Topics from the xVA desk

•xVA with Threshold and Independent Amount •Netting Intuition •Standalone/Incremental/Marginal xVA •Other challenges on the xVA desk

[email protected], Counterparty & Funding Risk, Danske , Markets 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 Amount may 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 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.

Source: www.danskebank.com/CI 2 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: Exposure(t)  max(min(H,V (t)),0)

•If in addition IA denotes the independent amount received, the exposure driving CVA is given by: Exposure(t)  max(min(H  IA,V (t)  IA),0)

Graphically this may be expressed by: Exposure Comparison, H=30, IA=10 60

50

40

30

20

10

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

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

Source: www.danskebank.com/CI 3 xVA with Threshold and Independent Amount (cont.)

Credit Valuation Adjustment - recap of LS-MC approach

•CVA is defined as the following (ignoring recovery R for simplicity): T CVA  E V (t)  (  t)dt 0 

•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 ~ CVA  E V (t)  (  t)dt 0 

•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.

Source: www.danskebank.com/CI 4 xVA with Threshold and Independent Amount (cont.)

Credit Valuation Adjustment - recap of LS-MC approach

•To reduce the dependency on the proxy the following alternative CVA calculation is used: T CVA E[ V (t)  (  t)dt] 0 T  E[ V (t) 1~  (  t)dt] 0 V(t)0 regression proxy T T  E[ Et [ c(u) du]1~  (  t)dt] 0 t  V (t)0 future cashflow T T  E[ c(u)1~  (  t)dudt] 0 t V (t)0 T  u   E[  1~  (  t)dt c(u)du] 0 0 V (t)0  CVA notional •We now only depend on the proxy close to zero.

Source: www.danskebank.com/CI 5 xVA with Threshold and Independent Amount (cont.)

Credit Valuation Adjustment – with Threshold

•To take a threshold H into account we can modify the CVA slightly: T CVA E[ max(min(V (t), H ),0) (  t)dt] 0 T H  E[ V (t) min(1, ) (  t)dt] 0 V (t) T H  E[ V (t)1~ min(1, ~ ) (  t)dt] 0 V (t)0 V (t) T T H  E[ Et [ c(u)du]1~ min(1, ~ ) (  t)dt] 0 t V (t)0 V (t)

•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 (us).

Source: www.danskebank.com/CI 6 xVA with Threshold and Independent Amount (cont.)

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) : T CVA E[ max(min(V (t)  IA, H  IA),0) (  t)dt] 0 T IA H  IA  E[ V (t) max(min(1 , ),0) (  t)dt] 0 V (t) V (t) T IA H  IA  E[ V (t)1~ max(min(1 ~ , ~ ),0) (  t)dt] 0 V (t)0 V (t) V (t) T T IA H  IA  E[ Et [ c(u)du]1~ max(min(1 ~ , ~ ),0) (  t)dt] 0 t V (t)0 V (t) V (t) •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

Source:unchanged. www.danskebank.com/CI 7 xVA with Threshold and Independent Amount (cont.)

Credit Valuation Adjustment – with Threshold and Independent Amount

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

Value Decomposition, H=30, IA=10 CVA Notional Comparison, H=30, IA=10 60 1.20

50 1.00

40 0.80

30 0.60

20 0.40

10 0.20

0 0.00 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 -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 CVA_Ntl CVA_Ntl(H) CVA_Ntl(H,IA)

Source: www.danskebank.com/CI 8 xVA with Threshold and Independent Amount (cont.)

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.

Exposure, Threshold = Infinite, IA = 0 Exposure, Threshold = 50M EUR, IA = 10M EUR 70,000,000 70,000,000 60,000,000 60,000,000 50,000,000 50,000,000 40,000,000 40,000,000 30,000,000 30,000,000 20,000,000 20,000,000 10,000,000 10,000,000 0 0 -10,000,000 -10,000,000 -20,000,000 -20,000,000 -30,000,000 -30,000,000 -40,000,000 -40,000,000

EPE ENE EE EPE ENE EE

Source: www.danskebank.com/CI 9 xVA with Threshold and Independent Amount (cont.)

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.

Exposure (stacked), Threshold = 50M EUR, IA = 10M EUR 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000,000 0 -10,000,000 -20,000,000 -30,000,000 -40,000,000

Positive Collateral Positive Exposure Positive IA Negative Collateral Negative Exposure Negative IA

Source: www.danskebank.com/CI 10 Netting Intuition

Expected Positive Exposure (EPE) is an on a portfolio

•Most quants have a developed intuition for what impacts the value of a on a single underlying. Often this intuition is derived from careful study of the Black-Scholes model 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 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.

Source: www.danskebank.com/CI 11 Netting Intuition (cont.)

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

•In practice a typical OTC derivatives netting set may consist of 1000’s of trades with exposure across different asset classes and 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: n Net NetNew Net New V (t)   X i (t) EPE (t)  EPE (t)  EPE (t) i1 and V New (t)  X (t) n1  EPE NewNet (t)  E V New (t) V Net (t) EPE Net (t)  EV Net (t)      Net Net   New Net Net  ENE (t)  EV (t)   E EV (t) V (t) V (t)    EE Net (t)  EV Net (t) EPE Net (t)  ENE Net (t)

Source: www.danskebank.com/CI 12 Netting Intuition (cont.)

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

•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)?

100 100 100 100

75 75 75 75

50 50 50 50

25 25 25 25

0 0 0 0 Net New New+Net Net New New+Net Net New New+Net Net New New+Net -25 -25 -25 -25

-50 -50 -50 -50

-75 -75 -75 -75

-100 -100 -100 -100

EPE EE ENE EPE EE ENE EPE EE ENE EPE EE ENE

Case A Case B Case C Case D

Source: www.danskebank.com/CI 13 Netting Intuition (cont.)

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: EPE Net (t; H )  Emax(min(V Net (t), H )0)  Emax(V Net (t),0)  max(V Net (t)  H,0)

•The last expression is identical to a so-called call spread on the value of the netting set with strikes EPE with Threshold H= 50 60

0 and H. 40

•From the Black-Scholes analysis we know the 20 behavior of call spreads (or digital options) around 0 -150 -100 -50 0 50 100 150 200 250 the two strikes. In particular their Vega/Gamma -20 sensitivity becomes important in understanding -40 changes in EPE (and ENE) driving Incremental -60 xVA. -80 max(min(V,H),0) Value Vega

Source: www.danskebank.com/CI 14 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: T T CVANewNet  CVANet E[ V NewNet (t)  (  t)dt] E[ V Net (t)  (  t)dt] 0 0 T Net New Net E[ V (t) V (t)1 Net New V (t)1 Net  (  t)dt] 0 V (t)V (t)0 V (t)0 T Net New  E[ V (t)1 Net New 1 Net V (t)1 Net New  (  t)dt] 0 V (t)V (t)0 V (t)0 V (t)V (t)0 T u   Net  E[  1~ Net ~ New 1~ Net  (  t)dt c (u)du] 0 0 V (t)V (t)0 V (t)0  Incremental CVA notional T u   New  E[  1~ Net ~ New  (  t)dt c (u)du] 0 0 V (t)V (t)0  CVA notional Source: www.danskebank.com/CI 15 Standalone/Incremental/Marginal xVA (cont.)

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

•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:

CVANewNet  CVANet|NewNet  CVANew|NewNet T  E[ V NewNet (t)  (  t)dt] 0 T Net New E[ V (t) V (t)1 Net New  (  t)dt] 0 V (t)V (t)0 T Net New  E[ V (t)1 Net New V (t)1 Net New  (  t)dt] 0 V (t)V (t)0 V (t)V (t)0 T u   Net  E[  1~ Net ~ New  (  t)dt c (u)du] 0 0 V (t)V (t)0   CVANet|NewNet T u   New  E[  1~ Net ~ New  (  t)dt c (u)du] 0 0 V (t)V (t)0   CVANew|NewNet

Source: www.danskebank.com/CI 16 Standalone/Incremental/Marginal xVA (cont.)

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

•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%

Net = EUR 100M 10Y IRS, We receive fixed New = EUR 200M 5Y IRS, We pay fixed 8,000,000 2,000,000

7,000,000 1,000,000 6,000,000 0 5,000,000 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 4,000,000 -1,000,000 3,000,000 -2,000,000 2,000,000

1,000,000 -3,000,000

0 -4,000,000 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 -1,000,000 -5,000,000 -2,000,000

-3,000,000 -6,000,000

Positive Negative Expected Positive Negative Expected

Standalone xVA Standalone xVA xVA (s.e.) xVA (s.e.) CVA 769,079 4,471 EUR CVA 66,174 851 EUR DVA -51,311 1,491 EUR DVA -80,461 325 EUR FCA 219,683 1,226 EUR FCA 17,732 227 EUR xVA Total 937,451 EUR xVA Total 3,445 EUR

Source: www.danskebank.com/CI 17 Standalone/Incremental/Marginal xVA (cont.)

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

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

Total Exposure (Net + New) Incremental Exposure (Net + New \ Net) 7,000,000 2,000,000

6,000,000 1,000,000 5,000,000 0 4,000,000 0.00 2.00 4.00 6.00 8.00 10.00 12.00

3,000,000 -1,000,000

2,000,000 -2,000,000

1,000,000 -3,000,000 0 0.00 2.00 4.00 6.00 8.00 10.00 12.00 -4,000,000 -1,000,000

-2,000,000 -5,000,000

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

Source: www.danskebank.com/CI 18 Standalone/Incremental/Marginal xVA (cont.)

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

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

Marginal Exposure (Net | Net + New), (New | Net + New) 8,000,000

6,000,000

4,000,000

2,000,000

0 0.00 2.00 4.00 6.00 8.00 10.00 12.00 -2,000,000

-4,000,000

-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

Source: www.danskebank.com/CI 19 Standalone/Incremental/Marginal xVA (cont.)

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

•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.

Net = 1B USD 5Y CCS vs 736M EUR, Pay USD3M vs Rec EUR3M - 8.13bp New = 1B USD 5Y IRS, Pay Fixed vs. Rec USD3M 80,000,000 30,000,000

60,000,000 25,000,000

20,000,000 40,000,000

15,000,000 20,000,000 10,000,000 0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 5,000,000

-20,000,000 0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 -40,000,000 -5,000,000

-60,000,000 -10,000,000

Positive Negative Expected Positive Negative Expected

Standalone xVA Standalone xVA xVA (s.e.) xVA (s.e.) CVA 3,873,850 15,117 EUR CVA 1,539,087 2,109 EUR DVA -850,636 4,111 EUR DVA -70,346 787 EUR FCA 1,072,246 4,102 EUR FCA 417,425 551 EUR xVA Total 4,095,460 EUR xVA Total 1,886,166 EUR Source: www.danskebank.com/CI 20 Standalone/Incremental/Marginal xVA (cont.)

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

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

New + Net Incremental: New + Net \ Net 80,000,000 25,000,000

60,000,000 20,000,000

40,000,000 15,000,000

20,000,000 10,000,000 0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 5,000,000 -20,000,000

0 -40,000,000 0.00 1.00 2.00 3.00 4.00 5.00 6.00

-60,000,000 -5,000,000

Positive Negative Expected Positive Negative Expected

Total xVA Incremental xVA PV (s.e.) PV CVA 4,605,043 11,767 EUR CVA 731,193 EUR DVA -736,914 4,177 EUR DVA 113,722 EUR FCA 1,270,623 3,226 EUR FCA 198,377 EUR xVA Total 5,138,751 EUR xVA Total 1,043,291 EUR

Source: www.danskebank.com/CI 21 Standalone/Incremental/Marginal xVA (cont.)

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

•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.

T = 1B USD 5Y CCS vs 736M EUR, Pay USD3M vs Rec EUR3M - 8.13bp, Quarterly USD Reset 15,000,000 (*) The USD notional is set to the current 10,000,000 spot at the beginning of each period and

5,000,000 the change in notional relative to previous period is paid/received. 0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 From an exposure point of view it has the -5,000,000 same effect as receiving collateral every 3

-10,000,000 months, or alternatively as considering the trade a string of 3M FX forwards. -15,000,000

Positive Negative Expected

Standalone xVA Incremental xVA PV (s.e.) PV CVA 528,295 12,368 EUR CVA -4,076,748 EUR DVA -102,935 3,451 EUR DVA 633,979 EUR FCA 142,970 3,293 EUR FCA -1,127,652 EUR xVA Total 568,330 EUR xVA Total -4,570,422 EUR Source: www.danskebank.com/CI 22 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 Fund contributions in check (it is also a counterparty exposure). •Balance-sheet and Leverage Ratio management, through trade-compression, novations, back-loading to CCPs.

Source: www.danskebank.com/CI 23 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 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 this presentation  Danske Bank, its affiliates, subsidiaries and staff may perform services for or solicit business from any issuer mentioned herein and may hold or 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 the 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 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 24 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 Bank  In the United States the presentation is intended solely to “U.S. institutional ” as defined in sec rule 15a–6  Any U.S. investor recipient of this presentation who wishes to purchase or sell any Relevant 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

Source: www.danskebank.com/CI 25