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 Credit & Funding Risk, Danske Bank, 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
max(V,0)
Source: www.danskebank.com/CI
0
max(min(H,V),0)
5
10 15 20 25 30 35 40 45 50
max(min(H-IA,V-IA),0)
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 ) 1V~ (t ) 0  (  t )dt ]
0

regression proxy
T
T
0
t
 E[  Et [ 
 E[ 
T
0

T
t
c
(u )
du ]1V~ ( t ) 0  (  t ) dt ]
future cashflow
c(u )1V~ ( t ) 0  (  t ) dudt ]
u

 E[    1V~ ( t ) 0  (  t )dt c(u )du ]
0  0

T
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
H
 E[  V (t ) min(1,
) (  t )dt ]

0
V (t )
T
H
 E[  V (t )1V~ ( t ) 0 min(1, ~  ) (  t )dt ]
0
V (t )
T
T
H
 E[  Et [  c(u ) du ]1V~ (t ) 0 min(1, ~  ) (  t )dt ]
0
t
V (t )
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 investor (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
IA H  IA
,
),0) (  t ) dt ]


0
V (t ) V (t )
T
IA H  IA
 E[  V (t )1V~ ( t ) 0 max(min(1  ~  , ~  ),0) (  t )dt ]
0
V (t ) V (t )
T
T
IA H  IA
 E[  Et [  c(u ) du ]1V~ ( t ) 0 max(min(1  ~  , ~  ),0) (  t )dt ]
0
t
V (t ) V (t )
T
 E[  V (t )  max(min(1 
•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.
Source:
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
IA
Source: www.danskebank.com/CI
-5
0
5
max(min(H-IA,V-IA),0)
10
15
20
Collateral
25
30
35
40
45
50
-50 -45 -40 -35 -30 -25 -20 -15 -10
CVA_Ntl
-5
0
CVA_Ntl(H)
5
10
15
20
25
30
35
40
45
50
CVA_Ntl(H,IA)
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
Source: www.danskebank.com/CI
ENE
EE
EPE
ENE
EE
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
Source: www.danskebank.com/CI
Positive Collateral
Positive Exposure
Positive IA
Negative Collateral
Negative Exposure
Negative IA
10
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-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 1000s 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:
V
Net
n
(t )   X i (t )
EPE Net  New (t )  EPE Net (t )  EPE New (t )
and
i 1

V New (t )  X n 1 (t )
 (t ) 
ENE (t )  E V (t ) 
EE (t )  E V (t )  EPE
EPE
Net
(t )  E V
Net
Net
Net

Net

Net
Source: www.danskebank.com/CI
EPE New Net (t )  E V New (t )  V Net (t )

Net
(t )  ENE Net (t )




 E  E  V New (t )  V Net (t ) V Net (t ) 
 
 
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
Net
New
New+Net
0
Net
New
New+Net
0
Net
New
New+Net
Net
-25
-25
-25
-25
-50
-50
-50
-50
-75
-75
-75
-75
-100
-100
-100
-100
EPE
EE
ENE
Case A
Source: www.danskebank.com/CI
EPE
EE
Case B
ENE
EPE
EE
Case C
ENE
EPE
New
EE
New+Net
ENE
Case D
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:


(t )  H ,0)
EPE Net (t ; H )  E max(min(V Net (t ), H )0)

 E max(V Net (t ),0)  max(V Net
•The last expression is identical to a so-called call
spread on the value of the netting set with strikes
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.
Source: www.danskebank.com/CI
EPE with Threshold H= 50
60
40
20
0
-150
-100
-50
-20
0
50
100
150
200
250
-40
-60
-80
max(min(V,H),0)
Value
Vega
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
0
0
CVA New Net  CVA Net E[  V New Net (t )   (  t )dt ]  E[  V Net (t )   (  t ) dt ]

 E[  V

T

E[  V Net (t )  V New (t ) 1V Net (t ) V New (t ) 0  V Net (t )1V Net ( t ) 0  (  t )dt ]
0
T
0
Net




(t ) 1V Net ( t ) V New (t ) 0  1V Net ( t )0  V New (t )1V Net (t )V New ( t ) 0  (  t )dt ]

u

 E[    1V~ Net ( t ) V~ New ( t ) 0  1V~ Net ( t )0  (  t )dt c Net (u ) du ]
0  0



T
Incremental CVA notional
u

 E[    1V~ Net ( t ) V~ New (t ) 0  (  t )dt c New (u )du ]
0  0



T
Source: www.danskebank.com/CI
CVA notional
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:
CVA New Net  CVA Net | New Net  CVA New| New Net
T
 E[  V New Net (t )   (  t )dt ]
0
T


E[  V Net (t )  V New (t ) 1V Net (t ) V New ( t ) 0  (  t )dt ]
0
T


 E[  V Net (t )1V Net ( t )V New ( t ) 0  V New (t )1V Net (t )V New ( t )0  (  t ) dt ]
0
u

 E[    1V~ Net ( t )V~ New (t ) 0  (  t )dt c Net (u )du ]
0
0



T
CVA Net|New Net
 E[    1V~ Net ( t )V~ New (t ) 0  (  t )dt c New (u )du ]
0
0



T
u
CVA New| 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%
New = EUR 200M 5Y IRS, We pay fixed
Net = EUR 100M 10Y IRS, We receive fixed
2,000,000
8,000,000
7,000,000
1,000,000
6,000,000
0
5,000,000
0.00
4,000,000
3,000,000
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
-2,000,000
2,000,000
-3,000,000
1,000,000
0
-1,000,000
1.00
-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
-4,000,000
-5,000,000
-2,000,000
-6,000,000
-3,000,000
Positive
Negative
Positive
Expected
Standalone xVA
Negative
Expected
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
3,445
EUR
Source: www.danskebank.com/CI
xVA Total
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
3,000,000
-1,000,000
2,000,000
-2,000,000
1,000,000
2.00
4.00
6.00
8.00
10.00
12.00
-3,000,000
0
0.00
2.00
4.00
6.00
8.00
-1,000,000
-2,000,000
10.00
12.00
-4,000,000
-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)
Source: www.danskebank.com/CI
(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
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.)
CVA
696,453
4,260 EUR
DVA
-32,058
FCA
200,430
xVA Total
864,825
Source: www.danskebank.com/CI
xVA (New | Net+New) (s.e.)
-195,977
2,197 EUR
1,453 EUR
-9,279
639 EUR
1,181 EUR
-53,361
591 EUR
-258,618
EUR
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
-40,000,000
-60,000,000
2.00
3.00
4.00
5.00
6.00
-10,000,000
Positive
Negative
Expected
Standalone xVA
Positive
Negative
Expected
Standalone xVA
xVA
CVA
1.00
-5,000,000
(s.e.)
xVA
3,873,850 15,117 EUR
(s.e.)
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
1,886,166
EUR
Source: www.danskebank.com/CI
xVA Total
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
-20,000,000
5.00
6.00
5,000,000
0
-40,000,000
0.00
-60,000,000
1.00
2.00
3.00
4.00
5.00
6.00
-5,000,000
Positive
Negative
Expected
Positive
Negative
Expected
Incremental xVA
Total xVA
PV
PV
(s.e.)
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
xVA Total
5,138,751
EUR
Source: www.danskebank.com/CI
xVA Total
198,377 EUR
1,043,291 EUR
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
spot at the beginning of each period and
the change in notional relative to previous
period is paid/received.
10,000,000
5,000,000
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
-5,000,000
-10,000,000
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.
-15,000,000
Positive
Negative
Expected
Standalone xVA
Incremental xVA
PV
CVA
Source: www.danskebank.com/CI
(s.e.)
PV
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
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 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.
Source: www.danskebank.com/CI
23
General disclaimer
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This presentation has been prepared by Danske Bank Markets (a division of Danske Bank A/S). It is provided for informational purposes
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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 investors as
defined in sec rule 15a6
 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
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Source: www.danskebank.com/CI
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