Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
New York, May 2012
Counterparty Credit Risk Workshop
CVA Computation,
Stress Testing and WWR
Dan Rosen
R2 Financial Technologies
dan.rosen@R2-financial.com
© 2006-2011 R2 Financial Technologies
Preface"
Counterparty risk is
“ probably the single most important variable in
determining whether and with what speed
financial disturbances become financial shocks,
with potential systemic traits”
Counterparty Risk Management Policy Group
(CRMPG 2005)
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Outline
Introduction
Computing CVA
WWR and CVA
An integrated market and credit risk model for WWR
Example
Concluding remarks
© 2006-2011 R2 Financial Technologies
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Pricing CCR: Credit Value Adjustment (CVA)
CVA is the market value of counterparty credit risk
CVA = Risk-free portfolio value – true portfolio value accounting
for counterparty defaults
CVA is an integral component of the value of derivatives
Now an integral part of accounting rules and Basel III
Prior to mid-2007, CVA was generally ignored or too small to be noticed
CVA is measured at the counterparty level
Ideally, it should be part of the trade valuation but needs calculated
separately because of portfolio effects (netting, collateral, etc.)
Desirable also to determine contributions of individual trades to the CP-level
CVA and to the total portfolio CVA – similar problem to EC allocation
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CCR Credit Risk Model Components
Exposures
Default
0
t
t
Recovery
8
© 2006-2011 R2 Financial Technologies
Calculating CVA
Bilateral CVA is the adjustment to the credit-risk-free value of the OTC
derivatives portfolio which reflects the credit risks faced by both CPs
Unilateral CVA: only the counterparty (but not the bank) can default
Simple approximation:
CVA = E A s A − EB sB
EA = PV expected exposure faced by Bank (B) to counterparty (A)
sA = market loss rate of A : essentially, PD X LGD (both risk-neutral)
EB = PV expected exposure
faced by A with respect to B
Exposures
sB = market loss rate of B
Default
0
t
t
Recovery
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Calculating CVA – Example
CVA = E A s A
Example: unilateral CVA
CVA as a spread: a trader wants a back-of-the envelope
approximation of a swap’s CVA
EPE is 5%
CP’s credit spread is 3%
CVA ≈ (0.03)(0.05) = 15 bps
The trader adds/subtracts 15bps from the leg of the swap as the
credit charge/CVA
(example from Gregory 2009)
© 2006-2011 R2 Financial Technologies
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Calculating CVA and Risk Neutral Pricing
CVA = E A s A − E B sB
CVA is a price not a risk measure
The market has been moving towards a risk-neutral (market implied)
approach
Practices of top-tier banks, accounting rules, Basel III
However pricing CVA is not as accurate as with standard (liquid)
derivatives – some issues include:
Limited liquid spreads
Limited arbitrage possibilities with “mispricing”
CVA hedging is much more complex – CVA will never be well-hedged
Heavy rebalancing costs (complex cross-asset credit contingent nature)
Some banks would rather look at it merely as a reserve (expected loss)
Banking book view real measure
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Basel III and CVA"
“Mark-to-market losses due to credit valuation
adjustments (CVA) were not directly
capitalised. Roughly two–thirds of CCR
losses were due to CVA losses and only onethird were due to actual defaults.”
Basel Committee on Banking Supervision (2009)
© 2006-2011 R2 Financial Technologies
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CVA Risk, Volatility and VaR
Banks that take a mark-to-market CVA are subject to market price
volatility
Need to hedge CVA market risks in addition to counterparty defaults
CVA market risks include changes in
Credit spreads of both counterparties (bi-lateral CVA)
Market risk factors that drive the underlying OTC derivative exposures
The recent financial crisis showed that CVA losses can be larger
than default losses
13
© 2006-2011
R2 Financial Technologies
©2009
13IACPM
5
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Remarks on Bilateral CVA
CVA = E A s A − EB s B
Adjusts for both the possibility that the
counterparty may default and that the bank itself may default
Always lower than unilateral CVA
Prices the “gain” that is realized on the contract when the bank
defaults while the exposure is negative
Only have to pay recovery, rather than contract value
Theoretically, the value is realized by bondholders, not shareholders
Occurs in bankruptcy, when shareholder value is zero
Difficult to realize in practice
Can lead to somewhat bizarre profits/losses
We “made a lot of money” on BCVA this quarter because our default
probability went way up
We “lost a lot” on BCVA because our PD went down
© 2006-2011 R2 Financial Technologies
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Bilateral CVA in the news
“Citigroup reported on Friday its first quarterly net profit in nearly
two years, the latest US bank to see an improvement in its
performance. It made a profit of 1.6 billion US dollars compared
with a loss of 5.1 billion a year earlier. Revenues rose 99% to 24.8 billion But it
gained from an accounting rule that allowed the bank to post a one-time gain of 2.5
billion USD.”
Source: en.mercopress.com
Citi’s Q1 2009 report
“Fixed income markets revenues of $4.7 billion reflected strong trading
performance, as high volatility and wider spreads in many products created
favorable trading opportunities. Interest rates and currencies and credit products
had strong revenue growth. Revenues also included (all reflected in Schedule B):
A net $2.5 billion positive CVA on derivative positions, excluding monolines,
mainly due to the widening of Citi’s CDS spreads. A net $30 million positive
CVA of Citi’s liabilities at fair value option.”
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Bilateral CVA – can we monetize our own default?
CVA = E A s A − EB s B
Go bankrupt
DVA = Debt value
adjustment
Not a very “popular choice”
Hedging
DVA is hard to hedge – cannot sell directly CDS protection on yourself!
Sell CDS on another counterparty or index who is highly correlated with bank
Buy back your own debt (not really a dynamic hedge) – where’s the cash?
This directly gives wrong-way risk to the protection buyer!
Unwinds or novations
An institution may realise a DVA gain if a trade is unwound in the future (e.g.
A banks unwinding transactions with a monoline)
Funding arguments
© 2006-2011 R2 Financial Technologies
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Outline
Introduction
Computing CVA
Exposures
Default
Migration
0
t
t
Recovery
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CVA Model – the real world is not so simple
Simple CVA approximation formula
In practice exposures and spreads are
time dependent
Exposures are stochastic and difficult to
model and compute – require MC simulation
Large portfolios of derivatives, hundreds of
market factors
Credit mitigation: netting, collateral, margins
Wrong-way risk: exposures and spreads
(defaults) may by correlated
CVA = E A s A − E B sB
Maximum Peak
Exposure (T)
Effective
EPE (T)
Effective
Expected
Exposure
Peak Exposure (95%)
Expected
Exposure
EPE
(T)
Source: de Prisco and Rosen (2005)
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© 2006-2011
R2 Financial Technologies
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Credit Losses and CVA
Unilateral CVA – the bank’s discounted
loss due to the CP default
L = 1 τ ≤T (1 − R ) E (τ ) D(τ )
{ }
CVA obtained by discounting losses and
applying the expectation
Exposures
Default
Migration
0
t
t
Recovery
T
CVA = (1 − R ) ∫ dP(t ) eˆ∗(t )
where
0
eˆ∗(t ) = Eˆ t [ D (t ) E (t ) ] ≡ E  D (t ) E (t ) τ = t 
is the risk-neutral discounted expected exposure at t, conditional on
the CP’s default at t
(everything is under the risk-neutral measure)
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Analytical CVA (Normal Approx.)
It is useful in practice to estimate EE, CVA
Vi (t ) = µi (t ) + σ i (t ) X i
an CVA contributions quickly outside of the simulation system
Analytical expressions can be derived for EE, CVA and CVA
contributions, for the case when trade values are normally distributed
Independent market and credit
eˆ∗(t ) = Eˆ t [ D (t ) E (t ) ] ≡ E  D (t ) E (t ) τ = t 
 µ (t ) 
 µ (t ) 
e (t ) = µ ( t ) Φ 
 + σ (t ) φ 

 σ (t ) 
 σ (t ) 
  µ (t ) 
 µ (t ) − H  
e(t ) = µ (t )  Φ 
 − Φ

 σ (t )  
  σ (t ) 
(Source Pykhtin and Rosen 2010)
© 2006-2011 R2 Financial Technologies
  µ (t ) 
 µ (t ) − H  
 µ (t ) − H 
+σ (t ) φ 
 −φ
 + H Φ 

(
t
)
(
t
)
σ
σ



 σ (t ) 
 
21
Calculating Exposures and CVA by Simulation
Banks use Monte Carlo simulation in practice to obtain the distribution of
counterparty-level exposures
The calculation of CVA and CVA contributions can be easily
incorporated to the MC simulation of the counterparty-level exposures
In general, exposures are simulated separately – implicitly assume that
they are independent of counterparties’ credit quality
Dependence of exposure on the counterparty’s credit quality can be
incorporated in the CVA calculation, if the trade values and credit quality
of the bank’s counterparties are simulated jointly (both right/wrong-way
risk and exposure-limiting agreements)
E.g. using a joint process with stochastic intensities, or a copula
methodology as in Garcia Cespedes et al
Need to make model computationally efficient"
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Calculating CVA – Simulation
CVA = E A s A
Unilateral CVA
T
CVA = (1 − R ) ∫ dP(t ) eˆ∗(t )
0
eˆ (t ) = Eˆ t [ D (t ) E (t ) ] ≡ E  D (t ) E (t ) τ = t 
∗
Independent market and credit risk
Counterparty PFE
35
Mean Exp o sure
CVA = (1 − R ) ∑ k e (t k )[ P (t k ) − P (t k −1 )]
∗
95 Percen tile
30
Co n d . EPE
25
∗
e (t k ) =
1
M
∑
M
j =1
( j)
( j)
D (t k ) E (t k )
Exposure (€ Million)
20
15
10
5
0
5475
5110
4745
4380
4015
3650
3285
2920
2555
2190
1825
1460
1095
730
365
0
Time (days)
23
© 2006-2011 R2 Financial Technologies
Bilateral CVA
The unilateral CVA formula in discrete time can be written as
CVA = (1 − RC ) ∑ E[ D(ti ) PFE (ti ) | ti −1 < τ C ≤ ti ] ⋅ P[ti −1 < τ C ≤ ti ]
ti ≤T
Similarly, bi-lateral CVA is given by
BCVA = (1− RC ) ∑ E[D(t i )PFE(t i ) | t i−1 < τ C ≤ t i , τ B > t i ]⋅ P [t i−1 < τ C ≤ t i , τ B > t i ]
t i ≤T
- (1− RB ) ∑ E[D(t i )NFE(t i ) | t i−1 < τ B ≤ t i , τ C > t i ]⋅ P [t i−1 < τ B ≤ t i , τ C > t i ]
t i ≤T
Subscripts: C indicates counterparty, B indicates bank (us)
Assumes no probability of simultaneous default
NFE = Negative Future Exposure (= max(-Value,0) )
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CVA Model Risk – the real world is not so simple
“Standard derivatives’ pricing accuracy” for CVA? – in addition to
computation, there are important practical issues
Spreads and liquidity
Generally liquid spreads are not available for all CPs – may only reflect
dealer quotes and not actual trades (specially in stressed times)
Mapping procedure (e.g. by rating-sector, statistical methods)
Term structure –most likely, only one (or two) point for a name (e.g. 5 year)
The term structure can have a significant impact on CVA calcualtions!
Accuracy of exposures’ calculations: risk factor evolution, derivatives’
approximations, mitigation and collateral assumptions
Correlations and WWR: are unobservable (specially “risk neutral”)
Impact not only on final price but also on sensitivities and hedging
Stress testing and model risk framework are vey important!
© 2006-2011 R2 Financial Technologies
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CVA Computation and Risk Management
CVA/counterparty risk is at the intersection of market and credit risk
Market risk: CP spreads and market factors driving exposures
Credit risk: default risk and event risk (CP migrations)
Need efficient CVA pricing (similar problem of other derivatives or
complex instruments" but this is more complex!)
Large number of valuations
Risk measurement and hedging
Sensitivities (spreads, market factors driving exposures, correlations)
Stress testing
Statistical measures: market VaR, credit VaR (e.g. IRC)
Economic and regulatory capital
Model risk (multiple model assumptions)
CVA and risk contributions/allocation
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CVA VaR
A proper market risk treatment of CVA requires to
incorporate it fully with the rest of the trading book
A full market simulation of CVA is very challenging
Portfolio loss distribution
10-day 99% VaR
0.12
Empirical
0.10
Normal
0.08
Probability
0.06
0.04
0.02
0.00
-140
-100
-60
-20
21
61
101
141
Thousands of market scenarios – each scenario
requires the calculation of the exposure profiles (e.g.1,000 scenarios and 100
time points) revalue the entire portfolio for each path at each time point
Brute-force nested Monte Carlo procedure is not generally fully feasible
Size of Loss (thousands USD)
Simulation with sensitivities (also required for hedging) is a practical
alternative".
However, calculation of CVA sensitivities is a computational challenge as well
Generally limited to a small number"
© 2006-2011 R2 Financial Technologies
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CVA Contributions
CVA is measured at the counterparty level
Portfolio effects from netting, collateral, margins
Desirable also to determine contributions of individual trades to the
CP-level CVA and to the total portfolio CVA
How much should a trader charge for the CVA of a new trade?
What should be CVA share of a given book or business unit?
This is a similar problem to computing capital allocation (in risk
management)
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CVA Contributions
CVA Contributions:
Stand-alone CVA
Incremental CVA – difference between portfolio CVA with and without
the trade
Marginal CVA – additive CVA contributions
Marginal contributions with a given CP give a clear picture how
much each trade contributes to the CP-level CVA
Once additive CVA contributions have been calculated for each CP, the
bank can calculate the price of CP credit risk for any collection of trades
without any reference to the CPs
e.g, the CVA contribution of a business unit or product is simply the
sum of CVA contributions of all trades booked by the business unit or of
the product type
Methodology described in Pykhtin and Rosen (2010)
© 2006-2011 R2 Financial Technologies
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Outline
Introduction
Computing CVA
Wrong-way risk (WWR)
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CVA and Wrong-Way Risk
In general, CP exposures are not independent of the CP’s credit quality
Wrong-way risk: exposure increases when the credit quality of the CP
deteriorates – i.e. exposures tend to be high when PDs are high
Also possible to have right-way risk, where exposures tend to be low when
PDs are high
Wrong-way risk positive
correlation between the exposure
of an instrument (or counterparty
portfolio) and the default of the
counterparty
© 2006-2011 R2 Financial Technologies
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CVA and Wrong-Way Risk
Two types of WWR
General WWR – the CP’s credit quality is for non-specific reasons
correlated with macroeconomic factors which also affect the value of
the derivatives
Example: correlation between declining corporate credit quality and high (or
low) interest rates causing higher exposures (e.g. portfolio of IR swaps)
Specific WWR – future exposure to a specific CP is highly correlated
with the CP’s PD
Arises generally through poorly structured
transactions – e.g. a company writing put options
on its own stock, derivatives collateralized by own
or related party shares, instruments for which there
exists a legal connection between the counterparty
and the underlying issuer
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Calculating CVA with Wrong-Way Risk
CVA = E A s A
Independent market-credit risk
CVA = (1 − R ) ∑ k e∗(tk )[ P (t k ) − P (t k −1 )]
Counterparty PFE
35
Mean Exp osure
95 Percen tile
30
Co nd. EPE
25
∑
j =1
20
D ( j ) (tk ) E ( j ) (tk )
Exposure (€ Million)
1
Expected exposure e∗(tk ) = M
M
15
10
5
0
5475
5110
4745
4380
4015
3650
3285
2920
2555
2190
1825
1460
1095
730
365
0
Time (days)
CVA with WWR
1.
CVAWWR = (1 − R ) ⋅ ∑k ∑ j D j (t k ) ⋅ E j (t k ) ⋅ P (ω j , t k −1 < τ < t k )
M
CVAWWR = (1 − R ) ⋅ ∑k eˆ∗ (t k ) ⋅ [P(t k ) − P(t k −1 )]
Expected exposure
conditional on default
eˆ∗ (t k ) = ∑ j D j (t k ) ⋅ E j (t k ) ⋅
M
© 2006-2011 R2 Financial Technologies
P (ω j , t k −1 < τ < t k )
P (t k ) − P(t k −1 )
37
Computing and Stress Testing Wrong-Way Risk
The codependence of exposures and counterparty defaults can have a
strong impact on CVA and capital calculations
Difficult to model and computationally intensive
Expensive exposure calculations
Systems in place often handle exposures and default simulations separately
Correlations (co-dependence) between exposures and default probabilities
are generally very difficult to estimate in practice
Brute force MC simulation is straightforward but not really feasible
Large number of valuations: sensitivities, CVA market VaR, CCR capital
We look for a robust method to calculate CVA and EC with WWR
Simple, easy to implement, and computationally efficient
Leverages existing exposure simulations, and PD estimates (real measure
and risk neutral)
Easy to stress test – model risk
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Outline
Introduction
Computing CVA
WWR and CVA
An integrated market and
credit risk model for WWR
CP exposures
(Simulated from market factors)
Credit factors defaults
Copula
Correlation parameters: ρ
Copula (X, Z)
Codependence of
exposures and credit events
Capital
Credit events
Exposures
(market
scenarios)
Yi = ρ Z + 1 − ρ ε i
Ei = f ( X )
Syst. factor
X
© 2006-2011 R2 Financial Technologies
 N −1 (PD ) − ρ z 

PD (Z ) = N 


1− ρ


Syst. factor
Z
Idiosyncratic
CP factors
39
CCR Capital/Alpha and CVA Methodology
(Garcia et al)
Framework defined by two key components:
Non-parametric sampling of exposures from pre-computed PFEs (unconditional)
Direct modelling of codependence of and exposures and systematic credit factors
(non-parametric exposures sampling also from joint market-credit factor scenarios)
Computationally efficient – fully leverages pre-computed PFE profiles
Minimizes work during the most computationally intensive step
Multiple capital calculations – model validation, sensitivities, stress testing
Maximum Peak
Exposure (T)
Effective
EPE (T)
EPE
(T)
Source: de Prisco and Rosen (2005)
© 2006-2011 R2 Financial Technologies
Effective
Expected
Exposure
Peak Exposure (95%)
Expected
Exposure
40
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CCR Capital/Alpha and CVA Methodology
(Garcia et al)
Framework defined by two key components:
Non-parametric sampling of exposures from pre-computed PFEs (unconditional)
Direct modelling of codependence of and exposures and systematic credit factors
(non-parametric exposures sampling also from joint market-credit factor scenarios)
Computationally efficient – fully leverages pre-computed PFE profiles
Minimizes work during the most computationally intensive step
Multiple capital calculations – model validation, sensitivities, stress testing
Model can be applied within general integrated market-credit risk models
Stress testing
Wrong-way risk and transparency of counterparty concentrations and factors driving
exposures
Market-credit correlations – contrast to industry studies, conservative assumptions, or
from a previously estimated market-credit risk model
© 2006-2011 R2 Financial Technologies
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Wrong-Way Risk – Correlated Market-Credit
General market-credit codependence framework
CP exposures
(Simulated from market factors)
Credit factors defaults
Copula
Correlation parameters: ρ
Capital
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Analytical CVA (Normal Approx.)
Analytical expressions including wrong-way risk
Vi (t ) = µi (t ) + σ i (t ) X i
Need to modify the approach to obtain CVA and the contributions
Y = Φ −1[ P (τ )]
to the CP EE conditional on the CP defaulting
2 ˆ
X i = bY
i + 1 − bi X i
For this purpose, we define a Normal copula
Same results – but instead of the unconditional expectations, standard
deviations and correlations of the trade values, we now use the
conditional ones
µˆ i (t ) ≡ E[ Vi (t ) | τ = t ] = µi (t ) + σ i (t )bi Φ −1[ P(t ) ]
σˆ i (t ) ≡ StDev[ Vi (t ) | τ = t ] = σ i (t ) 1 − bi2
ρˆ i (t ) =
ρi (t ) − bi β (t )
(1 − bi2 )[1 − β 2(t )]
43
© 2006-2011 R2 Financial Technologies
WWR Model at the Portfolio Level
Integrated market and
credit portfolio model
Simulated jointly for entire
portfolio
Copula (X, Z)
Codependence of
exposures and credit events
Credit events
Exposures
(market
scenarios)
Yi = ρ Z + 1 − ρ ε i
Ei = f ( X )
 N −1 (PD ) − ρ z 

PD(Z ) = N 


1− ρ


Syst. factor
X
© 2006-2011 R2 Financial Technologies
Syst. factor
Z
Idiosyncratic
CP factors
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Credit Model and Default Distribution
Yc = Φ −1 (Fc (t ))
Copula (X, Z)
Codependence of exposures
and credit events
Default if
Single factor model Yc = ρ c ⋅ Z + 1 − ρ c ⋅ ε c
Credit events
Example: constant hazard rate PD = 5%
mapped to a standard normal
© 2006-2011 R2 Financial Technologies
Syst. factor
Z
Idiosyncratic
CP factors
45
Market Risk Model
Mapping: Exposure Simulation “Market Factor”
Exposure scenarios are sorted in order of an
increasing “market factor”, which has a defined
value in each scenario and then mapped to a
standard Normal variable.
Example of factors: time-averaged exposure by
counterparty, total exposure; an aggregate
market factor; principal components, etc.
Copula (X, Z)
Codependence of exposures
and credit events
Exposures (market
scenarios)
Syst. factor
X
Ordered Scenarios (Market Factor)
4
TE
3
PC1
Market (-)
2
TE (Mov. Avg.)
PC1 (Mov. Avg.)
1
0
-1
-2
-3
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Joint Market-Credit Distribution
Copula model: bivariate normal with correlation ρ.
f (y,z) =


1
1
exp −
y 2 + z 2 − 2ρyz)
2 (
2π
 2(1− ρ )

ρ
© 2006-2011 R2 Financial Technologies
47
Market-Credit Model – Remarks
Scenario ordering and multiple time steps
Single scenario ordering
Market factor defines one ordering for all scenario paths – can be based on
a single-time step (e.g. 1-year), an aggregate or average (e.g. the timeaveraged exposures over 5 years), etc.
Model is simple and consistent with the Gaussian copula credit model
Preserves consistently the exposure scenario paths from the simulation
Time-dependent scenario ordering
Scenarios are ordered separately for every time point, based on a time
dependent factor X(t); e.g. the portfolio value in each time can be used to
generate an ordering for the scenarios
Presents a richer and more flexible codependence structure.
Looses path dependency, and does not preserve the joint distribution of
exposures through time (however, this does not affect CVA calculations)
Can be also extended to a dynamic setting
© 2006-2011 R2 Financial Technologies
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Market-Credit Model – Remarks
Specifying a market factor for correlating exposures with defaults
corresponds to defining a particular ordering of the exposure scenarios
Useful framework for stress-testing both:
The strength of market-credit correlations through the market-credit
correlation parameter
Different directions of the market-credit codependence, by considering
different ways of ordering the exposure scenarios.
Model extends naturally to multiple counterparties
© 2006-2011
Can be effectively used for bilateral CVA. – e.g.
R2
Define a single-factor copula model for the credit risk of both the
counterparty and the bank.
Two creditworthiness factors (Yc , YB ), each driven by credit factor Z
Market factor X drives both exposures consistently
X and Z again follow a jointly Normal distribution with given correlation
Financial Technologies
49
Exposure Simulation and “Market Factors”
CVA: each CP exposures can be mapped separately
General WWR measures by using a single model
for the portfolio
Portfolio capital calculations: map the entire simulation cube to the factor
Guarantees full consistency with the joint exposure distribution (described by
the exposure simulation cube)
Single market factor is sufficient with a single factor credit model (Basel II)
Simple extensions to multiple factors at the cost of
additional computation (and further approximation
of the joint exposures distribution)
© 2006-2011 R2 Financial Technologies
50
21
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Remarks on WWR – General and Specific
Integrated market and credit portfolio model
General WWR: while the CVA is the sum of CP CVAs, we must use
the same model and correlation parameters to compute CVA at the
portfolio level
Under a given model (market factor
and correlation levels) not all CPs will
simultaneously have WWR
Specific WWR and stress testing
of individual CPs can be done with
different set of parameters
(and correlating CP specific
factors as well)
Copula (X, Z)
Codependence of
exposures and credit events
Credit events
Exposures
Ei = f ( X )
Syst. factor
X
Yi = ρ Z + 1 − ρ ε i
 N −1 (PD ) − ρ z 

PD(Z ) = N 


1− ρ


Syst. factor
Z
Idiosyncratic
CP factors
51
© 2006-2011 R2 Financial Technologies
Remarks – Empirical Market-Credit Correlations
Historical time series: market index, implied credit driver and default rates (S&P data)
Historical Default Rates (1981-2007)
Correlation between all ratings and
investment only
Default rates = 78%
Implied Credit factors = 70%
4.5
4
Def Rate ALL (%)
Def Rate INV (%) X 10
3.5
3
2.5
2
Correlation between credit factor and
market index
All ratings
= 23%
Investment grade = 29% *
1.5
1
0.5
0
1981
1986
1991
1996
2001
2006
EQ Inde x vs. ALL Grades
EQ Index vs. Investment Grade
4.000
4.00
Equity Index
3.000
Equity Index
Systematic Credit factor
3.00
Default Rate (%)
2.00
1.000
1.00
0.000
0.00
1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007
1981
-1.000
-1.00
-2.000
-2.00
© 2006-2011 R2 Financial Technologies
Systematic Credit factor
default rate (%) X 10
2.000
1986
1991
1996
2001
2006
52
22
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Empirical Market-Credit Correlations
Market credit correlations estimated for various “market factors”
Market Factor 1 vs. Investment Gade
Market Factor 1 vs. ALL Grades
3.000
2.500
3.000
Syst. Credit - ALL
2.500
EQ Index (-)
2.000
2.000
Market Factor 1
1.500
1.500
1.000
1.000
EQ Index (-)
Market Factor 1
0.500
0.500
0.000
-0.5001981
Syst. Credit - INV
1986
1991
1996
2001
0.000
-0.5001981
2006
-1.000
-1.000
-1.500
-1.500
-2.000
-2.000
1986
1991
Market Factor 2 vs. ALL grades
2.000
1.500
2001
2006
2001
2006
Market Factor 2 vs. Investment Grade
3.000
2.500
1996
2.500
2.000
Syst. Credit - ALL
Syst. Credit - INV
EQ Index (-)
EQ Index (-)
1.500
Market Factor 2
Market Factor 2
1.000
1.000
0.500
0.500
0.000
1981
-0.500
1986
1991
1996
2001
0.000
1981
-0.500
2006
-1.000
-1.000
-1.500
-1.500
-2.000
-2.000
© 2006-2011 R2 Financial Technologies
1986
1991
1996
54
Outline
Introduction: WWR
CVA calculation and WWR
CCR capital, CVA VaR and Basel III
Integrated market-credit (WWR) model for CVA and CCR capital
Example
Portfolio Exposure Summary: Regulatory Capital
One-Year Equivalent Exposures
One-Year Equivalent Mean Exposures
6
800%
95% Percen ti le
700%
%5 Percen til e
Mean exp
5
4
500%
3
400%
2
Frequency
% Mean exposure
600%
300%
1
200%
0
100%
Alpha (99.9%): Base Case vs. Stressed Exposures
0%
Run of April, 2008 (26/06/2008)
Exposure ('000s)
Counterparty (Top 70 mean exposures)
Alpha
1.20
1.10
1.10
1.00
1.00
A48010615
Single Step
Multi-Steps 6EAD4TE
59 TS - MS / MS
Multi-Steps284500046
59 TS - MS / SS
9CRUP AR
90000001
Y18500690
10048092
280580040
6KHBNYC
P40010019
1.20
0.90
120,191
83,296
102,277
79,307
80,209
51,071
78,495
70,976
73,494
44,392
-1
-0.75
1.50 30. 75%
1.40
12. 23%
1.40 11. 26%
1.30
-0.5
-0.25
0
Correlation
0.25
0.5
0.75
0.031
69
31
0.032
1 M/ MS
€
69
31
S tat istics
Single
Step for CP E xposure > 1 Euro
Multi-Steps 12 TS - MS
31,092
Multi-Steps 59 TSMean:
- MS / MS
Multi-Steps
12 TS - MS / SS
Effective Counterparty Number
Multi-Steps 59 TS
- MS / SS
Median:
15, 947.26
40
0.032
STD:
Kurtosis*:
Skew*:
Maximum :
Largest(10):
Smallest(10):
Minimum:
9. 58%
1.20
28. 68%
1.10
1.20 4. 72%
7. 41%
1.00
1.10 3. 58%
0.90
15. 45%
33,411
4. 53
2. 21
158,032
59,223
8, 505. 34
8, 053. 13
30
20
10
0
No of CPs
Copyright R2 Financial Technologies 2007
0.80
-1
1 0.90
HI
32
-0.75
-0.5
-0.25
0
Correlation
0.25
0.5
0.75
1
0.80
0.80
-1
-0.75 as a -0.5
0
0.25
Alpha (99.9%)
function -0.25
of market-credit
correlation
Correlation
Correlation
Alpha Total (LTS/LTD)
-1
Alpha (99.9%) as a function of market-credit correlation
Base Case
0.91 -0.75
Correlation
-1
Stressed Exposures
Alpha Total (LTS/LTD)
Alpha Systematic
(LSS/LSD)
Single Step
Multi-Steps 12 TS - MS / MS
Base Case
Stressed
MS / SSExposures
Multi-Steps 59 TS - MS / MS
MS / SS
Alpha Systematic (LSS/LSD)
Single Step
Multi-Steps 12 TS - MS / MS
MS / SS
Multi-Steps 59 TS - MS / MS
MS / SS
1.00
0.91
0.94
0.97
0.96
0.99
1.06
1.03
1.10
1.04
1.09
1.06
1.05
0.5
0.75
-0.75
0.89
-0.5
0.97
0.89
0.92
0.95
0.94
0.97
0.93
0.94
0.99
0.93
0.97
0.93
1.00
0.92
0.93
0.93
0.98
0.94
0.98
0.93
0.96
-0.75
-0.25
-0.25
0.97
0.92
0.95
0.97
0.96
0.99
-1
1
-0.5
0.97
0
0.99
0.97
0.99
1.02
1.00
1.03
0.96
0.94
1.00
0.96
1.01
0.96
0.97
-0.5
0
1.04
0.25
1.06
1.04
1.07
1.10
1.06
1.09
1.01
1.00
1.06
1.01
1.06
1.01
1.02
-0.25
0.25
1.12
0
Correlation
1.08
1.11
0.25
0.5
0.5
1.13
1.12
1.15
1.18
1.21
1.25
1.21
1.24
1.08
1.08
1.14
1.08
1.13
1.18
1.16
1.23
1.16
1.21
1.18
1.25
0.5
0.75
0.75
0.75
1.26
1.21
1.22
1.16
Client
Confidential
1.13
Client Confidential
© 2006-2011 R2 Financial Technologies
No of CP No of Eff CP
70
1.30
1.00
Client Confidential
0.80
0.90
158,032
199,954
Multi-Steps 12 TS - MS / MS
154,454
238,365
Multi-Steps 12 TS - MS
/ SS
152,864
116,262
110,398
140,345
91,878
90,347
82,675
74,456
127,155
98,508
95,401
94,691
84,931
79,592
78,014
59,223
Alph a
A lp ha
1.30
TotalBase
Exposure:
Exposure >
Case
Run2,176,463
of April, 2008 (26/06/2008)
No Counterparties:
70
1€
Stress Exposures
- Systematic (99.9%)
No CP ExpAlpha
Zero or Neg:
100, 000 €
1.70
1.60
Sigma
1.60
1.50
15. 19%
Alpha
1.50
1.40
1.80
Alpha
5.65
(99.9%):
Comparison
Steps
Me an Ex posure
Sta tistics ('000s)Single vs Multiple
Portfolio S
ize Sta tistics
A verage Mean / V ol:
Base Case
Top 10 Ex posures ('000 Euros)
Stress Exposures
Alpha - Total
(99.9%)95th Percentile
Counterparty
Mean
5t h P ercentile
1.50
1.30
17.71%
No of Eff CP
Average % Vol:
1.60
1.40
Alpha - Systematic (99.9%)
Alpha - Total (99.9%)
1.70
1.30
1
1.41
1.30
1.29
1.33
1.30
1.44
1
1
1.42
1.63
1.42
1.39
1.43
1.42
1.74
Copyright
2008
1.29 R2 Financial Technologies
1.38
1.33
1.42
1.30
1.24
1.32
1.26
1.31
1.42
1.37
1.45
1.34
1.40
Copyright R2 Financial Technologies 2008
58
23
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Counterparty Level Exposures
Counterparty Report: Exposure and CVA
Run of December, 2009 (18/10/2010 CP level)
400
Mean Exposure
350
Cond. Mean Exposure
3.20
Hazard Rate
HR Volatility
3.10
Asset
Correlation
300
95 Percentile
CVA (€ Million)
Exposure (€ Million)
250
200
150
100
50
0
Mean Exposure
Cond Mean Exposure
95 Percentile
80
3.00
20
0.25
-0.50
1.36
1
0.25
0.50
5475
5110
4745
4380
4015
3650
3285
0.50
3.20
1
2920
CVA - WWR Stress Test (€ million)
-0.50
-0.25
1
0.25
2.93
3.00
3.06
3.13
0.75
2555
0
Correlation
2190
1.31
-0.25
1825
-0.75
2.87
40
1460
-1
2.80
60
1095
Correlation
CVA
42.88
35.50
59.35
2.81
3.44
730
Statistics (€ million)
Exposure
2.90
Actual Exposure
EPE
2.80
Effective EPE
Effective Maturity*
2.70
Total Capital
CVA
Mkt-2.60
Credit Corr.
-1
-0.75
CVA Unilateral
CVA Bilateral
CVA Uncorrelated
0
Time in Days
Copyright R2 Financial Technologies 2010
0.024
0.0500
0.2149
Exposure
Testing
365
5475
5110
4745
4380
4015
3650
3285
2920
2555
2190
1825
1460
1095
730
0
365
Statistics (€ million)
Exposure
Actual Exposure
304.64
EPE
158.03
Effective EPE
305.84
Effective Maturity*
1.57
Total Capital
8.51
CVA
Market-Credit Corr.
0.25
CVA Unilateral
3.13
CVA Bilateral
CVA Uncorrelated
3.06
Counterparty Info
CVA Stress
Counterparty
6MPEBTQ
No of3.40
Instruments
11
120
Sector
INDUSTRIALS
BBB3.30 Rating
PD
0.3059%
100
LGD
0.40
Exposure
(€ Million)
Counterparty Info
Counterparty
A48010615
No of Instruments
111
Sector FINANCIALS
Rating
BBB+
PD
0.1367%
LGD
0.40
Hazard Rate
0.024
HR Volatility
0.0500
Asset Correlation
0.2149
Time in Days 1
0.75
3.26
3.33
Copyright R2 Financial Technologies 2010
61
© 2006-2011 R2 Financial Technologies
Portfolio – Exposures
Portfolio Exposure Summary: Regulatory Capital
One-Year Equivalent Exposures
One-Year Equivalent Mean Exposures
6
300%
95% Percentile
5
%5 Percentile
4
% Mean exposure
Mean exp
200%
3
150%
2
100%
Frequency
250%
1
50%
0
0%
Exposure ('000s)
Counterparty (Top 70 mean exposures)
17.71%
Average Mean / Vol:
5.65
Mean Exposure Statistics ('000s)
Total Exposure:
2,176,463
No Counterparties:
Top 10 Exposures ('000 Euros)
Counterparty
Mean
95th Percentile
5th Percentile
Sigma
P40010019
158,032
154,454
127,155
98,508
95,401
94,691
84,931
79,592
78,014
59,223
199,954
238,365
152,864
116,262
110,398
140,345
91,878
90,347
82,675
74,456
120,191
83,296
102,277
79,307
80,209
51,071
78,495
70,976
73,494
44,392
15.19%
30.75%
12.23%
11.26%
9.58%
28.68%
4.72%
7.41%
3.58%
15.45%
1
A48010615
2
6EAD4TE
3
284500046
4
9CRUPAR
5
90000001
6
Y18500690
7
10048092
8
280580040
9
6KHBNYC
10
No CP Exp Zero or Neg:
Statistics for CP Exposure >
Mean:
Median:
STD:
Kurtosis*:
Skew*:
Maximum :
Largest(10):
Smallest(10):
Minimum:
Client Conf idential
© 2006-2011 R2 Financial Technologies
Portfolio Size Statistics
Exposure >
70
1 Euro
31,092
15,947.26
33,411
4.53
2.21
158,032
59,223
8,505.34
8,053.13
No of CP No of Eff CP
HI
1€
70
32
0.031
100,000 €
69
31
0.032
Effective Counterparty Number
40
No of Eff CP
Average % Vol:
30
20
10
0
No of CPs
Copyright R2 Financial Technologies 2010
62
24
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Counterparty Exposure, CVA, and WWR
Counterparty Report: Exposure and CVA
Run of December, 2009 (18/10/2010 CP level)
Counterparty Info
Counterparty
6MPEBTQ
No of Instruments
11
INDUSTRIALS
Sector
Rating
BBBPD
0.3059%
LGD
0.40
Hazard Rate
0.024
HR Volatility
0.0500
Asset Correlation
0.2149
1.60
Mean Exposure
100
1.50
Cond Mean Exposure
95 Percentile
1.40
(CVA € Million)
80
(€ Million)
42.88
35.50
59.35
2.81
3.44
120
60
1.30
1.20
40
1.10
20
1.00
0.25
1.36
-1
-0.75
-0.50
0
Time in Days
-0.25
1
0.25
0.50
0.75
1
Correlation
5475
5110
4745
4380
4015
3650
3285
2920
2555
2190
1825
1460
1095
730
0
1.31
365
Statistics (€ million)
Exposure
Actual Exposure
EPE
Effective EPE
Effective Maturity*
Total Capital
CVA
Mkt- Credit Corr.
CVA Unilateral
CVA Bilateral
CVA Uncorrelated
CVA Stress Testing
Exposure
Correlation
CVA
-1
1.08
-0.75
1.14
CVA - WWR Stress Test (€ million)
-0.50
-0.25
1
0.25
1.20
1.26
1.31
1.36
0.50
1.41
0.75
1.45
1
1.48
Copyright R2 Financial Technologies 2010
© 2006-2011 R2 Financial Technologies
63
Concluding Remarks
© 2006-2011 R2 Financial Technologies
70
25
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
CCR, CVA and Model Risk
CVA is pricing – do we want to achieve “pricing-level accuracy”?
Marginal contributions, real time, securitization of CVA"
But" how accurate can we / will we do it in the near future?
This is probably the Most complex “instrument” we have ever priced!
Complex risks
Modelling CCR and CVA requires integrated market-credit models:
10s or 100s of market factors driving exposures over very long horizons
Wrong-way risk
Complex underlying portfolio of derivatives and mitigation
Netting, collateral, margin calls".
Note we cant price very well simpler instruments like synthetic CDOs!
Simple portfolios, standardized instruments, 125 credits".
© 2006-2011 R2 Financial Technologies
71
CCR, CVA and Model Risk
Model risk framework"
Acknowledge the “heroic” assumptions in our CCR models and the
limitation of the information which we can reasonably extract from the
market and quoted prices
Effectively incorporate fundamental credit information, historical data
and expert judgment into the valuation and risk measurement
processes
Develop explicit model risk and stress testing approaches which can
help us understand better
The behaviour of instruments and portfolios,
“Knightean” uncertainties we could be facing
© 2006-2011 R2 Financial Technologies
72
26
Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Bio – Dan Rosen
Dan Rosen is the CEO and co-founder of R2 Financial Technologies and an adjunct professor of
Mathematical Finance at the University of Toronto.
Dr. Rosen lectures extensively around the world on financial engineering, enterprise risk and capital
management, credit risk and market risk. He has authored several patents and numerous papers on
quantitative methods in risk management, applied mathematics, operations research, and has coauthored
two books and various chapters in risk management books (including two chapters of PRMIA’s Professional
Risk Manger Handbook). In addition, Dr. Rosen is a member of the Industrial Advisory Boards of the Fields
Institute and the Center for Advanced Financial Studies at the University of Waterloo; the Academic Advisory
Board of Fitch; the Advisory Board of the IAFE; a founder former Regional Director and current steering
committee member of PRMIA in Toronto; and a member of the Oliver Wyman Institute. He is also one of the
founders of RiskLab, an international network of research centers in Financial Engineering and Risk
Management. Dr. Rosen was inducted in 2010 as a fellow of the Fields Institute for Research in Mathematical
Sciences, for his “outstanding contributions to the Fields Institute, its programs, and to the Canadian
mathematical community”.
Prior to co-founding R2 , Dr. Rosen had a successful ten-year career at Algorithmics Inc., where he held
senior management roles in research and financial engineering, strategy and business development, and
product marketing. In these roles, he was responsible for setting strategic direction, new initiatives and
alliances; the design and positioning of credit risk and capital management solutions, market risk tools,
operational risk, and advanced simulation and optimization, as well as their application to industrial settings.
He holds an M.A.Sc. and Ph.D. in Chemical Engineering from the University of Toronto.
© 2006-2011 R2 Financial Technologies
73
Selected Recent Publications
Rosen D. and Saunders D. 2012, CVA the Wrong-Way, Journal of Risk Management in
Financial Institutions
Pykhtin M., and Rosen D. 2011, Pricing Counterparty Risk at the Trade Level and CVA
Allocations, Journal of Credit Risk (also see Federal Reserve Research Paper Series)
Rosen D. and Saunders D. 2010, Computing and Stress Testing Counterparty Credit Risk
Capital, in Counterparty Credit Risk Modelling, (ed. E. Canabarro), Risk Books
Garcia Cespedes J. C., de Juan Herrero J. A., Rosen D., Saunders D. 2010, Effective
modelling of Counterparty Credit risk Capital and Alpha, Journal of Risk Model validation
De Prisco B., Rosen D., 2005, Modelling Stochastic Counterparty Credit Exposures for
Derivatives Portfolios, Counterparty Credit Risk (M. Pykhtin, Editor), Risk Books
Rosen D. and Saunders D. 2011, Structured Finance Valuation and Risk Management with
Implied factor Models, in Advances in Credit Derivatives, Bloomberg Publications
Rosen D. , 2010, Rethinking Valuations, in Rethinking Risk Measurement, RISK Publications
Nedeljkovic, J., Rosen D. and Saunders D. 2010, Pricing and Hedging CLOs with Implied
Factor Models, Journal of Credit Risk, Fall issue
Rosen D. and Saunders D. 2009, Valuing CDOs of Bespoke Portfolios with Implied MultiFactor Models, Journal of Credit Risk, Fall Issue
© 2006-2011 R2 Financial Technologies
74
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Dan Rosen
R2 Financial Technologies
Counterparty Credit Risk Workshop
PRMIA, May 2012
Selected Recent Publications
Rosen D. and Saunders D. 2010, Economic Capital, in Encyclopedia of Quantitative Finance
Rosen D. and Saunders D. 2010, Risk Contributions and Economic Credit Capital Allocation,
in Advances in Credit Derivatives, Bloomberg Publications (forthcoming)
Rosen D. and Saunders D. 2010, Measuring Capital Contributions of Systemic Factors in
Credit Portfolios, Journal of Banking and Finance
Rosen D. and Saunders D. 2009, Analytical Methods for Hedging Systematic Credit Risk
with Linear Factor Portfolios, Journal of Economic Dynamics and Control
Mausser H. and Rosen D. 2007, Economic Credit Capital Allocation and Risk Contributions,
in Handbook of Financial Engineering (J. Birge and V. Linetsky Editors)
Garcia Cespedes J. C., Keinin A., de Juan Herrero J. A. and Rosen D. 2006, A Simple
Multi-Factor “Factor Adjustment” for Credit Capital Diversification, Special issue on Risk
Concentrations in Credit Portofolios (M. gordy, editor) Journal of Credit Risk, Fall 2006
Rosen D., 2004, Credit Risk Capital Calculation, in Professional Risk Manager (PRM)
Handbook, Chapter III.B5, PRMIA Publications
Aziz A., Rosen D., 2004, Capital Allocation and RAPM, in Professional Risk Manager (PRM)
Handbook, Chapter III.0, PRMIA Publications
© 2006-2011 R2 Financial Technologies
75
www.R2-financial.com
© 2006-2011 R2 Financial Technologies
28