Modeling Credit Exposure for Collateralized Counterparties

Michael Pykhtin

Credit Analytics & Methodology

Bank of America

Fields Institute Quantitative Finance Seminar

Toronto; February 25, 2009

2

Disclaimer

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or its contents.

3

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

4

Margin agreements as a means of reducing counterparty credit exposure

5

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.

6

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

7

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

8

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.

9

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 ≡

10

Collateralized exposure and the margin period of risk

11

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

<= = < < >

12

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

13

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∆ = − − − − − −

14

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=

15

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 −

16

Semi-analytical method for collateralized EE

17

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

18

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

19

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

20

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

21

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−

22

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 ∂=

∂

23

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=

24

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

25

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

26

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

27

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

+ − += =

28

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%

29

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

30

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

31

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%

32

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