16
The Starting Point and Discounting

16.1 THE STARTING POINT

16.1.1 Basic Valuation

Before making a valuation adjustment, it is clearly necessary to define a valuation. Section 5.2.1 defined xVA as being an adjustment to what was defined as the ‘base value’. The first question will be to define this base value. Whilst there is no unique choice, an obvious choice for this is a collateralised transaction in a mature market, such as the interbank market, or the valuation defined by a central counterparty (CCP).

Another definition of base value could be in relation to the historical view of this valuation, prior to the birth of valuation adjustments. Pricing derivatives has always been relatively complex. However, prior to the global financial crisis (GFC), pricing of vanilla products was believed to be well understood, and most attention was paid to more complex products (‘exotics’). Credit, funding, and capital were ignored since their effects were viewed as negligible. The old-style framework for pricing financial instruments has undergone a revolution which is generally defined by the birth of xVA adjustments.

However, the concept of base valuation has also changed. Primarily, Interbank Offered Rates (IBORs) such as LIBOR (London Interbank Offered Rate) (Section 14.3.4) was traditionally seen as the appropriate discount rate for cash flows. For many years, LIBOR was seen as a good proxy for the risk-free rate, which was used to discount cash flows. It is important to note that both the time value of money and the concept of a risk-free rate are essentially theoretical constructs. Furthermore, traditional LIBOR discounting of risk-free cash flows, so standard for many years, was generally used with two key assumptions in mind:

LIBOR is (or at least is a very good proxy for) the risk-free interest rate; and
there are no material funding considerations that need to be considered – i.e. an institution can easily borrow and lend funds (including for margin purposes) at LIBOR.

The two above points are seen as being generally incorrect and so, irrespective of the move away from LIBOR as a benchmark, there has been a move away from using LIBOR as a discount rate.

Since concepts such as ‘risk free’ are becoming increasingly difficult to define, it is also important to define base value objectively and straightforwardly. Whilst, as mentioned above, there is no unique starting point, the base value would naturally conform to a number of principles:

it would be a transaction-level (not portfolio-level) calculation;
the calculation – at least for vanilla products – would be relatively simple (e.g. just requiring the discounting of cash flows), without any xVA adjustments (by definition); and
it would require only knowledge of the transaction in question and no other quantities (such as the identity of the counterparty).

Associated with the starting point in valuation is the role of an xVA desk. In general, banks are set up with trading desks broadly concerned with the base value, and xVA desks responsible for valuation adjustments. A trading desk will have expertise in a given asset class, such as interest rates, foreign exchange (FX), or commodities. The trading desk may price, value, and manage at a base value that may ignore any counterparty risk, funding, margin (collateral),¹ and capital effects. The xVA desk will act as a centralised resource to deal with some or all of these components. Whilst this may be a setup driven by historical development rather than the optimal structure, it does require the notion of base value and separate xVA adjustments.

It is also important to note that certain special cases of xVAs related to funding and margin simply involve changing discounting assumptions. In these situations, it may be a matter of choice to define the base value.

16.1.2 Perfect Collateralisation

In order to provide a starting point, we will define the concept of ‘perfect collateralisation’ as the case where a transaction can be valued without any further xVA adjustments. Such a concept is largely theoretical (except perhaps from the point of view of a CCP) but is a convenient base case. Perfect collateralisation would correspond to the following factors:

The transaction is covered by a symmetric (two-way) margin agreement based on the value of the underlying transaction, with zero threshold, minimum transfer amount, and rounding (i.e. the exact amount of required variation margin is received or posted at any time).
There is no requirement to provide overcollateralisation (e.g. initial margin).
Variation margin is transferred continuously (i.e. there is no delay in paying or receiving margin) and the underlying value does not change discontinuously.
The margin period of risk (MPoR) is also zero (i.e. the transaction can be closed out and replaced with no associated cost).
Margin can be reused and is not segregated.
A single type of margin is used with a known remuneration rate (typically this will be cash in the currency of the transaction).

Under the above assumptions, the amount of margin received or posted will be at all times identical to the value of the transaction and denominated in the same currency. If there were an xVA formula, then it would involve the expected future value (EFV), which is a relatively simple metric to calculate (Section 11.1.5). However, there is no need for an xVA adjustment, as in this situation a perfectly-collateralised transaction can be valued by discounting with a rate equal to the contractual return paid on the margin (Piterbarg 2010).

‘Collateral discounting’ is illustrated qualitatively in Figure 16.1, showing symmetry between the accumulation of the margin (collateral) amount versus the discounting of the cash flow. Since the cash flow is perfectly collateralised, the amount of margin at the end must be equal to the cash flow amount. Accordingly, the amount of margin currently held must be equal to the value of the cash flow. The only way to achieve this is for the discount rate used to value the cash flow to be equal to the margin remuneration rate. Note that this is the case even if this contractual rate is zero (i.e. there is no discounting) or negative (i.e. discount factors greater than zero). It also follows that if the margin is in a different currency, then the remuneration rate in this currency (converted to the currency of the cash flow) is the correct discount rate. It is also possible that the cash flows may be in different currencies, in which case the same argument will hold. This may be inconvenient since, for example, the margin against an FX transaction must be in only one currency. However, to allow margin to be posted in more than one currency will create optionality, as discussed in Section 16.2.3.

Schematic illustration of the concept of collateral discounting. — **Figure 16.1** Illustration of the concept of ‘collateral discounting’.

The only counterparty in the market that enjoys perfect collateralisation (or something very close to it) is a CCP. CCPs impose on their members (and their clients) margin requirements that are close to perfect collateralisation (e.g. the CCP can potentially make intraday margin calls). To the extent that this is not perfect (e.g. there are discontinuities in margin or valuation), then the CCP has initial margin to absorb losses. Note that CCP members are not perfectly collateralised primarily due to the need to fund financial resources (initial margin and default fund) paid to the CCP (against which they must also hold capital).

The other situation which is close to perfect collateralisation is a typical interbank transaction which is usually strongly two-way collateralised, although some valuation adjustments for counterparty risk, margin, and capital may still be relevant. Incoming bilateral margin rules (Section 7.4) will, on the one hand, improve this – for example, by requiring zero thresholds and incentivising or requiring cash margin. However, these rules also require bilateral initial margin posting, which reduces counterparty risk further but also creates extra funding costs and, consequently, the need for more valuation adjustments.

16.1.3 Collateral or OIS Discounting

The margin remuneration rate that defines the discounting rate need not be associated with any economic properties (such as being the ‘risk-free rate’); it merely needs to be deterministic and known. However, the standard remuneration rate specified in a margin agreement is the overnight index spread (OIS) rate (Section 14.3.3). This is not because OIS is the risk-free rate, but rather because (with daily calls) margin is only guaranteed to be held for one day (although in practice it can be held for much longer, which is an important consideration). Hence, in this situation ‘OIS discounting’ is the correct base valuation. Since OIS is a good proxy for the risk-free rate, it turns out that collateral discounting is close to the traditional concept of ‘risk-free valuation’. However, this is not a requirement: OIS is an appropriate discount rate because it is the margin remuneration rate, not because it is a proxy for the risk-free rate. In cases where another rate is referenced in the margin agreement, this is most obviously the appropriate discount rate. If it is not, then a separate – and potentially unnecessary – valuation adjustment will need to be made. This is discussed in more detail in Section 16.2.1.

Given the above, many transactions have seen a move from LIBOR to OIS discounting.

Even without the intervention of the GFC, OIS discounting should always have been the more correct way to approach valuation. However, prior to the crisis, the difference between this and traditional LIBOR discounting was not particularly material, as shown in Figure 14.8 in Chapter 14. There has been a gradual shift from LIBOR to OIS discounting for valuing collateralised transactions in recent years, although the extent of this move depends on the underlying product and region (see ISDA 2014).

Another driver for adopting OIS discounting is the move away from IBORs (Section 14.3.4). In this context, the appropriate discount rate will probably be the relevant risk-free rate as defined for the currency in question, assuming this aligns with the contractual remuneration rate. Note that such rates are not defined in a completely consistent fashion (e.g. some are secured rates and some are unsecured rates).

Prior to the IBOR transition discussed in Section 14.3.4, the use of LIBOR rates complicates valuation since there is one rate for discounting (OIS) and another for the projection of cash flows (LIBOR). This is often known as ‘dual curve’ pricing, which means that pricing and risk management of a single-currency interest rate swap involves multiple curves and basis risks.

Traditional interest rate curve building typically follows the following steps:

Select a set of liquid securities (cash deposits, futures, and swaps).
Make decisions on overlapping, interpolation, etc.
Fit a single curve via a ‘bootstrap’ procedure, which solves sequentially to fit market prices or a more complex algorithm.

Dual curve pricing complicates this process. LIBOR-OIS swaps are generally more liquid than OIS swaps, and so OIS and basis curves² have to be built simultaneously from these market prices together with standard LIBOR-based swaps. In this calibration, discounting is assumed based on OIS, whilst cash flows are projected in the relevant rate (OIS or LIBOR). This dual curve problem means that standard simpler bootstrap methods are not applicable. However, this remains a base valuation problem and does not need to be considered to be an xVA problem. More details on these issues can be found in, for example, Morini and Prampolini (2010), Kenyon (2010), and Mercurio (2010). The IBOR transition will generally remove these dual curve issues and potentially align the projection and discount rates via the chosen risk-free rate.

16.2 COLVA AND DISCOUNTING

16.2.1 Definition of ColVA

Suppose that a transaction denominated in one currency (Curr1) is collateralised in another currency (Curr2). This is a real situation in some markets, where local currency transactions are collateralised in USD, and there is, therefore, no reference for a local currency transaction collateralised in the local currency. Following the above discussion on collateral discounting, the obvious base valuation for this transaction is to discount the cash flows using the relevant remuneration rate (Curr2) converted back into the cash flow currency (Curr1) at the relevant forward FX rates (as observable from FX forwards and cross-currency basis swaps).

Suppose that, instead of the above, the base valuation was defined with respect to a discounting in Curr1. This is not necessarily incorrect – and it is an example of where there is no clear definition of base valuation – but it must be associated with a correction for the difference between the relevant rates in Curr1 and Curr2. This correction can be defined as a collateral value adjustment (ColVA).

A simple ColVA can, therefore, be defined as the difference between discounting at a rate given by images compared with using some base rate images :

(16.1)

It is possible to show that, for a set of fixed cash flows, the above is equivalent to (Appendix 16A):

(16.2)

where images represents discount factors according to the base rate, and the spread images represents the difference between the two rates. The term images represents the expected collateral balance in the future. For a perfectly-collateralised transaction, the ECB will be exactly equal to the EFV. This is a transaction-level quantity which is relatively easy to calculate (Section 11.1.5). This is not surprising since the ColVA is representing the difference between valuation using two different discounting rates. The above can be considered to be a discretised version of the following integral, which follows the general definition of an xVA calculation in Section 5.4.6:

(16.3)

where images is now the difference between the forward rates. The above integral can be calculated via other routes (see Green 2015).

To give an example of the above, consider the interest rate swap profile previously shown in Figure 15.23 (Section 15.6.1) with a notional amount of 100 million. This is a swap paying a rate which is slightly more than the par rate, and so the current value is negative. Suppose instead that the relevant margin rate was higher than the discount rate used (in the example shown, a flat spread of 10 bps is used). Discounting the cash flows at the higher rate leads to a more negative valuation (Table 16.1) since the floating payments to be received are larger for later dates, and the heavier discounting reduces their value by slightly more than those of the fixed payments to be paid. The same result can be achieved by using the ColVA formula. Here, the intuition as to the negative ColVA is that the ECB is positive, for which the received margin would have to be remunerated at a higher rate. The receive fixed swap shows the opposite behaviour (calculations shown in Spreadsheet 16.1).

Table 16.1 Illustration of ColVA calculation.

	Pay fixed	Receive fixed
Swap value discounted at base rate	−23,968	23,968
Swap value discounted at alternate rate	−33,401	33,401
Difference (Equation 16.1)	−9,433	9,433
ColVA formula (Equation 16.2)	−9,433	9,433

The definition of ColVA above may seem unnecessary and circular since it merely corrects for a base valuation that has been done with the ‘wrong’ discount rate. However, there may be situations where this is helpful from an operational point of view, where a given transaction is valued at a default rate images (without any knowledge of the margin agreement) and a ColVA adjustment is made elsewhere to capture the correct rate images (with knowledge of the margin agreement). Such situations could apply to the following cases:

Different currencies. Where the valuation and margin remuneration currencies are different – for example, a local transaction is by default valued by discounting in local currency and then an adjustment is made for the fact that the margin agreement requires (for example) USD cash.
Different remuneration rates. Where the valuation and margin remuneration rates are different – for example, a transaction is discounted in OIS, but the margin agreement actually specifies a rate of OIS plus or minus a spread and so a ColVA adjustment is made to capture the spread differential.

Note that the choice of whether to discount with the correct rate or make a separate ColVA adjustment is probably a matter of choice and may depend on systems constraints and organisation responsibilities.

16.2.2 Asymmetry

The treatment in Section 16.2.1 assumed that a transaction was strongly (strictly speaking, perfectly) collateralised. In this case, with zero thresholds and small minimum transfer amounts, the ECB will closely track the EFV of the transaction (note that since this scenario is not default related, there is no need to consider an MPoR concept). The EFV is quite easy to calculate, and the ColVA is also a transaction-level calculation in line with the fact that it is simply representing a change of discount rate. The above treatment would also – trivially – treat an uncollateralised transaction where the ECB, and therefore the ColVA, would be zero. Note that the above treatment is, by design, symmetric in that it requires the remuneration rate on margin posted and received to be equivalent, and also that the transaction is strongly collateralised.

There are, however, more complicated cases that may need to be dealt with:

Partially collateralised. In this case of undercollateralisation, the ECB will depend on the size of contractual terms such as thresholds which tend to decrease this value.
One-way collateralised. In a one-way margin agreement, margin will be posted in only one direction, and the ECB will only be either positive or negative.
Asymmetric remuneration rates. In this situation, the remuneration rates (or indeed eligible margin)³ would be different.

Whilst the above situations are relatively uncommon, they do give rise to a more complex ColVA formula, which can be written as an extension of Equation 16.3 in terms of collateral received adjustment (ColRA) and collateral posted adjustment (ColPA):

(16.4)

with images and images being the positive collateral (margin) balance and negative collateral balance, respectively, and images and images representing the relevant spreads when the collateral balance is positive and negative, respectively. In the special case of perfect collateralisation, the PCB and NCB equal the uncollateralised expected positive exposure (EPE) and expected negative exposure (ENE), respectively, and the spreads are equal (Figure 16.2). Since EPE + ENE = EFV, this then returns to the special (discounting) case of Equation 16.3.

Graph depicts the PCB and NCB for a fully-collateralised interest rate swap. — **Figure 16.2** Illustration of PCB and NCB for a fully-collateralised interest rate swap. Note that these terms are analogous to the EPE and ENE for an uncollateralised swap (Figure 15.24).

Table 16.2 Illustration of ColCA and ColBA calculation.

	Pay fixed
Difference (Equation 16.1)	−9,433
ColRA (Equation 16.4)	−24,226
ColPA (Equation 16.4)	14,799
ColRA + ColPA	−9,427

Table 16.2 shows the example of using the explicit ColVA expression for the fully-collateralised interest rate swap example shown previously. The ColRA (ColPA) term can be seen as the cost (benefit) of a higher remuneration rate when receiving (posting) margin. The sum of these two terms equals (with some Monte Carlo noise) the difference when discounting with a higher rate.⁴

In order to show the potential importance of the more complex ColVA formula, a real portfolio is used, and the following cases considered (Figure 16.3):

Strongly collateralised (‘strong’). A strong two-way margin agreement.
One-way collateralised in favour (‘receive’). A one-way margin agreement where the margin is only received. In this case, the PCB will be required.
One-way collateralised against (‘post’). A one-way margin agreement where margin is only posted. In this case, only the NCB will be required.
Partially collateralised (‘weak’). A two-way margin agreement with a relatively high bilateral threshold.

Graph depicts the collateral balance terms and EFV for the portfolio. — **Figure 16.3** Illustration of the collateral balance terms and EFV for the portfolio.

Table 16.3 Illustration of ColVA calculation.

	ColVA	ColRA	ColPA	Value
Valued at base rate	N/A			−14.900
Valued at adjusted rate	N/A			−14.689
Strong two-way	0.211	0.584	−0.373	−14.689
One-way (receive)	0.584	0.584	−	−14.316
One-way (post)	−0.373	−	−0.373	−15.273
Weak two-way	0.090	0.252	−0.162	−14.810

A symmetric spread of images is assumed. This could correspond to a situation where the margin remuneration is OIS minus 25 bps, and the transaction has been valued using OIS discounting. Alternatively, it could be considered similar to a situation where LIBOR (instead of OIS) discounting has been used.⁵ For all but the first cases above, it is necessary to use an exposure simulation to calculate the ECB.

The ECB terms are shown in Figure 16.3, and the associated ColVAs and valuations are shown in Table 16.3. Note that the ECB for the strongly-collateralised case is very close to the EFV, as before (any difference being only Monte Carlo noise). The current value of the portfolio valued with the base interest rate is -14.900. Normally, discounting a payable (liability) position with a lower rate (due to the negative spread of 25 bps) would be expected to lead to a more negative valuation. However, given that the profile of the portfolio over the entire maturity is predominantly positive, the discounted value at the adjusted rate is higher (less negative) due to the benefit of paying a lower return on the margin received over most of the lifetime. This more beneficial valuation under the adjusted rate can be seen to correspond exactly to the ColVA adjusted for the strong two-way agreement, which is 0.211. This figure, in turn, is made up of -0.373 of collateral posted and 0.584 of collateral received components.

In the one-way margin agreements, these cost and benefit terms exist in isolation and make the portfolio more (less) valuable when receiving (posting) margin under a one-way agreement. Finally, the two-way margin agreement with large thresholds has a smaller ColRA and ColPA due to less margin being received and posted, the net result being slightly positive.

16.2.3 Cheapest-to-deliver Optionality

Margining arrangements are historically quite flexible. A typical agreement will allow a range of cash and other assets to be posted as collateral. This range of eligible assets will comprise some or all of the following (in approximate decreasing order of likelihood):

cash in different currencies;
government bonds;
covered and corporate bonds;
equities;
mortgage-backed securities (MBSs); and
commodities (e.g. gold).

There will also be contractual haircuts specified for all of the above (cash in major currencies may be zero). This creates a choice for the giver of margin, who should pick the most optimal margin to post. This optimal choice will depend on:

the return paid on the margin (as usually specified by the OIS rate in the currency in question);
the haircut required in the margin agreement (generally only for non-cash securities);
the repo rate and associated haircut (for securities where such a market exists);
the availability of the margin in question (although if repo or reverse repo markets exist, then this is less of a concern as the assets in question can be readily acquired or lent).

The above optionality creates a valuation problem linked to the ability to optimise the margin posted by a given party, noting that their counterparty holds a similar option for the margin that they post. The market volatility experienced as a result of the GFC exposed – in dramatic fashion – the potential margin optionality value embedded within the contractual margin definitions. As a result, parties (especially the more sophisticated ones) began to value and monetise this embedded optionality. Collateral management, which used to be mainly a reactive back-office function, has moved on to become a proactive front-office process. Banks and some large financial institutions have become fairly optimal in managing the margin they post and the associated valuation issues. However, it is a great challenge to value and hedge the future impact of margin optionality, essentially monetising value that is effectively embedded in contractual terms.

For portfolios where the ECB is generally positive, the cheapest-to-deliver (CTD) valuation would be expected to be lower due to the counterparty posting margin requiring a higher return. Higher valuations would generally arise when the ECB is negative. The overall effect will depend on the CTD curve and the ECB profile. Asymmetric terms (such as in a one-way margin agreement) would also be expected to be important.

We will deal with optionality around different currencies of cash first. Assume that a party has to post a certain amount defined in a base currency. They can either:

post this currency directly and be remunerated at the relevant rate; or
exchange the amount via an FX transaction into an alternative currency and post this currency instead, receiving the remuneration at a different rate, which is then exchanged into their base currency again.

Return on cash margin is tied to the remuneration rate in the corresponding currency, such as the Euro Overnight Index Average (EONIA), the Sterling Overnight Index Average (SONIA) or Fed Funds. Assuming they have availability for all such currencies, a party should optimally choose to post margin in the currency which is remunerated at the highest rate. This is often called the CTD currency, which is the highest-yielding currency at a given time. The CTD currency can be calculated by comparing yields (implied by forward rates) earned in other currencies after exchanging them back into the base currency at the relevant forward FX rates. This adjustment is typically made by adjusting with cross-currency basis spreads which – for some currency pairs in particular – can be significant. The counterparty should be expected to follow the same optimal strategy in terms of their own optionality over posting. Clearly, this is a dynamic process, since the CTD currency may change through time.

The above represents a very challenging valuation problem due to the following factors:

It requires a model for the co-movement of the OIS (or other) curves in each currency that represents the remuneration rates and the underlying FX rates to convert between currencies. As noted in Section 15.4.2, exposure simulation models generally assume that these curves are perfectly correlated and do not model the basis between them. Even if such a model exists, the calibration of parameters such as the volatility of the basis between different remuneration currencies will be challenging.
It may be a path-dependent problem, since the amount of margin to be posted at a given time depends on the amount that was posted in the past. For example, there may be a difference between a party needing to post margin outright (where they have the optionality) and where margin needs to be returned (where the counterparty may have the optionality to request the return of a specific currency or asset). A key question here is whether margin can be substituted, since this will give greater value to the party posting because they can replace the posted margin with a more optimal choice (e.g. if the OIS in one currency widens with respect to another). Margin agreements not allowing substitution (or requiring consent)⁶ should have less optionality, but a party can still optimise to some degree.

A common simplification is to assume that the underlying remuneration curves are static and also that margin can be freely substituted. The former assumption means that the CTD currency at any point in time stays fixed over the lifetime of the portfolio. The latter point means that path dependency is unimportant since a party can either post CTD margin outright or substitute it for the current margin. In this simplified approach, it is possible to form a CTD curve, as illustrated in Figure 16.4. The projected margin return in each currency (forward rate for the curve in question) is converted into some base reference (currency 1 is used in this case), and then the maximum is calculated. The CTD curve is then a composite of all the admissible curves, with an FX adjustment derived from cross-currency basis spreads.

The above CTD curve can either be used to discount the collateralised transactions in question directly, or to define a spread in order to calculate the appropriate ColVA adjustment from Equation 16.2. A specific ColVA adjustment will only be required in asymmetric cases, as discussed in Section 16.2.2, and in other cases discounting will give the same result. Assuming that currency 1 is used as the base currency, this relevant ‘CTD basis’ is shown in Figure 16.5.

In order to illustrate the above, we consider a receive fixed interest rate swap under a strong (two-way) and also a one-way (in favour of the party) agreement. The ECB profiles for these two cases are shown in Figure 16.6. Note that the ECB for the two-way margin agreement is equal to the EFV. For the one-way agreement, there is only a PCB term as the margin is only received. This term is similar to the EPE of the equivalent uncollateralised swap, as noted previously.

Graphs depict the construction of a CTD curve. — **Figure 16.4** Construction of a CTD curve.

Under the above assumptions, the valuation results in Table 16.4 can be generated. The valuation using different curves for discounting is different, reflecting the different remuneration rates. The CTD valuation is the largest, which is expected since the ECB (two-way margin agreement) is negative, and therefore the party in question is expected to be posting margin and receiving a higher return. The ColVA calculation using the spread in Figure 16.5 and the two-way ECB profile in Figure 16.6 is the same as the difference in valuation (between currency 1 and CTD). Under the one-way margin agreement, the ColVA is negative, reflecting the fact that margin can only be received and the party in question is short optionality.

Graph depicts the CTD basis using currency one as the base currency. — **Figure 16.5** CTD basis implied from Figure 16.4 using currency 1 as the base currency.

Graph depicts the ECB for a swap under a two-way and one-way margin agreement. — **Figure 16.6** ECB for a swap under a two-way and one-way margin agreement.

Table 16.4 CTD valuations using discounting and ColVA adjustments.

	Value	ColVA
Value (currency 1 discounting)	38,594
Value (currency 2 discounting)	22,598
Value (currency 3 discounting)	20,398
Value (CTD discounting)	42,547
Difference (CTD vs currency 1)		3,953
ColVA (two-way)		3,953
ColVA (one-way)		−7,669

The above treatment, whilst relatively simple, makes two very important implicit assumptions:

It assumes that margin is always posted in full in the currency that earns the highest remuneration rate. This requires margin to be freely and immediately substituted into the CTD currency. In practice, it may be necessary for consent to be given for such a substitution. If the party wishing to switch currency finds it optimal, then the counterparty's optimal action is to refuse such consent. The assumptions here may also be jurisdiction specific: for example, substitution rights are generally viewed as enforceable under New York but not British law. However, anecdotal evidence suggests that there is a ‘gentleman's agreement’ not to refuse such requests. Even then, switching margin gives rise to settlement risk and may cause associated trading and hedging costs. It may not be optimal to substitute relatively small amounts of margin.
It captures only the intrinsic value of the margin optionality and does not price the time value of the optionality due to potential curve co-movements over time. Indeed, in many situations, the intrinsic value of the optionality is zero (i.e. the adjusted curves do not cross, as in Figure 16.4.

Since the above components have value for both parties, it is not clear whether the above approximations lead to a value which is too high or too low. It should also be noted that pricing via a CTD curve may result in complex risk management considerations, since even relatively small movements can result in dramatically different risk profiles (e.g. EUR exposure shifting to USD exposure on any given day).

As shown in Figure 16.7, the intrinsic CTD valuation method described above is common, although some banks do use a more sophisticated option-based valuation. This component is challenging to deal with as it requires a model for the joint evolution of all eligible currencies for the lifetime of the transactions in question. Note also that a more sophisticated representation of the substitution of margin is a path-dependent problem for a given margin balance; it must be known how much of the margin has already been posted (and would, therefore, need to be substituted) and how much needs to be posted (for which the optimal currency can be chosen). More sophisticated pricing of optionality has been discussed, for example, by Fuji and Takahashi (2011) and Piterbarg (2012 and 2013).

Note that there have been some problems in margin agreements where reference interest rates have become negative. Whilst remunerating margin at a negative return (i.e. the margin giver effectively pays the return) may appear unfair, flooring this rate will give rise to a much more complex pricing treatment, requiring the modelling of the dynamics of the remuneration curves.

16.2.4 Non-cash Margin

In the case of non-cash collateral, rates for transforming between cash and securities (e.g. repo rates) and associated haircuts must be considered. Suppose a party has to post margin of images and can choose between posting cash and securities both denominated in a given currency (with the multiple currency case being dealt with in Section 16.2.3). They can either:

post cash and earn the corresponding remuneration rate for the currency in question; or

Figure 16.7 Market practice around the discounting curve used for collateralised transactions.

Source: Solum FVA Survey (2015).
reverse repo this amount of cash into a larger notional amount of securities, where represents the current repo rate, and then post worth of such securities, where represents the haircut specified for these securities in the margin agreement.

The above implies that it is more efficient to post securities if the repo rate times images are higher than the remuneration rate on the cash. A given security will become more advantageous to post as its repo haircut increases and its margin agreement haircut decreases. This ratio will change as haircuts in the repo market change compared to the relatively static contractual haircuts in margin agreements. Note also that technical factors and balance sheet considerations may also be important: there may be a benefit in posting non-cash margin that cannot easily be repoed, and aspects such as the leverage ratio (Section 4.2.7) may also be relevant. Additionally, not all parties may have the same access to the repo market.

16.2.5 The End of ColVA

Dealing with pricing, valuation, and hedging when there is optionality over margin types is clearly a complex problem depending on many aspects such as the future exposure, OIS rates in different currencies, cross-currency basis swap spreads, haircuts, and substitution criteria. Even then, methods such as CTD valuation make inherent simplifications.

Many transactions are close to the theoretical ideal of OIS discounting, especially through interbank and centrally-cleared trades, as mentioned above. There is also clearly a push towards this standard of perfect collateralisation and OIS discounting through the following aspects:

Renegotiation. Market practice over recent years has been to renegotiate margin agreements such as credit support annexes (CSAs) bilaterally, often aiming to bring them closer to some of the perfect characteristics given in Section 16.1.2. This may involve having a more frequent exchange of margin (e.g. daily), reducing thresholds and minimum transfer amounts, and restricting the cash and securities that can be delivered.
Bilateral margin rules. The incoming bilateral rules discussed in Section 7.4 require frequent margin exchange, a zero threshold, and a minimum transfer amount of no more than €500,000. They also penalise certain types of margin through haircuts, in particular, when cash is posted in a currency different from that of the transaction (see Table 7.12). The US rules restrict variation margin in some cases to cash only. The requirement to negotiate new agreements to comply with the bilateral margin requirements has also catalysed a simplification of terms (such as variation margin in USD only). These simplifications restrict some of the embedded value and optionality inherent in margin agreements and tend to reduce ColVA.
Clearing mandate. Typically, CCPs require variation margin to be posted in the currency of the underlying transaction and do not allow netting across different currencies (Section 4.4.1). Cross-currency products are close to being centrally cleared, with one of the main hurdles to this being the currency of the underlying margin. For these reasons, CCPs consider the valuation of single-currency products such as swaps to be relatively simple, as they are – more or less – perfectly collateralised and OIS discounting is the valuation method used by the CCP.

The above comments generally apply to banks and other financial institutions who have already engaged in two-way margin agreements and are not exempt from the bilateral margin rules or clearing mandate. They are less relevant for end users who transact without a margin agreement or with a one-way agreement in their favour. However, through the pricing they receive from banks, such end users are also under pressure to move closer to the perfect collateralisation ideal.

However, this reduction of ColVA will not be absolute. Single-currency margin agreements create additional challenges, such as settlement risk. Many end users (e.g. pension funds) will struggle to move to post cash margin since they prefer to post directly the assets they hold. Problems will also remain with multicurrency products (e.g. cross-currency swaps).

Note also that collateral optionality adjustments may be present in other situations, such as those involving posting initial margin. Both bilateral markets and CCPs permit initial margin to be posted in a variety of different assets, and there is similar optionality to that described above for variation margin. However, in these situations, any optionality may be incorporated in the determination of the underlying funding costs, rather than being adjusted directly. This will be dealt with in Chapter 20.

16.3 BEYOND PERFECT COLLATERALISATION – XVA

16.3.1 Overview

We return to the problem of defining the starting point for valuation adjustments, as discussed previously in Section 5.2.1:

(16.5)

In collateral discounting (Section 16.1.3), the discount rate arises according to the remuneration rate on the margin and not for any other reason, such as that it is a good proxy for the risk-free rate. It therefore does not necessarily follow that collateral discounting would be the appropriate starting point for an uncollateralised transaction where, by design, there is no margin remuneration rate.

However, there are a number of reasons why it may be relevant to use the same collateral discounting (e.g. OIS) for all transactions, irrespective of whether or not they are collateralised:

Risk-free proxies. Margin remuneration rates do tend to be good proxies for the risk-free rate.
Backwards compatibility. This approach may be close to the historical approach used.
Operational. Using ‘risk-free’ discounting may be the easiest starting point as it requires (at most) knowledge only of the underlying margin agreement and not of any other components, such as the identity of the counterparty or the funding cost of the organisation. Related to this, client transactions and their hedges may lead to a completely flat book (in terms of market risk with respect to the base value) for the originating trading desk.
xVA desk setup. Related to the above point, the role of the xVA desk becomes better defined. An originating trading desk can value transactions at a base price which will ignore any counterparty risk, funding, and capital effects. The xVA desk will then act as a centralised resource and centre of expertise to deal with some or all of these components, and assume the related pricing, market volatility, and capital implications.

One could also argue that this is a reasonable starting point for all transactions, and additional xVA components can then be added on to this base case as required. This is not always the case in practice: for example, even after the general move to OIS discounting, some trading desks continued to use LIBOR discounting for uncollateralised transactions. Such choices – whilst in line with historical approaches – are often sub-optimal for the management of xVA.

One starting point could, therefore, be that all transactions be valued using risk-free rate discounting in the currency of the cash flows, with default choices for cross-currency trades. This has the advantage that the base valuation only requires knowledge of the cash flows and nothing else in line with the historical – pre-GFC – view of valuation (Section 5.1). However, this may also lead to some strange cases, as discussed in Section 16.2.1, with a transaction collateralised in a currency that is not the same as the transaction currency. In this situation, a purist approach might be to discount the transaction in the cash flow currency (even though this has no relevance for valuation) and then make a ColVA adjustment. However, it might be considered preferable (from an operational, pricing, valuation, and risk management perspective) to discount the transaction directly with the margin remuneration rate, which would then not require a separate valuation adjustment.

Another important point is that regulatory capital requirements for market risk treat the base value and xVA adjustments separately. This can, therefore, mean that different approaches to base valuation can have different capital impacts. This will be discussed further in Chapter 21.

From now on, the general view will be that all transactions should use the concept of perfect collateralisation as their base value and that xVA adjustments should then be made with respect to this value. There are two important cases where it may be desirable to deviate from this: one is the collateralised case discussed in Section 16.2.1, and another relates to funding and funding value adjustment, which will be discussed later in Section 18.2.3.

16.3.2 Definition of xVA Terms

Starting from the case of perfect collateralisation, it is useful to discuss xVA adjustments reflecting deviations from this ideal:

ColVA. Collateral adjustments due to deviations from a perfect collateralisation, in terms of margin type and remuneration, as discussed above.
CVA (credit value adjustment) and DVA (debt/debit value adjustment). The bilateral valuation of counterparty risk. CVA is in relation to the counterparty default, and DVA is related to a party's own default. Note that DVA will be defined as being specific to own default, as distinct from being a funding benefit.
FVA (funding value adjustment). Defines the cost and benefit arising from the funding effects of being uncollateralised, or partially uncollateralised. This may include contingent liquidity provisions, such as when a bank needs to hold a buffer of high-quality liquid assets against certain contractual clauses.
KVA (capital value adjustment). Defines the cost of holding capital (typically regulatory) over the lifetime of the transaction.
MVA (margin value adjustment). Defines the cost of posting initial margin over the lifetime of the transaction. Default fund contributions (for clearing members) could also be captured under this same definition.

Note that both FVA and MVA are funding costs, but the former is the cost of being undercollateralised, whereas the latter is the cost of being overcollateralised. From this point of view, it is probably easiest to treat them as separate terms.

Note that, like the problem in defining base value, there may be different ways to define certain valuation adjustments. For example, consider a transaction in USD for a bank that considers its funding costs to be in EUR. There are (at least) two possible similar ways to consider an FVA:

a single FVA calculated with respect to USD funding (i.e. raising cash in USD); or
an FVA calculated with respect to EUR funding and a collateral adjustment calculated with respect to transforming EUR cash into USD cash (ColVA).

The choice of which of the above to use may be a matter of preference and may also be driven by operational aspects. For example, an institution could fund itself directly in USD, or alternatively fund itself in EUR and use the FX market to convert this into USD. Overall, there should not be significant differences in the final result (in terms of the total xVA), as long as the approach chosen is consistent.

Clearly, different xVA adjustments will arise in different situations. Table 16.5 outlines the different components that are relevant in various common situations in over-the-counter (OTC) derivatives. Note that this is general and there can always be special cases: for example, if margin is hard to repo, then a collateralised transaction may be considered to have an FVA.

Table 16.5 Illustration of the various components of different types of margin arrangement.

	Uncollateralised	Collateralised	Collateralised with initial margin	Central clearing
Credit (CVA)	✓✓✓	✓✓	✓	✓
Funding (FVA)	✓✓✓	✓
Collateral (ColVA)		✓✓	✓
Capital (KVA)	✓✓✓	✓✓	✓	✓
Initial margin (MVA)			✓✓	✓✓

The above is obviously only a general qualitative treatment, but some of the choices in Table 16.5 are explained as follows:

Uncollateralised. Uncollateralised transactions have significant credit, funding, and capital adjustments, but contractually do not include any collateral or initial margin adjustments.
Collateralised. Collateralised transactions have a reduced CVA and little or no FVA due to the variation margin. They may have ColVA components, and the capital requirements will be generally lower but still material.
Collateralised with initial margin. Initial margin should – in theory – mean that there is no significant CVA contribution, although in practice this may not always be the case (Section 15.5.4). There may be ColVA adjustments, but these may be less impactful due to the inherent standardisation of the underlying margin agreement. Capital requirements will typically be lower due to the presence of initial margin. Initial margin costs through MVA will be an important consideration.
Central clearing. CVA may be considered insignificant, although the thinking is changing around this point (Section 17.6.7). There will be no FVA due to the variation margin requirements. The capital costs will be small, but again initial margin costs will be important.

One of the problems with xVA definition is that it may be important to consider the hedges of transactions. Since the originating transactions may be client transactions, a bank may naturally consider not only the xVA of this transaction but also the associated material adjustments for hedges. For example, an uncollateralised transaction with a client may have an associated MVA component due to the necessity to post initial margin on the hedge.