Topics from the xVAdesk · xVAwith Threshold and Independent Amount (cont.) Credit Valuation...

•xVA with Threshold and Independent Amount•Netting Intuition

Topics from the xVA desk

[email protected], Counterparty Credit & Funding Risk, Danske Bank, Markets

•Standalone/Incremental/Marginal xVA•Other challenges on the xVA desk

xVA with Threshold and Independent Amount

Threshold and Independent Amount

•If the ISDA (Master Agreement) is supported by a CSA (Credit Support Annex) the counterparty credit risk will be mitigated to a certain extend that depends on the specific details of the CSA.•The CSA stipulates that Collateral must be exchanged when the exposure of the derivatives portfolio covered exceeds a given Threshold, and when the difference between collateral exchanged and current exposure exceeds a given Minimum Transfer Amount.•In addition to the collateral exchanged to cover the exposure, an Independent Amountmay be exchanged, and sometimes delivered by both parties at the same time.•If the Threshold is zero (or very low), and Minimum Transfer Amount is very low, and

2Source: www.danskebank.com/CI

•If the Threshold is zero (or very low), and Minimum Transfer Amount is very low, and Collateral can be called for on a daily basis, the exposure is reduced significantly to be a matter of Close-Out risk (not covered further in this talk).•For any significant Threshold level the exposure below contributes to the CVA.•If an Independent Amount is received, only the exposure above contributes to the CVA.•Depending on the right to rehypothecate, the Independent Amounts delivered and received should be handled carefully for DVA and FCA.

xVA with Threshold and Independent Amount (cont.)

Threshold and Independent Amount

•With V(t) denoting the portfolio value at time t, and H the level of the threshold, the exposure driving CVA (and FCA) is given by:

•If in addition IA denotes the independent amount received, the exposure driving CVA is given by:

Graphically this may be expressed by:

)0)),(,max(min()( tVHtExposure

)0),)(,max(min()( IAtVIAHtExposure

Exposure Comparison, H=30, IA=10


Graphically this may be expressed by:

0

10

20

30

40

50

60

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50

Exposure Comparison, H=30, IA=10

max(V,0) max(min(H,V),0) max(min(H-IA,V-IA),0)


Credit Valuation Adjustment - recap of LS-MC approach

•CVA is defined as the following (ignoring recovery R for simplicity):

•If V can be computed in closed form or through a quick model we are done (but have to take the pain of deriving closed form expressions or implement quick models).

•We can do LS-MC on V to get a proxy (with the tilde), and evaluate CVA as:

T

dtttVECVA0

)()(

T

dtttVECVA )()(~


•This puts a high demand on the proxy, which needs to be very close for all states of the underlying variables, even in the extremes.

T

dtttVECVA0

)()(~


Credit Valuation Adjustment - recap of LS-MC approach

•To reduce the dependency on the proxy the following alternative CVA calculation is used:

])(1])([[

])(1)([

])()([

0 0)(~

0

proxyregression

0)(~

0

T

tV

T

tt

T

tV

T

dttduucEE

dtttVE

dtttVECVA


•We now only depend on the proxy close to zero.

])()(1[

])(1)([

0

notionalCVA

0 0)(~

0 0)(~

0 0)(

cashflowfuture

T u

tV

T

tV

T

t

tVtt

duucdttE

dudttucE


Credit Valuation Adjustment � with Threshold

•To take a threshold H into account we can modify the CVA slightly:

])())(

~,1min(1)([

])())(

,1min()([

])()0),),(max(min([

0 0)(~

0

0

T

tV

T

T

dtttV

HtVE

dtttV

HtVE

dttHtVECVA


•The dependence on the proxy is now stronger, but still most important around zero and around and above H!•Same trick can be applied to FCA and DVA, taking into account that Threshold may be different for counterparty and investor (us).

])())(

~,1min(1])([[

)(

0 0)(~

0

T

tV

T

tt dtt

tV

HduucEE

tV


Credit Valuation Adjustment � with Threshold and Independent Amount

•To take a threshold H and an independent amount IA received into account we can modify the CVA slightly more (assume H >> IA) :

])()0),~,~1max(min(1)([

])()0),)(

,)(

1max(min()([

])()0),,)(max(min([

0

0

T

T

T

dttIAHIA

tVE

dtttV

IAH

tV

IAtVE

dttIAHIAtVECVA


•The dependence on the proxy is now even stronger, but still most important around zero , around IA, and around and above H-IA!•If IA received may not be rehypothecated (used for funding) the exposure used for FCA is unchanged.

])()0),)(

~,)(

~1max(min(1])([[

])()0),)(

~,)(

~1max(min(1)([

0 0)(~

0 0)(~

T

tV

T

tt

tV

dtttV

IAH

tV

IAduucEE

dtttV

IAH

tV

IAtVE


Credit Valuation Adjustment � with Threshold and Independent Amount

•Graphically we can compare the impact on the CVA Notional:

20

30

40

50

60

Value Decomposition, H=30, IA=10

0.40

0.60

0.80

1.00

1.20

CVA Notional Comparison, H=30, IA=10


0

10

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50

IA max(min(H-IA,V-IA),0) Collateral

0.00

0.20

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50

CVA_Ntl CVA_Ntl(H) CVA_Ntl(H,IA)


Expected Positive/Negative Exposure � with Threshold and Independent Amount

•Example (i): 1B EUR 10Y IRS, Counterparty Pays Floating, Investor (us) Pays Fixed, IRS set ATM before XVA.

30,000,000

40,000,000

50,000,000

60,000,000

70,000,000

Exposure, Threshold = 50M EUR, IA = 10M EUR

30,000,000

40,000,000

50,000,000

60,000,000

70,000,000

Exposure, Threshold = Infinite, IA = 0


-40,000,000

-30,000,000

-20,000,000

-10,000,000

0

10,000,000

20,000,000

30,000,000

EPE ENE EE

-40,000,000

-30,000,000

-20,000,000

-10,000,000

0

10,000,000

20,000,000

30,000,000

EPE ENE EE


Expected Positive/Negative Exposure � with Threshold and Independent Amount

•Example (i): We can decompose the exposure into exposure captured by IA and collateral recevied/posted beyond the Threshold.

20,000,000

30,000,000

40,000,000

50,000,000

60,000,000

70,000,000

Exposure (stacked), Threshold = 50M EUR, IA = 10M EUR


-40,000,000

-30,000,000

-20,000,000

-10,000,000

0

10,000,000

20,000,000

Positive Collateral Positive Exposure Positive IA

Negative Collateral Negative Exposure Negative IA

Netting Intuition

Expected Positive Exposure (EPE) is an option on a portfolio

•Most quants have a developed intuition for what impacts the value of a call option on a single underlying. Often this intuition is derived from careful study of the Black-Scholesmodel and the background theory.•This intuition is less developed for options on the sum or difference of several underlyings, say basket or spread options, nor for best-of types of options.

•For analyzing Credit Valuation Adjustment and related xVAs, the ability to understand netting effects between different trades, or risk factors in a derivatives portfolio is crucial!•Most often this analysis is done on a before-and-after basis, hence the comparison is


•Most often this analysis is done on a before-and-after basis, hence the comparison is between the existing netting set and the netting set augmented with a new trade (alternatively reduced by a terminated trade). The resulting xVA impact is called the Incremental xVA.•When an xVA is decomposed into the contributions of different trades or risk factors we consider it a Marginal xVA.•Before analyzing how to compute these in the xVA framework, we consider some intuitive approaches for understanding the EPE and ENE impact.

Netting Intuition (cont.)


•In practice a typical OTC derivatives netting set may consist of 1000�s of trades with exposure across different asset classes and derivative types.•In a pricing situation it is close to impossible (within the time-frame given) to analyze the individual trades of the netting set, and their co-dependency structure in detail.•However, the derivative type and main risk factors of the single (or few) additional trade is known. Further analytics, such as exposure profiles (across time) for the netting set, the additional trade, and the augmented netting set allows heuristic reasoning for incremental xVA impacts.•Consider:


•Consider:

)()()()(

)()(

)()(

)()(

)()(

1

1

tENEtEPEtVEtEE

tVEtENE

tVEtEPE

tXtV

tXtV

NetNetNetNet

NetNet

NetNet

nNew

n

ii

Net

)()()(

)()()(

and

)()()(

tVtVtVEE

tVtVEtEPE

tEPEtEPEtEPE

NetNetNew

NetNewNetNew

NewNetNewNet



•Example (ii): Consider the following hypothetical exposure measures EPE, ENE, and EE observed for a single time point. What could be the possible reasons for the change from Net to (New+Net)?

25

50

75

100

25

50

75

100

25

50

75

100

25

50

75

100


-100

-75

-50

-25

0

Net New New+Net

EPE EE ENE

-100

-75

-50

-25

0

Net New New+Net

EPE EE ENE

-100

-75

-50

-25

0

Net New New+Net

EPE EE ENE

-100

-75

-50

-25

0

Net New New+Net

EPE EE ENE

Case A Case B Case C Case D


Expected Positive Exposure (EPE) with a Threshold behaves like a call spread.

•The expected positive exposure (EPE) in the presence of a Threshold H is given by:

•The last expression is identical to a so-called call spread on the value of the netting set with strikes 0 and H.

)0,)(max()0),(max(

)0)),(max(min();(

HtVtVE

HtVEHtEPENetNet

NetNet

40

60

EPE with Threshold H= 50


0 and H. •From the Black-Scholes analysis we know the behavior of call spreads (or digital options) around the two strikes. In particular their Vega/Gamma sensitivity becomes important in understanding changes in EPE (and ENE) driving Incremental xVA.

-80

-60

-40

-20

0

20

40

-150 -100 -50 0 50 100 150 200 250

max(min(V,H),0) Value Vega

Standalone/Incremental/Marginal xVA

Computing standalone, incremental, and marginal xVA in a single valuation.

•Let VNet(t) denote the value of the existing netting set (portfolio), and let VNew(t) denote the value of the new trade being priced. The corresponding cashflows are denoted cNet(t), and cNew (t),respectively.•Similarly, let CVANet and CVANew+Net denote the CVA of the existing netting set, and the augmented netting set, respectively.•We can then compute Incremental CVA by:

])(1)(1)()([

])()([])()([00

TNetNewNet

TNet

TNetNewNetNetNew

dtttVtVtVE

dtttVEdtttVECVACVA


])()(1[

])()(11[

])(1)(11)([

])(1)(1)()([

0

notionalCVA

0 0)(~

)(~

0

notionalCVA lIncrementa

0 0)(~

0)(~

)(~

0 0)()(0)(0)()(

0 0)(0)()(

TNew

u

tVtV

TNet

u

tVtVtV

T

tVtV

New

tVtVtV

Net

tV

Net

tVtV

NewNet

duucdttE

duucdttE

dtttVtVE

dtttVtVtVE

NewNet

NetNewNet

NewNetNetNewNet

NetNewNet

Standalone/Incremental/Marginal xVA (cont.)


•Adding back in the CVANet we can decompose CVANew+Net into Marginal CVA of Net and New denoted CVANet|Net+New , and CVANew|Net+New respectively:

NewNet

TNewNet

T

tVtV

NewNet

TNetNew

NetNewNewNetNewNetNetNew

dtttVtVE

dtttVtVE

dtttVE

CVACVACVA

])(1)(1)([

])(1)()([

])()([

0 0)()(

0

||


NetNewNew

NewNet

NetNewNet

NewNet

NewNetNewNet

CVA

TNew

u

tVtV

CVA

TNet

u

tVtV

tVtV

New

tVtV

Net

duucdttE

duucdttE

dtttVtVE

|

|

])()(1[

])()(1[

])(1)(1)([

0 0 0)(~

)(~

0 0 0)(~

)(~

0 0)()(0)()(



•Example (iii): Net = 100M EUR 10Y IRS, Counterparty Pays Floating, Investor (us) Pays Fixed, New = 200M EUR 5Y IRS, Counterparty Pays Fixed, Investor (us) Pays Floating, both IRSs set ATM before xVA. Assume CP CDS = 2.00% flat, Own CDS = Own Funding = 0.5%

-3,000,000

-2,000,000

-1,000,000

0

1,000,000

2,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

New = EUR 200M 5Y IRS, We pay fixed

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

Net = EUR 100M 10Y IRS, We receive fixed


-6,000,000

-5,000,000

-4,000,000

-3,000,000

Positive Negative Expected

-3,000,000

-2,000,000

-1,000,000

0

1,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00


Standalone xVA

xVA (s.e.)

CVA 66,174 851 EUR

DVA -80,461 325 EUR

FCA 17,732 227 EUR

xVA Total 3,445 EUR

Standalone xVA

xVA (s.e.)

CVA 769,079 4,471 EUR

DVA -51,311 1,491 EUR

FCA 219,683 1,226 EUR

xVA Total 937,451 EUR



•Example (iii): Computing Incremental Exposure and Incremental xVA.

-1,000,000

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Total Exposure (Net + New)

-4,000,000

-3,000,000

-2,000,000

-1,000,000

0

1,000,000

2,000,000

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Incremental Exposure (Net + New \ Net)


-2,000,000

Positive (Net+New) Negative (Net+New) Expected(Net+New)

-5,000,000

Positive(Net+New \ Net) Negative(Net+New \ Net) Expected(Net+New \ Net)

Incremental xVA

xVA (Net+New) (s.e.) xVA (Net+New \ Net) (s.e.)

CVA 500,476 3,012 EUR -268,604 2,832 EUR

DVA -41,337 1,000 EUR 9,974 802 EUR

FCA 147,068 847 EUR -72,614 750 EUR

xVA Total 606,207 -331,244 EUR



•Example (iii): Computing Marginal Exposure and Marginal xVA.

-4,000,000

-2,000,000

0

2,000,000

4,000,000

6,000,000

8,000,000

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Marginal Exposure (Net | Net + New), (New | Net + New)


-6,000,000

Positive(Net | Net+New) Negative(Net | Net+New) Expected(Net | Net+New)

Positive(New | Net+New) Negative(New | Net+New) Expected(New | Net+New)

Marginal xVA

xVA (Net | Net+New) (s.e.) xVA (New | Net+New) (s.e.)

CVA 696,453 4,260 EUR -195,977 2,197 EUR

DVA -32,058 1,453 EUR -9,279 639 EUR

FCA 200,430 1,181 EUR -53,361 591 EUR

xVA Total 864,825 -258,618 EUR



•Example (iv): Net=1B USD 5Y CCS vs. 736M EUR, Counterparty Pays EUR Euribor 3M �8.13bp, Investor (us) Pays USD Libor 3M. New = 1B USD 5Y IRS, Counterparty Pays USD Libor 3M Investor (us) pays Fixed 1.751%. Assume CP CDS = 2.00% flat, Own CDS = Own Funding = 0.5%. Notice that effectively, New+Net is a fixed-for-float CCS.

15,000,000

20,000,000

25,000,000

30,000,000

New = 1B USD 5Y IRS, Pay Fixed vs. Rec USD3M

20,000,000

40,000,000

60,000,000

80,000,000

Net = 1B USD 5Y CCS vs 736M EUR, Pay USD3M vs Rec EUR3M - 8.13bp


Standalone xVA

xVA (s.e.)

CVA 3,873,850 15,117 EUR

DVA -850,636 4,111 EUR

FCA 1,072,246 4,102 EUR

xVA Total 4,095,460 EUR

-10,000,000

-5,000,000

0

5,000,000

10,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00


-60,000,000

-40,000,000

-20,000,000

0

0.00 1.00 2.00 3.00 4.00 5.00 6.00


Standalone xVA

xVA (s.e.)

CVA 1,539,087 2,109 EUR

DVA -70,346 787 EUR

FCA 417,425 551 EUR




•Example (iv): New + Net, total and incremental Exposure and xVA

-20,000,000

0

20,000,000

40,000,000

60,000,000

80,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

New + Net

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

Incremental: New + Net \ Net


-60,000,000

-40,000,000


Total xVA

PV (s.e.)

CVA 4,605,043 11,767 EUR

DVA -736,914 4,177 EUR

FCA 1,270,623 3,226 EUR


Incremental xVA

PV

CVA 731,193 EUR

DVA 113,722 EUR

FCA 198,377 EUR


-5,000,000

0

0.00 1.00 2.00 3.00 4.00 5.00 6.00




•Example (v): Trade = 1B USD 5Y CCS vs. 736M EUR, Counterparty Pays EUR Euribor 3M - 8.13bp, Investor (us) Pays USD Libor 3M, Quarterly Reset of USD Notional(*)!•Consider replacing Net with Trade.

0

5,000,000

10,000,000

15,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

T = 1B USD 5Y CCS vs 736M EUR, Pay USD3M vs Rec EUR3M - 8.13bp,

Quarterly USD Reset

(*) The USD notional is set to the current spot at the beginning of each period and the change in notional relative to previous period is paid/received.


-15,000,000

-10,000,000

-5,000,000

0.00 1.00 2.00 3.00 4.00 5.00 6.00


Standalone xVA

PV (s.e.)

CVA 528,295 12,368 EUR

DVA -102,935 3,451 EUR

FCA 142,970 3,293 EUR

xVA Total 568,330 EUR

From an exposure point of view it has the same effect as receiving collateral every 3 months, or alternatively as considering the trade a string of 3M FX forwards.

Incremental xVA

PV

CVA -4,076,748 EUR

DVA 633,979 EUR

FCA -1,127,652 EUR

xVA Total -4,570,422 EUR

Other challenges on the xVA desk

Applying a diverse set of quantitative (and personal) skills.

•The xVA desk may have a mandate to cover both x = Credit, x = Debit, x = Funding, x = Collateral, and x = Capital.•The mandate may evolve dynamically, from pricing trades, to measuring and reporting risk of the derivative portfolio, to actively managing PnL through hedging activity.•It requires a combined set of skills in both Rates, FX, Inflation, Commodity, etc., and not least Credit.•The ability to interact constructively with Sales, Credit, Collateral Management, Legal, and other Trading desks is highly important!


•Other related challenges may come along:•Collateral optimization, how to post the collateral that is cheapest-to-deliver given the CCS market, the Repo market, and the LCR regulation.•CCP initial margin management, minimize the funding cost of posting IM across several CCPs, keeping Default Fund contributions in check (it is also a counterparty exposure).•Balance-sheet and Leverage Ratio management, through trade-compression, novations, back-loading to CCPs.

General disclaimer

This presentation has been prepared by Danske Bank Markets (a division of Danske Bank A/S). It is provided for informational purposes only and should be viewed solely in conjunction with the oral presentation provided by Danske Bank and/or Danske Markets Inc. It does not constitute or form part of, and shall under no circumstances be considered as, an offer to sell or a solicitation of an offer to purchase or sell any relevant financial instruments (i.e. financial instruments mentioned herein or other financial instruments of any purchase or sell any relevant financial instruments (i.e. financial instruments mentioned herein or other financial instruments of any issuer mentioned herein and/or options, warrants, rights or other interests with respect to any such financial instruments) (�Relevant Financial Instruments�)

The presentation has been prepared independently and solely on the basis of publicly available information which Danske Bank considers to be reliable. Whilst reasonable care has been taken to ensure that its contents are not untrue or misleading, no representation is made as to its accuracy or completeness, and Danske Bank, its affiliates and subsidiaries accept no liability whatsoever for any direct or consequential loss, including without limitation any loss of profits, arising from reliance on thispresentation

Danske Bank, its affiliates, subsidiaries and staff may perform services for or solicit business from any issuer mentioned herein and may hold long or short positions in, or otherwise be interested in, the financial instruments mentioned herein. The Equity and Corporate Bonds analysts of Danske Bank and undertakings with which the Equity and Corporate Bonds analysts have close links are, however, not permitted to invest in financial instruments which are covered by the relevant Equity or Corporate Bonds analyst or the research sector to which the analyst is linked

Danske Bank is authorized and subject to regulation by the Danish Financial Supervisory Authority and is subject to the rules and regulation of the relevant regulators in all other jurisdictions where it conducts business. Danske Bank is subject to limited regulation by

24

regulation of the relevant regulators in all other jurisdictions where it conducts business. Danske Bank is subject to limited regulation by the Financial Services Authority (UK). Details on the extent of the regulation by the Financial Services Authority are available from Danske Bank upon request

This presentation is not intended for retail customers in the United Kingdom or the United States This presentation is protected by copyright and is intended solely for the designated addressee. It may not be reproduced or distributed,

in whole or in part, by any recipient for any purpose without Danske Bank�s prior written consent

Source: www.danskebank.com/CI

Disclaimer related to presentations to U.S. customers

In the United States this presentation is presented by Danske Bank and/or Danske Markets Inc., a U.S. registered broker-dealer and subsidiary of Danske BankInc., a U.S. registered broker-dealer and subsidiary of Danske Bank

In the United States the presentation is intended solely to �U.S. institutional investors� as defined in sec rule 15a�6

Any U.S. investor recipient of this presentation who wishes to purchase or sell any Relevant Financial Instrument may do so only by contacting Danske Markets Inc. directly and should be aware that investing in non-U.S. financial instruments may entail certain risks. Financial instruments of non-U.S. issuers may not be registered with the U.S. Securities and Exchange Commission and may not be subject to the reporting and auditing standards of the U.S. Securities and Exchange Commission


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Topics from the xVAdesk · xVAwith Threshold and Independent Amount (cont.) Credit Valuation...

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