A Default Probability Mapping Model for Pricing and Managing CVA

A.Gigli, E. Renzetti August 2014

2

The views expressed in this presentation are those of the speaker and do not necessarily represent those of current employers.

Additional information is available upon request.

Information has been obtained from public sources believed to be reliable but the authors does not warrant its completeness or accuracy.

All information contained herein is as of the date referenced.

This material is for informational purposes only, should be viewed solely in conjunction with the oral briefing provided at the time of circulation, and is not intended as an offer or solicitation for the purchase or sale of any financial instrument.

DISCLAIMER

3

• Derivatives are financial contracts which allow to bet on the future value of a given underlying without holding it physically.

• Without taking into account Counterparty Risk, the value of the contract overtime depends on the moneyness of the bet and on the expected value of future cash flow(s)

• Martingale pricing theory tells us that under a risk-neutral measure Q it is possible to price a derivative contract according to

where is the expectation operator under the Q measure conditioned to the information available in t.

DERIVATIVE PRICING (NO CTP RISK)

)( 1tC )( 2tC )( 3tC )( TtC n Expected cash flow

Payement Dates 1t 2t 3t Ttn

)()( TVEtV Qt

QtE

V

4

• If the counterparty of a derivative contract defaults in then no future payments will be received by the derivative buyer after

• Introducing Counterparty risk, the value of the contract overtime depends also on the probability of actually receiving positive future cash flow(s) from the counterparty and

• CVA is defined as

DERIVATIVE PRICING (CTP RISK)

)( 1tC )( 2tC )( 3tC )( TtC n Expected cash flow

Payement Dates 1t 2t 3t Ttn

T

V̂

)()(ˆ tVtV

0)(ˆ)()( tVtVtCVA

5

• Credit value adjustment is the price (i.e. the market value) of counterparty credit risk, that is the premium to take into account of losses upon counterparty default when pricing a derivative contract.

• CVA can be calculated as the risk neutral expectation of the discounted loss over the life of the transactions with a given counterparty

where

is the counterparty-level exposure at the time of default is the counterparty time of default is the recovery rate is the value of the money market account at time t

CVA

)(1)(

EE

B

BREtCVA t

TQt

)(EERtB

CVA• Assuming constant recovery rate R, we can write

where:

is the risk-neutral cumulative probability of default (PD) between time 0 and time t

is the risk-neutral discounted expected exposure (EE) at time t conditional on the counterparty default at time t.

• If both exposure and money market account are independent then

tEEB

BEtEE

tdQtEECVA

tt

Q

T

|)(

)()(R-1)0(

0*

0

*

*)(tEE

t

t

EEB

BEtEEtEE 0** )()(

T

t tdQEECVA0

* )(R-1)0(

)(tQ

6

7

CVA: CREDIT VALUATION ADJUSTMENT

Looking at CVA formula in a naïve way

where:

- LGD (loss given default) is the percentage exposure we loose in

the case of counterparty default;

- EAD (exposure at default) is the expected derivative transaction

value at the time of default;

- DP (default probability) is the default probability assigned to

counterparty

• This relation holds only assuming independence between exposure and counterparty’s credit risk

T

t tdQEECVA0

* )(R-1)0(

LGD * EAD * DP

8

CVA AND RISK MANAGEMENT• Counterparty risk implies

• It also implies that changes in and for changes in the underlying value are different

• where

)()(ˆ tVtV

underlying

tCVA

underlying

tV

underlying

tV

underlying

tV

)()(ˆ)()(ˆ

)(ˆ tV )(tV

underlying

EADDPLGD

underlying

CVA

**

9

CVA AND RISK MANAGEMENT

Assuming no collateral agreement in place and assuming we can

approximate EAD(0) with

then

0)0(ˆ

)0(ˆ

Vunderlying

V

underlying

EAD

0)0(ˆ

)0(ˆ**

)(ˆ)(

Vunderlying

VDPLGD

underlying

tV

underlying

tV

0),0()0(ˆ AddonVMax

10

LGD, EAD, DPIn order to compute

And

we need:- EAD, which depends on a specific model assumptions;- LGD, which depends on several factors, both transaction specific (eg risk mitigation instruments in place) or not (eg country, sector or counterparty);- DP, which is derived from a market measure.

Several choices have to be made, few ones are both feasible and effective in order to price and manage CVA risk.

DP EAD LGD CVA

0)0(ˆ

)0(ˆ**

)(ˆ)(

Vunderlying

VDPLGD

underlying

tV

underlying

tV

11

DP(0,T)

• Risk Neutral DP(0,T) for a specific counterparty can be recovered from market prices, provided CDS quotes for that counterparty exists.

• If market data are not available, banks have at their disposal a statistical estimation of the default probabilities for homogeneous Rating groups, DP(0,T)*

• If a function mapping Statistical (or Real) probabilities into Risk-Neutral probabilities exists it would allow banks to price and manage CVA under a Measure closer to the Risk-Neutral one.

DP(0,T)* DP(0,T)

12

RISK NEUTRAL VS REAL PROBABLILITIES

• Assuming the Black-Scholes-Merton setting holds, the relation between Real measure P and the Risk Neutral measure Q can be defined by

where

is the Real measure for maturity T

is the RN measure for maturity T

is the cumulative normal distribution function,

is the market price of risk

is the correlation between issuer asset return and market return.

TPQ TT 1

()N

TQTP

13

RECOVERING RN PROBABILITIES FOR

• We propose to estimate the relation

using CDS implied default probabilities (as the Market RN probabilities) and Rating agencies default probabilities (as Statistical Default Probability)

• Once an estimate for and is obtained, we use the estimated function to recover Market RN probabilities starting from Bank XYZ statistical estimates of counterparty default probabilities

• In the following section

• We fit the model

• We show an application of the mapping function

TPQ TT 1

14

MODEL FITTING

15

S&P MATRIX

The S&P default probability Matrix estimated over 1981-2013 period:

16

5Y CDS SPREAD

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

40

80

120

160

AA

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

40

80

120

160

A

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

50

100

150

200

250

BBB

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

100

200

300

400

500

600

BB

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

200

400

600

800

1000

B

10/2

9/20

09

3/18

/201

0

8/5/

2010

12/2

3/20

10

5/12

/201

1

9/29

/201

1

2/16

/201

2

7/5/

2012

11/2

2/20

12

4/11

/201

3

8/29

/201

3

1/16

/201

40

500

1000

1500

2000

2500

3000

CCC

17

RECOVERING

• We estimate for qualitatively homogeneous issuers (labeled “R” in the following) accordingly to the S&P Rating scale.

• Using daily Risk Neutral DP recovered from CDS spread and Real DP* available in S&P matrix we can solve for the following

for maturity T = {1y, 2y, 4y, 5y, 7y, 10y} where

is the statistical default probability for issuers belonging to rating group “R” from S&P matrix, for maturity T

is the risk-neutral default probability observed for issuers belonging to rating group “R” on the market, for maturity T

Tmq RT

RT

RT ̂ˆˆ 1

TTT

RTq̂

RTm̂

RT̂

TTT

18

BOX-PLOT FOR DAILY ESTIMATES OF RT

19

A TERM STRUCTURE FOR RT̂

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In the following table we report the average over the period October 2010 –July 2014 for each maturity/rating group

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.97 0.70 0.63 0.57 0.54 0.55

A 0.70 0.57 0.57 0.53 0.50 0.49

BBB 0.49 0.38 0.40 0.38 0.39 0.40

BB 0.40 0.29 0.33 0.33 0.35 0.38

B 0.04 0.08 0.22 0.26 0.32 0.38

CCC -0.35 -0.07 0.20 0.25 0.35 0.45

AVERAGE ESTIMATE FOR

RT̂

R

T

21

MODEL EXTRAPOLATION

22

Let’s assume Bank XYZ has estimated the following internal default matrix

We can estimate the Risk Neutral DP(0,T), for any counterparty C belonging to rating group R, starting from the statistical estimate of the Real DP(0,T)* using estimates of

RECOVERING ACTIONABLE DP(0,T) FROM

TTPDTPD RT

RT

RT ̂,0ˆ),0(ˆ *1

RT̂

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.034% 0.100% 0.388% 0.626% 1.304% 2.763%

A 0.131% 0.357% 1.135% 1.685% 3.048% 5.545%

BBB 0.605% 1.458% 3.760% 5.100% 7.933% 12.205%

BB 1.667% 3.734% 7.922% 9.801% 13.050% 16.943%

B 6.458% 11.788% 18.856% 21.195% 24.584% 27.985%

CCC 16.458% 23.944% 30.245% 31.808% 33.790% 35.548%

R

T

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0 1 2 3 4 5 6 7 8 9 100%

10%

20%

30%

40%

50%

60%

70%

80%

AA A BBB BB B CCC

RISK NEUTRAL MEASURE DP(0,T)Using CDS data from July 28th 2014 for estimating we obtain the following Risk Neutral probability estimates

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.29% 0.77% 3.84% 6.65% 12.65% 21.77%

A 0.43% 1.42% 7.22% 11.73% 19.26% 28.40%

BBB 1.21% 3.02% 11.88% 17.68% 27.25% 37.85%

BB 2.30% 4.80% 15.91% 22.47% 31.06% 41.03%

B 3.62% 7.51% 21.44% 28.69% 38.30% 49.21%

CCC 6.86% 16.66% 36.61% 43.85% 56.71% 70.99%

R

T

24

0 1 2 3 4 5 6 7 8 9 100%

10%

20%

30%

40%

50%

60%

70%

80%

90%

AA A BBB BB B CCC

RISK NEUTRAL MEASURE DP(0,T)Using the average estimates of we obtain the following Risk Neutral probability estimates

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.75% 1.80% 8.08% 11.06% 21.64% 42.53%

A 1.06% 3.00% 12.80% 17.34% 29.06% 48.22%

BBB 2.16% 4.97% 16.36% 21.91% 34.93% 53.73%

BB 4.20% 8.48% 22.54% 29.09% 41.72% 59.18%

B 7.01% 14.33% 33.21% 41.69% 55.89% 73.02%

CCC 9.32% 21.15% 44.97% 53.82% 69.28% 85.20%

R

T

25

RECAP

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.034% 0.100% 0.388% 0.626% 1.304% 2.763%

A 0.131% 0.357% 1.135% 1.685% 3.048% 5.545%

BBB 0.605% 1.458% 3.760% 5.100% 7.933%12.205

%

BB 1.667% 3.734% 7.922% 9.801%13.050

%16.943

%

B 6.458%11.788

%18.856

%21.195

%24.584

%27.985

%

CCC16.458

%23.944

%30.245

%31.808

%33.790

%35.548

%

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.64 0.47 0.45 0.44 0.41 0.36

A 0.38 0.35 0.41 0.42 0.38 0.32

BBB 0.26 0.22 0.30 0.32 0.30 0.27

BB 0.13 0.08 0.21 0.24 0.24 0.23

B -0.28 -0.18 0.05 0.11 0.15 0.18

CCC -0.51 -0.18 0.09 0.14 0.22 0.29

Rating 1Y 2Y 4Y 5Y 7Y 10Y

AA 0.29% 0.77% 3.84% 6.65% 12.65% 21.77%

A 0.43% 1.42% 7.22% 11.73% 19.26% 28.40%

BBB 1.21% 3.02% 11.88% 17.68% 27.25% 37.85%

BB 2.30% 4.80% 15.91% 22.47% 31.06% 41.03%

B 3.62% 7.51% 21.44% 28.69% 38.30% 49.21%

CCC 6.86% 16.66% 36.61% 43.85% 56.71% 70.99%

10/29

/2009

1/14/2

010

4/1/20

10

6/17/2

010

9/2/20

10

11/18

/2010

2/3/20

11

4/21/2

011

7/7/20

11

9/22/2

011

12/8/

2011

2/23/2

012

5/10/2

012

7/26/2

012

10/11

/2012

12/27

/2012

3/14/2

013

5/30/2

013

8/15/2

013

10/31

/2013

1/16/2

014

0

50

100

150

200

250

BBB

10/29

/2009

1/14/2

010

4/1/20

10

6/17/2

010

9/2/20

10

11/18

/2010

2/3/20

11

4/21/2

011

7/7/20

11

9/22/2

011

12/8/

2011

2/23/2

012

5/10/2

012

7/26/2

012

10/11

/2012

12/27

/2012

3/14/2

013

5/30/2

013

8/15/2

013

10/31

/2013

1/16/2

014

0

100

200

300

400

500

600

BB

10/29

/2009

1/21/2

010

4/15/2

010

7/8/20

10

9/30/2

010

12/23

/2010

3/17/2

011

6/9/20

11

9/1/20

11

11/24

/2011

2/16/2

012

5/10/2

012

8/2/20

12

10/25

/2012

1/17/2

013

4/11/2

013

7/4/20

13

9/26/2

013

12/19

/2013

0

500

1000

1500

2000

2500

3000

CCC TPQ TTTT 1

TTT ˆˆˆ

TTPDTPD RT

RT

RT ̂,0ˆ),0(ˆ *1 DP for pricing and

managing CVA

Bank XYZmatrix

S&P DPmatrix

CDS implied DP

26

CONCLUSIONS

• CVA pricing and management requires Default Probability under a Risk Neutral Measure, which can be recovered from market data. If market data are not available for a specific counterparty, usually banks extrapolate a DP from the statistical estimation of the Default Probabilities at their disposal.

• If a model to map Real probabilities in the Risk Neutral space exists, it is possible to price CVA and compute CVA DV01 accordingly, using a RN measure

• In our application, we calibrated a term structure for for each rating class starting from CDS data and S&P default probabilities.

• Once we’ve estimated we use it to compute the risk neutral probabilities starting from statistical estimation of default probabilities for Bank XYZ customers.

TTT

TTT

27

For feedbacks please contact:

Andrea Gigliandrea.gigli@mpscs.it

Eros Renzettieros.renzetti@valuecuberesearch.com

The views expressed in this presentation are those of the speakers only.

Additional information is available upon request. Information has been obtained from public sources believed to be reliable but the authors does not warrant its completeness or accuracy.

All information contained herein is as of the date referenced. This material is for informational purposes only, should be viewed solely in conjunction with the oral briefing provided at the time of circulation, and is not intended as an offer or solicitation for the purchase or sale of any financial instrument.

THANK YOU