Modeling Credit Exposure for Collateralized Counterparties · 7 Exposure at Counterparty Level...
Transcript of Modeling Credit Exposure for Collateralized Counterparties · 7 Exposure at Counterparty Level...
Modeling Credit Exposure for Collateralized Counterparties
Michael Pykhtin
Credit Analytics & Methodology
Bank of America
Fields Institute Quantitative Finance Seminar
Toronto; February 25, 2009
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Disclaimer
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or its contents.
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Discussion Plan
� Margin agreements as a means of reducing counterparty credit exposure
� Collateralized exposure and the margin period of risk
� Semi-analytical method for collateralized EE
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Margin agreements as a means of reducing counterparty credit exposure
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Introduction
� Counterparty credit risk is the risk that a counterparty in an OTC derivative transaction will default prior to the expiration of the contract and will be unable to make all contractual payments.
– Exchange-traded derivatives bear no counterparty risk.
� The primary feature that distinguishes counterparty risk from lending risk is the uncertainty of the exposure at any future date.
– Loan: exposure at any future date is the outstanding balance, which is certain (not taking into account prepayments).
– Derivative: exposure at any future date is the replacement cost, which is determined by the market value at that date and is, therefore, uncertain.
� For the derivatives whose value can be both positive and negative (e.g., swaps, forwards), counterparty risk is bilateral.
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Exposure at Contract Level
� Market value of contract i with a counterparty is known only for current date . For any future date t, this value is uncertain and should be assumed random.
� If a counterparty defaults at time τ prior to the contract maturity, economic loss equals the replacement cost of the contract
– If , we do not receive anything from defaulted counterparty, but have to pay to another counterparty to replace the contract.
– If , we receive from another counterparty, but have to forward this amount to the defaulted counterparty.
� Combining these two scenarios, we can specify contract-level exposure at time t according to
( )i
V t0t =
( ) max[ ( ),0]i i
E Vτ τ=
( )i
E t
( ) 0i
V τ >
( ) 0i
V τ <
( )i
V τ( )
iV τ
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Exposure at Counterparty Level
� Counterparty-level exposure at future time t can be defined as the loss experienced by the bank if the counterparty defaults at time t under the assumption of no recovery
� If counterparty risk is not mitigated in any way, counterparty-level exposure equals the sum of contract-level exposures
� If there are netting agreements, derivatives with positive value at the time of default offset the ones with negative value within each netting set , so that counterparty-level exposure is
– Each non-nettable trade represents a netting set
( ) ( ) max[ ( ),0]i i
i i
E t E t V t= =∑ ∑
NS
NS
( ) ( ) max ( ), 0k
k
i
k k i
E t E t V t∈
= =
∑ ∑ ∑
NSk
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Margin Agreements
� Margin agreements allow for further reduction of counterparty-level exposure.
� Margin agreement is a legally binding contract between two counterparties that requires one or both counterparties to post collateral under certain conditions:
– A threshold is defined for one (unilateral agreement) or both (bilateral agreement) counterparties.
– If the difference between the net portfolio value and already posted collateral exceeds the threshold, the counterparty must provide collateral sufficient to cover this excess (subject to minimum transfer amount).
� The threshold value depends primarily on the credit quality of the counterparty.
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Collateralized Exposure
� Assuming that every margin agreement requires a netting agreement, exposure to the counterparty is
where is the market value of the collateral for netting set NSk at time t.
– If netting set NSk is not covered by a margin agreement, then
� To simplify the notations, we will consider a single netting set:
where VC (t) is the collateralized portfolio value at time t given by
NS
( ) max ( ) ( ), 0k
C i k
k i
E t V t C t∈
= −
∑ ∑
( )k
C t
{ }( ) max ( ), 0C C
E t V t=
( ) ( ) ( ) ( ) ( )C i
i
V t V t C t V t C t= − = −∑
( ) 0k
C t ≡
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Collateralized exposure and the margin period of risk
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Naive Approach
� Collateral covers excess of portfolio value V(t) over threshold H:
� Therefore, collateralized portfolio value is
� Thus, any scenario of collateralized exposure
is limited by the threshold from above and by zero from below.
( ) max{ ( ) ,0}C t V t H= −
( ) ( ) ( ) min{ ( ), }C
V t V t C t V t H= − =
{ }0 if ( ) 0
( ) max ( ), 0 ( ) if 0 ( )
if ( )
C C
V t
E t V t V t V t H
H V t H
<= = < < >
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Margin Period of Risk
� Collateral is not delivered immediately – there is a lag δtcol.
� After a counterparty defaults, it takes time δtliq to liquidate the portfolio.
� When loss on the defaulted counterparty is realized at time τ , the last time the collateral could have been received is τ −δt, where δt = δtcol + δtliq is the margin period of risk (MPR).
� Thus, collateral at time t is determined by portfolio value at time τ −δt .
� While δt is not known with certainty, it is usually assumed to be a fixed number.
– Assumed value of δ t depends on the portfolio liquidity
– Typical assumption for liquid trades is δ t =2 weeks
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Including MPR in the Model
� Suppose that at time t −δt we have collateral collateral C(t −δt) and portfolio value is V(t −δt)
� Then, the amount ∆C(t) that should be posted by time t is
– Negative ∆C(t) means that collateral will be returned
� Collateral C(t) available at time t is
� Collateralized portfolio value is
( ) ( ) ( ) max{ ( ) ,0}C t C t t C t V t t Hδ δ= − + ∆ = − −
( ) ( ) ( ) min{ ( ), ( )}C
V t V t C t V t H V tδ= − = +
( ) ( ) ( )V t V t V t tδ δ= − −
( ) max{ ( ) ( ) , ( )}C t V t t C t t H C t tδ δ δ∆ = − − − − − −
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Full Monte Carlo Algorithm
� Suppose we have a set of primary simulation time points {tk}for modeling non-collateralized exposure
� For each tk >δt , define a look-back time point tk −δt
� Simulate non-collateralized portfolio value along the path that includes both primary and look-back simulation times
� Given V(tk−1) and C(tk−1), we calculate
– Uncollateralized portfolio value V(tk −δ t) at next look-back time tk −δ t
– Uncollateralized portfolio value V(tk) at next primary time tk
– Collateral at tk :
– Collateralized value at tk :
– Collateralized exposure at tk :
( ) max{ ( ) ,0}k k
C t V t t Hδ= − −
( ) ( ) ( )C k k k
V t V t C t= −
{ }( ) max ( ), 0C k C k
E t V t=
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Illustration of Full Monte Carlo Method
�Simulating collateralized portfolio value
– Collateralized exposure can go above the threshold due to MPR and MTA
H
tk-1
Portfolio
Value
δ ttkδ t
( )C k
V t
1( )
C kV t −
( )k
V t1
( )k
V t −
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Semi-analytical method for collateralized EE
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Portfolio Value at Primary Time Points
� Let us assume that we have run simulation only for primary time points t and obtained portfolio value distribution in the form of M quantities , where j (from 1 to M) designates different scenarios
� From the set we can estimate the unconditional expectation µ (t) and standard deviation σ (t) of the portfolio value, as well as any other distributional parameter
� Can we estimate collateralized EE profile without simulating portfolio value at the look-back time points ?
( )( )jV t
( ){ ( )}jV t tδ−
( ){ ( )}jV t
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Collateralized EE Conditional on Path
� Collateralized EE can be represented as
where is the collateralized EE conditional on :
� Collateralized portfolio value is
� If we can calculate analytically, the unconditional collateralized EE can be obtained as the simple average of
over all scenarios j
( ) ( ) ( )EE ( ) E max{ ( ),0} ( )j j j
C Ct V t V t =
( )EE ( ) E[EE ( )]j
C Ct t=
( )EE ( )j
Ct
( )EE ( )j
Ct
( )EE ( )j
Ct
( )( )jV t
{ }( ) ( ) ( ) ( )( ) min ( ), ( ) ( )j j j j
CV t V t H V t V t tδ= + − −
( )( )j
CV t
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If Portfolio Value Were Normal…
� Let us assume that portfolio value V(t) at time t is normally distributed with expectation µ (t) and standard deviation σ (t).
� Then, we can construct Brownian bridge from to
� Conditionally on , has normal distributionwith expectation
and standard deviation
� Conditional collateralized EE can be obtained in closed form!
( )( )jV t
( )( )jV t tδ−
( ) ( )( ) (0) ( )j jt t tt V V t
t t
δ δα −= +
( )
2
( )( ) ( )j t t tt t
t
δ δβ σ −=
(0)V
( )( )jV t
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Illustration: Brownian Bridge
� Brownian bridge from to
� Conditionally on , the distribution of is
normal with mean and standard deviation
( )( )jV t
t0 t− δt
(0)V
H
( )( )jV t(0)V
( )( )jtα ( )( )j
tβ( )( )j
V t( )( )j
V t tδ−
( )( )jtα
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Arbitrary Portfolio Value Distribution
� We will keep the assumption that, conditionally on ,the distribution of is normal, but will replace σ (t)with the local quantity σ loc(t)
� Let us describe portfolio value V(t) at time t as
where is a monotonically increasing function of a standard normal random variable Z.
� Let us also define a normal equivalent portfolio value as
� To obtain σ loc(t) , we will scale σ (t) by the ratio of probability densities of W(t) and V(t)
( ) ( , )V t v t Z=
( , )v t Z
( ) ( , ) ( ) ( )W t w t Z t t Zµ σ= = +
( )( )jV t
( )( )jV t tδ−
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Scaled Standard Deviation
� Let us denote probability density of quantity X via and scale the standard deviation according to
� Changing variables from W(t) and V(t) to Z , we have
� Substitution to the definition of σ loc(t,Z) above gives
( )
loc
( )
[ ( , )]( , ) ( )
[ ( , )]
W t
V t
f w t Zt Z t
f v t Zσ σ=
( )
( )[ ( , )]
( , ) /V t
Zf v t Z
v t Z Z
φ=∂ ∂ ( )
( )[ ( , )]
( )W t
Zf w t Z
t
φσ
=
( )X
f ⋅
loc
( , )( , )
v t Zt Z
Zσ ∂=
∂
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Estimating CDF
� Value of corresponding to can be obtained from
� Let us sort the array in the increasing order so that
where j (k) is the sorting index
� From the sorted array we can build a piece-wise constant CDF that jumps by 1/M as V (t) crosses any of the simulated values:
( )( )jV t
[ ( )]
( )
1 1 1 2 1[ ( )]
2 2 2
j k
V t
k k kF V t
M M M
− −≈ + =
( )( ) 1 ( )
( )[ ( )]j j
V tZ F V t
−= Φ
( )jZ
( )( )jV t
[ ( )] ( )
sorted( ) ( )j k kV t V t=
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Estimating Derivative
� Now we can obtain corresponding to as
� Local standard deviation can be estimated as :
� Offset ∆k should not be too small (too much noise) or too large (loss of resolution). This range works well:
( )( )jV t
[ ( )] [ ( )][ ( )] [ ( )]
loc loc [ ( )] [ ( )]
( ) ( )( ) ( , )
j k k j k kj k j k
j k k j k k
V t V tt t Z
Z Zσ σ
+∆ −∆
+∆ −∆
−≡ ≈−
[ ( )] 1 2 1
2
j k kZ
M
− − = Φ
( )jZ
20 0.05k M≤ ∆ ≤
( )
loc ( )jtσ
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Back to the Bridge
� We assume that, conditionally on , has normal distribution with expectation
and standard deviation
� Collateralized exposure depends on , which is also normal conditionally on with the same standard deviation
and expectation given by
( )( )jV t tδ−
( ) ( )( ) (0) ( )j jt t tt V V t
t t
δ δα −= +
( ) ( )
loc 2
( )( ) ( )j j t t tt t
t
δ δβ σ −=
( )( )jV t
( )( )jV tδ
( )( )jtβ ( )( )j
tδα( ) ( ) ( ) ( )( ) ( ) ( ) ( ) (0)j j j jt
t V t t V t Vt
δδα α = − = −
( )( )jV t
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Calculating Conditional Collateralized EE
� Collateralized EE conditional on scenario j at time t is
� equals zero whenever , so that
� Since has normal distribution, we can write
{ }{ }( ) ( ) ( ) ( )EE ( ) E max min ( ), ( ) ,0 ( )j j j j
Ct V t H V t V tδ = +
{ } { }( )
( ) ( ) ( ) ( )
( ) 0EE ( ) 1 E min ( ), ( ) ( )
j
j j j j
C V tt V t H V t V tδ
> = +
( )EE ( )j
Ct ( )( ) 0j
V t <
{ } { }
{ }1
2 1
( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) 0
( ) 0
EE ( ) 1 min ( ), ( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( )
j
jk
j j j j
C
d
j j j
d d
V t
V t
t V t H t t z z dz
H t t z z dz V t z dz
δα β φ
δα β φ φ
∞
−∞− ∞
− −
>
>
= + +
= + + +
∫
∫ ∫
( )( )jV tδ
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Conditional Collateralized EE Result
� Evaluating the integrals, we obtain:
where
{ } [ ]{[ ] }
( )
( ) ( )
2 1
( ) ( )
2 1 1
( ) 0EE ( ) 1 ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
j
j j
C
j j
V tt H t d d
t d d V t d
δα
β φ φ
> = + Φ − Φ
+ − + Φ
( ) ( ) ( )
1 2( ) ( )
( ) ( ) ( )
( ) ( )
j j j
j j
H t V t H td d
t t
δα δαβ β
+ − += =
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Example 1: 5-Year IR Swap Starting in 5 Years
� Uncollateralized EE and the two thresholds we will consider
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
Exp
osu
re
[ %
of
no
tio
nal
]EE (no collateral) Threshold 0.5% Threshold 2.0%
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Forward Starting Swap and Small Threshold
� Collateralized EE when threshold is 0.5%
0.00%
0.05%
0.10%
0.15%
0.20%
0.25%
0.30%
0.35%
0.40%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
d E
xp
osu
re
[ %
of
no
tio
nal
]MPR = 0 Full Monte Carlo Semi-Analytical
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Forward Starting Swap and Large Threshold
� Collateralized EE when threshold is 2.0%
0.00%
0.10%
0.20%
0.30%
0.40%
0.50%
0.60%
0.70%
0.80%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
d E
xp
osu
re
[ %
of
no
tio
nal
]MPR = 0 Full Monte Carlo Semi-Analytical
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Example 2: 5-Year IR Swap Starting Now
� Uncollateralized EE and the two thresholds we will consider
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
Exp
osu
re
[ %
of
no
tio
nal
]EE (no collateral) Threshold 0.5% Threshold 2.0%
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Swap Starting Now and Small Threshold
� Collateralized EE when threshold is 0.5%
0.00%
0.05%
0.10%
0.15%
0.20%
0.25%
0.30%
0.35%
0.40%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
d E
xp
osu
re
[ %
of
no
tio
nal
]MPR = 0 Full Monte Carlo Semi-Analytical
33
Swap Starting Now and Large Threshold
� Collateralized EE when threshold is 2.0%
0.00%
0.10%
0.20%
0.30%
0.40%
0.50%
0.60%
0.70%
0.80%
0.0 1.0 2.0 3.0 4.0 5.0
Time [ years ]
Exp
ecte
d E
xp
osu
re
[ %
of
no
tio
nal
]MPR = 0 Full Monte Carlo Semi-Analytical
34
Conclusion
� Margin agreements are important risk mitigation tools that need to be modeled accurately
� Collateral available at a primary time point depends on the portfolio value at the corresponding look-back time point
� Full Monte Carlo method of simulating collateralized exposure is the most flexible approach, but requires simulating portfoliovalue at both primary and look-back time points
� We have developed a semi-analytical method of calculating collateralized EE that avoids doubling the simulation time
– Portfolio value is simulated only at primary time points
– For each portfolio value scenario at a primary time point, conditional collateralized EE is calculated in closed form
– Unconditional collateralized EE at a primary time point is obtained by averaging the conditional collateralized EE over all scenarios