Abstract:We address the problem of allocating the counterparty-level credit valuation adjustment (CVA) to the individual trades composing the portfolio. We show that this problem can be reduced to calculating contributions of the trades to the counterparty-level expected exposure (EE) conditional on the counterparty's default. We propose a methodology for calculating conditional EE contributions for both collateralized and non-collateralized counterparties. Calculation of EE contributions can be easily incorporated into exposure simulation processes that already exist in a financial institution. We also derive closed-form expressions for EE contributions under the assumption that trade values are normally distributed. Analytical results are obtained for the case when the trade values and the counterparty's credit quality are independent as well as when there is a dependence between them (wrong-way risk).
For years, the standard practice in the industry was to mark derivatives portfolios to market without taking the counterparty credit quality into account. In this case, all cash flows are discounted using the LIBOR curve, and the resulting values are often referred to as risk-free values.4 However, the true value of the portfolio must incorporate the possibility of losses due to counterparty default. The credit valuation adjustment (CVA) is, by definition, the difference between the risk-free portfolio value and the true portfolio value that takes into account the counterparty's default. In other words, CVA is the market value of counterparty credit risk.5
There are two approaches to measuring CVA: unilateral and bilateral (see Picoult, 2005 or Gregory, 2009). Under the unilateral approach, it is assumed that the counterparty that does the CVA analysis (we call this counterparty a bank throughout the paper) is default-free. CVA measured this way is the current market value of future losses due to the counterparty's potential default. The problem with unilateral CVA is that both the bank and the counterparty require a premium for the credit risk they are bearing and can never agree on the fair value of the trades in the portfolio. Bilateral CVA takes into account the possibility of both the counterparty and the bank defaulting. It is thus symmetric between the bank and the counterparty, and results in an objective fair value calculation.
Under both, the unilateral and bilateral approaches, CVA is measured at the counterparty level. However, it is sometimes desirable to determine contributions of individual trades to the counterparty-level CVA. The problem of calculating CVA contributions bears many similarities to the calculation of risk contributions and capital allocation (see Aziz and Rosen 2004, Mausser and Rosen 2007). There are several possible measures of CVA contributions. We refer to the CVA of each transaction on a stand-alone basis as the transaction's stand-alone CVA. Clearly, when the given portfolio does not allow for netting between trades, the portfolio-level CVA is given by the sum of the individual trades' stand-alone CVA. However, this is not the case when netting and margin agreements are in place. We refer to the incremental CVA contribution of a trade as the difference between the portfolio CVA with and without the trade.6 This measure is commonly seen as appropriate for pricing counterparty risk for new trades with the counterparty (see Chapter 6 in Arvanitis and Gregory, 2001 for details). One problem with incremental CVA contributions is that they are non-additive - the sum of the individual trade's CVA contributions does not add up to the portfolio's CVA. Hence neither stand-alone nor incremental contributions can be used effective contributions of existing trades in the portfolio to the counterparty-level CVA, in the presence of netting and/or margin agreements. For this purpose we require additive CVA contributions. In this case, we draw the analogy with the capital allocation literature and refer to these as (continuous) marginal risk contributions.
The marginal CVA contributions with a given counterparty give the bank a clear picture how much each trade contributes to the counterparty-level CVA. However, the use of CVA contributions is not limited to an analysis at a single counterparty level. Once the CVA contributions have been calculated for each counterparty, the bank can calculate the price of counterparty credit risk in any collection of trades without any reference to the counterparties. For example, by selecting all trades booked by a certain business unit or product type (e.g., all CDSs or all USD interest rate swaps), the bank can determine the contribution of that business unit or product to the bank's total CVA.
We show how to define and calculate marginal CVA contributions in the presence of netting and margin agreements, and under a wide range of assumptions, including the dependence of exposure on the counterparty's credit quality. The theory of marginal risk contributions, sometimes refer to as Euler Allocations (see Tasche 2008), is now well developed and largely relies on the risk function being homogeneous (of degree one). We show that this principle can be applied readily for CVA when the counterparty portfolio allows for netting (but does not include collateral and margins). We further extend this allocation principle for the more general case of collateralized/margined counterparties For the sake of simplicity, we assume the unilateral framework throughout the paper. However, an extension of all the results to the bilateral framework is straightforward.
The paper is organized as follows. In Section 2, we define counterparty credit exposure for both collateralized and non-collateralized cases. We show how counterparty-level CVA can be calculated from the profile of the discounted risk neutral expected exposure (EE) conditional on the counterparty's default. In Section 3, we introduce CVA contributions of individual trades and relate them to the profiles of conditional EE contributions. In Section 4, we adapt the continuous marginal contribution (CMC) method often used for allocating economic capital to calculating EE contributions for the case when the counterparty-level exposure is a homogeneous function of the trades' weights in the portfolio. This is the case when there are no exposure-limiting agreements, such as margin agreements, with the counterparty. When such agreements are present, the CMC method fails because the counterparty-level exposure is not homogeneous anymore. In Section 5, we propose an EE allocation scheme that is based on the CMC method, but can be used for collateralized counterparties. In Section 6, we show how to incorporate EE and CVA contribution calculations into exposure simulation process. In Section 7, we derive closed form expressions for EE contributions under the assumption that all trade values are normally distributed. We start with the case of independence between exposure and the counterparty's credit quality, and extend the results to incorporate dependence between them (wrong-way risk). We also provide an intuitive explanation to our closed-form results. In Section 8, we show several numerical examples that illustrate the behavior of exposure (and hence CVA) contributions for both, the collateralized and non-collateralized cases.
In this section, we review the basic concepts and notation for counterparty credit risk, credit exposures and CVA.
Counterparty credit risk (CCR) is the risk that the counterparty defaults before the final settlement of a transaction's cash flows. An economic loss occurs if the counterparty portfolio has a positive economic value for the bank at the time of default. Unlike a loan, where only the lending bank faces the risk of loss, CCR creates a bilateral risk: the market value can be positive or negative to either counterparty and can vary over time with the underlying market factors. We define the counterparty exposure of the bank to a counterparty at time as the economic loss, incurred on all outstanding transactions with the counterparty if the counterparty defaults at , accounting for netting and collateral but unadjusted by possible recoveries.
Consider a portfolio of derivative contracts of a bank with a given counterparty. The maturity of the longest contract in the portfolio is . The counterparty defaults at a random time with a known risk-neutral distribution .7 We further assume that the distribution of the trade values at all future dates is risk neutral.8
Denote the value of the th instrument in the portfolio at time from the bank's
perspective by . At each time , the counterparty-level exposure is determined by the values of all trades with the counterparty at time ,
. The value of the counterparty portfolio at is given
We start modeling collateral with a simplifying assumption: we incorporate the minimum transfer amount into the threshold and treat the margin agreement as having no minimum transfer amount. This approximation is rather crude, but it is very popular amongst banks because it greatly simplifies modeling.
We consider two models of collateral. In the instantaneous collateral model, we assume that collateral is delivered immediately and that the trades can be liquidated immediately as well. Under these simplifying assumptions, the collateral available to the bank
A more realistic collateral model must account for the time lag between the last margin call made before default and the settling of the trades with the defaulting counterparty. This time lag, which we denote by, is known as the margin period of risk. While the margin period of risk is not known with certainty, we follow the standard practice and assume that it is a deterministic quantity that is defined at the margin agreement
level.10 We assume that the collateral available to the bank at time is determined by the portfolio value at time according to
In the event that the counterparty defaults at time , the bank recovers a fraction of the exposure . The bank's discounted loss due to the counterparty's default is
The unilateral counterparty-level CVA is obtained by applying the expectation to Equation (7). This results in
We would like to develop a general approach to calculating additive contributions of individual trades to the counterparty-level CVA. We denote the contribution of trade by . We say that CVA contributions are additive when they sum up to the counterparty-level CVA:
Note first that, without netting agreements, the allocation of the counterparty-level EE across the trades is trivial because the counterparty-level exposure is the sum of the stand-alone exposures (Equation (2)) and expectation is a linear operator. Furthermore, when there is more than one netting set with the counterparty (e.g., multiple netting agreements, non-nettable trades), we can focus on first calculating the CVA contribution of a transaction to its netting set. The allocation of the counterparty-level EE across the netting sets is then trivial again because the counterparty-level exposure is defined as the sum of the netting-set-level exposures. Thus, our goal is to allocate the netting-set-level exposure to the trades belonging to that netting set. To keep the notation simple, we assume from now on that all trades with the counterparty are covered by a single netting set.
In this section, we develop the basic methodology to compute EE contributions and allocate portfolio-level EE for non-collateralized netting sets.
We derive EE contributions by adapting the continuous marginal contributions (CMC) method from the economic capital (EC) literature. EC is calculated at the portfolio level and then it is allocated to individual obligors and transactions. Under the CMC method, the risk contribution of a given transaction to the portfolio EC is determined by the infinitesimal increment of the EC corresponding to the infinitesimal increase of the transaction's weight in the portfolio (see Chapter 4 in Arvanitis and Gregory (2001) or Tasche (2008) for details). This follows from the fact that the risk function is homogeneous (of degree one) and the application of Euler's theorem.
A real function of a vector is said to be homogeneous of degree if for all , . If the function is piecewise differentiable, then Euler's theorem states that:
If denotes the vector of positions in a portfolio, and the corresponding economic capital, then Euler's theorem implies additive capital contributions
Consider now the calculation of EE contributions. Assume that we can adjust the size of any trade in the portfolio by any amount. Define the weight for trade as a scale factor that represents the relative size of the trade in the portfolio, . These weights can assume any real value, with corresponding to the actual size of the trade and being the complete removal of the trade. We describe adjusted portfolios via the vector of weights . For adjusted portfolios, we use the notations , , and for the exposure and EE at time and CVA. Furthermore, for convenience, denote by the vector representing the original portfolio.
When there is no margin agreement between the bank and the counterparty, the counterparty-level exposure is a homogeneous function of degree one in the trade weights:
We define the continuous marginal EE contribution of trade at time as the infinitesimal increment of the conditional discounted EE of the actual portfolio at time resulting from an infinitesimal increase of trade 's presence in the portfolio, scaled to the full trade amount:
Consider now a counterparty that has a single netting agreement supported by a margin agreement, which covers all the trades with the counterparty. As discussed in Section 2, the counterparty-level stochastic exposure is given by Equation (4), where the collateral available to the bank is given either by the instantaneous collateral model (Equation (5)) or by the lagged collateral model (Equation (6)). In what follows, we specify additive EE contributions for both models, starting with the simpler instantaneous collateral model.
We can derive additive contributions for this non-homogeneous case, which are consistent with the continuous marginal contributions as follows. First, notice that, while the exposure function in Equation (22) is not homogeneous in the vector of weights , the function
The first derivatives of the exposure with respect to the trade weights is given by
As the final step, we "allocate back" the contribution adjustment of the collateral threshold given by Equation (27) to the individual trades, so that Equation (28) can be written in terms of EE contributions only of the trades (as in Equation (11)):
We refer to the allocation scheme above as type A allocation. An alternative allocation scheme (type B) is obtained by bringing the weighting scheme of the threshold contribution now inside the expectation operator, so that instead of Equation (29) we now have:
We now apply the formalism developed for the continuous collateral model to the lagged collateral model. In this case, the counterparty credit exposure is obtained by substituting Equation (6) into Equation (4):
As defined for the instantaneous model, we rewrite the exposure given by Equation (33) as a homogeneous function in the extended vector of weights :
Banks commonly use Monte Carlo simulation in practice to obtain the distribution of counterparty-level exposures. Based on these simulations a bank can also compute the counterparty-level CVA. In this section, we show how the calculation of EE contributions can be easily incorporated to the Monte Carlo simulation of the counterparty-level exposure that banks already perform.
Consider first the case where the exposures are independent of the counterparty's credit quality. In general, banks implicitly assume that each counterparty's exposure is independent of that counterparty's credit quality when exposures are simulated separately. Let us now make this assumption explicitly. Then, conditioning on in the expectations in Equations (19), (29), (30) and (32) become unconditional, and these conditional expectations can be replaced by the unconditional ones.
The simulation algorithm for calculating counterparty-level CVA can be extended to calculate CVA contributions. For the ease of exposition, we assume that all the trades with the counterparty are nettable and that collateral (if there is any) can be described by the instantaneous model.
First, the counterparty-level CVA can be calculated in a Monte Carlo simulation as follows:
The calculation of EE and CVA contributions can be incorporated to this algorithm as follows. Consider, for example, the EE contributions given by Equation (32). The following calculations are added to Steps 2-4:
The algorithm above assumes independence between the exposure and the counterparty's credit quality. More generally, there may be dependence between them which can come from two sources:
Let us introduce a stochastic default intensity process without specifying its underlying dynamics.11 This intensity can be used as a measure of counterparty credit quality: higher values of the intensity correspond to lower credit quality. The counterparty-level exposure may depend either on the intensity value at time , or on the entire path of the intensity process
from zero to . We can use the intensity process to convert the
expectation conditional on default at time in Equation (21) to an unconditional expectation so that the conditional EE contribution becomes
As described for unconditional EE contributions, the calculations for conditional EE contributions can be performed during a Monte Carlo simulation of exposures. In this case, given the dependence of exposures on the counterparty credit quality, the intensity process needs to be simulated jointly with the market risk factors that determine trade values. This joint simulation is done path-by-path: simulated values of the intensity and of the market factors at time are obtained from the corresponding simulated values at the earlier time points ( .12 Assuming that we have already simulated the market factors and the intensity for times for all , the algorithm for computing CVA contributions for time can be expressed as follows:
It is also useful in practice to estimate EE and CVA contributions quickly outside of the simulation system. To facilitate such calculations, we derive analytical EE contributions, for the case when trade values are normally distributed. For simplicity, and to avoid dealing with stochastic
discounting factors, we assume that, at time the distribution of trade values is given under the forward (to time probability measure. Under this measure, the discounted conditional EE in Equation (9) can be written as
Assume that the value of trade at each future time is normally distributed with expectation and standard deviation under the forward to probability measure:
Since the sum of normal variables is also normal, the discounted portfolio value is normally distributed:
We first calculate counterparty-level EE and EE contributions assuming independence between exposures and counterparty credit quality. We obtain the results for the general case of a netting agreement with a margin agreement. The simpler case, with no margin agreement, is obtained as the limiting case when the threshold goes to infinity. For the clarity of exposition, we assume the instantaneous collateral model.
Similarly, we obtain the EE contributions corresponding to type A allocations as
We now lift the independence assumption to accommodate right/wrong-way risk. Note that the EE contributions obtained in the previous section are contributions of the trades in the portfolio to the counterparty-level unconditional discounted EE. We need to modify the approach to obtain the contributions to the counterparty-level EE conditional on the counterparty defaulting at the time when the exposure is measured. An obvious approach is to define an intensity process and compute the conditional EE contributions as the expectation over all possible paths of the intensity process (See Appendix 1), but this requires a Monte Carlo simulation. In this section, we develop an alternative, simpler approach that results in closed form expressions for the conditional EE contributions.
For this purpose, we define a Normal copula13 to model the codependence between the counterparty's credit quality and the exposures.14 Thus, we first map the counterparty's default time to a standard normal random variable :
If the portfolio contains trades with non-zero , the standard normal risk factor , which drives the portfolio value, also depends on :
In this section we briefly comment on the properties and interpretation of the analytical contributions derived in this section.
Netting & no margin
Equation (55) can be understood from the incremental viewpoint of the CMC method. According to Equation (17), the EE contribution of trade is determined by the infinitesimal change of the counterparty-level EE resulting from an infinitesimal increase of the weight of trade in the portfolio. The effect of an increase of the weight of a trade on the portfolio value distribution can be viewed as the sum of two effects:
If the weight of trade is increased by , the expectation of portfolio value changes by . Let us first ignore the change of the standard deviation and consider how a uniform shift of the entire distribution by affects the counterparty-level EE. Scenarios with positive portfolio value contribute the same amount to the exposure change, while scenarios with negative portfolio value contribute nothing. Therefore, the increment of the EE will be given by the product of the magnitude of the shift and the probability of the portfolio value being positive. It is straightforward to verify that . Thus, the first term in the right-hand side of Equation (55) describes the increment of the counterparty-level EE resulting from the infinitesimal uniform shift of the portfolio value distribution associated with an increase of the weight of trade .
The second term of Equation (55) describes the change of the width of the portfolio value distribution. The change of the standard deviation of the portfolio value resulting from increasing the weight of trade by can be calculated as
It appears that Equation (55) has simple linear dependence on and the product . However, this is only part of the true dependence. Since trade is part of the portfolio, depends on and depends on and the correlation of trade with the rest of the portfolio. Moreover, correlation is the correlation between the values of trade and the portfolio that includes trade itself. Because of this, depends on the ratio (see Equation (48)). Thus, unless trade represents a negligible fraction of the portfolio, the true dependence of EE contribution on trade parameters is non-linear.
Netting & margin
In this case, only the first two terms of Equation (52) allow interpretation from the incremental viewpoint of the CMC method: the first term can be explained as the effect of the uniform shift and the second term as the effect of the widening or narrowing of the portfolio value distribution. The third term results from the allocation of exposure when the portfolio value is above the threshold. An attempt to use the CMC method would give zero EE contribution from scenarios.
Equation (52) can be re-written as
If the value of trade is correlated with the counterparty's credit quality, its value distribution at time conditional on the counterparty's default at time differs from its unconditional value distribution. If the correlation is positive (right-way risk), the distribution shifts down; if the correlation is negative (wrong-way risk), the distribution shifts up. In both cases, the distribution becomes narrower. Under the normal approximation, the shift of the distribution is described by Equation (64), and the narrowing is described by Equation (65).
An interesting property of Equation (64) is its dependence on the counterparty's PD. To understand this, let us consider the bank entering into the same trade with an investment-grade counterparty A and with a speculative-grade counterparty B. We are interested in the trade value distribution conditional on the counterparty's default at the time of observation. For the case of wrong (right) way risk, the deterioration of the counterparty's credit quality to the point of default pushes trade values higher (lower). Since counterparty A is "further away" from default than counterparty B, the deterioration of credit quality to the point of default is larger for counterparty A. Therefore, trade values conditional on default of A are shifted more than trade values conditional on default of B. Note that this is not specific to the normal approximation, but is a general property not related to any model.
In this section, we present some simple examples that illustrate the behavior of exposure (and hence CVA) contributions. For ease of exposition, we assume that trade values are Normal, as well as market and credit independence. However, as discussed earlier in Section 7.2, the conclusions apply equally to the case of wrong-way risk by simply using conditional expectations, volatilities and correlations, instead of unconditional ones. We first present an example when there is no collateral agreement in place, and then show the impact of adding a collateral agreement to the portfolio.
As a first step to understand this behavior, consider Equations (54) and (55), which give the counterparty-level EE and the EE trade contributions, in the case when there is no margin agreement in place:
The EE contribution of instrument is a function of:
Both the counterparty-level EE and the trade contributions can be seen as the sum of two components: a mean value component (first term in the equations), and a volatility component (second term). These components weigh the mean value (or mean value contributions) and the volatility (volatility contribution), respectively, by the Normal distributions and density evaluated at the ratio (for the entire counterparty portfolio). Thus, the overall level of the counterparty portfolio's mean value and volatility determine how the individual instrument's mean and volatility contribution are weighted to yield the EE contributions. Figure 1 plots these weights as a function of . A low ratio weighs the volatility contribution much higher; while a high ratio weighs mean values much more. For example, if = -2, the volatility component weight is 2.4 times the mean value weight. In contrast, = 2 results in mean values being weighted 18 times the volatilities.
To illustrate the impact of various parameters on EE contributions, consider now the simple counterparty portfolio, which comprises of 5 transactions over a single step. Table 1 gives the individual trade's mean value, variance and volatility (in dollar values and % contributions). The portfolio has a mean value and variance of 10. We assume that trade values are independent.17 In this case, the portfolio's ratio = 3.16.
|σ 2 (%)||40%||30%||20%||10%||0%||100%|
The portfolio is constructed so that for each trade, its mean value and volatility are inversely related; thus the first instrument, P1, has the lowest mean (0) and largest volatility (2), while position 5 has the highest mean value (4), and lowest volatility (0). This may not only be reasonably realistic, but it will also help highlight some of the points below.
Using Equations (54) and (55), we compute the EE and contributions for the portfolio. The EE for the portfolio is 10.001, with most of this arising from the mean value component (9.992). The trade contributions to EE are fairly close to the contributions to the mean values in Table 1 (0.03%, 10.02%, 20.00%, 29.98%, 39.97%).
Now, we vary the overall mean value of the portfolio,, while leaving intact the volatility, , as well as the percent contributions of each instrument to the mean exposure and volatility in table 1. This allows us to express the trade contributions in terms of how deep in- or out-of-the-money the counterparty portfolio is (relative to its volatility). Figure 2 plots the EE, as well as its mean and volatility components, as functions of the portfolio's. For large negative portfolio mean values, the EE (red line) is zero. In this case, the mean value component of EE is actually negative, and the volatility component compensates for this to generate positive EEs. As increases beyond zero, the volatility component decreases and, once 2, the EE is completely dominated by the mean value.
Figure 3 shows the EE contributions for each of the 5 trades as a function of . There is a clear shift in dominance between the mean and volatility components as the portfolio's mean value increases. At one side of the spectrum, when the mean portfolio values are negative, trades 4 and 5, which have the largest (negative) mean values and lowest volatilities, produce very large negative EE contributions. The opposite occurs for trades 1 and 2 (with low negative means and large volatilities). As the portfolio's increases, trades 4 and 5 end up dominating the contributions, with the EE contribution converging to the mean value contributions themselves. For this particular symmetric portfolio, every trade contributes 20% of EE at = 0.506.
We consider now the case when there is a margin agreement, and demonstrate the impact of the collateral on the trade contributions. In very general terms:
As the threshold is increased, the EE reductions decrease, as expected. Also, the collateral thresholds become more effective at reducing EE as the portfolio is deeper in-the-money (i.e. when increases). For example, a normalized threshold of 2 does not reduce EE until the portfolio's mean value is positive. At a value of = 5, it reduces EE by about 60%.
Figure 5 plots the EE contributions for the case = 1, as a function of the standardized threshold, H/. At high threshold values, H/ > 4, trade contributions are essentially the uncollateralized contributions. Conversely at low H/ values, EE contributions are basically the mean value contributions. The presence of the collateral affects each instrument's contributions differently. In particular, a tighter threshold increases the percent contributions of trades P4 and P5 (which have the highest mean values) while reducing the contributions of P1 and P2 (the lowest mean values). These eventually converge in the limit to the mean value contributions.
Note finally that a trivial case arises when the ratio = 0. In this case, the EE contributions are independent of the threshold level H/ and equal the uncollateralized contributions.
Counterparty credit risk is usually measured and priced at the counterparty level. The price of the counterparty risk for the entire portfolio of trades with a counterparty is known as credit valuation adjustment (CVA). In this article we have proposed a methodology for allocation of the counterparty-level CVA to individual trades. These allocations are additive, so that one can aggregate the CVA allocations for any collection of trades with different counterparties. Thus, the contribution of all trades belonging to a certain class to the bank-level CVA can be calculated. Such a class can be defined as "all trades booked by a certain business unit", "all EUR interest rate swaptions", etc.
In this paper, we show that the calculation of CVA allocations can be reduced to the calculation of contributions of individual trades to the counterparty-level expected exposure (EE) conditional on the counterparty's default. To obtain conditional EE contributions, we adapt the continuous marginal contribution method which is often used for allocating economic capital. The method is directly applicable for CVA contributions only when the counterparty-level exposure is a homogeneous function of the trades' weights in the portfolio. This is the case when there are no collateral or margin agreements. We extend the methodology to deal with non-homogeneous exposures of the type encountered when the portfolios have margin agreements.
We further show how the calculations of conditional EE contributions can be incorporated into an existing exposure simulation process. In addition, the ability to make quick calculations of CVA allocations outside of the exposure simulation system may be also desirable. To facilitate such calculations, we derive closed form expressions for unconditional EE contributions under the assumption that trade values are normally distributed. By using unconditional EEs in the CVA calculations, one implicitly assumes that exposures are independent of the counterparty credit quality. To overcome this limitation, we extend the results for conditional EE contributions in the normal approximation, which incorporate dependence between the trade values and the counterparty's credit quality.
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Suppose we have a random variable and we want to calculate its expectation conditional on the counterparty defaulting at time (. We can express this expectation as
A2.1 Exposure Independent of Counterparty's Credit Quality
We present the derivation of the analytical EE contributions when exposures are correlated with the counterparty credit quality. Specifically, we show that we can use Equations (51)-(55), with the only difference that instead of using the unconditional expectations, standard deviations and correlations that specify the behavior of the trade values, we now use the conditional ones.
From conditional expectation to conditional random variables
The conditional expectation of a random variable can always be formulated as the unconditional expectation of a conditional random variable. For example, the counterparty-level conditional EE in Equation (59) can be represented as
Now we have formulated the conditional exposure model in exactly the same mathematical terms as the unconditional model of Section 7 and can use all the results of Subsection 7.1 after putting "hats" on the parameters!