Accounting for Dynamic Initial Margin in Credit Exposure Models

Leif Andersen (Bank of America) Michael Pykhtin (Federal Reserve Board)

Risk MindsAmsterdam

December 9, 2015

The opinions expressed here are our own, and do not necessarily reflect the views of Bank of America or Federal Reserve Board

2

Discussion Plan

Fundamentals of exposure modeling

Dynamic vs. static initial margin

Estimating initial margin on a simulation path

The scaling approach for expected exposure

The regression approaches

Fundamentals of exposure modeling

4

Credit Exposure and Margin Agreements

Credit exposure E(t) of a bank (B ) to a counterparty (C ) at a future time point t is given by B’s cost to close out the portfolio at t assuming C’s default preceding t– The close-out cost depends on the trades’ MTM value, and is, therefore,

uncertain for all future time points.

Margin agreement is a contract between two counterparties that requires one or both counterparties to post collateral– Credit Support Annex (CSA) to ISDA Master Agreement is most common

Collateral posted by C to B reduces B’s exposure to C’s default– Variation margin (VM) reduces the current exposure to a pre-specified

level, set by a threshold

– Initial margin (IM) protects B against the risk of exposure increase that is not matched by VM payments by C in the event of C’s default

5

Exposure in Presence of Variation Margin

Assuming a single netting set covered by VM (but no IM), B’s exposure to C can generally be represented as:

UP(t) is the value of net trade flows (TF) unpaid by both C (positive) and B (negative), accrued to the closeout time t– We use the term trade flows instead of trade cash flows to account for

physically settled trades (e.g., at the time of C’s default, B was supposed to enter into a swap with C by exercising a physically settled swaption)

VM(t) is the collateral actually available to B at closeout time t

– : at time t, B holds collateral in the amount |VM(t)|

– : at time t, B has posted collateral in the amount |VM(t)|

– : at time t, B neither holds nor has posted collateral

VM( ) 0t <

VM( ) 0t >

VM( ) 0t =

[ ]( ) ( ) UP( ) VM( )E t V t t t += + −

6

Model of Variation Margin

The classical model of collateral assumes that there is a delay δ , known as the margin period of risk (MPR), between the last margin call that C responds to and closing out the portfolio

Let us denote by C(t) collateral prescribed by the margin agreement for time t. – This is collateral B is entitled to hold given the market history up to t

Collateral available to B at time t is set equal to the prescribed collateral for time t − δ :

Suppose, CSA is bilateral with thresholds HC for C and HB for B. If we ignore minimum transfer amount and rounding, C(t) is

VM( ) ( )t C t δ= −

[ ] [ ]C B( ) ( ) ( )C t V t H V t H+ += − − − −

7

Model of Unpaid Trade Flows

A key assumption in modeling UP(t) is the times when B and Cstop making trade payments, given closeout at t after C’s default– Any possibility of B making a trade payment to C within the MPR would

result in a spike in the exposure profile because C would not post VM.

To avoid the spikes, we assume that no trade payments are made by either B or C within the MPR:

– where is the value at time t of the net trade flows scheduled to be paid by C (positive) and B (negative) between times s and u

For a bilateral CSA with zero thresholds, exposure becomes:

– For more realistic models of exposure under variation margin, see Andersen, Pykhtin and Sokol (2015)

UP( ) TF( ;[ , ])t t t tδ= −

[ ]( ) ( ) TF( ;[ , ]) ( )E t V t t t t V tδ δ += + − − −

TF( ;[ , ])t s u

Dynamic vs. static initial margin

9

Initial Margin

Traditionally, banks have required that certain counterparties, perceived as high risk, post initial margin (IM) – IM is posted at trade inception directly to the bank

– IM is calculated at a trade level (e.g., 10-day 95% VaR)

– IM is entirely deterministic (either stays fixed or amortizes according to a pre-specified schedule) and is returned at trade maturity

BCBS and IOSCO introduced new margin requirements for bilaterally cleared portfolios (BCBS-261):– IM is posted by both bank and counterparty in a segregated account

– IM is calculated at a netting set level as 99% stress VaR over time horizon h = 9 + a days, where a is the re-margining interval No diversification or offsetting across broad asset classes is allowed

– IM is calculated on a periodic basis and is posted/returned by both bank and counterparty, as dictated by the calculations

10

ISDA SIMM

It is critically important that both counterparties in an OTC derivative portfolio agree on the amounts of IM that either counterparty has to post

To facilitate this, ISDA is developing the Standard Initial Margin Model (SIMM) that any bank trading OTC derivatives could use

The SIMM is based on the standardized approach of the FRTB – Risk factor sensitivities to standard risk factors are calculated internally

– The sensitivities are multiplied by standard risk weights and aggregated across risk factors via a set of standard correlations

ISDA and BCBS’s Working Group on Margin Requirements (WGMR) are still discussing certain aspects of the SIMM…

11

The Challenge of Modeling Future IM

To calculate future exposure for CVA, capital or limits, one needs to model future IM requirements– Expected future IM is also needed for margin valuation adjustment (MVA)

In the presence of IM, exposure at future time point t is given by

Traditional IM is deterministic, so the calculations are trivial:– One has to aggregate the projected IM across all the trades in a given

netting set that are still alive at the time point of interest

The BCBS-IOSCO IM is a dynamic quantity that varies along the simulation time and across the paths

Direct application of the SIMM is unrealistic:– One would have to compute a set of future trade value sensitivities to

relevant risk factors at each simulation time along each simulated path!

[ ]( ) ( ) UP( ) VM( ) IM( )E t V t t t t += + − −

12

Model Set-Up for VM and IM

Applying the model we postulated for VM and unpaid trade flowsto a two-way CSA with zero thresholds, we have:

where we have used the notation for

The BCBS-IOSCO IM is generally defined as the netting set VaR for time interval δ and confidence level q , so we define:

– Neither stress calibration, nor restrictions on diversification are applied

– Therefore, this IM definition is less conservative than what Basel requires, thus resulting in a more conservative exposure!

[ ]( ) ( , ) IM( )E t V t t tδ += ∆ − −

IM( ) Q ( , )q tt V t t δδ − = ∆ − F

( , ) ( ) TF( ;[ , ]) ( )V t t V t t t t V tδ δ δ∆ − = + − − −

( , )V t tδ∆ −

Estimating IM on a simulation path

14

The Noise Curse

The most straightforward approach for calculating exposure is to use regression techniques to estimate IM on each simulation path and then subtracting it from the P&L realized on that path

However, there is a pitfall in this approach: because the IM confidence level is so high ( q = 99% ), only very few paths (1% on average) would result in non-zero exposure

Let us consider an example of a 1-year IR swap with quarterly payments with IR curve driven by a single lognormal factor

Because the IR model is single-factor, IM on a path can be calculated exactly by shifting the risk factor to the 99% confidence level and repricing the swap– Any regression approach for estimating IM on a path would add a

noticeable error to this idealistic calculation

15

Example: EE for 1-year IR Swap

EE is calculated without IM (left) and with IM (right) – The number of paths is the same in both cases: 5,000

0.000%

0.010%

0.020%

0.030%

0.040%

0.050%

0.060%

0.070%

0.00 0.25 0.50 0.75 1.00

Expe

cted

Exp

osur

e [

% o

f no

tiona

l ]

Time [ years ]

EE without IM

0.0000%

0.0001%

0.0002%

0.0003%

0.0004%

0.0005%

0.0006%

0.0007%

0.0008%

0.0009%

0.00 0.25 0.50 0.75 1.00

Expe

cted

Exp

osur

e [

% o

f no

tiona

l ]

Time [ years ]

EE with IM

The scaling approach for EE

17

Defining the Scaling Multiplier

We have defined exposure with IM as

Taking the expectation results in EE with IM

EE without IM is given by

The scaling multiplier is defined as

( )( ) ( , ) Q ( , )q tE t V t t V t t δδ δ −

+ = ∆ − − ∆ − F

( )EE( ) (1 ) ES ( , ) Q ( , )q t q tt q V t t V t tδ δδ δ− − = − ∆ − − ∆ − F FE

[ ]( )(0)EE ( ) ( , ) tt V t t δδ −+ = ∆ −

FE E

(0)

EE( )( )EE ( )

ttt

λ =

18

Local Normal Approximation (1)

The idea of the scaling approach is to calculate only EE without IM and then scale it down via the pre-computed multiplier

Suppose that portfolio value follows an Itô process

– Note that µ t and σ t can depend on history of any risk factors up to time t

Then, if the horizon δ is sufficiently small, the increment of portfolio value over δ conditional on is normally distributed

where X is a standard normal random variable

Note that we have not made any assumption on the distribution of portfolio value V(t)

( ) t t tdV t dt dWµ σ= +

( , ) t tV t t Xδ δδ µ δ σ δ− −∆ − = +t δ−F

19

Local Normal Approximation (2)

Substituting the expression for in the expressions for EE with and without IM results in

The resulting scaling multiplier is a constant that depends only on the IM confidence level q

This is a remarkable result, given that we only assumed that the portfolio value follows an Itô process and that δ is small– Gregory (2015) derives this result for a Gaussian process for V(t)

[ ] ( ) ( )1 1EE( ) ( ) ( ) 1tt q q qδσ δ φ − −−

= Φ −Φ − E

[ ](0)EE ( ) (0)tt δσ δ φ−= E

( ) ( )1 1

(0)

( ) 1 ( )EE( )( )EE ( ) (0)

q q qttt

φλ

φ

− −Φ − − Φ≡ =

( , )V t tδ∆ −

20

Local Normal Approximation (3)

EE scaling multiplier λ as a function of the IM confidence level q– At BCBS-IOSCO IM confidence level q = 99%, the multiplier is λ = 0.85%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

50% 60% 70% 80% 90% 100%

EE S

calin

g M

ultip

lier

Confidence Level

21

Is Local Normal Approximation Accurate?

While the Itô process assumption for portfolio value is quite general, the assumption of smallness of δ can be questioned

Suppose that portfolio value is a linear function of a risk factor that follows a geometric Brownian motion with volatility σ– While this specification is very restrictive, we only use it to test

the accuracy of the local normal approximation

Then, one can show that the scaling multiplier is given by

– Quantity ψ is the direction of the trade (has value +1 for long; −1 for short)

– Terms quadratic in have been neglected

( ) ( )( ) ( )

1 11 ( ) (1 )exp ( )

(1/ 2) (1/ 2)

q q qψ σ δ ψ σ δλ ψ

σ δ σ δ

− −− Φ Φ − − − Φ=

Φ − Φ −

σ δ

22

Deviations from Local Normal Approximation

EE scaling multiplier λ as a function of volatility σ– BCBS-IOSCO IM parameters of q = 99% and δ = 10 business days are used

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

0% 10% 20% 30% 40% 50% 60% 70%

EE S

calin

g M

ultip

lier

Relative Volatility

Norlmal Lognormal - Long Lognormal - Short

23

Trade Non-Linearity

Trade non-linearity can result in further deviations from the local normal approximation

The table shows scaling multiplier for a call option on a risk factor that follows geometric Brownian motion with volatility

– OTM options produce the largest deviations from the linear trade, but their contribution to exposure is small

More generally: a convex (concave) function of a risk factor tends to increase (decrease) the multiplier of the linear contract

Regression methods can be used to estimate the scaling multiplierfor specific netting sets

50%σ =

Moneyness Inf 2.00 1.25 1.00 0.83 0.67 0.50Multiplier 1.10% 1.15% 1.31% 1.45% 1.60% 1.85% 2.28%

Relative EE 100% 95% 76% 60% 45% 29% 13%

The regression approaches

25

Regression for Multiplier

Even at 10-day horizon, the scaling multiplier can sometimes deviate significantly from the local normal limit

We propose to use regression techniques to estimate the multiplier– Regression techniques may require more simulation paths than the daily

production exposure simulation for reasonably accurate estimates

– It may be optimal to run regressions (i) less frequently than daily and (ii) on a limited selection of netting sets

We use a single regression explanatory variable – This implicitly imposes the Markov assumption on portfolio value process

– Given that our target is the tail of conditional on , going beyond a single variable seems hopeless…

We have found that a hybrid of kernel and parametric regressions is the most promising approach

( )V t δ−

( , )V t tδ∆ − t δ−F

26

Regression Target Quantities

Regression is used to estimate IM conditional on portfolio value on path m

MVA calculations: quantities are averaged over paths mto obtain the time profile of expected IM

EE calculations: regression is further used to calculate conditional ES and, thus, EE conditional on via

– Note that we use regression to estimate tail expectation on a path rather than naively subtracting from realizations of The naive approach would produce excessive noise

– Quantities are averaged over paths m to obtain unconditional EE

( )mV t δ−

IM ( ) Q ( , ) ( )m q m mt V t t V tδ δ = ∆ − −

IM ( )m t

( )EE ( ) (1 ) ES ( , ) ( ) IM ( )m q m m mt q V t t V t tδ δ = − ∆ − − − E

( )mV t δ−

EE ( )m t

IM ( )m t ( , )mV t tδ∆ −

27

Cornish-Fisher Approximation

The quantile conditional on is calculated from the first four conditional moments via the Cornish-Fisher approximation

– is the standard deviation of conditional on

– and are conditional standardized skewness and kurtosis

The expected shortfall conditional on is calculated as the simple average of conditional quantiles:

( )

( ) ( )

1 1 2

21 3 1 1 3 1

( )Q ( , ) ( ) ( ) ( ) [ ( )] 16

( ) 3 [ ( )][ ( )] 3 ( ) 2[ ( )] 5 ( )24 36

mq m m m

m m

s tV t t V t t q q

t s tq q q q

δ δ σ

κ

− −

− − − −

∆ − − = Φ + Φ − −

+ Φ − Φ − Φ − Φ

1/2 (1 )1

1ES ( , ) ( ) Q ( , ) ( )N

q m m i m mq qi N

V t t V t V t t V tN

δ δ δ δ−+ −=

∆ − − ≈ ∆ − − ∑

( )mV t δ−

( )mV t δ−

( , )mV t tδ∆ −( )m tσ ( )mV t δ−

( )ms t ( )m tκ

28

Conditional Moments via Kernel Regression

The first four moments of are calculated via Nadaraya-Watson kernel estimator

Ignoring the first moment, the standard deviation is

The standard deviation is used to calculate skewness and kurtosis

[ ][ ][ ]

( ) 1

1

( ) ( ) ( , )( )

( ) ( )

M nh m i in i

m Mh m ii

K V t V t V t tt

K V t V t

δ δ δµ

δ δ=

=

− − − ∆ −=

− − −∑

∑

(2)( ) ( )m mt tσ µ=

( , )mV t tδ∆ −

(3) (4)

3 4

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

m mm m

m m

t ts t tt t

µ µκσ σ

= =

29

Improving Kernel Regression Results

Kernel regression often produces unstable results at the ends of the range of the conditioning variable

To remove this instability, we apply the following algorithm:

– Sort the moments in the increasing order of

– After throwing away the bottom x% and the top y% of , fit the remaining interior to a parametric function of

– Extend the parametric function for to the entire range of and use it to determine and from and

– After throwing away the bottom x% and the top y% of and , fit the remaining interior to a parametric function of

It is important to find a good fitting function for the interior of , while a constant may often suffice for and

( )mV t δ−

( )mV t δ−( ) ( )nm tµ

( )m tσ( )mV t δ−

( )mV t δ−(3) ( )m tµ (4) ( )m tµ( )ms t ( )m tκ

( )ms t ( )m tκ( )mV t δ−

( )m tσ ( )ms t ( )m tκ

( )m tσ

30

Example: Forward on Lognormal Factor (1)

Conditional moments at and 5,000 paths– Risk factor volatility is 50%, the time-zero value and the strike are 100

– The thresholds for the parametric fit are

– Note that corresponds to

2%x y= =

0

20

40

60

80

100

120

140

160

-100 400 900 1400

Standard Deviation

Kernel Regression Linear Fit

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

-100 400 900 1400

Skewness

Kernel Regression Constant Fit

2%y = ( ) 280V t δ− =

3 yearst =

31

Example: Forward on Lognormal Factor (2)

Time profile of expected IM at 5,000 paths– Cornish-Fisher approximation with 2 (i.e., local normal) and 4 moments

0

5

10

15

20

25

30

0 1 2 3 4 5Time [ years ]

Cornish-Fisher (2 moments) Cornish-Fisher (4 moments) True Normal True Lognormal

32

Example: Forward on Lognormal Factor (3)

Time profile of EE scaling multiplier at 5,000 paths– Cornish-Fisher approximation with 2 (i.e., local normal) and 4 moments

0.0%

0.5%

1.0%

1.5%

2.0%

0 1 2 3 4 5Time [ years ]

Cornish-Fisher (2 moments) Cornish-Fisher (4 moments) True Lognormal True Normal

33

Conclusion

We have presented analytical and numerical approaches to calculating EE in the presence of IM

Even with the knowledge of the exact IM on a path, the naive approach of subtracting IM from realized P&L results in a noise curse in exposure because of the high confidence level of 99%

We have introduced the concept of scaling downthe EE without IM via a pre-calculated multiplier– Local normal approximation, generally valid for small MPR, produces

a universal value of 0.85% for the EE scaling multiplier at q= 99%– Regression approaches can be used for estimating the multiplier

when deviations from the local normal limit are significant

Multiplier should not be applied to EE spikes resulting from trade cash flows – spikes should be treated separately– With BCBS-IOSCO IM, spikes can become major contributors to CVA!