19
KVA

19.1 OVERVIEW

Banks have historically charged capital to transactions, at least implicitly, by setting limits on capital usage or requiring a certain capital hurdle to be achieved. Whilst these actions discourage capital intensive transactions to a degree, they do not properly price in the lifetime costs associated with holding capital. Recent years have seen banks become increasingly focused on return on capital (ROC). Capital is a cost because investors require a return on their investment. High capital ratios make banks more resilient but potentially reduce their ROC (profit/capital). Whilst the Modigliani–Miller theorem (Section 14.1) suggests that a firm should be indifferent to how it is financed, the reality is that banks generally find capital expensive compared to debt.

Whilst businesses have always had ROC targets and hurdles, these have often been relatively easy to achieve and treated rather softly when compared with other components such as the net profit of a trading desk. ROC metrics have also often been based on ‘economic capital’ (Section 14.2.1) and not regulatory capital requirements. However, the size of regulatory capital components and the cost of raising new capital have created a greater need to price regulatory capital into transactions more explicitly. The greater focus on capital costs for banks can be seen to be driven by a number of components:

the requirement for banks to have higher capital ratios and a greater proportion of high-quality capital (Section 4.2.2);
additional capital requirements such as the credit value adjustment (CVA) capital charge (Section 13.3); and
other capital constraints such as the leverage ratio (LR) (Section 4.2.7), capital floors (Section 4.2.8), and bank stress tests (Section 4.2.10).

Whilst the above apply to all activities of a bank, derivatives are particularly expensive with respect to counterparty risk-related regulatory capital requirements and their relatively conservative treatment. Capital is seen as a scarce resource, and there is, therefore, a strong need to ‘match’ the profit in a transaction to the associated capital requirements.

Additionally, the more rigorous treatment of other xVA components, in terms of quantification and management, has led to similar approaches being applied to capital. For example, capital requirements for derivatives can be relatively volatile and, therefore, the correct lifetime cost of capital, instead of merely a current (spot capital) or approximate profile, may be considered.

In terms of pricing, it is clear that capital costs need to be factored in, at least to a degree. Capital value adjustment (KVA) is commonly used to define such costs. In one sense, KVA is just another way to express a minimum ROC hurdle. However, whilst ROC hurdles have been historically used as soft hurdles for derivatives businesses, more rigorous KVA charges are becoming more common.

In terms of valuation, including capital costs is more contentious. On the one hand, it can be seen as the cost of funding equity, and there is, therefore, an analogy with debt funding and funding value adjustment (FVA) (and margin value adjustment, MVA). On the other hand, it is possible to see capital not as a direct cost per se but as merely representing the ownership of a bank, with the owners being paid a dividend contingent on the bank's profitability. KVA accounting is, therefore, a subject of much debate.

19.2 CAPITAL VALUE ADJUSTMENT (KVA)

19.2.1 Return on Capital

Banks have used metrics such as return on risk-adjusted capital (RORAC) for many years as a benchmark of the performance of a trade or project. RORAC is often defined as:

(19.1)

Formulas such as the above often do not consider multiple periods, which is reasonable if the underlying is either of short maturity (e.g. one year or less) or if the capital is fairly constant over time.

For situations where the underlying time horizon is long and/or the amount of capital required is variable, it is necessary to consider multiple periods. In such situations, it is common to use an internal rate of return (IRR) formula to evaluate the profitability of a project. IRR is typically defined as an interest rate that makes the net present value of a project equal to zero. For example, IRR is the return that is the solution to the formula:

(19.2)

where images represents the cash flow at a time images .

In order to use the above formula to consider the ROC, it is necessary to consider the appropriate cash flows. At time zero images , there will be a profit images on the transaction minus the current amount of capital that will need to be held ( images ). The cash flows at any subsequent date will relate to the change (usually a release) in the amount of capital required, images . IRR is then the solution to:

(19.3)

It can be shown (Appendix 19A) that the above can be rewritten as being approximately:

(19.4)

where the term images can be interpreted as representing the future cost of capital for the period in question, and images can be seen as a discounting term. If we now consider IRR to be a pre-determined rate, then this formula is intuitive in that the profit required must be the amount of capital required over time multiplied by the cost of holding that capital (IRR) discounted at the cost of capital. However, in this simple representation, cash and capital were not treated differently, and there was only a single rate that represented a return and also gave rise to a discount factor.

19.2.2 KVA Formula

A KVA formula (e.g. Green et al. 2014) can be written as:

(19.5)

where images denotes the discounted expected capital profile, which is the amount of capital that is expected to be held, images , at a given time images in the future multiplied by an appropriate discount factor, and images is the cost of holding this capital (previously the IRR term in Equation 19.4). The formula is essentially integrating over all capital costs over the lifetime of the transaction or portfolio in question. It can be seen as defining the profit that is required to generate the required return on the capital deployed.

Note that the above formula is defined to be a negative value, which interprets KVA as a cost, although this is only a convention. It can also be written as an integral similar to CVA and FVA:

(19.6)

where the term images is as defined for CVA in Section 17.3.3. As discussed in Section 14.2.3, the capital cost ( images ) term is a relatively subjective parameter and may depend on aspects such as the nature of the business in question, the dividend policy of the bank, and the competition. Whilst numbers in the range 8–10% may often be mentioned, ‘efficiency’ and tax effects (since dividends are paid post-tax) are often included to make the actual gross number bigger (Green and Kenyon 2015 consider tax more explicitly and derive a tax valuation adjustment). The hurdle may also increase so as to penalise longer-dated transactions where more business and regulatory uncertainty exists. It may also be that the CC is actually not an input but rather an output in certain cases – for example, knowing the potential profit of a transaction, equating this to KVA and solving for the CC that gives this KVA.

The expected capital profile (ECP) term is a more quantitative challenge, similar to expected positive exposure (EPE) and expected negative exposure (ENE) terms required for CVA/DVA (debt value adjustment) and FVA calculations. However, this is also directly linked to the underlying regulatory capital requirements, both at the current time and in the future. It therefore requires a strong appreciation of the regulatory rules over the lifetime of the portfolio in question. This will be discussed in Section 19.2.3.

In Equation 19.5, unlike Equation 19.4, there are different rates for the cost of capital and the discount factor. The question on the discounting to apply and/or the use of survival probabilities will be discussed in Section 19.3.3.

KVA is likely to differ substantially between different banks for a number of fundamental reasons:

Capital requirements. Banks may be subject to different overall capital requirements due to regional rules or other aspects (e.g. the requirement for globally-systemically-important banks (G-SIBs) to hold additional capital).
Capital methodologies. Different banks may use different methodologies in capital calculations (e.g. internal models versus standardised formulas).
Return on capital. Banks may have different business models and, therefore, have differing ROC hurdles (e.g. a large commercial bank may accept making a low ROC on a relatively small derivatives business on the basis that this is provided as a service to clients).

Hence, the KVA calculation is highly subjective and may differ substantially between banks. It may also, like MVA, be computationally expensive due to the need to simulate market scenarios in order to calculate the ECP. This is particularly the case where a bank has internal model approval since this would involve a secondary Monte Carlo simulation. In practice, banks may sometimes hope that approximating using a projected capital profile (i.e. one in which there is only one market scenario) may be reasonable. Indeed, the error in doing this may be relatively insignificant compared to some of the other problems mentioned above, such as future changes in capital requirements and methodologies.

19.2.3 Capital Profiles

In order to calculate KVA, it is necessary to be able to generate capital profiles over time for the different methodologies in question. When calculating capital profiles, the first problem is the differing regulatory requirements and associated methodologies that exist. Capital is made up of a number of different components which may all need to be considered:

Credit risk. This is a charge for the counterparty default risk (Section 13.2).
CVA capital charge. The capital, introduced under Basel III, for the volatility of a bank's CVA (Section 13.3).
Leverage ratio. A minimum capital requirement based on a regulatory definition of leverage (Section 13.4.6).
Market risk. Market risk capital requirements may often be small, as derivatives transactions will generally be hedged. However, some residual requirements due to basis risks (for example) may exist. This has also been shown by Kenyon and Green (2014) to be potentially significant, where back-to-back hedges may create spurious capital charges due to being ineligible hedges, as discussed in Section 13.3.6. This may have less of an impact in the future, with more aligned regulatory methodologies such as standardised CVA (SA-CVA).
Prudent valuation (EBA 2013). Additional capital requirements for aspects such as the uncertainty of CVA calculations required in the European Union (EU).

Note that some capital components will not be explicitly considered since they are not easily ‘matched’ to a precise transaction. Examples would include minimum capital requirements for operational risk and any capital buffers. Such components can potentially be captured implicitly by increasing the CC term in the KVA formula (Equation 19.5). Of the above five components, most banks will rigorously capture the impact of the first two, with the other three being taken into account by some banks and in some situations.

Furthermore, given the timescales for implementation of regulation and changes to regulatory methodologies, it is important to consider the evolution of the underlying capital requirements over time. Some of the important aspects in this respect are:

Standardised approach for counterparty credit risk (SA-CCR). The new methodology for banks without internal models, which also applies to other regulations (Section 13.4.3).
Leverage ratio. The simple backstop to capital requirements using the above SA-CCR methodology (Section 4.2.7).
Fundamental Review of the Trading Book (FRTB). Changes to the market risk capital rules (Section 4.2.4).
FRTB-CVA. Changes to the CVA capital charge in line with the above market risk changes (Section 13.3).
Capital floors. Capital floors using standardised methodologies for banks with internal models (Section 4.2.8).
CVA capital exemption. The EU exemption for CVA capital charges, which may be removed at some point in the future (Section 4.2.5).
Hedging effectiveness. The impact of CVA hedges, which currently may produce adverse capital costs, but under future SA-CVA regulation may lead to substantial benefits for CVA hedges (Section 13.3.6).

KVA is clearly the most subjective xVA component.¹ This is amplified due to the difficulty of predicting future regulatory regimes and the behaviour of counterparties. Capital requirements can also be region specific, with different regulatory bodies differing in terms of the timescales for implementation and the precise rules applied (CVA capital exemption being a significant example of the latter). In some cases, changes in regulation may be known with certainty (implementation details and dates), and in others there may be some uncertainty. However, banks will typically attempt to calculate capital costs based on the best available information at the time in question. Clearly, there will be components that are too uncertain to quantify (such as the fact that the regulatory regime in a decade may be very different from the one now).

As an example to illustrate the above, Figure 19.1 illustrates the future capital costs that a bank may attempt to capture. Note that capital charges depend very much on the situation in question and this is only one possible example. There are changes to ECP driven by the introduction of SA-CCR, the possible removal of CVA capital charge exemption (EU banks) and the introduction of the new FRTB-related CVA capital charge methodologies. There is also a potential consideration due to the implementation of the LR. Since a bank only needs to meet this requirement overall, the question arises as to whether it is a necessary consideration. The dotted line shown in Figure 19.1 represents what will later be defined as a leverage ratio invariance capital charge (Section 19.2.6), and a bank is leverage ratio constrained when the normal capital requirement is below this line. Note that on removal of CVA capital charge exemption, the bank assumes it will have to hold more capital, and, therefore, the LR becomes less of a concern at this point.

Graph depicts the qualitative example of the approach to capturing regulatory capital costs over time. — **Figure 19.1** Qualitative example of the approach to capturing regulatory capital costs over time.

As a real example, the credit risk-related capital charges for a 1,000 notional uncollateralised 10-year maturity interest rate swap through time are shown in Figure 19.2.² All profiles decay to zero as the swap approaches maturity, with the current exposure method (CEM) approach being rather inelegant due to the simple add-ons used (Table 13.8). Note that the SA-CCR result is significantly higher than that of the internal model method (IMM). An implication of this is that an IMM bank may be more sensitive to the LR since its normal capital requirements will be lower.

The above example only showed a projected capital profile over time. In reality, it is necessary to model a particular transaction and also account for the volatility of market factors. This requires including capital calculations within the underlying exposure simulation in order to calculate the capital requirement at each point in the future. For methodologies such as SA-CCR, this is relatively straightforward since it is based on simple formulas. It is more of a problem for IMM approaches, which are themselves simulation based, and this, therefore, represents quite a significant computational problem.

Graph depicts the total credit risk capital charge through time for a ten-year interest rate swap of notional thousand using the CEM, SA-CCR and IMM methodologies. — **Figure 19.2** Total credit risk capital charge through time for a 10-year interest rate swap of notional 1,000, using the CEM, SA-CCR and IMM methodologies. The LR-implied capital charge assuming a 5% requirement is shown.

Figure 19.3 illustrates the evolution of the counterparty risk-related capital charges using SA-CCR and standardised CVA capital charge methodologies.³ Both projected⁴ and expected (ECP) values are shown. The capital is variable and can both increase and decrease, largely driven by the swap moving in- and out-of-the-money. The downward move in the capital is limited since it cannot become negative, and also due to the floor used in the SA-CCR approach (Equation 3.21). This makes capital quite convex and means that ECP is significantly higher than the (more easily calculated) projected profile. The variability of the future capital requirements potentially suggests the need to hedge counterparty risk capital requirements (discussed later in Section 21.3.2).

Figure 19.3 Simulations showing the evolution of counterparty risk capital for a 10-year interest rate swap with notional 1,000, using SA-CCR and SA-CVA capital charges. The dotted line shows the projected value, and the solid line the expected capital profile (ECP).

Figure 19.4 Expected capital profile (ECP) from Figure 19.3 broken down into credit (CCR) and market risk (CVA) charges.

The above graphs show the aggregate capital across both credit risk (CCR) and market risk (CVA). ECP showing the breakdown between the two capital charges is illustrated in Figure 19.4. Note that the CVA capital charge represents the largest contribution, as is often the case, and that CVA capital also has a different shape over time.

Note that the capital profile does not always amortise as in the above examples. For the case of a forward transaction, since there are no periodic cash flows, the capital will project approximately flat. Similar behaviour will also be seen for cross-currency swaps (although they will likely amortise slightly due to the interest rate cash flows).

The above examples are for a single transaction, but in reality ECP would need to be calculated for a bank's entire portfolio. Whilst the credit risk (CCR) capital charge is additive across netting sets, the CVA capital charge is a portfolio-level calculation in both the standardised and advanced versions (and the future basic CVA (BA-CVA)/SA-CVA methodologies). This will represent a potential computational challenge for pricing KVA on new transactions, especially in an IMM approach. Note also that, ideally, the impact of any capital-reducing hedges should be included in the ECP profile. As stated in Section 19.2.2, this requires certain subjective assumptions about how much capital relief can actually be achieved.

19.2.4 KVA Example

KVA calculations for the same swap discussed in Section 19.2.3 are now shown and compared to the CVA value, which has previously been discussed (Table 17.3). For KVA, the parameters are as used in Section 19.2.2, with a required ROC of 8%. KVA will be discounted with the cost of capital, which is discussed in more detail in Section 19.3.3.

It is first important to emphasise the need to capture the correct ECP rather than to approximate it using a projected profile based on just a single scenario (Figure 19.3). As shown in Table 19.1, KVA approximated via the projected profile is much smaller than true KVA.

Table 19.1 KVA calculations (in bps upfront) for the 10-year swap using the correct ECP compared to the more approximate projected capital profile.

	Using ECP	Using projection (approximation)
KVA	−38.8	−23.7

Table 19.2 CVA and KVA calculations for the 10-year swap on an upfront (basis points) or running (basis points per annum) basis.

	Upfront	Running
CVA	−17.9	−2.0
KVA (CCR)	−12.8	−1.4
KVA (CVA)	−26.0	−2.9
KVA (total)	−38.8	−4.3

Table 19.3 Standalone and incremental CVA and KVA calculations for the 10-year swap.

	Standalone	Incremental	Reduction
CVA	−17.9	−10.3	35%
KVA	−38.8	−27.8	29%

KVA is also shown broken down according to credit risk (CCR) and market risk (CVA) capital charges in Table 19.2. Note that full KVA is around double the CVA value.

Consider now the impact of the same swap but on an incremental basis on an existing portfolio. The portfolio is dominated by trades in a different currency, which gives rise to a reasonable netting impact. However, from the point of view of capital, the SA-CCR approach does not allow potential future exposures (PFEs) to offset across different currencies (Section 13.4.3). This reduces the portfolio effect for the purposes of calculating KVA because there is no netting effect with respect to the PFE term of the new transaction in SA-CCR, although there is a netting effect with respect to the replacement cost (RC) term (see Equation 13.16). Standalone and incremental EPE and ECP are shown in Figure 19.5, with the former showing a larger portfolio effect, as expected.

The result of the above profiles on CVA and KVA is shown in Table 19.3. The incremental reduction is smaller for KVA due to the aforementioned lack of netting when calculating the PFE term across different currencies in the SA-CCR methodology.

19.2.5 Implementation of KVA

It is quite natural to leverage the implementation of exposure simulation (Chapter 15) for CVA and FVA purposes in order to calculate KVA. For standardised regulatory methodologies – such as SA-CCR – ECP is a relatively simple function of the value of the underlying portfolio. It is, therefore, necessary to simply implement this methodology within the exposure simulation and call it repeatedly in each simulation path as required. This has been previously discussed in Section 15.1.1 (see Table 15.1) and is the approach implemented in Spreadsheet 19.1.

Graphs depict an incremental EPE (top) and ECP (bottom) profiles for the ten-year swap. — **Figure 19.5** Incremental EPE (top) and ECP (bottom) profiles for the 10-year swap.

For more complex capital methodologies, where the capital is not a simple function of the value of the portfolio at a given point in time, this leads to computation challenges. This applies for the following methodologies:

IMM (Section 13.4.5) . This is typically implemented using a Monte Carlo simulation and, therefore, would represent a ‘Monte Carlo within a Monte Carlo’ for KVA calculations. Adding to this problem is the fact that IMM implementations are often real-world based, but exposure simulations for CVA/FVA purposes are generally risk neutral (this has been previously discussed in Section 15.3.3).
Advanced CVA capital risk charge (Section 13.3.4) . This also requires a Monte Carlo simulation, although it will be replaced by SA-CVA.
SA-CVA (Section 13.3.5) . This does not require another simulation but is based on portfolio sensitivities that would need to be calculated in each underlying scenario within the exposure simulation. This is potentially a challenging and computationally insensitive problem.

There is also the question of whether or not KVA should be calculated using a risk-neutral simulation (as for CVA/FVA) or whether a real-world simulation is more appropriate. One answer to this depends on the likely management of KVA, most importantly whether or not it will be hedged (see later discussion in Section 21.3.2). The difference between KVA calculated with real-world and risk-neutral simulations can be significant in some cases. For example, Figure 19.6 shows EPE and ECP for a foreign exchange (FX) forward under these different assumptions. In the historical calibration, the volatility is flat, and EPE will follow a familiar ‘square root of time’ shape and the projected capital will be roughly flat. Using implied volatility will change the shape of EPE, making it lower in the short term and higher in the longer term. This means that ECP also increases over time.

The above differences between real-world and risk-neutral parameterisations are important. Since terms such as CVA and FVA are generally hedged and reflected in financial statements, risk-neutral calibrations are relevant. However, KVA is typically not (yet) charged directly to desks, hedged, or reflected in accounting statements. Hence, there is a question as to whether pricing KVA with risk-neutral parameters is relevant. Of course, a bank may have a view that KVA will be hedged at some point in the future, in which case a risk-neutral approach to capital calculations may be preferable.

19.2.6 The Leverage Ratio

Minimum regulatory capital requirements – often defined in terms of risk-weighted assets (RWAs) – have generally represented the binding balance sheet constraint in banks' derivatives portfolios. However, the introduction of the LR potentially adds a new constraint. As discussed in Section 4.2.7, a bank must comply with both minimum regulatory capital requirements and the LR, with the latter not being credit-quality sensitive. Furthermore, the calculation of derivatives exposures in the LR can be more penalising than the exposure for capital purposes (Section 13.4.6). High credit quality and/or collateralised counterparties and transactions with certain features (such as non-cash collateralisation) will, therefore, have relatively high implied LR capital requirements.

Since the LR is a binary condition, it is not completely clear how a bank would incorporate this constraint into pricing. A primary consideration is whether or not the bank has RWA requirements or the LR as their binding constraint. A broker-dealer is more likely to find that the LR is a binding constraint since they have a relatively high proportion of low-risk assets and large, more complex businesses such as repos, derivatives, and client clearing. On the other hand, a commercial bank is more likely to be, through its lending activities, RWA constrained and may not see the need to focus on the LR.

The pricing of KVA above follows the idea that the return on a given transaction or portfolio must be commensurate with the amount of underlying regulatory capital. Whilst a bank which naturally meets its LR requirement may see little reason to price in the LR, other banks may see this as important in order to create the right incentives (Section 5.4.1). An obvious way to do this is via an invariance approach (Section 5.4.3). This would require that the ROC is calculated on a capital amount which is consistent with a bank's LR staying constant. Since a bank must meet both standard capital requirements and the LR constraint, this would likely amount to pricing in the maximum capital requirement of the two at any given point in time. This may be expressed as:

Graphs depict an expected exposure (top) and projected capital charge (bottom) for a ten-year forward transaction using historical and implied volatility. The implied volatility is upwards sloping. — **Figure 19.6** Expected exposure (EPE) (top) and projected capital charge (bottom) for a 10-year forward transaction using historical and implied volatility. The implied volatility is upwards sloping.

(19.7)

where the second term simply arises from solving for capital in the LR formula (Equation 4.1). Note that the above requires that a bank needs to achieve a return on its counterparty risk regulatory capital requirements or meet LR invariance, whichever is the highest number. In the event that the regulatory capital requirements are the binding constraint, the LR of the bank will actually increase, and in the latter case it will stay the same.

The above may be more complex than net stable funding ratio invariance FVA pricing (Section 18.3.4), which may always be considered to be more conservative than traditional FVA pricing and where a comparison of different funding strategies is not, therefore, necessary. To illustrate this, the ECP profile is shown for three cases as the maximum of the minimum capital requirement (defined by CCR and CVA capital) and the implied LR capital (assuming a requirement of images ). The SA-CCR methodology is used for the LR capital calculations. Figure 19.7 shows three different cases:

Uncollateralised swap. This is the example of the 10-year swap previously shown in Section 19.2.3. Since the total capital requirement is relatively high, the LR invariance has very little impact on ECP.⁵
As above, but assuming a CVA capital exemption. This case is as above, but without pricing in CVA capital. In this situation, with a lower minimum capital requirement, the LR becomes more important, as discussed in Section 19.2.3.
Collateralised swap. This case is the same example, but assuming collateralisation where the counterparty has the right to post non-cash margin. Since such margin cannot be used to reduce the LR exposure (both current and PFE), it also has an important impact.

19.3 MANAGEMENT OF KVA

19.3.1 Current Treatment of KVA by Banks

Beyond the definition and calculation of KVA, there is difficulty in the underlying calculation and the treatment of KVA alongside components such as CVA and FVA. One aspect that is important to consider is that KVA is, unlike components such as CVA and FVA, not an expected future cost per se, but rather a profit that is expected (at some point) to be paid to shareholders, and possibly employees. Note that this shareholder profit is not guaranteed: equity investors may be paid lower returns in times of poor profitability and vice versa.

Although CVA and FVA have been embraced from not only a pricing but also a valuation perspective, the same is not yet true of KVA. From a pricing point of view, KVA is potentially the oldest valuation adjustment, as banks have long used the notional of capital hurdles in assessing new trades. However, there is currently no evidence of KVA – like CVA and FVA – being reflected as a valuation adjustment in the financial reporting of banks. This means that KVA is generally managed differently than terms such as CVA. Some of the important contrasting features (which are inter-related) are as follows:

KVA is generally not transfer priced to an internal xVA desk, and so not meeting a target KVA (ROC) will not create a loss.

Figure 19.7 Illustration of ECP profile using traditional regulatory capital requirements (RWA only) and also using the LR capital requirement (RWA and LR invariance). The cases shown are an uncollateralised trade (top), the same trade assuming no CVA capital (middle), and the same trade collateralised with non-cash margin (bottom).
KVA is not – anecdotally – fully seen in the entry price (‘clearing price’) of client trades.
Profits of transactions that relate to KVA are paid out immediately, irrespective of the lifetime of the transaction.

The above treatment creates different incentives when managing KVA, such as shown in Figure 19.8 (which is a more specific version of Figure 5.3, discussed previously). Here, CVA and FVA are priced at entry into a transaction and owned by the xVA desk. This desk is axed to pay out (the now different) CVA/FVA upon exit of the trade and will offset CVA/FVA charges (e.g. charged by another counterparty in a novation). In the event that it is possible to exit a trade with lower charges than CVA/FVA at that time, this will lead to a profit at the time of exit. This gives traders and salespeople the incentive to try and restructure client portfolios, as this can lead to real profit benefits.

However, whilst KVA may be priced into the transaction at entry, it is released as profit and is not part of the economic consideration upon exiting. This implies that it may only be possible to exit transactions at a loss (in order to pay the KVA charged) and raises the related problem of how a bank could incentivise staff to restructure portfolios so as to reduce regulatory capital costs.

The above has a number of implications for the pricing of KVA. One is that banks are more likely to incorporate structural features such as non-mandatory break clauses (Section 7.1.1) into KVA pricing, even though they would not be considered to mitigate CVA and FVA. This is because there is no ownership of KVA profit and loss (P&L) that is related to the decision as to whether or not to break. CVA and FVA actually change profit, but KVA only changes the required profit. Related to this is the fact that banks may also incorporate certain behavioural factors into their assessment of future capital requirements. For example, a client may be expected to unwind a transaction early, and this may lead to a reduced KVA charge. This will, of course, be related to the motive and past behaviour of the client: a hedge fund is likely to unwind transactions, whilst a corporate hedging debt issuance is not. The lack of ownership of KVA in a bank is why, as mentioned above, KVA is not fully part of entry pricing in the way that CVA and FVA generally are. The pricing of capital, despite the birth of KVA, represents one of the biggest divergences of pricing within over-the-counter derivatives (Figure 19.9).

Schematic illustration of CVA or FVA and KVA charges at entry and exit of a transaction. — **Figure 19.8** Illustration of CVA/FVA and KVA charges at entry and exit of a transaction.

Bar chart depicts the market view of the most significant causes of divergence in market prices. — **Figure 19.9** Market view of the most significant causes of divergence in market prices.

Source: Solum CVA Survey (2015). See Glossary for definitions.

19.3.2 Optimal KVA Management

The fundamental problem is that many derivatives, especially the most profitable ones, have maturities that are greater than the period (e.g. quarterly or annually) with which a bank remunerates shareholders and employees. Hence, when a bank makes a profit on a given transaction, it is typically paid out in the first year, and yet the transaction may exist for many years after this. Hence, long-dated transactions contribute to a large ROC in the first year and then represent a ‘drag’ on ROC for their subsequent lifetime, where the required capital does not have any associated return. Whilst ROC may change through the business cycle, large fluctuations are probably not desirable.

The question is whether or not KVA is a valuation adjustment or just a measure of profitability. Related to this is whether or not there should be a delay in recognition of profits, which represent the ROC. The debate around the treatment of KVA stems from whether or not capital is a cost. Consider a simple balance sheet and related funding with debt and equity (Figure 19.10). Debt repayments must be made to avoid default and are typically fixed. Equity dividends are discretionary and depend on the profitability of the firm. When ROC is low, equity holders can decide whether to accept this or redeploy their capital elsewhere. Whilst debt funding is a clear cost, there is a question as to whether equity funding is a cost or not.

A further problem is how to – if desired – withhold KVA profits and prevent them from being paid into the dividend (and bonus) stream. It is obviously not acceptable to arbitrarily reserve, which may be seen as an artificial way to prevent profits from being recognised. KVA clearly needs to be somehow held aside in the financial reporting of a bank, either by the explicit reporting of KVA (like FVA before) or some other method, such as within retained earnings.

Schematic illustration of the funding of a balance sheet with debt and equity. — **Figure 19.10** Simple illustration of the funding of a balance sheet with debt and equity.

With respect to the above, there seem to be (at least) three views regarding KVA management and accounting:

KVA is profit. In this view, KVA is a profit which should not be treated as a valuation adjustment or withheld in any way. Proponents of this approach probably do not consider that the term KVA is relevant and that it should instead be referred to as an ROC or hurdle rate.
KVA is retained earnings. In this view (Albanese et al. 2015), KVA is a profit that should be withheld but not hedged or treated in the same way as other valuation adjustments. Shareholders would be paid higher dividends if total KVA increased, and suffer lower dividends or possibly dilutions if KVA declined in value. Banks' derivatives businesses would accept some element of cyclicality, increasing and reducing hurdle rates with changes in their total KVA pool. In this approach, the total profit of a transaction would be realised eventually, although potentially in a heterogeneous fashion, suggesting that KVA should be part of retained earnings and not hedged.
KVA is a valuation adjustment. In this approach, KVA is treated as a valuation adjustment like CVA and FVA and is managed in a similar manner. For example, Green et al. (2014) state that ‘The most appropriate approach would be to manage KVA alongside CVA and FVA.’ This would also imply that KVA should be reported in financial statements, which would probably lead to the need to hedge KVA.

The first approach above will generate a high ROC in the first period (e.g. quarter/year) and then zero ROC thereafter. After the first year, capital costs will not be covered. This means that either the ROC in subsequent years will not be achieved or revenues will have to increase – probably unsustainably – in order to meet the required ROC. The more long-dated the transaction, the more acute this problem will be.

KVA – as in the second and third approaches above – can potentially correct the unsustainability problem and create more aligned incentives: the decay of KVA will release profits over the lifetime of trades/portfolios.

Consider the two latter approaches where KVA will be released over the lifetime of the transaction. In order to understand the implication of this, we use the interest rate swap from previous examples. The evolution of the regulatory capital requirements in three scenarios is examined: an average scenario corresponding to the expected future value (EFV) profile, and two more extreme scenarios represented by the 5% and 95% PFE. The exposure profile (seen before in terms of EFV/EPE/ENE for the opposite swap in Figure 15.24) and capital profiles are shown in Figure 19.11. In the average scenario (EFV), the capital decays to zero relatively predictably, whereas in the other scenarios the behaviour is quite different. In the 5% PFE scenario, the capital declines rapidly due to the transaction being OTM. In the 95% PFE scenario, the capital increases for the first two years, due to the transaction moving in-the-money (ITM), and then gradually declines to zero over the remaining lifetime.

Graphs depict an exposure (top) and capital (bottom) profiles for the thousand notional receive fixed interest rate swap. — **Figure 19.11** Exposure (top) and capital (bottom) profiles for the 1,000 notional receive fixed interest rate swap.

Graph depicts an evolution of KVA in the three scenarios shown in Figure 19.11. — **Figure 19.12** Evolution of KVA in the three scenarios shown in Figure 19.11.

The behaviour of KVA in the three scenarios is now examined (Figure 19.12). Whilst KVA decays to zero in all scenarios, meaning that it will be released at some point during the lifetime of the transaction, this again occurs in materially different ways. In particular, in the 95% PFE scenario, KVA initially becomes more negative as the underlying capital requirements increase.

The result of the above is very different ROC for each year over the life of the transaction, as shown in Figure 19.13. The three scenarios can be understood as follows:

EFV. In this average scenario, the ROC is approximately constant each year, with the average ROC being about 10%. Note that this is higher than the 8% return included in the KVA calculation for this transaction because of the convexity of capital (Figure 19.3).
5% PFE. In this scenario, the ROC is very high, especially in the early years, due to the low capital requirements for the OTM transaction. The average ROC is about 16%.
95% PFE. Here, there is a large negative ROC in the first year due to the more negative KVA driven by higher capital requirements for the ITM transaction. After this, there is relatively low positive ROC for the remainder of the transaction. The average ROC is only 3%.

The above example illustrates that whilst KVA represents the amount of profit that is required on average to generate a given ROC, the actual ROC will be extremely volatile, driven by the underlying volatility in regulatory capital. Whilst the above example is for a single transaction, it could reflect the general position for a bank in the case where its client business is quite directional (e.g. corporates wanting to pay the fixed rate in uncollateralised interest rate swaps to hedge floating borrowing). Whilst a bank may accept that ROC in a business such as derivatives may be cyclical and potentially balanced by other businesses, the variability is quite significant. Assuming this is not desirable, there are two obvious ways to mitigate the above volatility in the ROC:

Bar chart depicts an annual ROC in the three scenarios shown in Figure 19.11. — **Figure 19.13** Annual ROC in the three scenarios shown in Figure 19.11.

Balanced portfolio. A bank could aim to have a fairly balanced portfolio so that higher capital requirements on some transactions would be offset by lower requirements in others. For example, a bank predominantly receiving fixed in uncollateralised interest rate swaps would need to pay fixed in similar transactions. This may be easier for global banks compared to smaller and/or more regionally-focused ones.
Hedging. A bank could seek to ‘lock in’ a given ROC by hedging KVA, in a similar way to CVA and FVA. This would mean additional hedging profits (losses) as future capital requirements increase (decrease). For example, in a scenario where regulatory capital requirement dropped (due to the transaction moving OTM, as in the 5% PFE case above), profits would be offset by losses on KVA hedges, although the ROC would be maintained. In examples such as the 95% PFE above, KVA hedge profits would increase the ROC to the desired level. The initial profits and hedging P&L would be less relevant metrics as they would simply provide the mechanism to attain a given ROC. Clearly, a bank cannot hedge for the change in regulatory methodologies or a change in its required ROC.

It remains to be seen which of the above (or other) approaches is adopted by the industry or if banks follow different business models.

The topic of KVA hedging is discussed in more detail in Section 21.3.2.

19.3.3 Discounting

In the example in Table 19.1, ECP was discounted at the cost of capital, which led to KVA of -38.8 basis points. Without discounting at this higher rate, KVA is larger in absolute terms at -49.1 basis points. As with all xVA formulas, the precise assumptions to use depend on the underlying strategy and payoffs.

For KVA, discounting at cost of capital is appropriate if KVA is counted towards the regulatory capital requirements (see, for example, Albanese et al. 2016 and Kjaer 2018). This would then follow the IRR approach in Equation 19.4, where there is a single rate. For this to be appropriate, KVA-related profits would have to be withheld, either as retained earnings or via another accounting adjustment, such as an explicit KVA valuation adjustment. This would, therefore, apply to the second and third cases described at the start of Section 19.3.2. This point is also considered by Garcia Munoz et al. (2016), who consider that the use of KVA as capital creates a recursive problem.

The use of KVA as capital is illustrated in Figure 19.14, which shows the ECP of the previous interest rate swap example. The fact that KVA itself forms part of the regulatory capital can be seen to reduce the capital cost, which, in turn, improves the ROC and rationalises the lower initial KVA. In the example shown, KVA is reduced by around 20% in the case where it is not released and forms part of regulatory capital.

Graph depicts the ECP for the example shown in Section 19.2.4. Assuming that KVA is not released as a profit and which forms a part of the required regulatory capital. — **Figure 19.14** ECP for the example shown in Section 19.2.4, assuming that KVA is not released as a profit and, therefore, forms part of the required regulatory capital.

19.3.4 KVA Accounting

In general, individual xVA components initially developed as non-standard adjustments to prices but then developed into standard and rigorous charges with accounting implications. Given the increasing prominence of KVA, it is natural to ask whether or not it will eventually be seen as an accounting adjustment, like CVA and FVA before it.

It is hard to extract the precise magnitude of KVA in clearing prices, but there is a general consensus that KVA or a hurdle rate calculation is included in entry prices for derivatives. However, it appears as though only a portion of KVA is charged, or equivalently that the required ROC is quite low. At the time of writing, no bank appears to have taken a generic accounting reserve for KVA (although some banks have done so in certain specific cases). It is not market practice for KVA to be managed or reserved internally by banks, although some banks are clearly moving in this direction.

There is a debate over whether or not banks should and can report KVA in financial statements. One argument is that this can be justified via the exit price, since another bank will require KVA when entering a transaction. This would also be in line with an estimate of the cost of liquidating the business, although this is unlikely to be a desirable or practical strategy. The idea is also supported by the fact that capital costs measured by KVA (cost of funding equity) have similarities to the funding costs represented by FVA (cost of funding debt), as shown in Figure 19.10.

The potential benefits of KVA accounting are:

Return on capital. When KVA is only a hurdle, this implies that the ROC in the first year will be very strong since it will be based on the total profit of a transaction, but will then drop to zero for the remaining lifetime. Taking KVA as an accounting item would allow profits to be released over the lifetime of the transaction, thereby generating the required ROC.
Incentives. With capital hurdles, sales and trading in a bank will still gain from the entire profit of a transaction at inception. Taking KVA as an accounting item would allow profits to be deferred and released over the lifetime of a transaction, arguably creating the right incentive for front-office staff.
Management. Like other xVAs, KVA could be transfer priced to a central xVA desk and managed accordingly. Without this, a bank may suffer from extremely volatile return on capital numbers.

However, given that most banks have relatively long-dated derivatives books, the one-off adjustment that would be required to achieve the above would be large compared to the fairly significant FVA reported values discussed in Section 18.2.1. It is, therefore, difficult to see how a bank would satisfy its staff and shareholders by making such a fundamental change, especially if competitors were not also doing so.

On the other hand, there are arguments that KVA should not be an accounting adjustment as KVA represents profits and not costs. This is also supported by the lack of a standard and well-defined KVA calculation across the market, which negates the ability truly to define KVA in this fashion. In particular, the inputs for KVA are extremely subjective due to:

different ROC requirements for different banks;
different capital requirements (e.g. additional charges for G-SIB banks or the EU capital charge exemption);
contrasting approaches to pricing capital (e.g. banks pricing in the LR);
different (current) capital methodologies (e.g. banks pricing in economic instead of regulatory capital or the use of SA-CCR versus IMM); and
different approaches to pricing in future regulatory change (see Figure 19.1).

Some of the above have parallels with the accounting of FVA – for example, the choice of the cost and symmetry of funding (Section 18.3.2). However, KVA valuation is much more subjective than FVA, and it would be difficult to decide what the market-standard KVA calculation for exit price purposes should be.

Another problem with FVA accounting is that there are potential overlaps between KVA and other xVA terms. Regulatory capital could be used as funding, in which case KVA seemingly overlaps with FVA, or it could be used to absorb default losses, in which case it seemingly overlaps with CVA. We will discuss these aspects in Section 19.4.

19.4 KVA OVERLAPS

19.4.1 CVA and KVA

The calculation of CVA using risk-neutral default probabilities as implied by regulation and accounting standards is the theoretical cost of hedging counterparty risk (Section 3.1.7). Due to a credit risk premium (Figure 12.1), this is expected to be higher than both:

the actual loss experienced with respect to counterparty defaults; or
the CVA calculated using historical default probabilities (see discussion in Section 12.1.2).

If it were possible to hedge most of the variability associated with CVA, then the above would be of limited interest. However, hedging CVA (discussed later in Section 21.2) is not straightforward. This is particularly true for hedging credit risk, with single-name credit default swaps (CDSs) being especially illiquid, meaning – somewhat perversely – that hedging actual counterparty defaults is often not possible.

Another related problem is that regulatory and accounting frameworks are not harmonised. Whilst regulators make efforts to introduce risk-sensitive capital methodologies without double-counting (e.g. the subtraction of ‘incurred CVA’ in Basel capital rules, discussed in Section 13.4.1), inefficiencies inevitably exist. One example can be seen in the internal ratings-based (IRB) capital formula shown in Equation 13.1. Here, the subtraction of the probability of default (PD) term envisages that an expected loss has been taken as an accounting adjustment. However, in this framework the PD concerned would be a historical estimate, whereas the CVA accounting adjustment usually involves the higher risk-neutral PD. The PD subtracted is, therefore, too small.

Another example of the lack of harmonisation of regulatory and accounting CVA frameworks is the treatment of CVA hedges, and related capital relief, in capital methodologies. Under the current regulatory rules, CVA hedges may not reduce capital charges in line with economic risk reduction and may even increase capital in some situations (see Section 13.5.3). This has created some problems for banks managing CVA. For example, a large bank reported significant CVA losses associated with its capital reduction programme (Carver 2013).

For ease of exposition, it is useful to define two idealistic approaches to counterparty risk and related capital requirements:

Counterparty risk warehousing. In this approach, there is no CVA hedging, and there is the economic cost of counterparty defaults, which are provisioned for via an expected loss (EL), and associated capital requirements (KVA). CVA accounting – especially with risk-neutral parameters – is not desirable since this creates accounting volatility.

Table 19.4 Pricing for CVA and KVA under the two stylised frameworks and a more realistic partial hedging approach.

Result

Credit risk warehousing 0 1 EL + KVA

Fully hedged 1 0 CVA

Partial hedging 0–1 ?
Fully hedged. Here, CVA is hedged perfectly, and there is no residual risk and, consequently, no capital requirements.

			Result
Credit risk warehousing	0	1	EL + KVA
Fully hedged	1	0	CVA
Partial hedging	0–1	?

Whilst the above approaches are stylised, they have led to banks pricing in consideration of the overlap between CVA and KVA – for example, by charging the higher of the warehousing (EL + KVA) or hedging approaches (CVA). This is discussed by Morini and Prampolini (2015) and is similar to the approach where an ROC hurdle is defined according to Equation 19.1, but EL, and not CVA, is subtracted from the revenue as a cost. Such an approach may be a compromise to respond to the complaint (sometimes from salespeople in banks, or clients) that, by charging CVA and KVA, a bank is charging the cost of hedging counterparty risk and the capital requirement in relation to not hedging it.

More generally, Kenyon and Green (2014) describe a framework where a bank could be at some point between the two above extremes of credit risk warehousing and perfect hedging. In this framework, the cost of counterparty risk and capital can be defined as images , where images represents the amount of credit hedging that is being done and images accounts for the impact of CVA hedges on regulatory capital. The two stylised frameworks in this representation, together with a more realistic partial hedging approach, are shown in Table 19.4.

The first two representations above are stylised since they (at the very least) assume a perfect harmonisation of accounting CVA and capital requirements. In reality, accounting standards and the need for a CVA desk with an appropriate governance structure prevent a bank from completely warehousing counterparty risk. On the other hand, it is not possible to hedge perfectly, and regulatory capital definitions prevent a bank from achieving full capital relief. It is useful to consider two more practical situations:

Partial hedging (capital increasing). In this case, CVA hedges actually increase regulatory capital requirements , which, as explained in Section 13.3.6, is likely the current situation for many banks.
Partial hedging (capital reducing). Here, CVA hedges reduce regulatory capital requirements .

These scenarios are compared to the stylised cases in Figure 19.15. We note that this representation is only qualitative and does not consider certain factors, such as the cost of CVA accounting volatility and the associated benefit of reducing it through CVA hedges.

The representation in Figure 19.15 explains the reluctance of some banks to account for and actively manage CVA. This is because the increased hedging cost of moving away from the credit risk warehousing approach is undesirable, unless it is associated with a reduction in KVA images that is recognised within the bank. Indeed, at the current time, it is quite possible that the current value of images can be greater than unity due to the fact that market risk hedges actually consume, rather than reduce, CVA capital.

Schematic illustration of the comparison of CVA and KVA for partial hedging scenarios. — **Figure 19.15** Comparison of CVA and KVA for partial hedging scenarios.

However, looking to the future, the introduction of the FRTB-CVA capital rules may create different incentives. Under the more advanced SA-CVA approach (Section 13.3.5), most CVA hedges, which reduce P&L volatility, will achieve regulatory capital relief. In such a situation, a bank is more incentivised to follow a hedging approach. In this situation, the appropriate pricing approach would probably be images .⁶ In other words, the bank would charge CVA and seek to achieve a return on the residual capital requirement after factoring in the likely achievable capital relief. The value of images in this situation is probably counterparty specific (e.g. due to the liquidity of the single-name CDS market) and is probably, at best, an estimate (since a bank's CVA hedging strategy and its ultimate performance is not known precisely a priori).

Note that, from a KVA point of view, it is the situation in the future, not just at the current time, that is important. Hence, the future capital methodologies, KVA strategy, and values of images and images are important.

19.4.2 FVA and KVA

Another potential overlap is due to the ability to use capital as a source of funding which would – depending on the representation – lead to a reduction of either FVA or KVA. Albanese et al. (2015, 2016) consider a full funding cost environment, but where regulatory capital is used for funding which is also FVA reducing. The use of regulatory capital to reduce funding requirements is also discussed by Green et al. (2014).

In order to benefit from any such overlap, this would have to be part of the relationship between the xVA desk and the treasury in a bank. This would amount to calculating the holistic and incremental cost of funding a given transaction across both equity and debt funding. The relationship between the xVA desk and the treasury will be discussed in more detail in Section 21.3.1. This approach can be seen as optimisation of xVA (Section 21.3.4).