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test.html
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<div class="entities" style="line-height: 2.5; direction: ltr">In the wake of the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
200809
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
global financial crisis clearing through central counterparties CCPs has become mandatory for standardized derivatives other ones remaining under bilateral setup with higher capital requirements.
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
One
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
role of the CCPs
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
is to provide to their clearing members fully collateralized hedges of their market risk with their clients. But this comes at a cost to the clearing members which pass it to their corporate clients in the form of
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
crossvaluation adjustments addons. Bearing in mind that the risks of a hedge are by definition of the same magnitudes as the ones of the originating position and that standardized derivatives usable as hedging assets have to be traded through CCPs the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
footprint of not only bilateral but also centrally cleared trading is significant and should be analyzed in detail which is the topic of this paper.a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
BNP Paribas Stress Testing Methodologies
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
Models. This article represents the opinions of the author and it is not meant to represent the position or opinions of
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
BNP Paribas
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
or its members. dorinel.2.bastidebnpparibas.com corresponding author.b
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Laboratoire de Probabilits Statistique et Modlisation
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
LPSM Sorbonne Universit et Universit de Paris
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CNRS
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
UMR 8001.
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
stephane.crepeylpsm.paris.c
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Shanghai Jiao Tong University
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #feca74; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Shanghai
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">GPE</span>
</mark>
<mark class="entity" style="background: #feca74; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
China
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">GPE</span>
</mark>
. sdrapeausaif.sjtu.edu.cn. d
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Woldia University Mathematics Department
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #feca74; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Ethiopia
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">GPE</span>
</mark>
. mekonnentawldu.edu.et.More precisely the trades of a clearing member bank with a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
are partitioned between proprietary trades which are in effect hedges of the bilateral trading exposure of the bank and backtoback hedges of socalled cleared client trades through which nonmember clients gain access to the clearing services of a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. focus on the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
analysis of a bank only acting as a clearing member of
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
one
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
without proprietary trading. The present paper provides an integrated
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
analysis in the realistic situation of a bank dealing with many clients and CCPs through both proprietary also dubbed house accounts and client accounts.For the sake of tractability this is achieved in a stylized oneperiod setup finetuned to applications including risk assessment in the context of stress test exercises
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
or optimizing the porting of the portfolio of defaulted clearing members.The
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
first
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
type of application is motivated by the default in
<mark class="entity" style="background: #bfe1d9; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2020
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">DATE</span>
</mark>
of Ronin Capital a brokerdealer firm that had clearing exposures on both
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
services
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Fixed Income Clearing Corporation FICC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
GSD
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
segment
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
123
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
members and
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CME Futures
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
56
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
members of which
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
24
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
common with
<mark class="entity" style="background: #bfeeb7; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
FICC GSD
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PRODUCT</span>
</mark>
. If all members are assumed to be only exposed to these CCPs and their cleared clients we can illustrate these relationship by the network depicted in Figure
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
1.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. Any common member on those
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
CCPs needs to ensure conservative risk assessment that can be achieved in the proposed framework by accounting for common memberships on the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
CCPs. If such common memberships are ignored they can lead to lower loss estimates giving wrong risk view on potential losses.The
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
second
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
type of application is an illustration of the results of defaulted portfolio porting as it has been the case for the trader
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Einer Aas
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
on
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
NASDAQ OMX 4
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
that has defaulted on
<mark class="entity" style="background: #bfe1d9; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2018
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">DATE</span>
</mark>
with loss spillover effect on surviving members.
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Section 2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
sets the stage.
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Section 3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
develops the corresponding
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
analysis.
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Section 4
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
develops
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
applications in the above veins. Section 7 concludes. We consider a finite set of market participants also susceptible to serve as clearing members of CCPs. Derivative transactions can then be concluded between
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
individual participants or between a set of participants
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
5
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
pooled in the form of a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
and a clearing member of this CCP.CCPs are typically siloed into different services each devoted to a specific class of derivatives. We
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
first
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
consider a setup with a single
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
service
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
6
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
see Figure
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
where P and P represent the contractual cash flows from cleared and bilateral clients to a reference clearing member dubbed the bank hereafter hence promised in successive Figure
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
1.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
Network consisting of
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
CCPs in red
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
123
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
members for CCP1 seen on the left hand side and
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
56
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
members for CCP2 on the right hand side with
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
24
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
common members displayed as the group of members in the middle of the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
CCPs
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
155
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
members in total in blue and with
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
179
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
cleared clients in green).turns from the bank to the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
from the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
to other clearing members and from the latter to their own clients. As a consequence the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
is flat in terms of market risk as is also each of the clearing members. As reasserted in the wake of the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
200809
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
global financial crisis by the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Volcker
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
rule a dealer bank should be hedged as much as possible at least in terms of market risk
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
7
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. Jumptodefault risk on the other hand is hardly hedgeable in practice. Instead it is mitigated through netting and collateralization. Namely designated netting sets of transactions between
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
given counterparties
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
individual participants or a participant and the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
are jointly collateralized i.e. guaranteed against the default of
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
one
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
or the other party. The collateral or guarantee comprises a variation margin which tracks the marktomarket counterpartyriskfree value of the netting set between the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
parties and nonnegative amounts of initial margin posted by each party to the other which provide a defense against the risk of slippage of the value of the netting set away from its frozen variation margin during its liquidation period. In the case of transactions with a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
there is an additional layer of collateral in the form of the funded default fund contributions of the clearing members which is meant as a defense against extreme and systemic risk. For each participant variation margin is rehypothecable and fungible across all its netting sets. Initial margin is segregated at the netting set level. default fund contributions are segregated at the clearing member level.The general rule regarding the settlement of contracts of a defaulted netting set is that:Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
If a counterparty in default is indebted toward the other beyond its posted margin then this debt is only reimbursed at the level of this posted margin assuming
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
zero
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
recovery rate of the defaulted party for simplicity in this paper; otherwise the debt between the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
parties is fully settled.Here debt is understood on a counterpartyriskfree basis.This rule also applies to a netting set of transactions between a clearing member and a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. However in our stylized setup a
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
is nothing but the collection of its clearing members. Our
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
has no resources of its own in particular it cannot post any default fund contribution or "skininthegame"
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
8
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. As long as it is nondefault i.e. as long as at least one of its clearing members is nondefault our
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
can only handle the losses triggered by the defaults of some of its clearing members by redirecting these losses on the surviving ones. This participation of the surviving members to the losses triggered by the defaults of the other members corresponds in our framework to the usage by the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
of their default fund contributions both funded as already introduced above and unfunded. As will be detailed in equations below the funded default fund contributions are used in priority for covering losses triggered by the defaults of clearing members over their margins. The unfunded default fund contributions correspond to additional refills that can be required by the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
often up to some cap in principle without bounds in our model in case the funded default fund contributions of the surviving members are not enough. Assume that at time 0 all the banking participants including the reference clearing member bank in Figure
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
with no prior endowments enter transactions with their clients and hedge their positions both bilaterally between them and through the CCP.As seen above the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
and each bank are flat in terms of market risk. However as market participants are assumed to be defaultable with
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
zero
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
recovery in order to account for counterparty credit risk and its funding and capital consequences each banking participant requires from its corporate clients a pricing rebate considering conventionally the bank as the buyer with respect to the marktomarket counterpartyriskfree valuation of the deals. The corporate clients of the bank are assumed to absorb the ensuing prices via their corporate business which is their primary motivation for these deals.A reference probability measure Q with corresponding expectation operator denoted by
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
E
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
is used for the linear valuation of cash flows using the riskfree asset as our numraire everywhere. This choice of a numraire simplifies equations by removing all terms related to the assumed riskfree remuneration of all cash and collateral accounts. The funding issue is then refocused on the risky funding side of the problem i.e. funding costs in what follows really means excess funding costs with respect to a theoretical situation where the bank could equally borrow and lend at the riskfree rate.More precisely as suitable for
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
calculations Albanese et al. 2021 Remark
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
given a physical probability measure defined on the full model algebra A and equivalent to a reference riskneutral measure on the financial sub algebra B of A we take Q equal to the reference riskneutral measure on B and equal to the physical probability measure conditionally on B.Following the general
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
guidelines of the abovementioned pricing rebate required by the reference clearing member bank dubbed funds transfer price
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
FTP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
comes in
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
two
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
parts
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
first
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
the expected counterparty default losses and funding expenditures of the bank an amount that flows into the bank liabilities and which we refer to as contraasset valuation CA;
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
second
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
a cost of capital risk premium KVA which instead is lossabsorbing
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
9
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
and is also used by the management of the bank as retained earnings for remunerating the shareholders of the bank for their capital at risk within the bank. All in
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
one
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
the bank buys the deals from its clients at the
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
9
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
hence not a liability. .1 Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
At time 0 the amounts CA and KVA sourced from the corporate clients of the bank are deposited on reserve capital and capital at risk accounts of the bank.Let
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
denote an economic capital of the bank corresponding to the minimum level of capital at risk that the bank should hold from a regulatory i.e. solvency perspective. If
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
then the bank shareholders need to provide the missing amount ECKVA of capital at risk so that the actual level of capital at risk of the bank is maxEC KVA while shareholder capital at risk reduces to maxEC
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
KVA .23 Theoretical XVA AnalysisIn this section we detail each term in the equations above in the realistic setup of a bank involved into an arbitrary combination of bilateral and centrally cleared portfolios in a tractable oneperiod setup with period length T . In the oneperiod
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
model of Albanese et al. 2021 Section 3 there were no CCPs and the bank was assumed to have access to a fully collateralized backtoback hedge of its market risk ensuring by definition and for free to the bank a cashflow
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
P
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
MtM at time
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
irrespective of the default status of the bank and its client. There P denoted the contractual cash flows from the assumed unique client to the bank and
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
MtM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
was the corresponding counterpartyriskfree value. In the present paper we reveal the mechanism of such a fully collateralized hedge of the market risk of the bank which can be achieved through central clearing but at a certain cost that we analyze. We use the terms client for cleared clients and counterparty for bilateral counterparties. Given disjoint sets of indices I 0 C and B for the clearing members including the reference bank labeled by
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
0
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
and for the respective cleared and bilateral netting sets of the bank with its individual clients and counterparties We denote by J
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
0
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
shortened as J and J i i I 0 the survival indicator random variables of the bank and of the other clearing members at time
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
Q J
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
0
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
the default probability of the bank J max i J i the survival indicator random variable of the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
i.e. of
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
at least one
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
clearing member P i
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
MtM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
i E P i and
<mark class="entity" style="background: #c887fb; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
IM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">NORP</span>
</mark>
i i I the contractual cash flows variation margin and initial margin from the clearing member i to the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
corresponding to the cleared clients account of the member i P i
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
MtM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
i E P i and
<mark class="entity" style="background: #c887fb; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
IM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">NORP</span>
</mark>
i i I the contractual cash flows variation margin and initial margin from the clearing member i to the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
corresponding to the house account of the clearing member i DF i i I the default fund contribution posted by the clearing member i to the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
J b b B the survival indicator random variable of the counterparty of the bilateral netting set b of the reference bank P b VM b and
<mark class="entity" style="background: #c887fb; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
IM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">NORP</span>
</mark>
b the associated contractual cash flows variation margin and initial margin from the corresponding counterparty to the bank and
<mark class="entity" style="background: #c887fb; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
IM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">NORP</span>
</mark>
b the initial margin from the bank to the counterparty J c c C the survival indicator random variable of the client of the cleared trading netting set c of the bank and P c MtM c E P c
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
10
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
and
<mark class="entity" style="background: #c887fb; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
IM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">NORP</span>
</mark>
c the associated contractual cash flows variation margin and initial margin from the corresponding client to the bank
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
11
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
L the loss of the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
i.e. the loss triggered by the defaults of its clearing members beyond their posted collateral
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
12
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
which is borne by the surviving members if any J the proportion of these losses allocated to the reference clearing member bank. Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
i P i P i 0 the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
is flat in terms of market risk c P c P 0 by definition of cleared trades and of their mirroring trades and b P b P 0 the reference bank is flat in terms of market risk. Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
yields the clearing conditions regarding the contractually promised cash flows which applies to each banking participant written there for the reference bank and to the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
is monitoring the default cash flows. We need
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
one
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
more condition regarding the funding side of the problem
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Assumption 3.2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
The bank can use the amounts CA and maxEC KVA on its reserve capital and capital at risk accounts for its variation margin borrowing purposes. Funds needed beyond CA maxEC KVA for variation margin posting purposes are borrowed by the bank at its credit spread above
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
OIS
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. The initial margin and default fund contributions instead must be borrowed entirely by the bank but this can be achieved at some blended funding spread . The rationale for funding variation margin but not initial margin from
<mark class="entity" style="background: #bfeeb7; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CAmaxEC KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PRODUCT</span>
</mark>
is set out before
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Equation 15
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
in
<mark class="entity" style="background: #feca74; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Albanese et al
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">GPE</span>
</mark>
.
<mark class="entity" style="background: #bfe1d9; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
2017
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">DATE</span>
</mark>
. The motivation for the assumption is provided in Albanese et al. 2020 Section 5 along with numerical experiments suggesting that can be several times lower than .
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
Proof. On the bilateral trades of the bank and their hedges the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Treasury
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
of the bank receives b VM b of variation margin from the counterparties and has to post an aggregated amount b
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
MtM
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
b of variation margin. The assumption stated before the lemma then leads to
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
.
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Lemma 3.2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
On the bank survival event
<mark class="entity" style="background: #f0d0ff; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
J 1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">WORK_OF_ART</span>
</mark>
the counterparty default losses C and the funding expenses F of the bank are given by C where L and F6 Proof. On the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
survival event
<mark class="entity" style="background: #f0d0ff; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
J 1 the CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">WORK_OF_ART</span>
</mark>
receives by
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Assumption 2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
i Using the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
clearing condition Assumption
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
0 which is
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
5
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. On the bank survival event J 1 ⊆
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
J
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
1 by the respective
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Assumptions 2.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
and
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
the bank receives from its clients and counterparties c Subtracting
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8
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
from
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9
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
we obtain c. On top of this comes the participation L of the bank to the
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CCP
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
default losses which yields
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4
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. Moreover in view of
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Lemma 3.1 and Assumption 3.2
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>
the risky funding expenses of the bank are given by
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
6
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. Let E denote the expectation with respect to the bank survival measure Q associated with Q i.e. for any random variable Y E0 Y. Under a costofcapital
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XVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
approach the bank charges its future losses to its corporate clients at a CA level making
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JC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
F CA the trading loss of the shareholders of the bank Q centered. In addition given a target hurdle rate h assumed in
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
0 1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
and typically of the order of
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10
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
the management of the bank ensures to the bank shareholders dividends at the height of h times their capital at risk
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
KVA cf. 2 where we model
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EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
as
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ES JC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
F CA the expected shortfall of the trading loss
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
JC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
F CA computed under the bank survival measure at a quantile level
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13
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
of 99.75 i.e. under the primal and dual representations of the expected shortfall
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
14
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
<mark class="entity" style="background: #bfeeb7; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
VaR
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PRODUCT</span>
</mark>
a denoting the Q valueatrisk lower quantile of level a of
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11
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
which for atomless also coincides
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15
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
with E . Note that in view of the dual representation an expected shortfall of a centered random variable is nonnegative. E Accordingly as detailed after the definition D Hence in view of
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
4
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
and
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
6
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
C i.e. E J CF CA
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
0
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
as desired
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16
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. The terminal cash flows of the form 1J×. . . in
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
12
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
or
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
13
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
are thus consistent with the desired shareholder centric perspective. They can also be interpreted as the amounts of reserve capital and risk margin lost by the bank shareholders as their property is transferred to the liquidator of the bank if the bank defaults. Due to these terminal cash flows the above definition is in fact a fixpoint system of equations. The split of the underlying CA equation
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
13
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
into the collection of equations
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
12
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
is motivated by both interpretation and numerical considerations. From an interpretation viewpoint it is useful to provide the more granular view on the costs of the bank provided by the split of the global CA amount between on the one hand bilateral and centrally cleared trading default risk components BCVA and CCVA and on the other hand bilateral and centrally cleared trading funding risk components BMVA and
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CMVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
for segregated initial margin whereas the FVA cost of funding variation margin is holistic in nature can only be apprehended at the level of the bank balancesheet as a whole via the feedback impact of CA
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
maxEC KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERCENT</span>
</mark>
into the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
FVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. From a numerical viewpoint the collection
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
12
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
of smaller problems may be easier to address than the global equation
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
13
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. Each of the smaller problems can also be handled by a dedicated desk of the bank namely the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
CVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
desk for the BCVA and CCVA and the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Treasury
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
of the bank for the BMVA CMVA and the
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
FVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
. Passing in the above equations to the bank survival measure Q based on Lemma
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
3.3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
shows that the corresponding fixed point problem is in fact wellposed and yields explicit formulas for all the quantities at hand. P4 All the above XVA numbers are nonnegative. Proof. By the result recalled after
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
11
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
EC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
is nonnegative as an expected shortfall under Q of the random variable JC F CA which is centered under Q and therefore under Q by
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
10
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
. The
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
first
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORDINAL</span>
</mark>
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
four
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
formulas in
<mark class="entity" style="background: #e4e7d2; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
14
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">CARDINAL</span>
</mark>
directly follow from
<mark class="entity" style="background: #ff8197; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Definition 3.1
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LAW</span>
</mark>
and
<mark class="entity" style="background: #ff9561; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
Lemma 3.3
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">LOC</span>
</mark>
which also implies that
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
E hEC KVA
<mark class="entity" style="background: #7aecec; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
hEC
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">ORG</span>
</mark>
KVA . As h is nonnegative this
<mark class="entity" style="background: #aa9cfc; padding: 0.45em 0.6em; margin: 0 0.25em; line-height: 1; border-radius: 0.35em;">
KVA
<span style="font-size: 0.8em; font-weight: bold; line-height: 1; border-radius: 0.35em; vertical-align: middle; margin-left: 0.5rem">PERSON</span>
</mark>