Credit valuation adjustment in Basel III is studied from the perspective of the mathematics involved. A bank covers mark-to-market losses for expected counterparty risk with a CVA capital charge. The CVA is known as credit valuation adjustments. In this paper it will be argued that CVA and conditioned value at risk (CVaR) have a common mathematical ancestor. The question is raised why the Basel committee, from the perspective of CVaR, has selected a specific parameterization. It is argued that a fine-tuned supervision, on the longer run, will be beneficial for counterparties with a better control over their spread.CVA, CVaR, statistical methodology.

Geurdes, Han / J. F.

English

Research Papers in Economics

Counterparty credit Risk
On the mathematical form of CVA in Basel III
J.F. Geurdes
Logica plc , Prof Keesomlan 14, Amstelveen, Netherlands
han.geurdes@logica.com
Abstract
Credit valuation adjustment in Basel III is studied from the perspec-
tive of the mathematics involved. A bank covers mark-to-market losses
for expected counterparty risk with a CVA capital charge. The CVA is
known as credit valuation adjustments. In this paper it will be argued
that CVA and conditioned value at risk (CVaR) have a common math-
ematical ancestor. The question is raised why the Basel committee,
from the perspective of CVaR, has selected a speciﬁc parameterization.
It is argued that a ﬁne-tuned supervision, on the longer run, will be
beneﬁcial for counterparties with a better control over their spread.
Keywords: CVA, CVaR, statistical methodology
1 Introduction
1.1 Measures of counterparty risk
The reservation of capital is based on what to expect in the future. In Basel III
the Credit Valuation Adjustment (CVA) capital charge takes a prominent role
[1] in the treatment of mark-to-market counterparty risk. Banks with IMM
approval for counterparty credit risk need to calculate additional capital charge
by modeling the impact of changes in the counterparties’ credit spreads. The
capital charge calculation for each counterparty is based on the CVA formula
of page 31 of the framework document and is represented below in equation (1).
We may ask if a presently suﬃcient decrease in counterparty spread can be
trusted for the future such that it will, rightfully, not add to the required
capital charge. Our method enables a comparison of counterparties and a
’trusworthiness’ factor to be incorporated in the CVA. The method on which
’trust’ can be made numerical is e.g. to compare historical series of spread
and to use the variance of the spread. An alternative method to compare is2 J.F. Geurdes
to employ Fourier analysis on the ﬂuctuations of diﬀerent spread. If principal
wavenumber components of two series are more or less equal, both counter-
parties suﬀer from the same economic processes. If one counterparty has a
greater mean amplitude than the other, one can conclude that this counter-
party is less in control over his spread. Therefore it could be wise to reserve
capital despite of a suﬃcient decrease in spread in the present period. The
proposed parametrization of ﬁne-tuning on counterparty risk is based on the
method of conditional value at risk [2]. Of course other assesments of trust, like
a panel of experts, can be employed too to classify trust in counterparty spread.
The CVA is deﬁned in the framework [1] as:
CV A = LMKT
T X
i=1
max
n
0,e
−ˆ si−1ti−1 − e
−ˆ siti
o ￿EEi−1Di−1 + EEiDi
2
￿
(1)
Here, EEi is the expected exposure [4] to the counterparty at time (bucket)
ti, with t0 = 0 and i = 1,2,...T. The EEi is obtained in the new capital con-
servation mechanism of the committee and refers to a required countercyclical
capital buﬀer [3]. Moreover, Di is a default risk-free discount factor at ti. It
is somewhat unclear what is intended here. The LMKT is the so-called loss-
given-default of the counterparty and must be based on the spread of a market
instrument of the couterparty. It is a market assesment of the loss and, hence,
does not depend on time buckets ti. The ﬁler max(0,y) in equation (1) equals
y when y > 0 and vanishes if y ≤ 0. Note that LMKT does not change for the
measurement period from t0 to tT. But that does not imply that this factor
will be constant for a possible next period. The ˆ si will be discussed below.
The diﬀerence with Basel II [4] appears to be substantial because in Basel
II supervision the committee only subscribed banks to employ the following
maturity measure, M.
M =
Ptk≤1year
k=1 Eff (EEk)∆tk Dk +
Pmaturity
k∈{tk>1year} EEk ∆tk Dk
Ptk≤1year
k=1 Eff (EEk)∆tk Dk
(2)
Conceptually, M is the (eﬀective) credit duration of counterparty exposure [4].
1.2 Mathematical resemblance with CVaR
The ﬁrst thing that catches the eye in the Basel III CVA formulation is the
use of the max(0,y) operator. Its systematics resembles the way in which the
β-CVaR [5] is obtained. The basis for β-CVaR is the expression
Fβ(  x,α) = α +
1
(1 − β)T
T X
i=1
[f(  x,  yi) − α]
+ (3)Basel III math 3
with [t]
+ = t when t > 0 and [t]
+ = 0 when t ≤ 0. Furthermore, f(  x,  y) is a
general measure of loss associated to a decision vector   x ∈ X ⊂ Rn. The   x
refer to the ’portfolio’ while   yi ∈ Y ⊂ Rm refer to the uncertainties that arise
from market parameters. Note that the CVaR is a general scheme.
As can be seen in equation (3) the   y are supposed to be uniformly distributed
with 1/T, for bucket numbering i = 1,2,..,T. In the deﬁnition, the index β is
the probability level of the loss, i.e. there is a probability β that the loss will
not exceed α. Later on we will make this more explicit.
In order to establish a relation between the CVA deﬁned in (1) and the CVaR,
we can make the following comparison: the EEi together with the ti can be
assigned to the   x from (3), the ˆ si =
si
LMKT in (1) naturally are assigned to the
  yi from (3). The si refer to the credit spread of the counterparty at, what the
committee calls, ’tenor’ ti. When the α is selected zero we acknowledge a form
[gi]+ employed in (1). Or,
f(  x,  yi) ∼
￿
e
−ˆ si−1ti−1 − e
−ˆ siti
￿￿EEi−1Di−1 + EEiDi
2
￿
= gi (4)
On the point of using a maximum operator max(0,gi) one can see the resem-
blance beween CVA and CVaR because [gi]+ = max(0,gi). Hence, the methods
to optimize CVaR can, in principle, be applied to CVA as well. The mathe-
matical statistics apparently allows an optimizing approach. For the ease of
comparing the CVA in equation (1) with CVaR we may write
CV A = LMKT
T X
i=1
max(0,gi) (5)
with gi as deﬁned in (4).
2 Factor dynamics in the CVA
Here we study the dynamics of CVA. Recall that the factor ˆ si is deﬁned by
ˆ si =
￿
si
LMKT
￿
and si refers to the credit spread of the counterparty. Now be-
cause, ti > ti−1 ≥ t0 = 0, it follows that, when the (weighted) credit spread of
the counterparty, or the, ˆ si at ti is greater than ˆ si−1, then, e−ˆ si−1ti−1−e−ˆ siti > 0
and the mean
￿
EEi−1Di−1+EEiDi
2
￿
> 0 times the e−ˆ si−1ti−1 − e−ˆ siti adds to the
CV A. The weight factor is the diﬀerence between the exponentials.
Furthermore, if ˆ si < ˆ si−1 then
￿
EEi−1Di−1+EEiDi
2
￿
still may add to the CV A
provided
δti−1 >
 
ˆ si−1
ˆ si
− 1
!
ti−1 (6)4 J.F. Geurdes
given ti = ti−1 + δti−1. This means that for relative large time steps a certain
amount of diminishing of credit spread for the counterparty appears to still
add to the capital charge CVA. This is due to the fact that in the initial di-
minishing of credit spread over time, the spread of the counterparty can still
increase in a, more uncertain, larger time period.
Only when ˆ si < ˆ si−1 and equation (6) is not valid does the term gi in time
period ti not contribute to the CVA capital charge. Hence, a lower CVA cap-
ital charge turns out to depend on a suﬃcient decreasing series {ˆ si}
I1
i=I0 with
I0 < I1 in the set {1,2,....,T}, that is a consecutive series of decreasing spread
periods. Moreover, when e−ˆ si−1ti−1 − e−ˆ siti ≈ 0 there is litte contribution to
the CVA. No doubt, when δti is relatively small a prudent control will then
require a relatively large T.
3 CVA and CVaR
For comparison with CVaR the CVA is made dependent on α. Hence,
CV A(α) = α + LMKT
T X
i=1
max(0,gi − α) (7)
gi deﬁned as in (4). As can be seen from equation (1) the Basel III CVA has,
α = 0.
Subsequently we note that the CVaR is deﬁned by
φβ(  x) = (1 − β)
−1
Z
f(  x,  y)≥αβ
f(  x,  y)P(d  y) (8)
with αβ a parameter depending on α and β. The CVaR can be obtained as a
minimalization problem: φβ(  x) = min{Fβ(  x,α)|α ∈ R}, with Fβ(  x,α) as in
(3), or more generally
Fβ(  x,α) = α + (1 − β)
−1
Z
  y∈Rm[f(  x,  y) − α]
+P (d  y). (9)
Here, P(  y) is a, Lebesgue, probability measure over the vector   y, similar as used
by Komogorov [6]. In case of equation (3), the Lebesgue measure ’collapses’ to
a discrete space with uniform density, ρ(  yi) = 1/T for   yi ∈ Rm,i = 1,2,3,,,T.
Hence, according to [7] the minimum can be found through the expression
∂
∂α
Fβ(  x,α) = 1 + (1 − β)
−1 (Ψ(  x,α) − 1) (10)
with
Ψ(  x,α) =
Z
f(  x,  y)≤α
P(d  y) (11)Basel III math 5
This is equal to the expression that the random variable f(  x,  y) does not ex-
ceed a treshold α. Hence, Ψ(  x,α) = β, or in words, the probability that f(  x,  y)
does not exceed a treshold α is equal to β ∈ (0,1). Having, ∂
∂αFβ(  x,α) = 0, the
minimum of Fβ(  x,α) is found and hence, the minimum CVaR is determined
as φβ(  x) with the probability that a test random variable does not exceed a
value α with a probability β.
Interestingly, introducing a Heaviside θ function as correction term
αLMKT
PT
i=1 θ(gi − α) to equation (7) does not lead to a minimum for terms
similar to Fβ(  x,α). This is true when we reformulate CV A′(α) = CV A(α) +
αLMKT
PT
i=1 θ(gi − α) in terms of Gβ(  x,α) in CVaR terms like
Gβ(  x,α) = Fβ(  x,α) + α(1 − β)
−1
Z
  y∈Rm θ(g(  x,  y) − α)P(d  y) (12)
with Fβ(  x,α) from equation (9). Because
Z
  y∈Rm θ(g(  x,  y) − α)P(d  y) = 1 − Ψ(  x,α) (13)
with, f(  x,  y) in equation (11) replaced by g(  x,  y), it follows that
∂
∂α
Gβ(  x,α) = 1 − α(1 − β)
−1
Z
  y∈Rm δ(g(  x,  y) − α)P(d  y). (14)
It is highly likely that g(  x,  y) is unequal to α given g(  x,  y) and   x ∈ X,   y ∈
Y . Simply stated there appears not to exist trivial compensation terms for:
∆(α) = −αLMKT
PT
i=1 θ(gi − α) in equation (7). This conclusion can be
supported by noting the fact that, generally, max(0,z) = zθ(z). Hence, for
numerical purposes there appears no CVaR trivial way from CV A(α) to sums
without ∆(α).
4 Conclusion & discussion
4.1 CVar and CVA resemblance
As can be seen from comparing the two formula’s the Basel III CVA method-
ology resembles the CVaR. Moreover, it was shown that a CVaR minimum
exists for α ∈ R. As can be seen from the deﬁnition (1), the Basel III CVA
is connected to the CVaR with the use of α = 0 and there is no CVaR triv-
ial compensation term for ∆(α) deﬁned previously. If for proper choice of
T and β we have LKMT = 1
(1−β)T, then max(0,gi − α) = [gi − α]+, with,
gi =
￿
e−ˆ si−1ti−1 − e−ˆ siti
￿ ￿
EEi−1Di−1+EEiDi
2
￿
warrants a CVaR reformulation of
CVA.6 J.F. Geurdes
In this interpretation it makes sense to view the ˆ si as belonging to the   y
’market’ variables. Moreover, expected exposure, EEi, can be seen as belong-
ing to the state variables   x. For a certain period, LMKT and Di also belong
to   y. Concerning the time-buckets they also can be assigned to the state vari-
ables in the sense that they can, directly, be inﬂuenced by the bank. More in
particular, the δti appear state variables. Note that a bank can negotiate with
the supervisor the δti of the reports and in this way the time-buckets certainly
can be viewed as state variable.
4.2 Proof that CV A(α) > CV A(0) for α < 0
In Figure 1 below the direction of the diﬀerence is already made plausible. Let
us deﬁne the set I+ as I+ = I(gi > 0) = {i|gi > 0} and I− = I(gi ≤ 0) =
{i|gi ≤ 0} such that I+ ∪ I− = {1,2,3,...T}. This implies that one may write
CV A(0) = LMKT
X
i∈I+
gi (15)
If we note that [g]+ = gθ(g), it follows for α < 0, that, CV A(α) − CV A(0) =
−|α| + LMKT
X
i∈I−
(|α| + gi)θ(|α| + gi) + LMKT|α|
X
i∈I+
(|α| + gi) (16)
This is true because, when gi > 0 then gi + |α| > 0 too. Now because
LMKT|α|
X
i∈I+
(|α| + gi) = card
￿
I
+
￿
LMKT|α|, (17)
with card denoting ’cardinality’of the set, then if card(I+)LMKT > 1, it
follows that CV A(α) − CV A(0) > 0, because
LMKT
X
i∈I−
(|α| + gi)θ(|α| + gi) ≥ 0. (18)
4.3 Financial consequence
The question now arises whether the selection α = 0, implicitly present in the
Basel III deﬁnition of risk capital required for counterparty risk (1), represents
the best optimal value given the previous obvious subdivision in state and
market variables. In other words, granted the structural resemblance between
CVaR methodolgy and the required CVA, we may ask what the reasons for
the Basel committee were to select the α = 0 parameterization in their CVA
required capital charge. The application of equation (7) enables to have α < 0.
Looking at the discussion of the dynamics of factors in section 2, one sees thatBasel III math 7
Figure 1: The CV A(α) for α < 0 and the Basel
III CV A(0) (horizontal line) are given where α ∈
{−0.01,−0.02,−0.03,−0.04,−0.05,−0.06,−0.07,−0.08,−0.09,−0.1}. The
si, EEi and Di are pairwise randomly generated from the interval (0,1) and
LMKT = 2. With the present parametrisation the gap in the lower |α| region
requires to reserve 3 - 4 × the amount of Basel III CVA8 J.F. Geurdes
α < 0 is interesting for unstable counterparty spread. Despite the fact that
e.g. in the ti the spread of the counterparty is diminished related to ti−1, one
could still want to add gi + |α| because of (expected) ﬂuctuation in spread in
some future. The latter selection enables a ﬁne-tuning in discriminating stable
and unstable counterparties basing onself on the variance of the couterparty
spread, i.e. the V ar(si).
From ﬁgure 1 we present numerical proof of the fact that for α < 0 the
CV A(α) is larger than the CV A(0) used in the Basel III framework. Use
was made of parametrizations LMKT = 2, T = 103 and si, EEi together with
Di pairwise randomly generated (in a VBA program). The α are taken from
the set {αn|n = 1,2,...,10,αn = αn−1 − ∆α,α0 = 0,∆α = 0.01}. Hence, the
introduction of α < 0 enables a ﬁne-tuning in the CVA in terms of more or
less creditworthy counterparties.
4.4 Measure of trust
An alternative to the variance of the spread of the counterparty is a Fourier
analysis on spread ﬂuctuations. When characteristic frequencies for two given
counterparties are approximately equal but there is a substantial diﬀerence in
spread-amplitude a measure of trust in a counterparty is obtained that can
be expressed in terms of α < 0. Use of comparing characteristic frequencies
will enable to judge whether counterparties are subjected to similar economic
ﬂuctuations.
4.5 Question of supervision
In view of possible negotiating processes on time-buckets and the creditworthi-
ness of counterparties it can be questioned whether the present Basel III CVA
form with α = 0 is the best possible capital charge in a regulatory framework
for more resilient banking. Diﬀerentiating on a numerical measure of trust will
avoid having too low capital reservation for certain counterparties and will
have the eﬀect that, because of less required capital reserve, banks will prefer
to do business with counterparties with a better control over their spread.
References
[1] Basel III: A global regulatory framework for more resilient banks and
banking systems, Basel Committee on Banking Supervision, december
2010.Basel III math 9
[2] Messages from the academic literature on risk measurement for the trading
book, Basel Committee on Banking Supervision, januari 2011.
[3] Position paper on a countercyclical capital buﬀer,www.c-
ebs.org/getdoc/715bc0f9-7af9-47d9-98a8-778a4d20a880/CEBS-position-
paper-on-a-countercyclical-capital-b.aspx.
[4] Basel II: International Convergence of Capital Measurement and Capital
Standards, Basel Committee on Banking Supervision, june 2006, Annex
4.
[5] R.T. Rockafellar and S. Uryasev, Optimization of conditional value at
risk, Preprint: University of Florida, Dept. of Industrial and Systems
Engineering, PO Box 116595, 303 Weil Hall, Gainesville, FL 32611-6595,
E-mail: uryasev@ise.uﬂ.edu, URL: http://www.ise.uﬂ.edu/uryasev
[6] A. Kolmogorov, Grundbegriﬀe der Wahrscheinlichkeitsrechnung, Julius
Springer, Berlin, (1933).
[7] A. Shapiro and Y. Wardi, Nondiﬀerentiability of the Steady-StateFunction
in Discrete Event Dynamic Systems, IEEE Trans. on Aut. Contr 39.
1701-1711 (1994).

A global regulatory framework for more resilient banks and banking systems,

Grundbegriﬀe der Wahrscheinlichkeitsrechnung,

International Convergence of Capital Measurement and Capital Standards, Basel Committee on Banking Supervision,

Messages from the academic literature on risk measurement for the trading book, Basel Committee on Banking Supervision,

Nondiﬀerentiability of the Steady-StateFunction in Discrete Event Dynamic Systems,

Optimization of conditional value at risk, Preprint:

Position paper on a countercyclical capital

On the mathematical form of CVA in Basel III.

On the mathematical form of CVA in Basel III.

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