Modeling Credit Exposure for Collateralized Counterparties Michael - - PowerPoint PPT Presentation

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

This document is NOT a research report under U.S. law and is NOT a product of a fixed income research department. Opinions expressed here do not necessarily represent opinions or practices of Bank of America N.A. The analyses and materials contained herein are being provided to you without regard to your particular circumstances, and any decision to purchase or sell a security is made by you independently without reliance on us. This material is provided for information purposes only and is not an offer or a solicitation for the purchase or sale of any financial instrument. Although this information has been obtained from and is based

• n sources believed to be reliable, we do not guarantee its accuracy. Neither Bank of

America N.A., Banc Of America Securities LLC nor any officer or employee of Bank

• f America Corporation affiliate thereof accepts any liability whatsoever for any

direct, indirect or consequential damages or losses arising from any use of this report

• r 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 t t = ( ) max[ ( ),0]

i i

E V τ τ = ( )

i

E t

( )

i

V τ > ( )

i

V τ < ( )

i

V τ ( )

i

V 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 ( ), 0

k 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 ( ) ( ), 0

k

C i k k i

E t V t C t

∈

  = −    

∑ ∑

( )

k

C t

{ }

( ) max ( ), 0

C C

E t V t = ( ) ( ) ( ) ( ) ( )

C i i

V t V t C t V t C t = − = −

∑

( )

k

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 ( ) ( ) 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 δt depends on the portfolio liquidity – Typical assumption for liquid trades is δt =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 H δ δ = − + ∆ = − − ( ) ( ) ( ) min{ ( ), ( )}

C

V t V t C t V t H V t δ = − = + ( ) ( ) ( ) V t V t V t t δ δ = − − ( ) max{ ( ) ( ) , ( )} C t V t t C t t H C t t δ 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 −δt) at next look-back time tk −δt – 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 H δ = − − ( ) ( ) ( )

C k k k

V t V t C t = −

{ }

( ) max ( ), 0

C 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 δt tk δt ( )

C k

V t

1

( )

C k

V t − ( )

k

V t

1

( )

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 ?

( )( ) j

V t

( )

{ ( )}

j

V t t δ −

( )

{ ( )}

j

V 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

• ver all scenarios j

( ) ( ) ( )

EE ( ) E max{ ( ),0} ( )

j j j C C

t V t V t   =  

( )

EE ( ) E[EE ( )]

j C C

t t =

( )

EE ( )

j C t ( )

EE ( )

j C t ( )

EE ( )

j C t ( )( ) j

V t

{ }

( ) ( ) ( ) ( )

( ) min ( ), ( ) ( )

j j j j C

V t V t H V t V t t δ = + − −

( )( ) j C

V 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 distribution

with expectation and standard deviation

Conditional collateralized EE can be obtained in closed form!

( )( ) j

V t

( )(

)

j

V t t δ −

( ) ( )

( ) (0) ( )

j j

t t t t V V t t t δ δ α − = +

( ) 2

( ) ( ) ( )

j

t t t t t t δ δ β σ − = (0) V

( )( ) j

V t

20

Illustration: Brownian Bridge

Brownian bridge from to Conditionally on , the distribution of

is normal with mean and standard deviation

( )( ) j

V t

t t−δt

(0) V H

( )( ) j

V t (0) V

( )( ) j t

α

( )( ) j t

β

( )( ) j

V t

( )(

)

j

V t t δ −

( )( ) j t

α

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 Z µ σ = = +

( )( ) j

V t

( )(

)

j

V t t δ −

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 Z t Z t f v t Z σ σ =

( )

( ) [ ( , )] ( , )/

V t

Z f v t Z v t Z Z φ = ∂ ∂

( )

( ) [ ( , )] ( )

W t

Z f w t Z t φ σ = ( )

X

f ⋅

loc

( , ) ( , ) v t Z t Z Z σ ∂ = ∂

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:

( )( ) j

V t

[ ( )] ( )

1 1 1 2 1 [ ( )] 2 2 2

j k V t

k k k F V t M M M − − ≈ + =

( )

( ) 1 ( ) ( )[

( )]

j j V t

Z F V t

−

= Φ

( ) j

Z

( )( ) j

V t

[ ( )] ( ) sorted

( ) ( )

j k k

V 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:

( )( ) j

V t

[ ( )] [ ( )] [ ( )] [ ( )] loc loc [ ( )] [ ( )]

( ) ( ) ( ) ( , )

j k k j k k j k j k j k k j k k

V t V t t t Z Z Z σ σ

+∆ −∆ +∆ −∆

− ≡ ≈ −

[ ( )] 1 2

1 2

j k

k Z M

−

−   = Φ    

( ) j

Z 20 0.05 k M ≤ ∆ ≤

( ) loc ( ) j t

σ

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

( )(

)

j

V t t δ −

( ) ( )

( ) (0) ( )

j j

t t t t V V t t t δ δ α − = +

( ) ( ) loc 2

( ) ( ) ( )

j j

t t t t t t δ δ β σ − =

( )( ) j

V t

( )( ) j

V t δ

( )( ) j t

β

( )( ) j t

δα

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) (0)

j j j j

t t V t t V t V t δ δα α   = − = −  

( )( ) j

V 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 C t

V t H V t V t δ   = +  

{ }

{ }

( )

( ) ( ) ( ) ( ) ( ) 0

EE ( ) 1 E min ( ), ( ) ( )

j

j j j j C V t

t V t H V t V t δ

>

  = +  

( )

EE ( )

j C t ( )( ) j

V t <

{ }

{ }

{ }

1 2 1

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) 0 ( ) 0

EE ( ) 1 min ( ), ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( )

j j k

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

∞ −∞ − ∞ − −

> >

= + +       = + + +        

∫ ∫ ∫

( )( ) j

V t δ

27

Conditional Collateralized EE Result

Evaluating the integrals, we obtain:

where

{ }

[ ]

{

[ ]

}

( )

( ) ( ) 2 1 ( ) ( ) 2 1 1

( ) 0

EE ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

j

j j C j j

V t

t 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 t d d t t δα δα β 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 ] Expecte Exposure [ % of notional ] 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 ] Expected Exposure [ % of notional ] 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 ] Expected Exposure [ % of notional ] 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 ] Expecte Exposure [ % of notional ] 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 ] Expected Exposure [ % of notional ] 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 ] Expected Exposure [ % of notional ] 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 portfolio value 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