Derivatives risks as costs in a one-period network model

Dorinel Bastide; Stéphane Crépey; Samuel Drapeau; Mekonnen Tadese

doi:10.3934/fmf.2023014

Article Contents

2023, Volume 2, Issue 3: 283-312. Doi: 10.3934/fmf.2023014 open access

This issue Previous Article Exploiting arbitrage requires short selling Next Article Optionality as a binary operation

Derivatives risks as costs in a one-period network model

1.
BNP Paribas Stress Testing Methodologies & Models, France
2.
LaMME, Université d'Evry/Université Paris-Saclay CNRS UMR 8071, France
3.
LPSM, Sorbonne Université and Université Paris Cité, CNRS UMR 8001, France
4.
Shanghai Jiao Tong University, Shanghai, China
5.
Woldia University, Mathematics Department, Ethiopia

^*Corresponding author: Stéphane Crépey
^*Corresponding author: Stéphane Crépey

Received: July 2022

Revised: November 2022

Early access: April 4, 2023

Published online: April 4, 2023

This article is not meant to represent the position or opinions of BNP Paribas or its members..

Abstract / Introduction Full Text(HTML) Figure(8) / Table(12) Related Papers Cited by

Abstract

In counterparty credit risk complete markets, collateral and capital requirements would be indifferent to banks. The quantification by banks of market incompleteness based on various XVA metrics ([11]) has emerged as the unintended consequence of the FRTB banking reform ([26]) and of the more demanding regulatory capital requirements ([38]). The related risks are in fact reckoned today as the major risks for banks, well ahead market risk ([35, Figure 65 page 67]). The XVA metrics have been introduced and traditionally used by investment banks for pricing and collateral/capital optimization purposes. We demonstrate in this paper that they can be fruitfully used for risk management, suggesting a sound approach to regulatory requirements. We present a one-period cost-of-capital XVA setup encompassing bilateral and centrally cleared trading in a unified framework, with explicit formulas for most quantities at hand. We illustrate possible uses of this framework for running stress test exercises on financial networks with one and two clearinghouses from a clearing member's perspective or for optimizing the porting of the portfolio of a defaulted clearing member using Monte-Carlo technique with corresponding confidence errors in elliptical models. A continuous-time extension of this approach is provided in the companion paper [7].

Keywords:

Mathematics Subject Classification: Primary: 91-10, 91B05, 91G15, 91G45; Secondary: 62P05, 91G70.

Citation:

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Figure 1. Network consisting of two CCPs (in red), 123 members for CCP1 seen on the left hand side, and 56 members for CCP2 on the right hand side, with 24 common members displayed as the group of members in the middle of the two CCPs (155 members in total, in blue), and with 179 cleared clients (in green)

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Figure 2. Promised cash flows between market participants. The reference clearing member bank is on the left

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Figure 3. Financial network composed of 1 CCP, its 20 members (labeled by B) and one client per member

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Figure 4. CCVA and KVA w.r.t. credit factors correlation and credit and portfolio variation factors correlation

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Figure 5. CCVA and KVA w.r.t. market factors correlation and credit and portfolio variation factors correlation

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Figure 6. Decreasing absolute $ {\rm{nom}}_i$ per member for CCP1

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Figure 7. Decreasing absolute $ {\rm{nom}}_i$ per member for CCP2

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Figure 8. The 1-CCP, former 20-member financial network with 19 members post CM0 default. Defaulted CM0, labeled "B0" in the presented network, is represented as pale dashed node with pale dashed links to reflect former exposures to its client and toward the CCP. The optimal porting of CM0 portfolio with CM1, labeled "B1", is outlined with bold links to reflect the new exposures for CM1

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Table 1. XVA definitions, cf. Section 2.2 (with $\mathcal{C}, $ $\mathcal{F}$ and ${\mathcal{L}}$ given by Lemma 3.5)

XVA	Expression	Full name and description
${\rm KVA}$	$ \begin{array}{c} {\mathbb{E}^\star}\Big( J h ({\rm EC}-\mathrm{KVA})^+ +(1-J){\rm KVA} \Big), \\ \mbox{ where }{\rm EC}= {\mathbb{ES}}\big(J( \mathcal{C}+\mathcal{F}-{\rm CA})\big) \end{array}$	capital valuation adjustment
${\rm CA}$	${\rm CVA}+{\rm MVA}+{\rm FVA}$	contra-asset valuation
${\rm CVA}$	${\rm BCVA}+{\rm CCVA}$	credit valuation adjustment
${\rm BCVA}$	$\begin{array}{c}{\mathbb{E}^\star}\Bigg( J\displaystyle\sum_b (1-J_b)(\mathcal{P}_b-{\rm VM}_b-{\rm IM}_b )^+ \\ + (1-J){\rm BCVA} \Bigg)\end{array}$	credit valuation adjustment for bilateral exposures
${\rm CCVA}$	$\begin{array}{c}{\mathbb{E}^\star}\Bigg(J\displaystyle\sum_c (1-J_c )(\mathcal{P}_c - {\rm MtM}_c -{\rm IM}_c )^+\\ + \mu{\mathcal{L}} +(1-J){\rm CCVA} \Bigg)\end{array}$	credit valuation adjustment for clearing activity exposures
${\rm MVA}$	${\rm BMVA}+{\rm CMVA}$	margin valuation adjustment
${\rm BMVA}$	${\mathbb{E}^\star}\left( J \widetilde{\gamma}\displaystyle\sum_b \overline{\rm IM}_b +(1-J){\rm BMVA} \right)$	margin valuation adjustment for bilateral exposures
${\rm CMVA}$	${\mathbb{E}^\star}\Big( J\widetilde{\gamma}\big( {\rm IM}+ \overline{\rm IM} + \rm{DF} \big) +(1-J){\rm CMVA} \Big)$	margin valuation adjustment for clearing activity exposures
${\rm FVA}$	$\begin{array}{c}{\mathbb{E}^\star}\Bigg( J\gamma\Big(\displaystyle\sum_b ({\rm MtM}_b-{\rm VM}_b) -{\rm CA}\\ - \max({\rm EC}, {\rm KVA}) \Big)^+ +(1-J){\rm FVA}\Bigg)\end{array}$	funding valuation adjustment

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Table 2. XVA explicit formulas (with $\mathcal{C}, $ $\mathcal{F}$ and ${\mathcal{L}}$ given by Lemma 3.5)

XVA	Explicit formula
${\rm CCVA}$	${\mathbb{E}} \left( \displaystyle\sum_c (1- J_c )(\mathcal{P}_c - {\rm MtM}_c - {\rm IM}_c )^+ +\mu{\mathcal{L}} \right)$
${\rm CMVA}$	$\widetilde{\gamma}\big({\rm IM} +\overline{\rm IM} + \rm{DF} \big)$
${\rm BCVA}$	${\mathbb{E}} \left( \displaystyle\sum_b (1-J_b)(\mathcal{P}_b-{\rm VM}_b-{\rm IM}_b )^+ \right)$
${\rm BMVA}$	$\widetilde{\gamma}\displaystyle\sum_b \overline{\rm IM}_b$
${\rm EC}$	${\mathbb{ES}}\big(J( \mathcal{C} - {\rm CVA} )\big)$
${\rm FVA}$	$\displaystyle\frac{\gamma}{1+\gamma} \left(\displaystyle\sum_b ({\rm MtM}_b-{\rm VM}_b) -({\rm CCVA}+{\rm CMVA}+{\rm BCVA} +{\rm BMVA})- {\rm EC} \right)^+$
${\rm KVA}$	$\displaystyle\frac{h}{1+h} {\rm EC}$

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Table 3. CCVA and CMVA definitions with several CCPs (also, $\mathcal{C} $ and $\mathcal{F}$ are now given by Proposition 3.9, as also ${\mathcal{L}}^{ccp}$)

XVA	Expression	Full name and description
${\rm CCVA}$	$\begin{array}{c}{\mathbb{E}^\star}\Bigg(J\displaystyle\sum_{ccp, c}(1-J_c)(\mathcal{P}^{ccp}_c - {\rm MtM}^{ccp}_c-{\rm IM}^{ccp}_c)^+\\+ \displaystyle\sum_{ccp}\mu^{ccp} {\mathcal{L}} ^{ccp} +(1-J){\rm CCVA} \Bigg)\end{array}$	credit valuation adjustment for clearing activity exposures
${\rm CMVA}$	${\mathbb{E}^\star}\left( J\displaystyle\sum_{ccp} \widetilde{\gamma}\big({\rm IM}^{ccp}+\overline{\rm IM}^{ccp} + \rm{DF}^{ccp}\big)+(1-J){\rm CMVA} \right)$	margin valuation adjustment for clearing activity exposures

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Table 4. CCVA and CMVA explicit formulas with several CCPs (also, $\mathcal{C} $ and $\mathcal{F}$ are now given by Proposition 3.9, as also ${\mathcal{L}}^{ccp}$)

XVA	Explicit formula
${\rm CCVA}$	${\mathbb{E}} \left( \displaystyle\sum_{ccp, c} (1- J_c )(\mathcal{P}^{ccp}_c - {\rm MtM}^{ccp}_c -{\rm IM}^{ccp}_c )^+ +\sum_{ccp}\mu^{ccp} {\mathcal{L}} ^{ccp} \right)$
${\rm CMVA}$	$\displaystyle\sum_{ccp} \widetilde{\gamma}\big({\rm IM}^{ccp}+\overline{\rm IM}^{ccp} + \rm{DF}^{ccp}\big)$

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Table 5. Member characteristics and portfolio parameters, ordered by decreasing member size

cm id	0	1	2	3	4	5	6	7	8	9
DP (bps)	50	60	70	80	90	200	190	180	170	160
size	-242	184	139	105	-80	-61	-46	35	26	-20
vol (%)	20	21	22	23	24	25	26	27	28	29
cm id	10	11	12	13	14	15	16	17	18	19
DP (bps)	150	140	130	120	110	100	90	80	70	60
size	-15	-11	-9	-6	5	-4	-3	2	2	-1
vol (%)	30	31	32	33	34	35	36	37	38	39

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Table 6. XVAs calculation configuration

One-period length $T$	5 years
Liquidation period at default $\Delta_l$	5 days
Portfolio variations correlation $\rho^{cr}$	30%
Credit factors correlation $\rho^{mkt}$	20%
Correlation between credit factors and portfolio variations ${\rho^{wwr}}$	20%
IM covering period (MPoR) $\Delta_s$	2 days
IM quantile level	95%
Funding blending ratio $\widetilde{\gamma}/\gamma$	25%
SLOIM calculation²⁹ for DF Cover-2	VaR 97%
Funded DF allocation rule	$\propto$ SLOIM
${\mathcal{L}}$ allocation rule ($\ni\mu$)	$\propto$ $DF_i$
Quantile level used for clearing members EC calculation	99.75%
Hurdle rate $h$ used for KVA computations	10.0%
Number of Monte-Carlo simulation (for CCVA and KVA computations)	10M
Number of batches (for KVA computations)	100

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Table 7. Initial XVA costs: estimates, [value-at-risk underlying the KVA estimate] and (95% confidence level errors)

cm id	CMVA	CCVA	KVA (99%)	KVA (99.75%)
0	0.0687	0.0778 (0.3%)	0.2734 [0.1396] (0.6%)	0.5138 [0.2923] (1%)
1	0.0656	0.0805 (0.4%)	0.3407 [0.1422] (0.7%)	0.7022 [0.3806] (1.1%)
2	0.0604	0.0635 (0.4%)	0.2725 [0.1132] (0.7%)	0.5624 [0.3039] (1.1%)
3	0.0544	0.0503 (0.5%)	0.2191 [0.0903] (0.9%)	0.4549 [0.2439] (1.5%)
4	0.0485	0.0356 (0.5%)	0.1654 [0.0625] (0.9%)	0.3565 [0.185] (1.2%)
5	0.0834	0.0252 (0.5%)	0.1339 [0.0507] (0.8%)	0.2856 [0.1513] (1.2%)
6	0.0623	0.021 (0.5%)	0.1085 [0.0429] (0.8%)	0.2284 [0.1211] (1.3%)
7	0.0467	0.0187 (0.5%)	0.0883 [0.0365] (0.8%)	0.1836 [0.098] (1.1%)
8	0.0341	0.0146 (0.5%)	0.0685 [0.0282] (0.9%)	0.1432 [0.0755] (1.5%)
9	0.0256	0.0113 (0.5%)	0.0549 [0.0223] (1%)	0.1157 [0.06] (1.6%)
10	0.0187	0.009 (0.5%)	0.0429 [0.0173] (1%)	0.0908 [0.0467] (1.7%)
11	0.0132	0.007 (0.7%)	0.0328 [0.0132] (1.3%)	0.0701 [0.0354] (2.2%)
12	0.0104	0.006 (0.6%)	0.0279 [0.0111] (1.3%)	0.0598 [0.0299] (2.3%)
13	0.0066	0.0042 (0.9%)	0.0198 [0.0077] (1.9%)	0.0438 [0.0206] (3.3%)
14	0.0052	0.0037 (0.9%)	0.0174 [0.0066] (1.8%)	0.0389 [0.0177] (3.1%)
15	0.0039	0.0032 (1.3%)	0.0151 [0.0054] (2.5%)	0.0351 [0.0146] (4.3%)
16	0.0027	0.0025 (1.4%)	0.012 [0.0042] (2.7%)	0.0285 [0.0113] (4.4%)
17	0.0017	0.0018 (2%)	0.0088 [0.0029] (4%)	0.0218 [0.0078] (6.4%)
18	0.0015	0.0019 (2%)	0.0093 [0.0029] (4.1%)	0.0233 [0.008] (6.5%)
19	0.0007	0.0011 (3.7%)	0.006 [0.0015] (6.9%)	0.017 [0.0041] (9.7%)

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Table 8. Stress test (ST) extreme quantile, $1.5\times$ ST extreme quantile and RST probability to breach 1.5 times the $99.9^{th}$ quantile loss level, for each member, based on $10$M simulations (in parentheses: corresponding 95% confidence intervals)

cm id	$99.9\%$	$1.5\times\, \, 99.9\%$	RST scenario probability
0	4.9022 (-0.9%, 0.9%)	7.3534	0.0387% (5%)
1	6.7852 (-0.9%, 1%)	10.1778	0.0428% (4.4%)
2	5.4194 (-1%, 1%)	8.1291	0.0435% (4.4%)
3	4.3402 (-0.9%, 0.9%)	6.5103	0.0437% (4.3%)
4	3.3843 (-0.9%, 1%)	5.0764	0.0452% (4.5%)
5	2.7516 (-0.9%, 1%)	4.1274	0.044% (4.6%)
6	2.1804 (-1%, 0.9%)	3.2706	0.0437% (3.9%)
7	1.7481 (-1%, 1%)	2.6221	0.0434% (4.2%)
8	1.3452 (-1%, 1.2%)	2.0178	0.0435% (4.1%)
9	1.0714 (-1%, 1%)	1.6071	0.0433% (4.2%)
10	0.8362 (-1%, 1.1%)	1.2544	0.0429% (4.4%)
11	0.6325 (-0.9%, 1%)	0.9487	0.0427% (4.4%)
12	0.5330 (-1%, 0.9%)	0.7995	0.0431% (4.2%)
13	0.3688 (-0.9%, 0.9%)	0.5533	0.0428% (4.6%)
14	0.3186 (-1%, 1%)	0.4779	0.0429% (4.4%)
15	0.2626 (-0.9%, 1.1%)	0.394	0.0427% (4.3%)
16	0.2037 (-1.1%, 1.2%)	0.3055	0.0427% (4.4%)
17	0.1403 (-1.1%, 1%)	0.2105	0.0434% (4%)
18	0.1458 (-1%, 1%)	0.2187	0.0427% (4.1%)
19	0.0753 (-1%, 0.9%)	0.113	0.0431% (4%)

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Table 9. Economic Capital 20 worst scenarios details for member 1 in decreasing order of total loss where column with header $\mu$ indicates allocated coefficient loss to member 1 and $n$ is the number of defaults within the scenario

Rank	Loss	n	$\mu$	Defaulters	Losses triggered by defaulters
1	17.22	2	0.23	cm0, 7	842.25, 0
2	12.68	6	0.31	cm0, 2, 5, 9, 11, 14	300.22, 0, 92.28, 36.03, 20.83, 0
3	12.16	5	0.29	cm0, 2, 5, 14, 15	335.56, 0,112.91, 0, 9.67
4	11.82	7	0.33	cm0, 3, 5, 7, 8, 9, 14	394.96, 0, 0, 0, 0, 0.65, 0
5	11.05	5	0.26	cm0, 5, 6, 10, 15	465.41, 0, 0.59, 0, 0.05
6	10.99	1	0.21	cm0	566.93
7	10.74	1	0.19	cm2	608.58
8	9.23	2	0.23	cm0, 7	451.35, 0
9	9.1	1	0.21	cm0	469.81
10	8.83	5	0.31	cm0, 2, 5, 8, 12	300.42, 14.38, 0, 0, 0
11	8.57	3	0.23	cm0, 6, 16	408.40, 1.88, 0
12	8.41	3	0.22	cm0, 16, 17	429.53, 0, 0
13	8.22	12	0.51	cm0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17	81.53, 0, 32.12, 22.44, 17.66, 0, 0, 9.95, 6.19, 4.84, 2.63, 1.51, 0
14	8.09	2	0.23	cm0, 7	395.87, 0
15	7.87	1	0.21	cm0	406
16	7.86	8	0.31	cm0, 4, 6, 7, 8, 9, 12, 14	275.27, 0, 0, 0, 0, 0, 0, 0
17	7.83	2	0.22	cm0, 9	391.07, 0
18	7.49	12	0.55	cm0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18	51.85, 0, 27.98, 25.76, 20.82, 0, 0, 8.41, 4.26, 5.33, 3.21, 1.62, 0
19	6.85	7	0.36	cm0, 2, 3, 7, 8, 12, 17	0, 84.82, 70.26, 29.05, 20.74, 0, 2.06
20	6.7	3	0.22	cm0, 10, 11	330.03, 0.39, 0.34

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Table 10. Quantile loss levels (confidence errors) for $90\%$ and $99.9\%$ confidence levels across members for the example with 2 CCPs and 155 members including 24 common members. Legend for column headers: Ⅰ. Member Id, Ⅱ. DP (%), Ⅲ. Size on CCP1, Ⅳ. Volatility on CCP1, Ⅴ. Size on CCP2, Ⅵ. Volatility on CCP2, Ⅶ. $90^{th}$ Perc. stand-alone, Ⅷ. $90^{th}$ Perc., Ⅸ. $99.9^{th}$ Perc. stand-alone, Ⅹ. $99.9^{th}$ Perc

Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ	Ⅵ	Ⅶ	Ⅷ	Ⅸ	Ⅹ
3	0.1	19.9	21	-97.48	23	0.0755 (-1.5%, 1.5%)	0.0794 (-1.1%, 1.1%)	4.0067 (-1.8%, 1.7%)	3.6185 (-1.8%, 1.8%)
4	0.1	80.79	24	-18.79	22	0.0687 (-1.5%, 1.4%)	0.0739 (-0.9%, 1%)	3.4324 (-2%, 2%)	3.0625 (-1.5%, 1.9%)
9	3.1	-31.58	29	17.74	23	0.0478 (-1.3%, 1.3%)	0.0624 (-0.7%, 0.7%)	1.9749 (-1.8%, 2.3%)	1.6109 (-2%, 2.1%)
12	0.1	17.97	21	-16.75	24	0.0351 (-0.9%, 0.9%)	0.0423 (-0.6%, 0.6%)	1.0592 (-1.8%, 1.9%)	0.8008 (-2%, 2.3%)
13	0.1	-14.9	22	15.81	25	0.0337 (-0.9%, 0.9%)	0.0403 (-0.6%, 0.6%)	0.9738 (-1.8%, 1.9%)	0.7319 (-2%, 2.2%)
14	0.2	12.34	23	-14.93	26	0.0323 (-0.9%, 0.9%)	0.0385 (-0.6%, 0.6%)	0.8805 (-1.7%, 1.9%)	0.6518 (-1.5%, 1.5%)
15	0.1	-10.23	24	14.09	27	0.0311 (-0.8%, 0.9%)	0.0364 (-0.6%, 0.6%)	0.8245 (-1.8%, 1.8%)	0.6175 (-2.1%, 1.8%)
17	0.3	-7.03	26	-13.3	28	0.0295 (-0.8%, 0.8%)	0.0337 (-0.5%, 0.6%)	0.6991 (-1.8%, 1.8%)	0.5259 (-2.1%, 1.7%)
19	0.2	-4.83	28	12.56	29	0.0278 (-0.7%, 0.8%)	0.0309 (-0.5%, 0.5%)	0.622 (-1.8%, 1.8%)	0.4808 (-1.8%, 2%)
22	3.9	2.75	20	-11.86	30	0.0301 (-0.7%, 0.7%)	0.0315 (-0.6%, 0.6%)	0.524 (-1.9%, 2%)	0.4439 (-1.6%, 2.2%)
26	0.1	1.3	24	11.2	20	0.0159 (-0.7%, 0.7%)	0.0164 (-0.6%, 0.6%)	0.3061 (-1.8%, 1.8%)	0.2695 (-2%, 2.1%)
27	0.1	1.07	25	-10.57	21	0.0157 (-0.7%, 0.7%)	0.016 (-0.7%, 0.6%)	0.2958 (-1.7%, 1.7%)	0.2649 (-2%, 2%)
28	1.5	0.89	26	9.98	22	0.0163 (-0.7%, 0.7%)	0.0166 (-0.7%, 0.6%)	0.2831 (-1.8%, 2.1%)	0.2533 (-2.1%, 2%)
31	0.1	-0.51	29	-9.42	23	0.0151 (-0.7%, 0.7%)	0.0151 (-0.6%, 0.7%)	0.2723 (-1.7%, 1.8%)	0.2547 (-1.9%, 1.8%)
34	0.1	0.29	21	8.89	24	0.0147 (-0.6%, 0.7%)	0.0147 (-0.6%, 0.6%)	0.2539 (-1.7%, 1.8%)	0.2469 (-2%, 1.8%)
35	0.1	-0.24	22	-8.4	25	0.0145 (-0.6%, 0.7%)	0.0144 (-0.6%, 0.6%)	0.2482 (-1.8%, 1.7%)	0.242 (-2%, 1.8%)
36	0.1	0.2	23	7.93	26	0.0142 (-0.6%, 0.6%)	0.0141 (-0.7%, 0.6%)	0.2427 (-1.8%, 1.6%)	0.2375 (-2%, 1.8%)
39	0.1	-0.11	26	-7.48	27	0.0138 (-0.6%, 0.6%)	0.0138 (-0.6%, 0.6%)	0.2361 (-1.8%, 1.7%)	0.2328 (-1.8%, 1.7%)
40	0.5	0.09	27	7.07	28	0.0138 (-0.7%, 0.6%)	0.0138 (-0.7%, 0.7%)	0.225 (-2%, 1.6%)	0.2218 (-2%, 1.4%)
44	0.1	0.04	20	-6.67	29	0.0132 (-0.6%, 0.6%)	0.0131 (-0.6%, 0.6%)	0.2224 (-1.8%, 1.7%)	0.2217 (-1.8%, 1.7%)
49	0.1	-0.02	25	6.3	30	0.0129 (-0.6%, 0.6%)	0.0129 (-0.6%, 0.7%)	0.2168 (-1.8%, 1.7%)	0.2163 (-1.8%, 1.7%)
50	0.1	0.01	26	-5.95	20	0.0081 (-0.6%, 0.6%)	0.0081 (-0.6%, 0.6%)	0.1366 (-1.7%, 1.7%)	0.1364 (-1.7%, 1.7%)
51	0.1	-0.01	27	5.61	21	0.008 (-0.6%, 0.6%)	0.008 (-0.6%, 0.6%)	0.1355 (-1.8%, 1.7%)	0.1352 (-1.8%, 1.7%)
55	0.1	-0.01	20	-5.3	22	0.008 (-0.6%, 0.6%)	0.008 (-0.6%, 0.6%)	0.134 (-1.7%, 1.7%)	0.1338 (-1.8%, 1.8%)

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Table 11. Total $\Delta{\rm XVA}*$ aggregated over survivors corresponding to the different surviving ${\rm CM}*$, i.e. for $*$ other than 0, assuming an instant default of CM0 at time 0. In parenthesis, the contributions to $\Delta{\rm XVA}*$ of ${\rm CM}*$ itself

Surv. member $*$	Total $\Delta{\rm CMVA}*$	Total $\Delta{\rm CCVA}* $	Total $\Delta{\rm KVA}*$	Total ${\rm FTP}*$
1	0.0768 (0.0295)	-0.0511 (-0.0038)	-0.9182 (-0.1709)	-0.8926 (-0.1452)
2	0.0921 (0.0428)	-0.047 (0.0028)	-0.8262 (-0.0866)	-0.7811 (-0.0411)
3	0.1054 (0.0576)	-0.0394 (0.0088)	-0.7307 (-0.0185)	-0.6647 (0.0479)
19	0.1298 (0.0818)	-0.0253 (0.039)	-0.6785 (0.2818)	-0.574 (0.4026)
18	0.1417 (0.0939)	-0.0192 (0.0379)	-0.5995 (0.2693)	-0.477 (0.401)
17	0.1549 (0.107)	-0.0138 (0.0377)	-0.5547 (0.2702)	-0.4137 (0.4149)
16	0.1688 (0.1208)	-0.0088 (0.037)	-0.4435 (0.2665)	-0.2835 (0.4243)
15	0.1814 (0.1334)	-0.0032 (0.0363)	-0.3841 (0.2622)	-0.2059 (0.4319)
4	0.1525 (0.1022)	-0.0364 (0.0159)	-0.2671 (0.0427)	-0.151 (0.1608)
14	0.1903 (0.1426)	0.0035 (0.0353)	-0.3146 (0.253)	-0.1208 (0.4309)
13	0.2061 (0.1582)	0.008 (0.035)	-0.2108 (0.2583)	0.0033 (0.4515)
12	0.2171 (0.1692)	0.0101 (0.0335)	-0.1767 (0.2458)	0.0506 (0.4485)
8	0.234 (0.1881)	0.02 (0.0264)	-0.1305 (0.1778)	0.1235 (0.3924)
11	0.2285 (0.1807)	0.0147 (0.0326)	-0.101 (0.2415)	0.1422 (0.4548)
7	0.2327 (0.1876)	0.02 (0.0235)	-0.0949 (0.1501)	0.1578 (0.3612)
10	0.2385 (0.1908)	0.0188 (0.0311)	-0.0259 (0.2316)	0.2314 (0.4535)
9	0.2478 (0.2003)	0.0205 (0.029)	0.0442 (0.2158)	0.3125 (0.4451)
6	0.2687 (0.2225)	0.0215 (0.0205)	0.2937 (0.1399)	0.5839 (0.3828)
5	0.2728 (0.2274)	0.0189 (0.0163)	0.3527 (0.0922)	0.6444 (0.3359)

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Table 12. Standard deviation across surviving members $*$ of the $\Delta{\rm XVA}*$ for the example with 1 CCP and 20 members, assuming an instant default of CM0 at time 0

$\Delta {\rm CMVA}$	$\Delta{\rm CCVA} $	$\Delta {\rm KVA}$
0.0593	0.0251	0.3557

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