ss + ds when both the bank and the counterpartysurvive, ξ Cs + ds when the counterparty defaults but the bank survives, ξ Bs + ds when the bank defaults but the counterparty survives and ξ BCs + ds when bothsimultaneously default. Equation (47) shows the financing condition at time s + ds considering these scenarios discriminated by indicator functions. Seethat the indicator functions are expressed in terms of the first to default time τ = min ( τ C , τ B ), where τ C and τ B are the times to default of respectivelythe counterparty and the bank. ξ Ss + ds { τ>s + ds } + ξ Cs + ds { s<τ ≤ s + ds,τ = τ C } + ξ Bs + ds { s<τ ≤ s + ds,τ = τ B } + ξ BCs + ds { s<τ ≤ s + ds,τ = τ B ,τ = τ C } = 0 (47)42 { s<τ ≤ s + ds,τ = τ C } = { τ B >s } { s<τ C ≤ s + ds } = { τ B >s } d { τ C ≤ s } { s<τ ≤ s + ds,τ = τ B } = { τ C >s } { s<τ B ≤ s + ds } = { τ C >s } d { τ B ≤ s } (48)Under the assumption that simultaneous defaults cannot happen (theyhappen one after another), the last term of equation (47) can be neglectedand the indicator functions can be simplified according to equation (48). ξ Ss + ds { τ>s + ds } + ξ Cs + ds { τ B >s } { s<τ C ≤ s + ds } + ξ Bs + ds { τ C >s } { s<τ B ≤ s + ds } = 0 (49) ξ Ss + ds = ˆ V s + ds + dC s − M s (1 + r Cs ds ) − ˆ F s (1 + r Fs ds ) ξ Cs + ds = H Cs + ds − ˆ F s (1 + r Fs ds ) ξ Bs + ds = H Bs + ds − ˆ F s (1 + r Fs ds ) (50)The value of the financing condition at time s + ds for the three scenariosis shown in equation (49) where ξ Ss + ds , ξ Cs + ds and ξ Bs + ds are given by equation(50). For the first scenario, the position on derivatives, ˆ V , takes the valueat s + ds plus the payments in the interval s to s + ds provided by dC s .The term, M s , associated with the collateral and the term, ˆ F s , associatedwith the financing, simply accrue the corresponding interest along the inter-val. For the other two scenarios, the terms H Cs + ds and H Bs + ds are given byequation (51) and account for the position on derivatives minus the collat-eral amount at time s + ds considering the default events. See that equation(51) corresponds to equation (19) but evaluated at time s + ds with the al-ready mentioned considerations of the evolution of derivatives and collateralthrough the period from time s to s + ds . H Cs + ds = R Cs (cid:16) ˜ V s + ds + dC s − M s (1 + r Cs ds ) (cid:17) + + (cid:16) ˜ V s + ds + dC s − M s (1 + r Cs ds ) (cid:17) − H Bs + ds = (cid:16) ˜ V s + ds + dC s − M s (1 + r Cs ds ) (cid:17) + + R Bs (cid:16) ˜ V s + ds + dC s − M s (1 + r Cs ds ) (cid:17) − (51)Equation (52) transforms the indicator functions of equation (49) in termsof the differential of the indicator function (the unconditional default condi-tion). 43 { s<τ C ≤ s + ds } = { τ C ≤ s + ds } − { τ C ≤ s } = d { τ C ≤ s } { s<τ B ≤ s + ds } = { τ B ≤ s + ds } − { τ B ≤ s } = d { τ B ≤ s } { τ>s + ds } = { τ>s } − { s<τ ≤ s + ds } = { τ>s } − d { τ ≤ s } (52)In order to expresses the joint survival indicator in terms of the sur-vival of the bank and the counterparty, the third line of equation (52) isreplaced in (53) and the result follows assuming that the joint default con-dition ( d { τ C ≤ s } d { τ B ≤ s } ) is neglected. { τ>s + ds } = { τ C >s + ds } { τ B >s + ds } = (cid:16) { τ C >s } − d { τ C ≤ s } (cid:17) (cid:16) { τ B >s } − d { τ B ≤ s } (cid:17) ≈ (cid:16) { τ>s } − { τ C >s } d { τ B ≤ s } − { τ B >s } d { τ C ≤ s } (cid:17) (53) ξ Ss + ds { τ>s } + ( ξ Cs + ds − ξ Ss + ds ) { τ B >s } d { τ C ≤ s } +( ξ Bs + ds − ξ Ss + ds ) { τ C >s } d { τ B ≤ s } = 0 (54)Replacing equation (53) in (49) yields equation (54) where the indicatorfunctions no longer depend on s + ds . Now the dependence with respect to s + ds is removed from ξ s + ds terms by using differentials. Replacing ˆ F s =ˆ V s − M s in ξ Ss + ds , see equation (50), adding and subtracting the terms r ∗ s M s ds and r ∗ s ˆ V s ds , where r ∗ is the arbitrary rate of the principle of invariance andknowing that d ˆ V s = ˆ V s + ds − ˆ V s , yields equation (55). ξ Ss + ds = d ˆ V s − r ∗ s ˆ V s ds + dC s + M s ( r ∗ s − r Cs ) ds + ˆ F s ( r ∗ s − r Fs ) ds (55)( ξ Cs + ds − ξ Ss + ds ) { τ B >s } d { τ C ≤ s } = ( H Cs + M s − ˆ V s ) { τ B >s } d { τ C ≤ s } ( ξ Bs + ds − ξ Ss + ds ) { τ C >s } d { τ B ≤ s } = ( H Bs + M s − ˆ V s ) { τ C >s } d { τ B ≤ s } (56) H Cs + ds d { τ C ≤ s } = ( H Cs + dH Cs ) d { τ C ≤ s } = H Cs d { τ C ≤ s } ˆ V s + ds d { τ C ≤ s } = ( ˆ V s + d ˜ V s ) d { τ C ≤ s } = ˆ V s d { τ C ≤ s } ds · d { τ C ≤ s } = 0 dC s · d { τ C ≤ s } = 0 (57)The rest of the terms of equation (54) are presented in equation (56).They were obtained from equation (50) assuming the simplifications shownin equations (57). These simplifications are based on the fact that the product44f two differential terms are negligible higher order infinitesimals. If theseterms are replaced in equation (54), the final transformed expression of thefunding condition is given by equation (58). ξ Ss + ds { τ>s } + ( H Cs + M s − ˆ V s ) { τ B >s } d { τ C ≤ s } +( H Bs + M s − ˆ V s ) { τ C >s } d { τ B ≤ s } = 0 (58) (cid:82) ∞ t p r ∗ t,s (cid:16) d ˆ V s − r ∗ s ˆ V s ds + dC s (cid:17) { τ>s } − p r ∗ t,s ˆ V s d { τ ≤ s } + p r ∗ t,s ˆ V s d { τ ≤ s } + (cid:82) ∞ t p r ∗ t,s M s ( r ∗ s − r Cs ) { τ>s } ds + p r ∗ t,s ˆ F s ( r ∗ s − r Fs ) { τ>s } ds + (cid:82) ∞ t p r ∗ t,s (cid:16) H Cs + M s − ˆ V s (cid:17) { τ B >s } d { τ C ≤ s } + (cid:82) ∞ t p r ∗ t,s (cid:16) H Bs + M s − ˆ V s (cid:17) { τ C >s } d { τ B ≤ s } = 0 (59)If equation (55) is replaced in (58), the resulting expression is multipliedby p r ∗ t,s , the term p r ∗ t,s ˆ V s d { τ ≤ s } is added and subtracted and the whole expres-sion is integrated from time t to infinity, expression (59) is obtained. d { τ ≤ s } = { s<τ ≤ s + ds,τ = τ C } + { s<τ ≤ s + ds,τ = τ B } + { s<τ ≤ s + ds,τ = τ C ,τ = τ B } ≈ { τ B >s } d { τ C ≤ s } + { τ C >s } d { τ B ≤ s } (60)Taking expectation, E t ( · ), conditioned on time t in equation (59) andreplacing equation (38) with r = r ∗ on the first line of (59) would collapse thefirst, second and fourth terms of the integrand into − ˆ V s { τ>s } . In addition,if the differential d { τ ≤ s } of the fifth term of the integrand is replaced byequation (60) , this term cancels the − ˆ V s component inside the parenthesisof the third and fourth lines of equation (59), the CVA and DVA terms.Finally, the second line of equation (59) can be replaced by the collateraland funding adjustments of equation (17). The final result is presented inequation (61). − ˆ V s { τ>s } + E t (cid:104)(cid:82) ∞ t p r ∗ t,s dC s { τ>s } (cid:105) + ColV A r ∗ t + F V A r ∗ t + E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Cs + M s (cid:17) { τ B >s } d { τ C ≤ s } (cid:105) + E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Bs + M s (cid:17) { τ C >s } d { τ B ≤ s } (cid:105) = 0 (61) The first to default indicator condition between s and s + ds is expressed as the sumof the first to default of the counterparty plus the first to default of the bank plus the jointdefault which is neglected. See equation (48) to understand the final result.
45n order to calculate the second term of equation (61), dC s is expressedin terms of V s discounted by the rate r ∗ s through equation (13) applied to therate r ∗ and equation (38) assuming τ → ∞ . This yields equation (62): dV r ∗ s − r ∗ s V r ∗ s ds + dC s = 0 (62)The calculation of the second term of equation (61) is shown in equation(63) and it is obtained by replacing equation (62) in this second term andadding and subtracting p r ∗ t,s V r ∗ s d { τ ≤ s } . E t (cid:104)(cid:82) ∞ t p r ∗ t,s dC s { τ>s } (cid:105) == − E t (cid:104)(cid:82) ∞ t p r ∗ t,s ( dV r ∗ s − r ∗ s V r ∗ s ds ) { τ>s } − p r ∗ t,s V r ∗ s d { τ ≤ s } + p r ∗ t,s V r ∗ s d { τ ≤ s } (cid:105) = V r ∗ t { τ>t } − E t (cid:104)(cid:82) ∞ t p r ∗ t,s V r ∗ s d { τ ≤ s } (cid:105) (63)Once more, replacing equation (38) with r = r ∗ in the second line ofequation (63) would collapse the first, second and third terms of the integrandinto − V r ∗ t { τ>t } yielding the third line of equation (63).ˆ V t { τ>t } = V r ∗ t { τ>t } + ColV A r ∗ t + F V A r ∗ t + E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Cs + M s − V r ∗ s (cid:17) { τ B >s } d { τ C ≤ s } (cid:105) + E t (cid:104)(cid:82) ∞ t p r ∗ t,s (cid:16) H Bs + M s − V r ∗ s (cid:17) { τ C >s } d { τ B ≤ s } (cid:105) (64)Finally, if equation (60) is replaced in (63) and the result replaced inequation (61), the final expression (64) equal to equation (16) to be provedfollows. C Improved solution for XVA calculation
This appendix shows the hypotheses and transformations carried out to ob-tain equations (25) to (27) of section 9 after solving the circular dependenceand choosing V r sys s as exit price, from the general equations (16) to (19) ofthe invariance principle with default risk (see section 6).The starting point is the equation (16) of the invariance principle withdefault risk. The rate r ∗ , which can be arbitrarily chosen without affectingthe pricing equation (16), is set to the funding rate, r F , so that the term F V A r ∗ t turns to zero. This breaks the circular dependence as the right hand46ide of equation (65) no longer depends on ˆ V and equation (16) becomes(65). ˆ V t { τ>t } = V r F t { τ>t } + ColV A r F t + CV A r F t + DV A r F t (65)If equation (44) is substituted in equation (65), the ColV A r F t term com-bines with the last adjustment term of equation (44) according to equation(66). See that the third line of this equation is obtained by adding and sub-tracting the integrand term p r F t,s M s ( r syss − r Fs ) { τ>s } ds to the second line andtaking into account that F r sys s = V r sys s − M s . The fourth line of equation (66)is obtained by replacing the definitions given by equation (26). Replacingthis calculation in equation (65) yields equation (67). ColV A r F t + E t (cid:104)(cid:82) ∞ t p r F t,s V r sys s ( r syss − r Fs ) { τ>s } ds (cid:105) == E t (cid:104)(cid:82) ∞ t p r F t,s M s ( r Fs − r Cs ) { τ>s } ds + p r F t,s V r sys s ( r syss − r Fs ) { τ>s } ds (cid:105) = E t (cid:104)(cid:82) ∞ t p r F t,s M s ( r syss − r Cs ) { τ>s } ds + p r F t,s F r sys s ( r syss − r Fs ) { τ>s } ds (cid:105) = − ColV A syst − F V A syst (66)ˆ V t { τ>t } = V r sys t { τ>t } − ColV A syst − F V A syst + CV A r F t + DV A r F t (67) CV A r F t = E t (cid:104)(cid:82) ∞ t p r F t,s (cid:16) H Cs + M s − V r F s (cid:17) { τ>s } d τ C ≤ s (cid:105) = E t (cid:104)(cid:82) ∞ t p r F t,s (cid:16) H Cs + M s − V r sys s (cid:17) { τ>s } d τ C ≤ s (cid:105) == − E t (cid:104)(cid:82) ∞ t p r F t,s (1 − R Cs )( V r sys s − M s ) + { τ>s } d τ C ≤ s (cid:105) = − CV A syst (68)
DV A r F t = E t (cid:104)(cid:82) ∞ t p r F t,s (cid:16) H Bs + M s − V r F s (cid:17) { τ>s } d τ B ≤ s (cid:105) = E t (cid:104)(cid:82) ∞ t p r F t,s (cid:16) H Bs + M s − V r sys s (cid:17) { τ>s } d τ B ≤ s (cid:105) = − E t (cid:104)(cid:82) ∞ t p r F t,s (1 − R Bs )( V r sys s − M s ) − { τ>s } d τ B ≤ s (cid:105) = − DV A syst (69)Setting the rate r ∗ = r F and the exit price ˜ V s = V r sys s for the CVA andDVA terms of equation (67) and replacing V r F s in terms of V r sys s accordingto equation (44) yields equations (68) and (69). See that when V r F s is re-placed according to equation (44), the term V r sys t continues to appear within47he parenthesis of the second line of equations (68) and (69) but the adjust-ment term disappears. Equation (70) shows that the contribution of thisadjustment term for the CVA equation is zero due to the fact that d { τ C ≤ s } is measurable with respect to time s getting inside the expectation and theproduct of differentials, dsd τ C ≤ s , is negligible. The transformation of theterms inside the parenthesis of the second line of equation (68) are explainedby equation (71), where the term ( V r sys s − M s ) + is added and subtracted sothat combining positive and negative parts cancels the term M s − V r sys s . Thesame approach is carried out for the DVA equation. The final expression forthe parenthesis inside the CVA and DVA integrands is presented in equation(72). E t (cid:18)(cid:90) ∞ t E s (cid:20)(cid:90) ∞ s p r F s,u V sysu ( r sysu − r Fu ) { τ>u } du (cid:21) · d { τ C ≤ s } (cid:19) = 0 (70) H Cs + M s − V r sys s = R Cs ( V r sys s − M s ) + + ( V r sys s − M s ) − +( V r sys s − M s ) + − ( V r sys s − M s ) + + M s − V r sys s (71) H Cs + M s − V r sys s = − (1 − R Cs )( V r sys s − M s ) + H Bs + M s − V r sys s = − (1 − R Cs )( V r sys s − M s ) − (72)ˆ V t { τ>t } = V r sys t { τ>t } − ColV A syst − F V A syst − CV A syst − DV A systsyst