This chapter will describe the consideration of funding and its potential impact on price and valuation through funding value adjustment (FVA). The nature of funding costs and benefits will be described, together with underlying formulas and examples. The recent debate around the use of FVA in pricing and valuation and the link between FVA and debt value adjustment (DVA) will be addressed.
FVA can be broadly thought to be a consideration of funding for uncollateralised – and partially-collateralised – transactions. FVA, therefore, is a consideration alongside credit value adjustment (CVA) in many situations. From a quantification point of view, FVA is similar in many ways to CVA, and many of the components to calculate the two are shared. Transactions with large CVA (and DVA) components are also likely to have significant funding components.
Assets create funding costs, whereas liabilities provide funding benefits. In-the-money (ITM) positions and expected positive exposure (EPE) will, therefore, relate to funding costs, whilst out-of-the-money (OTM) positions and expected negative exposure (ENE) will lead to funding benefits.
FVA was not given much consideration prior to 2007 because unsecured funding for institutions such as banks was trivial and could be achieved at more or less ‘risk-free’ rates. However, such funding costs have increased considerably, which has led to their inclusion in pricing and valuation. This means that transactions – especially those that are uncollateralised – are now typically treated including the party's own funding as a component of their price. This is the role of FVA, although its use in valuation through accounting statements has been more controversial.
Despite the increased use of margining, a significant portion of transactions such as over-the-counter (OTC) derivatives remain uncollateralised (Figure 2.7). This arises mainly due to the nature of the counterparties involved, such as corporates and sovereigns, without the liquidity and/or operational capacity to adhere to frequent margin calls. In general, funding costs (and benefits) in derivatives portfolios can be seen as arising from the following situations:
Undercollateralisation. Transactions that are undercollateralised give rise to funding costs (and potentially benefits). This includes transactions that are completely uncollateralised (no margin agreement), but also cases of partial collateralisation (e.g. a two-way margin agreement with a material threshold). One-way margin agreements are also a special case, since one party is collateralised whilst the other is not.
Non-rehypothecation and segregation. Even if a party can receive margin, there is a question of whether or not this can be used. If the margin cannot be rehypothecated or reused and/or must be segregated, then this will prevent it from offsetting any funding position.
Note that FVA – as a definition – will not include margin value adjustment (MVA), which defines the funding of initial margin requirements. Whilst FVA represents the cost of being undercollateralised, MVA represents the cost of overcollateralisation. Generally, FVA is related to variation margin, whilst MVA (Chapter 20) is related to initial margin. FVA also considers debt funding, with capital value adjustment (KVA) (Chapter 19) being the cost of funding equity in relation to capital requirements.
In some sense, FVA is not a particularly new concept. Prior to the global financial crisis (GFC), it was common to use rates such as LIBOR (London Interbank Offered Rate) to discount cash flows, not because it was the risk-free rate (which is, in any case, a theoretical construct), but because it was a good approximation of a bank's short-term unsecured funding costs. Post-GFC, banks have simply realised that funding costs have increased and they cannot be as reliant on short-term (e.g. LIBOR) funding, and they have, therefore, sought to incorporate these higher costs through FVA.
18.2 FVA AND DISCOUNTING
18.2.1 Market Practice
Whilst collateral discounting (Section 16.1.3) is now seen as the correct valuation of collateralised transactions, the aftermath of the GFC created a parallel problem regarding the valuation of uncollateralised transactions. Banks started to incorporate funding costs into transactions by discounting at a funding rate. For example:
Transactions secured with collateral are valued using a discount curve based on the overnight index spread. Transactions not secured with collateral are valued using a discount curve based on Euribor/Libor plus a spread that reflects market conditions. (Rabobank Interim Report 2015)
The difference between a standard (collateralised) valuation and one incorporating funding became known as FVA. FVA is generally seen by banks as an internal cost of financing their uncollateralised derivatives portfolios, and has correspondingly been reported in financial statements alongside CVA and DVA components. For example:
… a fair-value adjustment was applied to account for the impact of incorporating the cost of funding into the valuation of uncollateralised derivatives. (Barclays Annual Report 2012)
FVA reporting has become standard amongst the largest banks, with many other banks following, and the total FVA reported now amounts to many billions of US dollars. Note that, whilst the rationale for FVA seems to stem from economic value considerations, its reporting in financial statements is justified by fair value and exit price considerations. Whilst it is clear that banks consider FVA to arise from their own internal costs of funding uncollateralised derivatives, there is some reference to market pricing. Although the exit price concept is not completely compatible with the view of FVA as an internal cost, it is acceptable to report FVA if the market practice is to incorporate this into pricing. Hence, the inclusion of FVA (and indeed any other xVA component) in financial statements can be supported via a self-fulfilling prophecy. For example:
The Firm implemented a Funding Valuation Adjustments (“FVA”) framework this quarter for its OTC derivatives and structured notes, reflecting an industry migration towards incorporating the cost or benefit of unsecured funding into valuations. For the first time this quarter, we were able to clearly observe the existence of funding costs in market clearing levels. As a result, the Firm recorded a $1.5B loss this quarter. (J.P. Morgan, Fourth Quarter 2014)
FVA adjustments only consider uncollateralised (or partially collateralised) trades due to the special case of collateral discounting (Section 16.1.3). For example:
The adjustment this quarter is largely related to uncollateralized derivatives, as … [c]ollateralized derivatives already reflect the cost or benefit of collateral posted in valuations. (J.P. Morgan, Fourth Quarter Earnings Presentation 2013)
Finally, as mentioned above, funding costs can arise from the inability to reuse (rehypothecate) margin, as well as merely from situations where margin is not received:
In general, FVA reflects a market funding risk premium inherent in the uncollateralized portion of derivative portfolios, and in collateralized derivatives where the terms of the agreement do not permit the reuse of the collateral received. (Citigroup Third Quarter 2014)
18.2.2 Source of Funding Costs and Benefits
One of the difficult aspects in defining FVA in relation to derivatives is that it may be considered to be related to the following:
receiving or posting margin;
paying or receiving cash flows; or
holding assets or liabilities (defined by the value).
The above components are not mutually exclusive. For example, a strongly-collateralised transaction would seem to have funding considerations due to the need to post or receive variation margin. However, this will be offset by the cash flow payments or the transaction value, leading to no overall adjustment, as mentioned above.
Most banks probably see the economics of FVA as being driven by their margin requirements. This is because banks generally hedge most of their market risk, and so cash flow payments and valuations will be approximately offset by the equal and opposite values from hedges. Many banks, therefore, define the source of funding costs and benefits to be the posting and receipt of margin. However, this alone is not completely accurate and may misrepresent some components and special cases.
From this point of view, a high-level view of funding is represented in Figure 18.1. Since banks aim to run mainly flat derivatives books, market risk will generally be hedged, and so there will not be a large mismatch with respect to cash flows. Generally, there is a mismatch between uncollateralised and collateralised derivatives, with the former usually being client transactions and the latter their associated hedges. A trading desk or business unit will need to have an ability to fund itself via its internal treasury and/or the market. Increasingly, the xVA desk may intermediate this relationship, as shown in Figure 18.1.
Figure 18.1 Illustration of the source of funding costs (and benefits) within a bank.
Historically, trading desks would carry such funding costs on an accrual basis, but as they have increased in magnitude, a more rigorous treatment has been seen as being important. The xVA desk may, therefore, facilitate better pricing, valuation, and risk management of FVA.
In the above example, and as previously shown in Figure 7.4, when a client trade is ITM from the bank's point of view, then they may not receive margin (collateral) but will be required to post margin on the equivalent OTM hedge(s) (due to being transacted with other banks bilaterally or in an exchange/CCP environment). Such margin must be funded for a reasonable period, either for reasons of good liquidity management or as a result of the net stable funding ratio (NSFR) (Section 4.3.4). The return paid on the margin will typically be the overnight indexed spread (OIS) rate (Section 7.2.4). Hence, unless the bank can fund the margin it posts at the OIS rate, there will be an associated cost. The funding rate will be the spread of the funding rate of the margin over its remuneration rate (typically the OIS rate). Note that when the trade moves in the opposite direction (i.e. the client is ITM), the reverse effect occurs and the bank receives funding. Note that this funding benefit requires the reuse of the margin (e.g. if required, rehypothecation must be allowed).
The above explanation is commonly used by banks to justify FVA. For example:
Those [funding] costs arise when the trade is hedged with a collateralised transaction, meaning that when the dealer is in-the-money on the first trade it does not receive any margin from its client, but would be out-of-the-money on the hedge and required to post margin as a result.1
Although useful for explanation purposes, the above analogy must not be taken too literally, as it may suggest the wrong FVA in certain situations. For example:
The bank does not hedge the transactions in question. Here, there may likely still be an FVA adjustment, even though there is no movement of margin.
Profit margin. The profit that a bank makes on a transaction will not be a component of the value of the hedges and, therefore, will not be posted as margin, and yet an FVA will likely still include this amount.
Intermediation/novation. Here a bank might effectively step in to a portfolio with an upfront payment, but any hedges executed would likely be at par (zero value) and, therefore, not require margin posting. This does not mean that there would be no FVA incorporated in the price.
Restriking a portfolio. If a bank restrikes the value of a client portfolio, involving paying or receiving an upfront amount in cash, but again without impacting the hedges, then there are still FVA considerations (indeed, FVA may be one reason for doing this).
Change of margin terms. If the bank and client change their credit support annexe (CSA) terms (e.g. move from a one-way to a two-way CSA), then there will be FVA considerations, even though the hedges are again not impacted.
The reason for the above problems is that they involve cash flows or valuation differences that are not captured by considering only margin movements. It is therefore important to capture these as consistently and easily as possible. Another consideration is currency: funding needs to take into account the currency of the transaction and/or margin, as different currencies can have rather different funding costs.
With the above in mind, there are the following options – in order of sophistication – in terms of defining the funding costs of a business (hereafter known as the ‘funding profile’):
Uncollateralised value. This is the simplest way to calculate funding costs and is based on the total net value of all uncollateralised transactions. However, it misses components such as thresholds and the correct impact of novations and restructurings, which involve cash payments.
Total margin posted. This will correctly assess the cost of margin in each currency and also account for thresholds on collateralised trades. However, it will still incorrectly state the funding related to novations and restructurings. It will also not charge funding on a positive value on a client trade that does not lead to a margin posting requirement (e.g. a client swap with a positive value which is hedged with a collateralised par swap).
Collateralised value. This involves assessing the value of all collateralised transactions. This will be an approximation to the above, but may be misleading in terms of funding currency (e.g. a euro swap collateralised in US dollars) and would also miss the impact of thresholds on collateralised trades.
Total value minus total margin. This approach will capture all effects and is the most accurate metric. Strictly speaking, it should include all margin posted as a negative amount and all rehypothecable margin received as a positive amount (in practice, this may simply mean that any initial margin received should be ignored due to segregation).
The last approach above requires knowledge of the total valuation of all transactions and the total amount of margin. Simpler approaches may require only information on the total net margin amount or total net valuation.
Which of the above options to choose for calculating FVA is an important consideration. Considering only the uncollateralised value will be simpler but will mean that collateralised transactions (of which there will likely be a greater volume) will not need to be assessed. However, this will lead to certain assumptions which may not be correct.
The difficulty in defining FVA is that it relates to values, cash flows, and margin flows, but these are not mutually exclusive. Hence, it may be appropriate to focus primarily on one of these components but to capture the others. Depending on the nature of the underlying business of the party concerned, this might be done in different ways. For example, since banks generally hedge their market risk, cash flows and values tend to net, leading to margin flows being the primary measure of the funding of derivatives, as depicted in the example in Figure 18.1. However, this may lead to an incorrect conclusion, such as that an ITM transaction has a lower funding cost because the hedge will be a par transaction.
Taking the final definition above (total value minus total margin) as the best representation of funding is actually relatively simple and intuitive. Unsecured derivatives assets require funding since this is a value that has not been realised and, therefore, has to be funded. By contrast, unsecured derivatives liabilities represent funding benefits. Recalling the relationship given previously in Equation 11.7 and assuming the reuse of margin:
(18.1)
Funding costs (benefits) arise when the above term is positive (negative). The analogy in Figure 18.1 can still be used, but any valuation differences between transactions (e.g. where the client transaction has a positive value, reflecting a profit, whereas the hedge is a par transaction) and cash payments (e.g. upfront cost of novating into a transaction) must be accounted for. Alternatively, the hedge should be assumed to be an idealistic hedge (i.e. exactly the opposite of the transaction at all times) performed under a perfect margin agreement (according to the terms at the start of Section 16.1.2).
Yet another way to think of this funding cost (benefit) is that a party would receive (pay) the positive exposure (negative exposure) in cash if a transaction was terminated and hence there would be a funding position in order to maintain the transaction.
The key point is that, regardless of how it is defined, exposure drives a funding cost, just as it drives CVA. By symmetry, negative exposure drives a funding benefit analogous (or perhaps identical) to DVA (Figure 11.24).
18.2.3 Definition of FVA
In order to understand FVA, consider a single, uncollateralised receiver interest rate swap transaction as an example. Figure 18.2 shows the cash flows and associated funding considerations for a payer interest rate swap, assuming an upwards-sloping yield curve. In the early stages of the swap, the fixed cash flows being paid are expected (on a risk-neutral basis) to be greater than the floating ones received. This creates a positive value (an asset) which needs to be funded and is real since cash is being paid out. The value increases cumulatively for the first five payment dates and then reduces as the projected floating payments start to exceed the fixed ones.2
Figure 18.2 (Top) Illustration of the funding needs on a payer interest rate swap which arises due to the future cash flow differential. The grey bars show the net projected funding cost (based on a risk-neutral valuation). (Bottom) The cumulative effect over time of the cash flow differential and resulting funding profile.
The funding profile that arises is the same as the expected future value (EFV) of the transaction (for example, see Figure 15.24), as will be proved below. The corresponding receiver swap would have precisely the opposite profile, creating a funding benefit overall.
Intuitively, it would be expected that FVA be calculated by multiplying the future cost of funding by the funding profile above. This will be illustrated in Section 18.2.4.
18.2.4 Symmetric FVA Formula
As noted in Section 18.2.1, a simple way to include funding costs in a valuation is to discount using the cost of unsecured funding (note that this rate is implicitly symmetric, meaning that the party assumes it can both borrow cash and invest excess cash at this rate). It is important to understand what assumptions this corresponds to and under which conditions it is valid. As shown in Appendix 18A, the FVA discounting approach can be shown to be equivalent to a CVA-like formula:
(18.2)
For reasons that will become clear, this will be referred to as symmetric FVA. It is driven by EFV, which is the discounted expected amount that has to be funded (if positive), or that will create a funding benefit (if negative). is the funding spread for the time with respect to the rate used for valuation (e.g. OIS). Note that the above formula is equivalent to the difference between discounting at a higher rate minus discounting at the base rate (see Appendix 18A).
The above can be written more intuitively in terms of a forward funding spread:
(18.3)
Compared to the CVA formula (Equation 17.3), the main differences in Equation 18.3 are that the funding spread of the bank () replaces the counterparty credit spread, and (discounted) EFV replaces EPE. Note that the funding spread should be the difference between the cost of funding and the rate used for discounting (most obviously the OIS rate).
Equation 18.3 represents the overall economic value adjustment associated with respect to funding, without showing costs and benefits explicitly. This is not problematic in the completely uncollateralised and symmetric case,3 but it does not give any insight into other cases such as partial collateralisation. However, a very simple transformation can produce a more intuitive formula. Recalling the definitions of EPE and ENE from Section 11.1.5, , and so the above formula can be decomposed into two terms. Whilst at first glance this is unnecessarily complex, it does generalise the FVA formula and provide additional intuition. FVA is, therefore, typically defined as being driven by both funding costs (FCA) and benefits (FBA):
In certain situations, this might complicate things unnecessarily, since EFV is relatively easy to compute, depending mainly on forward rates, whilst EPE/ENE are more complex to quantify and depend on factors such as volatilities. However, the EPE and ENE components above are readily calculated from a CVA/DVA framework, although these may differ in the event of non-rehypothecable margin, as will be explained later. Furthermore, the above representation can be extended to understand other situations.
The formula above does not include survival probabilities. It is possible to include these in a similar way to bilateral CVA (Section 17.3.3) – we will refer to this as ‘contingent’ FVA. Survival adjustments, as for CVA, should be considered based on aspects such as close-out assumptions (e.g. should the FVA decline as a counterparty approaches default?), as discussed in Section 6.3.4. Contingent FVA assumes that FVA vanishes at the default of one or either party: this is consistent with a risk-free close-out process using the base value. If a party thought that funding costs and benefits could be included in the close-out amount, then it would be more appropriate to ignore survival probabilities. Such assumptions may even be asymmetric – i.e. a party may assume that it would charge for funding costs in a close-out process but not pay funding benefits (Section 6.3.4). It is also important to consider that survival adjustments will create credit spread sensitivity in the FVA number. As shown in Figure 18.3, market practice is divided on which survival probabilities to adjust for in FVA calculation. Anecdotally, the Totem results (Section 5.3.5) show a similar divergence in market practice.4
Figure 18.3 Market practice around including survival probabilities in FVA computation.
Source: Solum CVA Survey (2015).
Note that symmetric FVA is a trade-level quantity in this specification, and total FVA would simply be summed over all relevant trades. This is simpler than CVA, which is a counterparty-level (or, strictly speaking, netting set-level) calculation. This means that the FVA of a new transaction can be calculated on a standalone basis as there is no portfolio effect.
FVA can be either positive or negative depending on the relative size of the FCA and FBA terms above. This is consistent with the fact that discounting at a higher rate may lead to a higher or lower valuation depending on the nature of the trade (e.g. moneyness and cash flow timing).
Table 18.1 Upfront FVA value for interest rate swaps. The credit spreads and loss given defaults are the same as the previous example in Table 17.2. The cost of funding is the same as the party's own credit default swap curve.
Pay fixed
Receive fixed
Non-contingent
Contingent
Non-contingent
Contingent
Discounting approach
−6.8
6.8
FCA
−15.5
−14.0
−8.7
−7.9
FBA
8.7
7.9
15.5
14.0
FVA
−6.8
−6.1
6.8
6.1
Table 18.1 shows FVA for the interest rate swaps considered previously. The pay fixed swap has negative FVA due to positive EFV (Figure 15.24). The receiver swap has equal and opposite FVA. Note that the discounting approach (i.e. discounting at a rate with and without the cost of funding) corresponds exactly to ‘non-contingent’ FVA, as shown above.
The equivalence between the discounting and (non-contingent) FVA approach is illustrated in Figure 18.4. Note that the funding spread in FVA must be defined as the difference between the base rate used for valuation and the cost of funding.
The base rate above is, therefore, most obviously the margin rate, to be consistent with ‘perfect collateralisation’ (Section 16.1.2), although this is not necessary. For example, historically, some banks have used LIBOR discounting as a starting point (Figure 18.5). The funding spread used in Equation 18.3 must then be defined with respect to LIBOR. This is often consistent with internal policies of basing funding costs as a spread over LIBOR. However, the move away from LIBOR (Section 14.3.4) will make this increasingly less common and eventually obsolete.
The definition of the funding spread above requires that a ‘funding’ currency must be nominated and an associated curve or spread curve must be defined. This may be a single currency (e.g. if a bank primarily raises the bulk of its marginal funds in one currency and converts this pool of funds as needed to other currencies via FX markets), or there may be several curves defined from issuance levels in different currencies. One approach to calculating FVA is to use separate funding curves and EFV calculations for different currencies. Alternatively, a single currency could be used, with the associated cost of funding in other currencies being determined within the exposure simulation via the appropriate implied ‘risky FX forwards’ (Section 15.4.3).
Figure 18.4 Illustration of equivalence between discounting at own cost of funding versus applying a symmetric FVA adjustment. Note that this adjustment can be positive or negative.
Figure 18.5 Market practice for valuing uncollateralised transactions.
Source: Solum CVA Survey (2015).
18.2.5 CVA/DVA/FVA Framework
There is a generally agreed upon double-counting (e.g. Tang and Williams 2010) of DVA and FBA illustrated in Figure 18.6, meaning that only one (at most) should be considered. It is possible to see both DVA and FBA as the benefit of a negative exposure: the former because, in default, some of this is not paid, whilst the latter is a funding benefit. Morini and Prampolini (2010) show that the explicit inclusion of DVA leads to a duplication of the funding benefit in a transaction and, therefore, would be equivalent to discounting cash flows twice.
It is clear from banks' FVA reporting practices that they consider there to be an overlap between FVA (as it is typically referred to) and DVA. For example:
The adjustment this quarter is largely related to uncollateralized derivatives receivables, as … [e]xisting DVA for liabilities already reflects credit spreads, which are a significant component of funding spreads that drive FVA. (J.P. Morgan Fourth Quarter Earnings Presentation 2013)
Figure 18.6 Illustration of the link between FBA and DVA.
FVA is considered the primary adjustment applied to derivative liabilities. The extent to which DVA and FVA overlap is eliminated from DVA. (RBS Annual Report 2016)
The FVA applies to both assets and liabilities, but largely relates to uncollateralized derivative assets given the impact of the Bank's own credit risk, which is a significant component of the funding costs, is already incorporated in the valuation of uncollateralized derivative liabilities through the application of debit value adjustments. (Toronto Dominion Annual Report 2016)
The double-counting of DVA and funding benefits is therefore generally agreed upon.
More rigorously, Burgard and Kjaer (2011a, 2011b) derive a framework for the economic cost of a derivative, including credit and funding. In addition to a CVA term, as given in Equation 17.7b, they derive a funding cost adjustment (FCA), which can be written as:
(18.5)
where is a risky discount factor including survival probabilities as defined in Section 17.3.3, and represents the funding spread at time . The above formula is the contingent (including survival probabilities) integral representation of the FCA summation term in Equation 18.4b where is the risk-free interest rate, and are the default intensities of the party and counterparty, respectively.
In the Burgard and Kjaer framework, there is another term that is referred to as DVA:
(18.6)
where is the loss given default of the party making the calculation. The above term can be seen as a funding benefit which is monetised by using excess cash to buy back bonds, with the FCA being a symmetric term that arises from the equivalent cost of issuing bonds to generate cash. The DVA term in Equation 18.6 can equivalently be seen as the previously-defined funding benefit (Equation 18.4c) under the following two assumptions:
Same spread. Equivalence between the risk-neutral default probability derived from the credit default swap (CDS) market and the cost of funding, which would be more obviously linked to a bond yield spread. This amounts to and is the case in the Burgard and Kjaer framework, since they consider the existence of bonds only and there is, therefore, a single credit spread that can be seen as defining own credit risk and funding costs.
Portfolio effect. DVA should be applied at the netting set level with respect to the close-out process in a default. A funding benefit would not have this additivity since it does not relate to the default process. In the funding strategy considered by Burgard and Kjaer,5 this condition is met since all terms (CVA, FCA, and DVA) are additive at the netting set level (Burgard and Kjaer 2015).
To avoid a double-counting of funding benefits, there are two obvious frameworks for treating counterparty risk and funding consistently. These will be known as:
CVA and symmetric funding (CVA + FCA + FBA).
Bilateral CVA and asymmetric funding (CVA + DVA + FCA).
The symmetric funding case would be more consistent with Basel III capital rules where DVA cannot be recognised, but inconsistent with accounting requirements (e.g. IFRS 13) where DVA must be included (Section 5.3.3). This may also be more closely aligned to the treatment of funding in a bank where a treasury department considers both funding costs and benefits, and considers that two hedged transactions with the same margin terms (but potentially different counterparties) have a net-zero funding cost, as their benefits and costs cancel perfectly.6 It also allows the xVA desk to consider CVA but ignore DVA, which may be considered to be a benefit that is hard to monetise. Not surprisingly, the majority of market practitioners could be seen to adopt this framework for pricing (CVA + symmetric funding) fairly early in the development of FVA (Figure 18.7).
The symmetric funding version of FVA fits the simple discounting approach mentioned in Section 18.2.4 for completely uncollateralised transactions, as shown by Piterbarg (2010) and also considered by Fries (2011). One way to see this simplification is that the quantity is largely model independent, unlike the components and . Under these assumptions, uncollateralised derivatives can be valued via an appropriate choice of discount factor, rather than via the need to calculate an explicit FVA adjustment. This has been useful for banks to implement funding adjustments in a relatively straightforward manner (see the first quote in Section 18.2.1). This symmetry also means that FVA is additive across transactions, meaning that pricing can be done at a transaction level.
Figure 18.7 Market practice on pricing CVA, DVA, and funding.
Source: Deloitte Solum CVA Survey (2013).
In the symmetric funding framework, the simpler discounting approach can be used, and there is no obvious need to implement the more complex formulas in Equation 18.4. Collateralised transactions can be discounted at the margin rate and uncollateralised ones at the cost of funding. However, given that EPE (and ENE) need to be calculated for CVA purposes, it may be that this formula approach is preferred. A disadvantage of the discounting approaches (without a separate FVA adjustment) is that they make the following implicit assumptions about the nature of funding for a bank – either in general or with respect to individual trades:
Trades are assumed to be completely uncollateralised, and there is no ability to represent partially-collateralised (e.g. non-zero-threshold) or one-way collateralised trades.
Costs and benefits are completely symmetric so that funding benefits are equal and opposite to funding costs. Whilst this is true to some degree (e.g. two exactly opposite trades under the same margin terms must represent a zero-funding cost strategy), it might not always be the case at a high level.
There are no additional constraints on the funding strategy of a bank. In reality, this may not be the case due to aspects of the liquidity coverage ratio (LCR) and NSFR requirements.
A simple discounting approach is not appropriate in cases of partial collateralisation (e.g. a margin agreement with a threshold or one-way margin agreement). In such cases, the formula in Equation 18.4 can be used together with the relevant EPE and ENE modelling, similar to those required for CVA computation. For example, Burgard and Kjaer (2013) show that a one-way margin agreement (where the counterparty does not post) only leads to a CVA and FCA term. Not surprisingly, the funding benefit (or DVA) is absent due to the need to post margin. More generally, evaluating funding using the two explicit terms in Equation 18.4 will capture effects such as high thresholds and one-way collateralisation. However, such cases will require calculations to be made at the margin set level for all transactions covered by the same margining terms (this portfolio-level calculation is similar to CVA).
To properly account for aspects such as partial collateralisation, it is necessary to model EPE/ENE consistently with the margin terms in question. This has been discussed already for CVA computation in Section 17.5. There are caveats with respect to modelling EPE/ENE for CVA and FVA purposes, which were discussed previously in Section 11.4.2. It may, therefore, be necessary to change certain assumptions for modelling EPE/ENE for FVA compared to calculating CVA in these cases:
Margin period of risk. One important difference is regarding the MPoR. In CVA/DVA calculations, the MPoR reflects the relevant time horizon to consider over the default and close-out period (Section 7.2.3), and is typically taken to be 10 business days or more. For FVA purposes, the equivalent time horizon should only be the time taken to receive margin in a normal (not default) scenario and would therefore be assumed to be much shorter. Note that small funding costs theoretically occur in the zero-threshold, two-way CSA case, since OIS discounting alone requires continuous margin posting and zero minimum transfer amounts. However, market practice is generally to ignore such impacts. In other words, for a zero-threshold margin agreement, CVA but not FVA may be calculated.
Rehypothecation. Whilst non-reusable margin will still reduce CVA, it will not mitigate funding and so would need to be ignored in an EPE simulation for FVA, but not CVA.
Wrong-way risk. Another important distinction between CVA and FVA modelling is the treatment of wrong-way risk, which is generally conditioned on default and therefore of relevance for CVA. However, there may be situations of wrong-way funding risk (Section 18.3.7).
Following Equation 18.2, it is possible to write a more general formula for FVA:
(18.7)
where represents the value of a given trade at the future date , and represents the total amount of reusable margin held (probably variation margin only, since any initial margin will probably be segregated). As noted above, this formula would now need to be calculated at the counterparty level (strictly speaking, the margin set level if there is more than one agreement). The above can be decomposed into FCA and FBA, although this is not required. The term inside the expectation will become total EFV when there is no margin.
Note that the above has used the complete definition of funding as being ‘total value minus total margin’, as discussed in Section 18.2.2. However, it is possible to ignore strongly-margined counterparties (zero threshold and small minimum transfer amounts) in this representation, where as a general rule will always hold as a good approximation.
Table 18.2 shows CVA, FCA, and FBA in the case of a one-way margin agreement. Whilst there is a small CVA when receiving margin, FCA is zero. A one-way agreement in favour of the party (i.e. the party has an infinite threshold and the counterparty has a zero threshold), therefore, has FBA and no FCA, whilst the reverse is true when the agreement is in favour of the counterparty.
The above formulas, together with appropriate modelling of EPE and ENE, can also be used in cases where there is a material threshold in the margin agreement. For example, Figure 18.8 shows FCA and FBA for a receive fixed interest rate swap as a function of the threshold in the margin agreement. A positive exposure will create a funding cost only up to the threshold amount, after which margin would be taken, and the exposure above the threshold would essentially be capped, reducing the FCA term. Correspondingly, the negative exposure defining the funding benefit would also be capped (at a potentially different threshold), reducing the FBA term. FCA and FBA increase (in absolute terms) monotonically with increasing threshold, with their sum tending – although not completely monotonically – to uncollateralised FVA for high thresholds.
Table 18.2 Upfront CVA and FVA (non-contingent) values for interest rate swaps for cases of one-way collateralisation, compared to the uncollateralised (UC) case. Assumptions are as for Table 18.1, but the MPoR for calculating FCA and FBA is zero.
Pay fixed
Receive fixed
UC
One-way (in favour)
One-way (against)
UC
One-way (in favour)
One-way (against)
CVA
−29.9
−1.2
−30.8
−17.2
−0.8
−18.4
FCA
−15.5
–
−16.0
−8.7
–
−9.4
FBA
8.7
9.4
–
15.5
16.0
–
Total
−36.6
8.2
−46.8
−10.3
15.2
−27.8
Figure 18.8 Upfront FCA and FBA for a receive fixed interest rate swap as a function of the threshold (which is the same for the party and counterparty). Assumptions are as for Table 18.1, but the MPoR for calculating FCA and FBA is zero.
Table 18.3 shows CVA and FVA for two-way margin agreements where the margin can and cannot be rehypothecated (or generally reused). In cases where reuse is not possible, CVA is mitigated but FCA remains. This situation is relatively uncommon,7 but it is one of the cases used in Totem (Section 5.3.5). Note that in the case of the receiver swap, total CVA and FVA in the case of no reuse is more negative than the uncollateralised case.
Table 18.3 Upfront CVA and FVA (non-contingent) values for interest rate swaps for cases of collateralisation with and without reuse of margin, compared to the uncollateralised (UC) case. Assumptions are as for Table 18.1, but the MPoR for calculating FCA and FBA is zero.
Pay fixed
Receive fixed
UC
Two-way
Two-way (no reuse)
UC
Two-way
Two-way (no reuse)
CVA
−29.9
−3.0
−3.0
−17.2
−3.2
−3.2
FCA
−15.5
–
−15.5
−8.7
–
−8.7
FBA
8.7
–
–
15.5
–
–
Total
−36.6
−3.0
−18.5
−10.3
−3.2
−11.9
Note that there are other funding strategies that can give rise to different results. For example, Burgard and Kjaer (2012) discuss a strategy where a party can completely hedge out all its own and its counterparty's default risks, and there is no FCA term. However, Burgard and Kjaer argue that the funding strategy required for this result is not practical since a party would have to trade freely in and out of its own bonds of different seniorities as a delta hedge (effectively buying back junior bonds and issuing senior ones). Another way to recover a CVA and DVA result is to assume (again impractically) that the transactions can be repoed (with zero haircut and at the margin remuneration rate) so as to be ‘self-funding’. In this case, an asset would be repoed for an amount of cash equal to its value and, therefore, there would be no FCA. This point is important since some transactions (such as the purchase of treasury securities) do not have analogous funding costs due to the presence of an active repo market. However, transactions such as derivatives cannot be repoed and so this argument does not hold.
It should be noted that the form of the Totem xVA consensus pricing (Section 5.3.5) makes it possible to extract information of xVA terms and potentially determine the assumptions being made by a contributing bank. For example, suppose the quotes in Table 18.4 were seen, and assume a symmetric FVA framework. It is possible to back out the xVA terms as follows:
FCA can be determined from the first bilateral agreement with segregation (since in this case CVA only would be mitigated, as discussed in Section 11.4.2).
The unilateral agreement in favour of the bank would lead to only FBA (or DVA).
The unilateral agreement in favour of the counterparty would lead to CVA and FCA. These three quotes allow determination of CVA, FCA, and FBA, as shown.
Finally, these can be checked against the uncollateralised transaction, which is the sum of these three terms.
Table 18.4 Hypothetical upfront xVA prices in bps for different types of collateralisation.
Collateralisation
Interpretation in symmetric FVA framework
Price
Bilateral (counterparty posts to segregated account)
FCA
−25.5
Unilateral, in favour of bank
FBA
15.4
Unilateral, in favour of counterparty
CVA + FCA
−62.3
Uncollateralised
CVA + FCA + FBA
−46.9
Implied xVA values in symmetric FVA framework
CVA
−36.8
FCA
−25.5
FBA
15.4
Check (sum of three above terms)
−46.9
The above hypothetical values would be consistent with a bank that was pricing using CVA, FCA, and either FBA or DVA (but not both). Furthermore, by looking at the opposite transaction (e.g. a receive versus pay fixed swap), if FCA and FBA values for the opposing transactions were equal and opposite, then this would confirm the use of symmetric FVA and mean that the bank was categorically not pricing in DVA. Anecdotally, this can be seen for a number of the Totem contributors.
18.2.6 The FVA Debate
To some, FVA is problematic, since including a party's individual funding cost in the price breaks the price symmetry of CVA/DVA and goes against basic foundations of valuation and the ‘law of one price’. FVA also suggests that parties may be unable to agree on a price due to their unique funding costs, and that arbitrage opportunities will, therefore, be present in the market.
Early on in the use of FVA, Hull and White (2012a) put forward a view that FVA should not be considered in the pricing or valuation of derivatives. Specifically, they state ‘we argue that FVA should not be considered when determining the value of the derivatives portfolio, and it should not be considered when determining the prices the dealer should charge when buying or selling derivatives’. In general, their arguments stem from very well-established principles in finance, such as the risk-neutral valuation principle and the Modigliani–Miller theorem. For example, Hull and White (2014) argue that the use of FVA in pricing would lead to arbitrage opportunities whereby an end user could buy options from a bank and sell the same options to a bank with a higher funding cost. This could be countered by taking the view that there is no ‘market’ for uncollateralised derivatives (Kenyon and Green 2014).
Some authors have countered the Hull and White arguments (for example, see Carver 2012, Castagna 2012, and Laughton and Vaisbrot 2012, with a response in Hull and White 2012b), in general, by arguing that inefficiencies make Hull-and-White-type arguments not perfectly valid, and that FVA is real. Some of the arguments can be circular – for example:
Against FVA. Discounting should be at the risk-free rate since this is required by the risk-neutral valuation principle.
For FVA. Funding costs show that the standard risk-neutral valuation principle is incorrect because it assumes risk-free borrowing and lending.
The Hull and White argument can be seen to relate to a setup where there is CVA, DVA, and FCA, but where FCA is cancelled by another term, which they call DVA2.8 DVA2 is a benefit that arises because a party may default on its general funding liabilities (e.g. bonds), as opposed to DVA, which is the benefit of defaulting on derivatives liabilities. Hence, FCA can be seen as a cost to shareholders, with the equivalent gain accruing to creditors (via DVA2), and so the firm as a whole would not see a cost from FCA. This leads to a CVA and DVA framework, compared to the CVA and symmetric FVA one (Table 18.5).
It is important to realise that the debate around the reality of DVA and DVA2 stems primarily from the shareholder or bondholder views introduced in Section 17.3.2. The consideration of shareholder value leads to the CVA and FVA framework, whereas the total firm value (shareholders and creditors) leads to CVA and DVA. A new deal may, therefore, increase the total firm value but reduce shareholder value due to the related funding costs. Andersen et al. (2016) illustrate that the CVA and FVA framework can be seen as a pricing result where the objective is to maximise shareholder value. Albanese et al. (2013, 2015) also agree that incremental FVA can be included in entry prices under the assumption that these are set with shareholders' interests in mind.9 Burgard and Kjaer (2012) note that derivatives funding strategies can result in windfalls or shortfalls to bondholders in the case of a firm's default.
Table 18.5 Valuation adjustments in the no-FVA and FVA regimes.
No FVA (e.g. Hull and White 2012a)
FVA (e.g. Burgard and Kjaer 2011a)
Terms
CVA + DVA + FCA + DVA2 = CVA + DVA
CVA + FCA + FBA = CVA + FVA
Interpretation
Total firm value (shareholders and creditors)
Shareholder value
There is probably little or no theoretical debate around the use of FVA in pricing subject to the shareholder point of view being relevant. In practical terms, FVA pricing is clearly seen in clearing prices and Totem consensus pricing (Section 5.3.5). The remaining debate is more in relation to valuation; one argument (Albanese et al. 2015) is that, for accounting purposes, bilateral CVA (CVA and DVA) should be used.10 Andersen et al. (2016) also support this view. This requires the argument that fair value should represent the combined value to both shareholders and creditors. This does lead to potential issues with fair value being defined as exit price: one solution being to argue (probably unrealistically) that the best price would come from a party with negligible funding costs. This accounting approach would also involve a bank posting a net profit or loss due to the incremental funding and DVA adjustments. Withholding or prematurely releasing profits may be undesirable, as discussed in Section 5.4.5. Since exit price represents another party's entry price, the shareholder-driven view that seems to be adopted by most banks is not surprising.
Even in the total firm value view of funding, there would still be a component of FVA, which is the component of the funding cost that is not related to the credit risk of the party concerned. As explained in Morini and Prampolini (2011), this funding would be a market funding risk premium or liquidity component (Section 14.3.2). Assuming the CDS spread is a ‘pure credit spread’, then the liquidity premium component of the credit spread could be estimated from bond yields or the ‘CDS-bond basis’ (CDS spread minus the bond yield spread), although this term can become negative. Indeed, Hull and White (2014) state that FVA is ‘justifiable only for the part of a company's credit spread that does not reflect default risk’. Note that some theoretical studies treat funding costs as exogeneous and may assume implicitly that the bond-CDS basis is zero, and so will not identify any specific components as being attributable to default risk and liquidity effects.
An important question that remains is the correct definition of fair value and whether this should focus only on shareholders, or rather look at the combined view of shareholders and bondholders. In the former case, an FVA accounting adjustment is relevant (as is currently market practice), but in the latter, there need be no adjustment since FVA (apart from the liquidity component mentioned above) represents an internal transfer from shareholders to bondholders.
There are other debates around the implementation of FVA that relate to the funding strategy assumed and whether excess cash can always be recycled, as is the case in the symmetric FVA framework. This is discussed in Section 18.3.
18.2.7 Funding Costs and FVA Accounting
As discussed in Section 14.3.2, a funds transfer pricing (FTP) framework will probably define a fixed curve for transfer pricing funding costs, which may be determined from the treasury, potentially via the xVA desk. Such a framework usually represents an assessment of the average funding cost of the balance sheet of the party in question, rather than being specific to a counterparty. To take a quote from a market practitioner:
Kok [ING Bank] also argues there is a double-count between CVA and FVA, because poor-quality derivatives counterparties could drive up a bank's funding costs.11
Potentially related to the FVA debate (Section 18.2.6) is the fact that FVA may not capture the true incremental cost of funding a transaction given its type and counterparty, but will rather capture a static cost based on the average balance sheet cost. This is probably a necessary approximation given the situation, but it may lead to ad hoc adjustments for special cases (e.g. triple-A counterparties). This point is made by Hull and White (2014), who state that a bank should consider its incremental funding cost when entering into a transaction and, if appropriately considered, this cost should be zero. Morini (2014) argues that this is not completely correct as it is based on three crucial assumptions:
that the market has instantaneous efficiency;
that the funding of a deal happens after the market knows about the deal; and
that the effect of a new deal on the funding costs of a bank is linear.
In other words, the market may not react appropriately and efficiently, for example, when a bank is raising funds against business with high credit quality counterparties.
FVA is generally seen by banks as an internal cost of financing their uncollateralised derivatives portfolios (which involves their own cost of funding). Calculating FVA using a bank's own cost of funding does not sit well with the accounting concept of exit price (which would involve another party's funding costs). Accounting standards require that fair value should not be entity specific. For example, FAS 157 emphasises that ‘fair value is a market-based measurement, not an entity-specific measurement’, which implies that fair value should be based on the assumptions that market participants would use in pricing. This is why many banks use a ‘blended market cost of funds’ (Figure 18.9) within their FVA valuation, which represents an average level of banks funding charges that would be relevant from an exit price point of view. It is not unreasonable for a bank to use its own cost of funding if this can be shown to be representative of average market funding cost levels. If taken literally, then this leads to further problems: in exiting a derivative with a larger FCA (FBA), it is optimal to find a counterparty with a lower (higher) cost of funding. This would imply that funding spreads would be transaction specific and change with market factors.
Lou (2015) has suggested a point of view that removes this problem by pricing funding at the counterparty's funding spread. The rationale for this is that a counterparty should be indifferent to collateralising a derivative as long as the margin earns an interest rate commensurate with the cost of its own debt. This argument does not seem to be borne out in practice: for example, it has not been the norm for multilateral development banks with close to zero funding costs to move willingly to two-way margin agreements, nor are they charged zero FCA.
Not surprisingly, for pricing, some banks do use their own funding costs (Figure 18.9). This obviously creates a mismatch with the accounting treatment, especially if the bank has a relatively high or low funding cost, but this mismatch should converge over time.12
Figure 18.9 Market practice on determining funding costs for FCA valuation and pricing.13
Source: Solum FVA Survey (2014).
An example of the scaling of a bank's funding costs to meet market levels is as follows (emphasis added):
The FFVA [FVA] incorporates a scaling factor which is an estimate of the extent to which the cost of funding is incorporated into observed traded levels … The effect of incorporating this scaling factor at 31 December 2014 was to reduce the FFVA by £300m. (Barclays Annual Report 2014)
Another accounting problem arises from the treatment of the overlap between DVA and FBA, as banks may price the funding benefit component of FVA but then have to report a DVA benefit under accounting standards.
Let us also use the case of J.P. Morgan to illustrate an important aspect of banks reporting FVA.14 J.P. Morgan reported an FVA charge of $1.5bn in its fourth quarter earnings in 2013 (as discussed below, this charge is likely to be more akin to what is defined here as FCA) and noted that future FVA/DVA volatility was expected to be significantly lower as a result. In order to understand this statement, assume that J.P. Morgan used an effective funding spread of 60 bps in the calculation. This would, therefore, imply a sensitivity of -$25m per basis point.15 In the same period, J.P. Morgan's DVA had changed by $536m (a loss), since their CDS spread had tightened from 93 bps to 70 bps. This suggests an opposite $23m per basis point sensitivity for DVA.16 Hence, if J.P. Morgan's credit spread widens by 1 bp, then they can expect to lose approximately $25m due to increased funding cost, but gain $23m in DVA benefit.17 It therefore appears as if FVA is being used to partially cancel the impact of DVA.
For most banks, accounting FVA is mainly applied to derivatives assets (receivables) and therefore constitutes mainly FCA (for example, see comments as the start of Section 18.2.5). Assuming the bank believes that it should be reporting FBA instead of DVA, there is a question of whether or not this will be treated imperfectly, or whether there are further adjustments within FVA for this. For example, whilst some banks treat this problem imperfectly, others report using terms such as ‘incremental FBA’ (Solum FVA Survey 2014), which is defined as the difference between FBA and DVA. This would lead to the following:
The above definition of FVA can be seen as adding FCA and then effectively removing the DVA component and replacing it with FBA, and it is partially justified by evidence such as Totem submissions (see discussion in Section 18.2.5: some banks' Totem quotes can be clearly shown to contain no DVA).
Given the presence of FVA in accounting statements alongside CVA, it has been common practice for the xVA desk of a bank to own the profit and loss volatility of FVA and to manage it alongside that of CVA. However, whilst FVA has become a relatively standard adjustment in the financial reports of banks, it is not yet part of market risk capital rules and is, therefore, not recognised as part of the value of a derivative by regulation. Not surprisingly, given the number of banks reporting FVA in their accounting statements, the Basel Committee has launched an FVA project to determine its position with respect to this adjustment.18 There is also the question of whether or not a bank would need to derecognise DVA from its equity (Section 5.3.3) in the event that it was – effectively – not reporting DVA.
18.3 ASYMMETRIC FVA
18.3.1 Overview
Recall that the assumption for the symmetric FVA introduced in Section 18.2.4 is that funding is symmetric: required cash has to be funded, but available cash is assumed to earn an equivalent funding rate, either by buying back bonds or by recycling cash for other funding needs. Note that this symmetry is inconsistent with the NSFR, which assigns 100% RSF (required stable funding) to net derivatives assets, but 0% ASF (available stable funding) to net derivatives liabilities (and also the potential 100% RSF charge for 20% of derivatives liabilities and the ineligibility of non-cash variation margin, as discussed in Section 4.3.4). This does not mean that a bank cannot use a symmetric FVA approach, but just that this approach will tend to be at odds with the bank maintaining a strong NSFR. We will illustrate this with an NSFR invariance FVA (Section 18.3.4).
More generally, the form of FVA is linked to a funding strategy, and alternative funding strategies will lead to different FVA terms and overall economic values to shareholders. In particular, there is a potential alternative (or extension) to FVA which treats funding asymmetrically.
The key defining point for the correct funding strategy is the rates at which a business can borrow and lend funds, either via its own internal treasury or externally with the market. There are a number of possibilities and resulting outcomes:
borrow and lend at the overnight rate (no FVA);
borrow and lend at the same unsecured rate (symmetric FVA);
borrow at an unsecured rate but lend at the overnight rate (asymmetric FVA); or
borrow at an unsecured rate but lend at a shorter-term unsecured rate (partially asymmetric FVA).
This section will deal with the latter two possibilities, where FVA is asymmetric or partially asymmetric. If it is not possible to monetise net funding benefits at the same rate as net funding costs, there will be an asymmetry in the funding. Note that funding must, in this situation, be calculated for the aggregate portfolio level. This is because there is a difference between funding benefits that offset funding costs and those that create outright funding benefits. Asymmetric funding is illustrated in Figure 18.10.
Figure 18.10 Illustration of the impact of asymmetric funding assumptions. The total funding requirement (or benefit) of the portfolio is depicted.
To understand the motivation behind the asymmetric funding considerations above, recall that derivatives variation margin is paid on a daily basis. Therefore, the assumption that this can be used to buy back own bonds (for example) is potentially aggressive, since the variation margin may need to be returned one day later. On the other hand, margin posted against long-dated transactions could be required for a long period of time and – on a conservative basis – may give rise to long-term funding requirements.
If there is asymmetry over funding, then the funding profile of the portfolio in question is important. Asymmetry would tend to be more of a problem if this profile is a liability – and not asset heavy (Figure 18.11). However, due to the nature of client trading activity (e.g. long-dated cross-currency swaps with more ENE against short-dated transactions with more EPE), it is possible for the profile to change sign. At the time of writing, many banks have an asset-heavy derivatives book, but there are definitely banks with all of the characteristics shown in Figure 18.11. To some extent, the form of a bank's funding profile may be changeable (e.g. by restructuring or incentivising certain client trading). However, this may be limited due to certain systematic factors (e.g. a low interest rate environment coupled with predominantly receiving the fixed rate on uncollateralised trades). Note that asset-heavy portfolios are still sensitive to asymmetric funding assumptions, although less so than those with larger relative liability components.
18.3.2 Asymmetric FVA
Albanese and Iabichino (2013) and Albanese et al. (2015) propose the view that excess cash for derivatives books is an unstable source of funding and should be assumed to earn only the risk-free lending rate.19 Burgard and Kjaer (2012) also assume asymmetry regarding unsecured borrowing and lending, where unsecured lending may be assumed to yield only the risk-free rate,20 whilst borrowing will require the unsecured term funding rate. In this setup, FVA should be considered at the ‘funding set’ level, defined as a collection of transactions which can be combined from the point of view of funding. In the funding set, reusable variation margin received can be freely rehypothecated to meet the funding requirements of other transactions. However, this rehypothecation cannot be done outside the funding set. An obvious interpretation of a funding set is that it is the entire OTC derivatives book of the party in question.21 A positive cash position in the funding set is not rehypothecated across funding sets but invested at the risk-free rate. This assumes that a net benefit on a funding set cannot be used to reduce funding costs across any other activity (different funding set) of the bank.
In this situation, a bank would have only an FCA term, which would be a portfolio-level (funding set) calculation:
(18.8)
where is the value of trade j, and is the value of rehypothecable22 margin k (both discounted). now represents the appropriate funding spread for borrowing. The summation above needs to be across all trades and margin agreements in the funding set. However, it may be necessary first to sum over counterparty-level information to capture:
counterparty survival probabilities (if desired); and
thresholds in margin agreements.
Note that the calculation of the above term requires a consistent simulation of the total funding set (e.g. the entire OTC derivatives book), although, as mentioned previously, it may be relevant to leave out strongly collateralised counterparties where the trade value will be cancelled by the margin held.
This is clearly quite challenging in terms of processing time and memory requirements and could mean that pricing one transaction would involve resimulation of the entire funding set. To understand this intuitively, note that, under asymmetric funding assumptions, it is necessary to know if a future funding benefit reduces overall funding requirements in the funding set or increases benefits (in the former case, there is a benefit, and in the latter, there is not). This was illustrated in a stylised fashion in Figure 18.10. The larger the asymmetric region in Figure 18.11, the greater the deviation will be between the symmetric and asymmetric FVA approaches.
More generally, it is not necessary to assume that lending is done at the risk-free rate and there is an equivalent FBA formula:
(18.9)
where the only difference is the collateralised ENE term and the funding spread for lending. With respect to this spread, there are three possibilities:
Symmetric:
Asymmetric:
Partially asymmetric:
An immediate consequence of the (partially) asymmetric assumption is that FCA (and FBA, if relevant) will not be additive across transactions or counterparties (note that FCA is negative):
(18.10)
In order to give an example of this, an asset-heavy portfolio is used with exposure profiles, as shown in Figure 18.12. The funding spread is as used in the previous example in Section 18.2.4.
For the partially asymmetric case, the lending spread is assumed to be half the borrowing spread. Table 18.6 shows FVA for this portfolio in the three different cases. Since the portfolio is asset heavy, asymmetric FVA is only slightly less negative than symmetric FCA. Fully asymmetric FVA is about 18% larger than symmetric FVA, with partially asymmetric being in between the two. Note that asymmetric FVA is different even though EFV for the portfolio is always positive at all maturities.
Table 18.6 Different FVA calculations for the portfolio characterised in Table 18.12.
Symmetric (portfolio level)
Partially asymmetric
Asymmetric
FCA
−181,984
−181,984
−181,984
FBA
27,384
13,769
–
FVA
−154,600
−168,215
−181,984
18.3.3 FVA Allocation
Like CVA, it is important to be able to allocate FVA to transaction level for pricing and valuation purposes. One advantage of the symmetric framework is that FVA is additive across transactions: there is no difference between reducing funding cost and accruing funding benefits. In the symmetric framework, the allocation is trivial as all calculations can be made at the trade level. The only exception to this is partial collateralisation – such as one-way margin agreements – where FVA would need to be calculated at the counterparty (margin agreement) level to account for these features. In symmetric cases, this approach (Equation 18.4) will correctly represent overall funding costs across transactions. For example, consider the total funding cost of a non-collateralised transaction hedged via a partially-collateralised transaction represented via the sum of the relevant FVAs. Suppose an uncollateralised receiver swap has an overall funding benefit. By symmetry, the payer hedge will have an equal and opposite funding cost. However, this cost will be smaller due to the ability to receive collateral above the threshold. Hence, the combination of the two transactions has an overall funding benefit. This can be seen as the benefit from receiving margin above the threshold on the hedge but not posting margin on the uncollateralised transaction, minus the cost from posting on the hedge and not receiving.
However, in an asymmetric FVA framework, funding costs and benefits are assessed differently. There is the difference between a funding benefit that will reduce an existing funding cost and one that will add to existing funding benefit. When pricing a transaction, it will be important to know if the existing portfolio has a net cost or benefit at a given tenor and the impact a new transaction has on this. Incremental pricing of FVA would be similar to that for CVA (Section 17.4.1), except that FVA would then need to be considered at the overall portfolio level. This would lead to very significant computational requirements for pricing new transactions and allocating FVA to existing ones, as, for example, discussed by Albanese and Iabichino (2013).
Pricing asymmetric FVA, therefore, requires a full simulation approach at the portfolio level. It will generally (but not always) lead to a more negative (higher-cost) FVA than a symmetric approach. To illustrate this, consider a typical funding-beneficial trade, a cross-currency swap paying the lower rate of interest which, therefore, has a large ENE (and, therefore, potentially large FBA) component. The standalone EPE/ENE of this can be seen in the previous Figure 15.27 (bottom). Incremental FVA for this new transaction is shown in Table 18.7 and the relevant exposure profiles in Figure 18.13. The cross-currency swap has a large funding benefit term under symmetric FVA due to large ENE, which causes portfolio EFV to reduce (Figure 18.13, top). However, this EFV reduction is a result of an increase in EPE but a larger reduction (more negative) in ENE (Figure 18.13, bottom). Hence, an asymmetric FVA framework, where ENE is not relevant, shows only a cost due to the increase in EPE and FCA. In a symmetric FVA world, there is, therefore, a benefit of 26,231, whereas in an asymmetric one, there is a cost of -15,166. Note that incremental FVA in the symmetric case is exactly equal to standalone FVA (but this is not true for FCA and FBA).
Whilst asymmetric FVA is always worse (more negative) than symmetric FVA, this may not be the case for incremental charges (although it almost always is). Asymmetric FVA can give more beneficial pricing for risk-reducing trades where there is a large loss in the funding benefit, and the framework without FBA is more beneficial. This difference will be large for a liability-heavy portfolio where the loss in FBA benefit can be more substantial. This is illustrated in Table 18.8 and Figure 18.14. Symmetric FVA leads to an incremental cost resulting from the loss of funding benefit. On a standalone basis, this is equivalent to the cost of FCA being larger (in absolute terms) than FBA. In an asymmetric FVA framework, there is no loss of funding benefit and the overall incremental effect is positive due to a reduction of FCA.
Table 18.7 Incremental pricing of a five-year cross-currency swap for symmetric and asymmetric FVA regimes.
Standalone
Incremental
Symmetric
Asymmetric
FCA
−64,391
−15,155
−15,166
FBA
90,622
41,397
–
FVA
26,231
26,231
−15,166
Figure 18.13 EFV (top) and EPE and ENE (bottom) profiles for the portfolio and total (portfolio plus cross-currency swap).
Table 18.8 Incremental pricing for a risk-reducing trade according to the profiles in Figure 18.14.
Standalone
Incremental
Symmetric
Asymmetric
FCA
−46,225
9,770
9,770
FBA
10,332
−45,664
–
FVA
−35,893
−35,893
9,770
Figure 18.14 EFV (top) and EPE and ENE (bottom) profiles for a risk-reducing trade (portfolio is pay fixed, and new trade is receive fixed).
18.3.4 NSFR Invariance
The examples in Section 18.3.3 illustrate that it may be relevant for banks to adapt their FVA pricing so as to avoid paying excessively for funding benefits. This will be further amplified with the implementation of the NSFR (Section 4.3.4). The NSFR contains elements that will skew the funding requirements of a derivatives business:
Net derivatives payables have a 0% ASF weight. This means that a bank with a derivatives book that provides funding overall will not gain any benefit from this in terms of an increase in its NSFR. This is broadly consistent with the completely asymmetric representation in Equation 18.8.
20% of total standalone derivatives liabilities (payables) have 20% ASF. This means that even funding benefits have an associated funding cost under NSFR (the intention of this is to reflect the large variability in derivatives exposures and the fact that a funding benefit can quite quickly become a cost).
Margin received can only reduce the funding cost of a derivative if it is cash in the currency of the transaction.23 This means that collateralised transactions, which are generally thought of as having little or no FVA, may still have a large contribution to the NSFR if, for example, bonds are held as collateral.
The concept of invariance pricing was introduced in Section 5.4.3. Not surprisingly, NSFR invariance FVA pricing (i.e. pricing funding to maintain a given NSFR) requires asymmetric FVA. If the funding cost for the high-quality liquid assets (HQLAs) is different from the standard funding cost, then there should be a (scaling) adjustment made. However, even transfer pricing via asymmetric FVA will deteriorate a bank's NSFR. To price to maintain the NSFR (‘NSFR invariance’), the following terms would need to be included:
an adjustment for the increase in liabilities in relation to the 20% charge;
an adjustment for non-cash variation margin; and
a scaling factor corresponding to the bank's desired NSFR.
This is illustrated in Table 18.9 for the cross-currency swap example above (Table 18.7), assuming that the bank's desired NSFR is 120% and 20% of derivatives liabilities are given an RSF charge. A further assumption is that the bank's cost of funding HQLAs is the same as its normal unsecured funding cost. Note that NSFR invariance approximately doubles the cost of the trade compared to an asymmetric FVA framework.
Since NSFR applies only at the bank level, there is the question of whether a bank would allow derivatives to be partly subsidised by other activities or require them to be NSFR compliant on a standalone basis, as the NSFR invariance price above implies. Some banks, especially those with relatively small derivatives businesses, are likely to allow this cross-subsidisation and may be more competitive on certain trades as a result.
For allocation for valuation purposes, asymmetric FVA would require a calculation of marginal FVAs following the discussion in Section 15.2.2, whereas symmetric FVA is a trivial allocation. Table 18.10 shows marginal FVA for symmetric and asymmetric cases for the four transactions discussed previously for marginal EPE (see Figure 15.31). Note that the difference allocation is quite different in the two cases and not completely predictable. For example, whilst asymmetric FVA is generally more negative, the swaption has a less negative contribution due to having very little ENE.24
Table 18.9 NSFR invariance incremental pricing for the five-year cross-currency swap compared to the symmetric and asymmetric FVA regimes.
Symmetric
Asymmetric
NSFR invariance
FCA
−64,391
−15,166
−18,199
FBA
90,622
–
–
Liability RSF
–
–
9,935
Total
26,231
−15,166
−28,134
Table 18.10 Marginal FVA in symmetric and asymmetric cases for the transactions considered previously in Section 15.6.2.
Symmetric
Asymmetric
7Y Payer IRS
−50,895
−59,786
5Y Payer IRS
−17,947
−18,152
5Yx5Y Long Payer Swaption
−85,758
−76,380
5Y USDJPY XCCY
26,231
−42,832
Total
−128,369
−197,150
18.3.5 Funding Strategies
It is important to align FVA with the funding costs actually experienced. In the illustration in Figure 18.1, FVA appears as the cost of all the future funding charges experienced by the xVA desk from the treasury. This means that the daily cost of funding charged to the xVA desk will be offset by a profit resulting from the reduction in FVA held. Changes in FVA due to market variables can also be hedged (this will be discussed in more detail in Section 21.2).25 However, a misalignment between FVA charges and the funding costs experienced (e.g. symmetric versus asymmetric) will lead to profits or losses (probably the latter) being experienced by the xVA desk, which is probably not in line with its intended mandate.
A symmetric FVA approach is clearly desirable since it makes pricing and valuation easy and generally leads to more competitive pricing. However, there is a question of whether symmetric FVA is consistent with paying for funding benefits, when this may not align with the funding strategy and the NSFR. Banks with asset-heavy portfolios may feel that they are close to a symmetric FVA regime, but this may also incentivise the liability-heavy transactions, such as the example above (Table 18.7), that may create a portfolio with more FBA, which may be difficult to monetise.
An asymmetric funding policy used internally in a bank creates an incentive to maintain a derivatives book that is asset heavy to simplify pricing and valuation and to be able to pay funding benefits.26 This could be achieved, for example, by executing large zero-coupon swaps or novating into uncollateralised derivatives positions with a positive value, which can be seen as lending to the counterparty at an unsecured rate (rather than lending internally at a lower rate). It is, therefore, important to consider the incentives that arise from the treatment of funding and the consequences for pre-deal FVA.
The asymmetric framework arises from the view that excess funding benefits (in the form of variation margin, for example) do not represent term funding due to the daily nature of the margining process. However, if the funding costs in Figure 18.1 seem to be based on the current funding requirement of a business, this may, therefore, implicitly offset short-term funding benefits (e.g. an OTM short-dated transaction) against long-term funding costs (e.g. an ITM cross-currency swap). It may, therefore, be desirable for the treasury to charge funding across the entire EFV profile and incorporate this into their Asset Liability Management (ALM) process. This will be discussed in more detail in Section 21.3.1.
The process in Figure 18.1 may also treat funding costs as fixed, irrespective of aspects such as the type of counterparty. This means that the funding of a new transaction will be priced at something like the average funding cost of the bank, and FVA charges would not be sensitive to the counterparty type. In reality, it is the incremental cost of funding new business that is relevant. Trading with poor-quality counterparties will likely drive up funding costs and vice versa. To create the right incentives, an xVA desk may incorporate this into pricing in an ad hoc manner by charging less FCA to high-quality counterparties and vice versa. However, this will potentially misalign with the funding costs experienced from the treasury. The true economic cost of funding a new transaction is very hard to assess as it is driven by the change in the funding cost of the balance sheet that this creates. The xVA and treasury setup is only approximating the true effect by considering the average cost of funding that balance sheet at the current time.
18.3.6 LCR Costs
Contingent FVA could occur due to rating triggers within the margin agreement and the requirement to pre-fund with HQLAs any outflows in relation to a downgrade (e.g. three notches, Section 4.3.3) of a bank's own credit rating. In order to assess FVA in such a situation, it is necessary to define rating transition probabilities. These can be estimated from historical default data, although such estimates are long-term averages of mainly corporations and the rating changes cannot be hedged.
For example, suppose a given trade is uncollateralised, but the bank is contractually obliged to post variation margin in the event that it is downgraded. The FVA representation – which would amount to an LCR invariance price since it would charge the funding cost of new HQLAs – would have the following components:
the current cost of HQLAs that would need to be held against a negative value (this would reduce the funding benefit component arising from an ENE-like term in the event that the downgrade trigger is not breached); and
an actual funding cost when the trigger is breached, leading to variation margin being posted (or some modelling of the dependency between the rating process and the funding cost).
Note that the above would be one sided and a cost, since only the downgrade of the bank must be considered (no benefit can be achieved as a result of the counterparty being downgraded). The adjustment for this contingent FVA (CFVA) could be written as:
(18.11)
where is the hypothetical variation margin that would have to be pre-funded under the LCR, and is the funding spread for the HQLA cost (conditional on no ratings trigger). The term is the actual variation margin that would be posted in the event of the downgrade, and is the funding spread in this downgraded state. The term gives the probability of the contingent outflow (e.g. three-notch downgrade). Assuming the threshold for contingent margin posting is zero, and will be equivalent to the value of the transaction and the expectation term becomes an ENE. Furthermore, since the above two terms are of similar form, it is possible to use a blended funded spread:
(18.12)
Note that it would be unrealistic for this funding spread to be higher than the base funding spread (assuming the funding of HQLAs is lower), since the latter value should include the possibility of rating downgrades. This leads to:
(18.13)
In a symmetric FVA framework, this can, therefore, be seen as a reduction in the funding benefit term, which occurs intuitively due to the need to post variation margin if the rating trigger occurs, or holding HQLA against the amount if not:
(18.14)
In an asymmetric FVA framework, the CFVA components need to be included, with any LCR component related to ratings triggered captured at the counterparty level and aggregated to the full portfolio level.
An example of this is given in Table 18.11, with the same assumptions used in Section 18.3.2.27 The HQLA funding cost is assumed to be 50% of the standard funding cost, and the annual probability of the rating trigger being breached is 5%. In the symmetric case with LCR costs, total FVA is the sum of FCA, FBA, and CFVA cost, or equivalently FCA and modified FBA. In this example, CFVA cost is the same in both frameworks, as the example is for a single counterparty. In a full portfolio case, this would not be the case for asymmetric FVA.
The above calculations become more involved when there are multiple rating triggers (e.g. different threshold levels in the margin agreement depending on the rating). In such cases, there are multiple possible rating states, with different HQLAs and funding costs. In the event of a downgrade, there is a potential increase in funding cost due to the need to post variation margin, but a potential reduction in HQLA cost since this is no longer a contingent component.
Table 18.11 Total FVA adjustments with and without LCR costs related to a downgrade trigger. Note that with LCR costs, the total FVA can be seen as either FCA + Modified FBA or FCA + FBA + CFVA Cost.
Symmetric (portfolio level)
Asymmetric
Base
With LCR cost
Base
With LCR cost
FCA
−197,150
−197,150
−197,150
–
FBA
68,781
68,781
–
–
CFVA cost
–
−42,714
–
−42,714
Modified FBA
–
26,065
–
–
FVA
−128,369
−171,085
−197,150
−239,866
18.3.7 Funding and Wrong-way Risk
In the context of funding, wrong-way risk (WWR) is a positive relationship between an institution's funding spread and other market variables. There may also be WWR funding implications, although these may be less severe and more easily captured compared to CVA WWR. This is because, unlike CVA, funding is not related to default events. CVA WWR can be the result of causal relationships between defaults and other market variables that may be hard to capture and model (e.g. the FX example discussed in Section 17.6.4). On the other hand, WWR in FVA may occur primarily as a result of more observable macro-economic relationships.
An obvious example of WWR in FVA is the correlation between funding spreads and interest rates. Consider the interest rate swap example presented in Section 17.6.3. Table 18.12 shows the symmetric and asymmetric FVA calculations for positive and negative correlations between interest rates and funding spreads. A positive correlation causes the swap to be more ITM when funding spreads are high, leading to higher FCA and lower FBA. The reverse is true for the negative correlation case, and the receive fixed swap shows the opposite behaviour. Note also that in the symmetric FVA case, WWR has a larger impact due to simultaneously changing FCA and FBA in the same direction (e.g. FCA becomes more negative and FBA becomes less positive).
WWR in funding may also be important for FX products where a devaluation in a bank's local currency may be linked to an increase in funding spreads. This is discussed in more detail by Turlakov (2012).
Table 18.12 Upfront (non-contingent) FVA value for interest rate swaps with a correlation between interest rates and funding spreads.
Pay fixed
Receive fixed
−50%
No WWR
+50%
−50%
No WWR
+50%
FCA
−11.5
−15.5
−21.8
−4.7
−8.7
−12.0
FBA
11.9
8.7
4.2
21.3
15.5
11.3
Symmetric FVA
0.4
−6.8
−17.6
16.7
6.8
−0.7
Asymmetric FVA
−11.5
−15.5
−21.8
−4.8
−8.7
−12.0
NOTES
1 Wood, D. (2012). Putting the fun in funding valuation adjustment. Risk (6 September). www.risk.net.
2 Note that the reduction in this case occurs exactly halfway through the profile. This is for illustration purposes only and the true profile depends on the precise shape of the yield curve.
3 Meaning we consider funding costs and benefits to be associated with the same funding spread, which will be discussed more in Section 18.3.
4 The Totem submissions include counterparties of different credit quality, and so by examining the contributions across counterparties, it is possible to see if survival probabilities are being used.
5 Burgard and Kjaer (2013) call this ‘Strategy 1’ and do consider alternative strategies such as discussed in Section 18.3.
6 Aside from the time delay in receiving margin, which is usually ignored in such situations.
7 It is uncommon contractually, although it could occur also when receiving securities which cannot be repoed or used as margin in another agreement.
8 Albanese et al. (2015) make a similar argument around what they call ‘funding debt adjustment’ (FDA) and Elouerkhaoui (2016) discusses a similar term, the ‘fair-value option’ (FVO).
9 Note that Albanese et al. (2015) also propose a different version of FVA which is asymmetric and will be discussed in Section 18.3.
10 With some additional adjustments for the ‘first-to-default’ effect.
11 Becker, L. and N. Sherif (2015). FVA: How six smaller banks do it. Risk (2 April). www.risk.net.
12 For example, banks with high funding costs will realise a day one profit if they charge more FCA than they value. However, this will be lost in a negative carry over time as they borrow from the treasury. The net result is to encourage the right trades.
13 The question focused on FCA due to the potential overlap of FBA and DVA.
14 This analysis is based on market information and the statements of J.P. Morgan and is not their own analysis.
15 The loss of $1.5bn divided by the assumed funding spread of 60 bps.
16 The loss of $536m divided by the change in CDS of −23 bps.
17 Ignoring any basis between their own CDS and funding costs.
19 Furthermore, in Burgard and Kjaer (2012), symmetric funding arises when a zero-coupon bond is a sole funding instrument, whilst asymmetric funding is a consequence of assuming that a single bond with recovery is freely tradable.
20 Of course, unsecured lending can yield more than the risk-free (or OIS) rate, but this then involves taking additional credit risk.
21 In an earlier version of Albanese et al. (2015) they state ‘we propose that excess collateral on OTC books should be considered as an unstable source of funding, not fungible with bank debt’.
23 For more detail, see BCBS (2014b), Paragraph 35. www.bis.org.
24 It is a physically-settled swaption and so there is some ENE arising after the exercise date, but this is small.
25 Any changes in the funding spread of the bank will be more difficult to hedge.
26 Noting that paying funding benefits will potentially lead to trades which are antithetic to maintaining an asset-heavy book.
27 The portfolio considered is the one used in Section 18.3.2, including the cross-currency swap; the total EPE and ENE can be seen in Figure 18.13 (total).
is the funding spread for the time 
with respect to the rate used for valuation (e.g. OIS). Note that the above formula is equivalent to the difference between discounting at a higher rate minus discounting at the base rate (see Appendix 18A).
) replaces the counterparty credit spread, and (discounted) EFV replaces EPE. Note that the funding spread should be the difference between the cost of funding and the rate used for discounting (most obviously the OIS rate).
, and so the above formula can be decomposed into two terms. Whilst at first glance this is unnecessarily complex, it does generalise the FVA formula and provide additional intuition. FVA is, therefore, typically defined as being driven by both funding costs (FCA) and benefits (FBA):
is a risky discount factor including survival probabilities as defined in Section 17.3.3, and 
represents the funding spread at time 
. The above formula is the contingent (including survival probabilities) integral representation of the FCA summation term in Equation 18.4b where 
is the risk-free interest rate, 
and 
are the default intensities of the party and counterparty, respectively.
is the loss given default of the party making the calculation. The above term can be seen as a funding benefit which is monetised by using excess cash to buy back bonds, with the FCA being a symmetric term that arises from the equivalent cost of issuing bonds to generate cash. The DVA term in Equation 18.6 can equivalently be seen as the previously-defined funding benefit (Equation 18.4c) under the following two assumptions:
and is the case in the Burgard and Kjaer framework, since they consider the existence of bonds only and there is, therefore, a single credit spread that can be seen as defining own credit risk and funding costs.
is largely model independent, unlike the components 
and 
. Under these assumptions, uncollateralised derivatives can be valued via an appropriate choice of discount factor, rather than via the need to calculate an explicit FVA adjustment. This has been useful for banks to implement funding adjustments in a relatively straightforward manner (see the first quote in Section 18.2.1). This symmetry also means that FVA is additive across transactions, meaning that pricing can be done at a transaction level.
represents the value of a given trade at the future date 
, and 
represents the total amount of reusable margin held (probably variation margin only, since any initial margin will probably be segregated). As noted above, this formula would now need to be calculated at the counterparty level (strictly speaking, the margin set level if there is more than one agreement). The above can be decomposed into FCA and FBA, although this is not required. The term inside the expectation will become total EFV when there is no margin.
will always hold as a good approximation.
is the value of trade j, and 
is the value of rehypothecable22 margin k (both discounted). 
now represents the appropriate funding spread for borrowing. The summation above needs to be across all trades and margin agreements in the funding set. However, it may be necessary first to sum over counterparty-level information to capture:
is the hypothetical variation margin that would have to be pre-funded under the LCR, and 
is the funding spread for the HQLA cost (conditional on no ratings trigger). The term 
is the actual variation margin that would be posted in the event of the downgrade, and 
is the funding spread in this downgraded state. The term 
gives the probability of the contingent outflow (e.g. three-notch downgrade). Assuming the threshold for contingent margin posting is zero, 
and 
will be equivalent to the value of the transaction 
and the expectation term becomes an ENE. Furthermore, since the above two terms are of similar form, it is possible to use a blended funded spread:
