Stress Testing for Counterparty Risk
Colin Burke April 2014
[email protected]
Introduction • This presentation focuses on stress testing counterparty risk with examples relevant to: • Inflation swaps
• Counterparty recovery rates • Credit default swaps and presents some research in progress on the use of filtering for stress testing
•Important: All views in this presentation are the views of the author and do not necessarily correspond with those of the Lloyds Banking Group 2
Counterparty Credit Risk Counterparty credit risk refers to the risk that a counterparty will default on a transaction or series of transactions before the maturity of the longest derivative or repo-style trade transacted with that counterparty Credit exposure can be bilateral - at some times, for a given set of transactions, counterparty A may owe counterparty B a monetary amount (counterparty B has a credit risk to counterparty A) while at other times, for the same transactions, counterparty B may owe a monetary amount to counterparty A. Related to this is that the monetary amount, the credit exposure, is driven by market factors (interest rates, FX rates etc).
Fixed rate
“Bank”
Variable rate (LIBOR)
“Counterparty”
Counterparty Risk for an Interest Rate Swap
Counterparty Credit Risk – modelling approaches (i) CCR exposure models generally consist of the following components: 1. A scenario generator (a way to simulate future market rates) 2. One or more product pricers (i.e. a mechanism to price each derivative on each scenario). The output is a hypothetical MTM. Many derivatives can be priced using closed form expressions but some cannot and numerical methods must be used (Bermudan swaptions are the prime example). 3. A mechanism to age trades. For many trade types, this just means decreasing the remaining maturity but for some trade types such as options this means deciding whether the option has been exercised and modelling whatever is the result of the exercise. 4. A way of aggregating MTM's for each scenario (only applicable to portfolio simulation) 5. A way of modelling collateral (generally only applicable to portfolio simulation) 6. A statistical analysis – i.e. forming quantiles and conditional averages
Correlation enters here
Counterparty Credit Risk – modelling approaches (ii)
Notes : Adaptiv not used for reg. Cap. See arrows : Adaptiv Panorama is no longer used: it is just a staging area for AA : Reference Point no longer used (flow through)
Counterparty Credit Risk: The use of model output (i) 1. Potential Future Exposure (PFE) is the maximum exposure estimated to occur on a future date at a high level of statistical confidence. PFE is normally used when measuring counterparty credit risk exposure against counterparty credit limits and as such PFE is then calculated at portfolio level. 2. Expected Exposure (EE) is the mean (average) of the distribution of positive exposures at any particular future date before the longest-maturity transaction in the netting set matures. An expected exposure value is typically generated for many future dates up until the longest maturity date of transactions in the netting set. 3. Expected Positive Exposure (EPE) is the weighted average over time of Expected Exposures, (EE) where the weights are the proportion that an individual Expected Exposure represents of the entire time interval. When calculating the minimum regulatory capital requirement for IMM purposes, the average is taken over the first year or, if all the contracts in the netting set mature before one year, over the time period of the longest-maturity contract in the netting set. PFE, EE and EPE illustrated 7
6
Exposire
5
4
PFE EE EPE
3
2
1
0 0
0.5
1
1.5
2
2.5
3
Remaining maturity (years)
3.5
4
4.5
5
Counterparty Credit Risk: The use of model output (ii)
The main exposure approaches for counterparty risk are listed below: Purpose
Possible approaches
Limits management
PFE add-ons, portfolio simulation
Pillar 1 capital
Regulatory add-ons, Standardised Approach, Internal Models Method (IMM) using EPE
Pillar 2 capital
Regulatory add-ons, EE
Pricing
CVA
PFE
Stressing the evolution of market variables
Calibration (parameterisation) typically needs to specify some or all of the following: Drift – the trend of market variables
Volatility- the future uncertainty of market variables
Mean reversion - many market variables tend to be pulled back to some average level
Correlation between risk factors - the co-movement between market variables
We are going to look a stressing drifts, volatilities and correlations and relating these to macro variables. We also need to distinguish between the time horizons for our stress tests: Medium and long term horizons (planning and risk appetite etc). Drifts are key here Short (sometimes very short) risk horizons (e.g. wrong way risk). We might need to change the way parameters e.g correlations, are thought of
Stressing inflation: Some inflation products Introducing 2 inflation swap products (one with simplified pricing): Zero Coupon Inflation Swaps Year-on-year inflation swaps The present value of the ZCIIS is just the discounted differences of the pay and receive cash flows:
I t , T n T ,T 12 DF t , T PV ZCIIS t N 1 R 0 I base
(1)
Where 1 for a receive fixed type and 1 for a pay fixed type Defining the rank l such that: Tl 1 t Tl and discounting the cash flow amounts leads to the valuation:
I t , Tl PVYYIIS t N l R 1 nTl 1 , Tl 12 DF t , Tl I T l 1
I t , Ti nTi 1 , Ti 12 DF t , Ti N R 1 i I t , T i l 1 i 1 I 1
(2)
Stressing inflation: Inflation Modelling
We model the inflation index spot as a log-normal process and both curves are modelled using Hull-White processes. That leads to the system of stochastic differential equations:
dg t g t a g g t dt g dB g t dr t r t a r r t dt r dBr t dI t I t dt I t dB t I I
(1)
Where BI t , Br t and B g t are three correlated Brownian motions. In our model, the expected inflation index is given as a function of the current inflation index at time t and the inflation growth rate at time T view from time t, that is:
I t , T I t 1 g t , T
T t
(2)
Stressing inflation: What should we stress (focus on)?
The zero-coupon peak exposure is highly sensible to a variation in the Inflation index parameters. This is due to the fact that at maturity, the growth inflation term and the discount factor are one therefore the PV is a linear function of the inflation index at maturity:
PVZCIIS T I T
(1)
N 1 R n T0 ,T 12 n T ,T 12 N 1 R 0 I base
(2)
Where:
The Year-on-year depends mainly on the inflation growth parameters, this is due to the fact that the term
I t , Ti in the relation (2) depends only on the I t , Ti 1
growth rate as the inflation index cancelled:
I t , Ti I t 1 g t , Ti i I t , Ti 1 I t 1 g t , Ti 1 Ti 1 t T t
(3)
Stressing inflation: RPI history Inflation index growth 300
250
Index Value
200
150
100
50
0 1950
1960
1970
1980
1990 Year
2000
2010
2020
Stressing inflation: filtering The basic idea is when a data series can be seen as a slowly varying trend plus a higher frequency cycle – filtering out the slowly varying trend leaves the cycle. The filter operates as follows: let
{xt } be the raw data and { yt } be the oscillatory or cyclic part
of the series. We suppose that
xt y t z t In which zt is the slowly varying trend which is a solution of: N
N
2
arg min xi z i z i 1 2 z i z i 1 zt
i 1
2
i 2
is a smoothing parameter and is related to the cut-off frequency of the Butterworth filter via 1 0 2 41 cos 0 In which 0 is the angular frequency corresponding to a gain of 50% in the Butterworth filter.
Think of trends relating to drifts and cycles to volatility Relating cycles to the marco-economy
Filter realisation 1.2
1
Gain
0.8
Ideal One possible realisation
0.6
0.4
0.2
0 0.00001
0.0001
0.001
0.01
0.1 Frequency
1
10
100
1000
10000
Filtered RPI cycle 15
10
Index trend
5
0 1950
1960
1970
1980
1990
-5
-10
-15 Year
2000
2010
2020
Monetary policy ??? Filtered Inflation returns (trend) 0.16
0.14
0.12
Trend rate
0.1
0.08
0.06
0.04
0.02
0 1950
1960
1970
1980
1990 Year
2000
2010
2020
Filtered RPI returns (cycle) 12.0%
10.0%
8.0%
Cycle rate
6.0%
4.0%
2.0%
0.0% 1950
1960
1970
1980
1990
-2.0%
-4.0%
-6.0% Year
2000
2010
2020
Stressing inflation: relating to macro-variables Filter inflation and GDP 12.0%
• Lead/lags? •-30% correlation • Pick your econometric model? • judgemental? •…..
10.0%
8.0%
Filtered rate
6.0%
4.0%
Filter GDP growth Filtersed Inflation 2.0%
0.0% 1950
1960
1970
1980
1990
-2.0%
-4.0%
-6.0% Year
2000
2010
2020
0.009
Hodrick Prescott and Loess Smoothing of log Inflation returns (monthly data, swap market implied). Not annualised
0.008 0.007
Filtered return
0.006 0.005 loess
0.004
hp
0.003 0.002 0.001 0 0
5
10
15
20 years
25
30
35
40
Stressing inflation: How reactive is the drift?
0.01
Mean (drift) estimation. Swap market data (not annualised)
0.008 0.006 0.004
HP
0.002
Regimes of the mean 0 0
5
10
15
20
25
30
35
40
-0.002 -0.004
-0.006
Years
Deflation at the end of 2008
Stressing Counterparty Recovery rates and Credit Default Swaps
We will examine stressing: • Recovery rates • Sovereign spreads and correlations
Senior Unsecured Bond Recovery Rates 60
50
Recovery
40
30
20
10
0 1985
1990
1995
2000 Date
2005
2010
2015
Stressing Credit Default Swaps: recovery rates
Counterparty filtered recovery rates (fast cycle) and US GDP
Filtered Recovery and US GDP rates 15
0.03
Highly correlated (60%)
0.02 10 0.01 0
Filetered Rate
5 -0.01 0
-0.02 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 -0.03
-5 -0.04 -0.05 -10 -0.06 -15
-0.07 Year
Filtered recovery rates Filtered US GDP
Counterparty recovery rates (slow trend) and US GDP
Recovery rate trend 60
50
rate
40
30
20
10
0 1985
1990
1995
2000 date
2005
2010
2015
Be wary of structural breaks…. Credit Default Swaps UK: CDS 1Y Plot 160
140
No slowly varying trend.. …so previous filter will not work
120
Spreads
100
80
60
40
20
0 2007-04-28
2007-11-14
2008-06-01
2008-12-18
2009-07-06
2010-01-22
Dates
2010-08-10
2011-02-26
2011-09-14
2012-04-01
2012-10-18
In the 2008-? crisis, there was (arguably) no structural breaks in inflation, key FX rates etc but were breaks in credit spreads, LIBOROIS spreads etc
Spreads
UK: CDS 1Y Plot and Kalman filtered (predicted) spreads 250
200
150
CDS_1Y Predicted_CDS_1Y
100
Kalman filtering gives better results but the modelling becomes more complex. Std errors in the mean reversion are very high
50
0 2007-04-28 2007-11-14 2008-06-01 2008-12-18 2009-07-06 2010-01-22 2010-08-10 2011-02-26 2011-09-14 2012-04-01 2012-10-18
-50 Dates
Notes: an affine model (CIR) is used for hazard rates under the risk neutral and physical measures
Sovereign Correlations under stress: default (asset) correlations Sovereign correlations
180
160
140
CDS spread
120
100 Germany France
80
60
40
20
0 28-Apr2007
14-Nov2007
01-Jun2008
18-Dec2008
06-Jul-2009
22-Jan2010 Date
10-Aug2010
26-Feb2011
14-Sep2011
01-Apr2012
18-Oct2012
Sovereign Correlations under stress: joint default
To calculate CDS implied asset correlations we proceed as follows: 1. We obtain a history of 1,3,5,7 and 10 years CDS spreads for senior unsecured debt from MarkIT 2. We bootstrap survival probabilities for each day 3. We convert the one year default probability into a standardised asset return 4. We compute the first differences of the asset returns 5. We compute the linear correlation between each asset return time series
Sovereign Correlations under stress: joint default Dec2007 to Jul2012 Germany France Germany 1 0.8048 France 0.804753 1 Italy 0.009683 0.0134 Austria 0.034111 0.0229 Spain -0.00467 0.0086 UK 0.007678 -0.001 US -0.0029 -0.002
Jul2010 to Jul 2012 Germany Germany 1 France 0.819911 Italy 0.053659 Austria 0.105344 Spain 0.019096 UK 0.046651 US -0.01355
France 0.8199 1 0.0457 0.0826 0.035 0.0288 -0.003
Italy 0.00968 0.01339 1 0.32951 0.37394 0.11823 0.08704
Austria Spain UK 0.0341 -0.0047 0.0077 0.0229 0.0086 -0.001 0.3295 0.3739 0.1182 1 0.3138 0.221 0.3138 1 0.2834 0.221 0.2834 1 0.1587 0.1622 0.1622
Italy 0.05366 0.0457 1 0.56542 0.7738 0.415 0.27711
Austria Spain 0.1053 0.0191 0.0826 0.035 0.5654 0.7738 1 0.5578 0.5578 1 0.3719 0.4406 0.2552 0.245
US -0.003 -0.002 0.087 0.1587 0.1622 0.1622 1
UK 0.0467 0.0288 0.415 0.3719 0.4406 1 0.17
US -0.014 -0.003 0.2771 0.2552 0.245 0.17 1
General Wrong Way Risk: The Basel III Rule (i) "Banks must identify exposures that give rise to a greater degree of general wrong-way risk. Stress testing and scenario analyses should be designed to identify risk factors that are positively correlated with counterparty credit worthiness. Such testing needs to address the possibility of severe shocks occurring when relationships between risk factors have changed. Banks should monitor general wrong way risk by product, by region, by industry…" "For all counterparties, banks should produce, at least monthly, exposure stress testing of principal market risk factors (eg, interest rates, FX, equities, credit spreads, and commodity prices) in order to proactively identify, and when necessary, reduce outsized concentrations to specific directional sensitivities. Stressed market movements have an impact not only on counterparty exposures, but also on the credit quality of counterparties. At least quarterly, banks should conduct stress testing applying stressed conditions to the joint movement of exposures and counterparty creditworthiness."
31
2/ 20 07 03 /0 1/ 20 08 31 /0 1/ 20 08 28 /0 2/ 20 08 31 /0 3/ 20 08 28 /0 4/ 20 08 28 /0 5/ 20 08 25 /0 6/ 20 08 23 /0 7/ 20 08 20 /0 8/ 20 08 18 /0 9/ 20 08 16 /1 0/ 20 08 13 /1 1/ 20 08 11 /1 2/ 20 08 13 /0 1/ 20 09 10 /0 2/ 20 09 10 /0 3/ 20 09 07 /0 4/ 20 09 08 /0 5/ 20 09 08 /0 6/ 20 09 06 /0 7/ 20 09
03 /1
An Example: Morgan Stanley and 3M LIBOR
0.06
0.04
0.02
0 0.09
0.08
0.05 0.07
0.06
0.05
0.03 LIBOR CDS
0.04
0.03
0.02
0.01 0.01
0
32
An Example: Morgan Stanley and 3M LIBOR (ii) 1. For illustration a Gaussian model of interest rates is used (generalises easily to log-normal models) 2. Plot interest rate first difference versus asset value first difference 0.2
0.004
0.003 0.1 0.002
0.001
04 /1
2/ 20 04 07 /0 1/ 20 01 08 /0 2/ 20 29 08 /0 2/ 20 01 08 /0 4/ 20 29 08 /0 4/ 20 29 08 /0 5/ 20 26 08 /0 6/ 20 24 08 /0 7/ 20 21 08 /0 8/ 20 19 08 /0 9/ 20 17 08 /1 0/ 20 14 08 /1 1/ 20 12 08 /1 2/ 20 14 08 /0 1/ 20 11 09 /0 2/ 20 11 09 /0 3/ 20 08 09 /0 4/ 20 11 09 /0 5/ 20 09 09 /0 6/ 20 07 09 /0 7/ 20 09
0
-0.1
0 asset value first diff interest rate first diff
-0.2
-0.001
-0.002 -0.3 -0.003 -0.4 -0.004
-0.5
-0.005
33
An Example: Morgan Stanley and 3M LIBOR (v) – DCC
MS asset value: GARCH process 3
2.5
Volatility
2
Estimated GARCH process for Morgan Stanley asset values
1.5
1
0.5
03/07/2009
03/06/2009
03/05/2009
03/04/2009
03/03/2009
03/02/2009
03/01/2009
03/12/2008
03/11/2008
03/10/2008
03/09/2008
03/08/2008
03/07/2008
03/06/2008
03/05/2008
03/04/2008
03/03/2008
03/02/2008
03/01/2008
03/12/2007
0
date
Period
correlation
DCC Sep -08 Conventional Sep-08
63% 29% 13%
We could of course look at exposure to inflation swaps under deflation 34
Conclusion • This presentation focused on stress testing counterparty risk with examples relevant to: • Inflation swaps
• Counterparty recovery rates • Credit default swaps Look at your portfolio decomposition/pricing models to decide what ti stress
Introduced low pass filtering as a calibration technique for drift and volatility and related filtered data to macro-variables for stress testing. Will not work in the presence of structural breaks or other “fast” processes (e.g. large jumps) though there are methods around this. However, even in a crisis, not all relevant variables undergo discontinuities. Currently researching the use of multi-dimensional filters for the correlation case 35
