Pricing Counterparty Risk at the Trade Level and CVA Allocations

Finance and Economics Discussion Series
Divisions of Research & Statistics and Monetary Affairs
Federal Reserve Board, Washington, D.C.
Pricing Counterparty Risk at the Trade Level and CVA
Allocations
Michael Pykhtin and Dan Rosen
2010-10
NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary
materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth
are those of the authors and do not indicate concurrence by other members of the research staff or the
Board of Governors. References in publications to the Finance and Economics Discussion Series (other than
acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

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Pricing Counterparty Risk at the Trade Level and CVA Allocations1

Michael Pykhtin2 and Dan Rosen3

November 2009

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).



1 The opinions expressed here are those of the authors and do not necessarily reflect the views or policies of their
employers.
2 Federal Reserve Board, Washington, DC, USA. michael.v.pykhtin@frb.gov.
3 R2 Financial Technologies and The Fields Institute for Research in Mathematical Sciences, Toronto, Canada.
dan.rosen@R2-financial.com and drosen@fields.utoronto.ca

1

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1. Introduction
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.

4 More precisely, LIBOR rates roughly correspond to AA risk rating and incorporate the typical credit risk of large
banks.
5 See Canabarro and Duffie (2003) or Pykhtin and Zhu (2007) for an introduction to counterparty credit risk and
CVA.
6 Sometimes these are referred to as discrete marginal contributions.

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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.

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2. Counterparty credit risk and CVA
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 E(t) of the bank to a counterparty at time t as the economic
loss, incurred on all outstanding transactions with the counterparty if the counterparty defaults at
t , accounting for netting and collateral but unadjusted by possible recoveries.
2.1 Counterparty exposures
Consider a portfolio of N derivative contracts of a bank with a given counterparty. The
maturity of the longest contract in the portfolio is T . The counterparty defaults at a random time
τ with a known risk-neutral distribution P(t) ≡ Pr[τ ≤ t] .7 We further assume that the
distribution of the trade values at all future dates is risk neutral.8
Denote the value of the ith instrument in the portfolio at time t from the bank’s
perspective by V (t) . At each time t, the counterparty-level exposure E(t) is determined by the
i
values of all trades with the counterparty at time t, {V (t)}N . The value of the counterparty
i
i 1
=
portfolio at t is given by
N

V (t) = ∑V (t)
(1)
i
i 1
=
When netting is not allowed, the (gross) counterparty-level exposure E(t) is
N

E (t ) = ∑ max{0,V (t
(2)
i
}
)
i 1
=
For a counterparty portfolio with a single netting agreement, the (netted) exposure is

E(t) = max{V (t), }
0
(3)

7 The term structure of risk neutral probabilities of default can be obtained from credit default swaps spreads quoted
for the counterparty on the market for different of different maturities. See, for example, Schönbucher (2003).
8 See, for example, Brigo and Masetti (2005).

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When the netting agreement is further supported by a margin agreement, the counterparty
must provide the bank with collateral whenever the portfolio value exceeds a threshold. As the
portfolio value drops below the threshold, the bank returns collateral to the counterparty.
Collateral transfer occurs only when the collateral amount that needs to be transferred exceeds a
minimum transfer amount. The counterparty-level (margined) exposure is given by

E(t) = max{V (t)− C(t), }
0
(4)
where C(t) is the collateral available to the bank at time t.
Counterparty portfolios with a combination of multiple netting agreements and trades
outside of these agreements can be modeled in a straightforward way by a combination of
Equations (2)-(4).
2.2 Models of Collateral
We start modeling collateral with a simplifying assumption: we incorporate the minimum
transfer amount into the threshold H 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 is

C(t) = max{V (t) − H , }
0
(5)
The instantaneous collateral model is attractive because of its simplicity, but is rarely used in
practice because its assumptions materially affect the exposure distribution.9 However, we use
this model to show the simple, intuitive interpretation of our results for collateralized netting.
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 byδt , 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 t is determined by the portfolio value at time t − δt
according to

C(t) = max{V (t − δt) − H , }
0
(6)

9 When the threshold is not too small, the instantaneous collateral model works reasonably well for expected
exposure. See Pykhtin (2009).

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We refer to this more realistic model as the lagged collateral model. While more difficult to
implement, it is often used by banks to obtain results, which have more practical value.
2.3 Credit losses and CVA
In the event that the counterparty defaults at time τ , the bank recovers a fraction R of the
exposure E(τ ) . The bank’s discounted loss due to the counterparty’s default is

L = 1
(1 − R) E(τ )D(τ )
(7)
{τ ≤T}
where 1
is the indicator function that takes value 1 when logical variable A is true and value 0
{ }
A
otherwise, D(t) is the stochastic discount factor process at time t, defined according to
D(t ) = B B , with B the value of the money market account at time t.
0
t
t
The unilateral counterparty-level CVA is obtained by applying the expectation to
Equation (7). This results in
T

CVA
(1 R) d (
P t) ˆe (
∗
=
−
∫
t)
(8)
0
where ˆe (
∗ t) is the risk-neutral discounted expected exposure (EE) at time t, conditional on the
counterparty’s default at time t:

∗
ˆ
ˆe (t) = E [D(t)E(t)] ≡ E D(t)E(t) τ =t
t


(9)
Throughout this paper we use “star” to designate discounting and “hat” to designate conditioning
on default at time t. Note that we have not made so far any assumptions on whether the exposure
depends on the counterparty’s credit quality.
3. CVA Contributions from EE Contributions
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 i by CVA .
i
We say that CVA contributions are additive when they sum up to the counterparty-level CVA:
N

CVA = ∑CVA
(10)
i
i 1
=

10 The margin period of risk depends on the contractual margin call frequency and the liquidity of the portfolio. For
example, δ t = 2 weeks is usually assumed for portfolios of liquid contracts and daily margin call frequency.

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Note that the recovery rate R and the default probabilities P(t) are defined at the counterparty
level in Equation (8). Thus, the problem of calculating CVA contributions reduces to that of
calculating contributions of individual trades to the portfolio conditional discounted EE, ˆe (
∗ t) , at
i
each future date. To obtain additive CVA contributions, then the conditional discounted EE
contributions must sum up to the portfolio conditional discounted EE:
N

ˆe (
∗ t)
ˆe (
∗
= ∑
t)
(11)
i
i 1
=
and the CVA contribution of trade i can be calculated from its EE contribution according to
T

CVA
(1 R) d (
P t) ˆe (
∗
=
−
∫
t)
(12)
i
i
0
Thus, from now on we focus on defining and calculating EE contributions.
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.
4. Additive EE Contributions for Non-collateralized Netting Sets
In this section, we develop the basic methodology to compute EE contributions and
allocate portfolio-level EE for non-collateralized netting sets.
4.1 Continuous Marginal Contributions and Euler Allocation
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.

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A real function f (x) of a vector x = (x , ... , x ) is said to be homogeneous of degree β
1
N
if for all c > 0 , f (c x)
cβ
=
f (x) . If the function f ( )
⋅ is piecewise differentiable, then Euler’s
theorem states that:
N
f
∂ ( )

β ⋅ f ( ) = ∑
x
x
⋅ x
(13)
i
i 1
=
x
∂ i
The risk measures most commonly used, such as standard deviation, value-at-risk (VaR) and
expected shortfall, are homogeneous functions of degree one ( β = 1) in the portfolio positions.
Thus, Euler’s theorem is applied to allocate EC and compute risk contributions across portfolios.
If x denotes the vector of positions in a portfolio, and EC(x) the corresponding
economic capital, then Euler’s theorem implies additive capital contributions
N

EC(x) = ∑ EC (x)
(14)
i
i 1
=
where the terms
E
∂ C(x)

EC (x) =
⋅ x
(15)
i
i
x
∂ i
are referred to as the marginal capital contributions of the portfolio.
4.2 Continuous Marginal EE Contributions for netted exposures without collateral

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 α for trade i as a scale factor that
i
represents the relative size of the trade in the portfolio, V (α ,t) = α V (t) . These weights can
i
i
i i
assume any real value, with α = 1 corresponding to the actual size of the trade and α = 0 being
i
i
the complete removal of the trade. We describe adjusted portfolios via the vector of weights
α = (α ,K,α ) . For adjusted portfolios, we use the notations E(α,t) , ˆ
e∗(α,t ) , and CVA(α) for
1
N
the exposure and EE at time t and CVA. Furthermore, for convenience, denote by = (1,K
1
,1)
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:

E(cα,t) = cE(α,t)
(16)

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The intuition behind Equation (16) is simple: if the bank uniformly doubles the size of its
portfolio with the counterparty by entering into exactly the same trade with the counterparty for
each existing trade, the bank’s exposure doubles.
We define the continuous marginal EE contribution of trade i at time t as the infinitesimal
increment of the conditional discounted EE of the actual portfolio at time t resulting from an
infinitesimal increase of trade i’s presence in the portfolio, scaled to the full trade amount:
ˆe (
∗ t,
∗
1 + δ ⋅ u ) − ˆe (
∗ t)
∂ ˆe (
∗ t,α)

ˆe (t) = lim
i
=

(17)
i
δ→0
δ
α
∂ i
α=1
where u describes a portfolio whose only component is one unit of trade i. Since the portfolio
i
exposure is homogeneous in the trades’ weights, the EE contributions defined by Equation (17)
automatically sum up to the counterparty-level conditional discounted EE by Euler’s theorem
(Equation (13)).
We can derive an expression for the marginal EE contributions as follows. First,
substitute Equation (9) into Equation (17) and bring the derivative inside the expectation. This
results in


∗
E
∂ (α,t)

ˆ
ˆe (t) = E D(t)

(18)
i
t 
α
∂


i
α=1 
where exposure of the adjusted portfolio (with weight vector α = (α ,K,α ) ) is given by
1
N
N



E(α,t) = max ∑α V (t), 0
(19)
i
i
 i 1
=

Calculating the first derivative of the exposure with respect to the weight α and setting all
i
weights to one, we have:
∂ E(α,t)
∂
∂V (α,t)

=
max{V (α,t), }
0
=
1
= V (t)1

(20)
{V ( ,t) 0}
i
{V (t) 0}
α
∂
α
∂
α
>
>
∂
α
i
i
i
α=1
α=1
α=1
Substituting Equation (20) into Equation (18), we obtain the EE contribution of trade i:
