Measuring and Dynamically Hedging Counterparty Credit Exposure …ieor.columbia.edu/files/seasdepts/industrial-engineering... · Measuring and Dynamically Hedging Counterparty Credit

Measuring and Dynamically Hedging Counterparty Credit Exposure and Risk

PRESENTED TO: Financial Engineering SeminarDepartment of Industrial Engineering and Operations Research.Columbia University

BY: Evan Picoult, Managing DirectorRisk ArchitectureCitigroupNew York, New York

DATE: Monday, January 24th, 2005

PLACE: New York City

o COUNTERPARTY RISK - Counterparty of a forward or derivative defaults prior to finalsettlement of cash flows and the contract (portfolio) has a positive economic value.

BASIC DEFINITIONS

o LENDING RISK - Borrower defaults - an accrual (non MTM) perspective

- Or the loan portfolio’s economic value also decreases because of:- decrease in the credit quality of the obligor and/or- increase in general market spreads.

o ISSUER RISK (Specific Risk) - Issuer of security defaults.

- Or the security’s market value also decreases because of:- decrease in the credit quality of issuer and/or- increase in general market spreads.

o SETTLEMENT RISK - In an exchange: you pay but do not receive what you are owed.

TYPE OF CREDIT RISK CAUSE OF ECONOMIC LOSS

- Or the economic value of derivatives with counterpartyalso decreases because of:

- decrease in credit quality of counterparty and/or- increase in general market spreads

How to measure this?

a.k.a. Pre-settlement risk

Types of Credit Risk

Page 2E 2Evan Picoult, Citigroup January, 2005


Contents

• PORTFOLIO SIMULATION OF A COUNTERPARTY’S EXPOSURE PROFILE.

• ECONOMIC CAPITAL FOR LOAN CREDIT RISK– Default only perspective.– Loss of economic value perspective.

• ECONOMIC CAPITAL FOR COUNTERPARTY RISK - DEFAULT ONLY:– Full coherent simulation of potential exposure and default.– Approximation using incoherent simulation with expected positive exposure profile

scaled up by factor α .

• ECONOMIC CAPITAL FOR COUNTERPARTY RISK - ECONOMIC LOSS:– Defining the Credit Value Adjustment (CVA) for credit risk of counterparty’s portfolio. – Simulating default, recovery and changes in CVA over time.

• DYNAMICALLY HEDGING COUNTERPARTY RISK

BASIC QUESTION: WHAT ARE THE CONSEQUENCES OF CREDIT EXPOSURE DEPENDING ON THE POTENTIAL FUTURE STATE OF MARKET RATES?

MEASURINGCOUNTERPARTY CREDIT EXPOSURE

FORMS OF CREDIT RISK


Potential Exposure Of A Single Transaction

EXPOSURE PROFILE OF SINGLE TRANSACTION AND MARKET RATE SCENARIOS

Example 1: Forward FX, We buy GBP and sell US$ for settlement in two years at 1.5000 US$/GBP.

Random path of forward FX rate for a fixed settlement date, over life of forward transaction in scenario 1.

Profile of market value of forward FX transaction over its life, for scenario 1.

Exposure Profile of transaction for scenario 1.

We only have exposure when the contract has a positive value to us.

Random Scenario 1 for Forward FX Rate

1.250

1.375

1.500

1.625

1.750

0 3 6 9 12 15 18 21 24Time (months)

Forw

ard

Exc

hang

e R

ate

Forward FX Replacement Cost for Scenario 1

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

0 3 6 9 12 15 18 21 24Time (Months)

Rep

lace

men

t C

ost

(% N

otio

nal)

Forward FX Exposure Under Scenario 1

0%

5%

10%

15%

20%

0 3 6 9 12 15 18 21 24

Time (months)

Pote

ntia

l Ex

posu

re (

% N

otio

nal)


Potential Exposure Of A Single TransactionEXPOSURE PROFILES AND MARKET RATE SCENARIOS Example 2: I.R. Swap, We pay fixed and receive floating every 6 months for three years.

Two Random scenariosof the path of six month LIBOR over the 36 month life of swap.

Profile of market valueof swap over its life for each scenario.

Exposure Profile of swap over its life for each scenario .

Scenario 2: Random change in 6 month LIBOR

5%

6%

7%

8%

9%

10%

11%

0 6 12 18 24 30 36Time (months)

Inte

rest

Rat

e

Scenario 2 : Profile of IR Swap Value

-2%-1%0%1%2%3%4%5%6%7%

0 6 12 18 24 30 36

Time (Months)

Pote

ntia

l Ex

posu

re (

% N

otio

nal)

Scenario 2: IR Swap Exposure Profile

0%

1%

2%

3%

4%

5%

6%

7%

0 6 12 18 24 30 36

Time (Months)

Pote

ntia

l Ex

posu

re (

% N

otio

nal)

Scenario 3 : Profile of IR Swap Value

-2%-1%0%1%2%3%4%5%6%7%

0 6 12 18 24 30 36

Time (Months)

Pote

ntia

l Ex

posu

re (

% N

otio

nal)

Scenario 3: IR Swap Exposure Profile

0%

1%

2%

3%

4%

5%

6%

7%

0 6 12 18 24 30 36

Time (Months)

Pote

ntia

l Ex

posu

re (

% N

otio

nal)

Scenario 3: Random change in 6 month LIBOR

5%

6%

7%

8%

9%

10%

11%

0 6 12 18 24 30 36Time (months)

Inte

rest

Rat

e


Potential Exposure Of A Single Transaction

Three Exposure Profiles for a two year US$/GBP forward FX transaction. At threeconfidence levels:

- 99% CL Exposure Profile

- 97.7% CL Exposure Profile

- Expected Positive Exposure Profile

99.0% CL Profile.

97.7% CL Profile.

Expected Profile

Forward FX Exposure Profiles at Three Confidence Levels

0%

10%

20%

30%

40%

50%

60%

0 3 6 9 12 15 18 21 24

Time (Months)

Expo

sure

Pro

file

(% N

otio

nal)

Three Exposure Profiles for a three year fixed/floating US$ interest rate swap. At three confidence levels:

- 99% CL Exposure Profile

- 97.7% CL Exposure Profile

- Expected Positive Exposure Profile99.0% CL Profile.

97.7% CL Profile.

Expected Profile

Int. Rate Swap Exposure Profiles at Three Confidence Levels

0%

1%

2%

3%

4%

5%

6%

7%

0 6 12 18 24 30 36

Time (Months)

Expo

sure

Pro

file

(% N

otio

nal)

On the basis of thousands of such simulations we can represent the potential exposure over time statistically, at different confidence levels:

Statistical Picture Of Potential Exposure


Potential exposure for a counterparty with multiple transactionsTWO METHODS FOR MEASURING COUNTERPARTY EXPOSURE (CE):

SIMPLE “ADD-ON” METHOD

CE TRANSACTION = CURRENT MTM + “WORST CASE” POTENTIAL INCREASE IN VALUE

= CURRENT MTM + NOTIONAL PRIN. * CREDIT EXPOSURE FACTOR

CE CP PORTFOLIO = Σ CE TRANSACTION

PORTFOLIO SIMULATION METHOD

CE CP PORTFOLIO = THE EXPOSURE PROFILE OF COUNTERPARTY

COUNTERPARTY EXPOSURE PROFILE

0

25

50

75

100

125

150

0 6 12 18 24 30 36 42 48 54 60

TIME (months)

POTE

NTI

AL

REP

LAC

EMEN

T C

OST

($

mm

)

Potential increase in value per unit of notional principal.

Potential exposure to a counterparty, at a high C.L., over lifetime of transactions with counterparty.

Assumes:

- No additional transactions

- Contractual cash flows set and settle over time.

- All legally enforceable riskmitigant agreements are taken into account.


MEASURINGCOUNTERPARTY CREDIT EXPOSURE

PORTFOLIO SIMULATION METHOD


Counterparty Exposure Portfolio Simulation

O DETAILED CONTRACT TERMS AND CONDITIONS.

CreditAdmin O TABLES OF LEGAL AGREEMENTS

- NETTING- MARGIN

O COLLATERAL

CreditAdmin O TABLES OF DEFAULT TRANSACTION PROFILES

FX FX DEBT I.R. EQ. COMM. COLLATERAL PRODUCT PROCESSOROPT SEC. DER. DER. DER. SYSTEM SYSTEMS

Detailed T&Cof Transaction

COUNTERPARTYCREDIT DATA BASE

COUNTERPARTY’S:- TRANSACTION DETAILS.- RISK MITIGANT DATA.

COUNTERPARTY’S:EXPOSURE PROFILE.

CE SERVER(analytical engine)

ANALYTICALENGINE

TABLES OF HISTORICAL VOLATILITIES AND CORRELATIONS

DAILY FEEDS OF CURRENT MARKET DATA MARKET DATA


General Method To Measure Counterparty’s Exposure Profile:

1) SIMULATE A PATH, P, OF MARKET RATES OVER TIME, M(t)P

- Start with current market rates.- Simulate a scenario (or path) of market rates at many future dates, over many years,

using tables of volatilities and correlations.

2) FOR SIMULATED PATH, P, MEASURE THE POTENTIAL MARKET VALUE OVER TIME OF EACH TRANSACTION WITH COUNTERPARTY K.

- Start with feed of transaction details and legal information.- For each simulated scenario, calculate the potential market value of each contract

at many future dates, using the contract’s terms and conditions, revaluation formula and the simulated state of the market.

- For each simulated scenario, at each future point in time, transform thepotential market value of each contract into the potential exposure of the portfolio through aggregation rules that take risk mitigants and legal context into account.

- i.e. For the counterparty K, for path M(t)P derive Exposure(t)K,P

3) THEN FOR SIMLUATED PATH, P, DERIVE COUNTERPARTY K’S POTENTIAL EXPOSURE OVERTIME

Loop

ove

r tho

usan

ds o

f pat

hs P

.

4) AFTER SIMULATING THOUSANDS OF POTENTIAL PATHS OF MARKET RATES, M(t)PCALCULATE EXPOSURE PROFILE OF COUNTERPARTY: THE POTENTIAL EXPOSURE AT SOME HIGH CONFIDENCE LEVEL, AT A SET OF FORWARD DATES


The Counterparty’s Exposure Profile:

THE EXPOSURE PROFILE EXPOSURE PROFILEPotential replacement cost of portfolio of contracts, over time, calculated at some confidence level, assuming:- no additional transactions- netting and margin taken

into account

COUNTERPARTY EXPOSURE PROFILE

0

25

50

75

100

125

150

0 6 12 18 24 30 36 42 48 54 60

TIME (months)

POTE

NTI

AL

REP

LAC

EMEN

T C

OST

($

mm

)

Exposure Profile LimitExposure Profile for credit line limits is typically calculated at a very high CL (e.g. 97.7%).

Exposure profile can be calculated at other CL, including expected positive profile or even a negative profile – how much one’s firm may owe counterparty in the future.


ECONOMIC CAPITAL

MEASUREMENT ISSUES FOR LOAN PORTFOLIO

Evan Picoult, Citigroup January, 2005 Page 13 Page 13

EC Definition• Economic Capital (also called “Economic Risk Capital” or “Risk Capital”) is a

measure of risk.

• Risk in this context means the potential unexpected loss of economic value over one year, calculated at a very high confidence level (99.97% CL).

• Thus EC measures risk from an insolvency or debt holders perspective (potential loss of value) rather than from an equity investment perspective (undiversified volatility of returns).

• Here is an example of EC for a loan portfolio:

Probability Distribution of Potential Credit Loss for a Portfolio of Many Obligors

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0-20-40-60-80-100-120-140-160Potential Credit Loss ($mm)

Prob

abili

ty o

f Cre

dit L

oss

Economic capital

= Unexpected Loss

= Loss at very high CL– Expected loss.

Expected loss should be covered by reserves and/or pricing.

Expected LossLoss at high CL

Economic Capital


Probability Distribution of Potential Credit Loss for a Portfolio of Many Obligors

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0-20-40-60-80-100-120-140-160Potential Credit Loss ($mm)

Prob

abili

ty o

f Cre

dit L

oss

The probability distribution of potential credit loss, and the ratio UL/EL, depends on the composition of the portfolio and the definition of credit loss.

Expected Loss (EL)Loss at a very high CL (e.g. 99.9%)

Economic Capital for Credit Risk to cover Unexpected Loss (UL)

EXAMPLE: EC FOR CREDIT RISKDefinition Of Economic Capital

Economic Capital For Credit Risk = A measure of risk: The unexpected loss, at a high confidence level, in excess of the expected loss.


Economic Loss - Loan Portfolio - Default Only Analysis

ASSUME SOURCE OF CREDIT RISK IS DEFAULT AND RECOVERY ONLY.

• FACTORS NEEDED TO SIMULATE LOSS DISTRIBUTION:

- Credit exposure per obligor

- Probability distribution of exposure at default, for contingent credit.

- Probability of default and correlations of probability of default

- Probability distribution of loss given default (LGD) (i.e. 1 – recovery%).

• There are several very different ways of modeling the potential loss distribution due to default and recovery.

• A robust method will model and capture the relative degree of risk diversification or risk concentration in the portfolio.



Drivers of EC

Idiosyncratic risk Systemic Risk

Uncertainty in the relative number of obligors that will default.

UL/EL tends to decrease as number of obligors increases.

UL/EL tends to decrease if diversify exposures across many industries / countries.

Uncertainty in relative size of exposure at default.

UL/EL tends to decrease as size of exposures becomes more uniform.

Uncertainty in LGD Relevant for portfolios with:• Few obligors• Inhomogeneous exposure

Economic Loss - Loan Portfolio - Default Only Analysis

- Probability distribution of migration of PD (internal risk rating).

- Volatilities and correlations of change in spread, given rating.

ASSUME SOURCE OF CREDIT RISK IS ECONOMIC LOSS

• FACTORS NEEDED TO SIMULATE LOSS DISTRIBUTION:

- Credit exposure per obligor

- Probability distribution of exposure at default, for contingent credit.

- Probability of default and correlations of probability of default.

- Probability distribution of loss given default (LGD).

Components of default only perspective

Component of long term simulation of obligor’s spread

• THERE ARE SEVERAL VERY DIFFERENT WAYS OF MODELING THE POTENTIAL LOSS DISTRIBUTION DUE ECONOMIC LOSS.


ECONOMIC CAPITAL FOR COUNTERPARTY RISK

DEFAULT ONLY PERSPECTIVE


Economic Loss - Counterparty Risk - Default Only Analysis

• EC For Counterparty Risk vs. EC For Loan Risk From A Default Only Perspective.

LOAN PORTFOLIO COUNTERPARTY CREDIT RISK

Default and recovery √ √DRIVERS OF WIDTH OF LOSS DISTRIBUTION

Inter-counterparty portfolio w.r.t. default. √ √

Inter-counterparty portfolio w.r.t. exposure √

TYPES OF DIVERSIFICATION BENEFITS

Variable exposure √

Intra-counterparty portfolio √

• Full simulation of EC from a default only perspective:

– For a given path of the market over time, the risk free value of each contract and the corresponding conditional exposure of each counterparty can be fully specified.

– For each path we can therefore simulate thousands of scenarios of default and recoveries, exactly as we would for a loan portfolio from a default only perspective.

– We can then loop over thousands of potential scenarios of changes in market rates.


Economic Loss - Counterparty Risk - Default Only AnalysisEC BY FULL SIMULATION: GENERAL METHOD, FIVE STEPS

1) SIMULATE A PATH, P, OF MARKET RATES OVER TIME M(t)P

Same as for Exposure Profile.

5) AFTER SIMULATING THOUSANDS OF POTENTIAL PATHS OF MARKET RATES, M(t)PCALCULATE FULL LOSS DISTRIBUTION AND DERIVE THE FULL SIMULATION ECONOMIC CAPITAL FOR COUNTERPARTY RISK.

2) FOR SIMULATED PATH P, MEASURE THE POTENTIAL MARKET VALUE OVER TIME OF EACH TRANSACTION WITH FOR COUNTERPARTY K. Same as for Exposure Profile.

3) FOR SIMLUATED PATH P, DERIVE COUNTERPARTY K’S POTENTIAL EXPOSURE OVER TIME.i.e. For each counterparty K, for path M(t)P derive Exposure(t)K,P Same as for Exposure Profile.

Loop

ove

r tho

usan

ds o

f pat

hs P

.

4) USING THE SET OF EXPOSURE PROFILES {Exposure(t)K,P } FOR ALL COUNTERPARTIES K, GENERATED BY MARKET PATH P:

CALCULATE THE POTENTIAL LOSS DISTRIBUTION BY SIMULATING THOUSANDS OF SCENARIOS OF DEFAULT AND RECOVERY FOR THE SET OF COUNTERPARTIES K.

Loop

ove

r tho

usan

ds o

f sce

nario

s of

def

ault

and

reco

very

.



IS IT POSSIBLE TO DEFINE A “LOAN EQUIVALENT” FOR COUNTERPARTY EXPOSURE?

• A “Loan Equivalent” is the fixed exposure profile, per counterparty, that would generate the same economic capital as the actual varying potential exposure.

Note:

• The use of a fixed exposure profile per counterparty to calculate total EC for the credit risk of all counterparties entails the incoherent summation of each counterparty’s potential exposure and has the same potential error as the simple, incoherent summation of the potential exposure of each transaction with a counterparty to get the counterparty’s total potential exposure.

HOW GOOD AN APPROXIMATION IS THE EXPECTED EXPOSURE PROFILE AS A LOAN EQUIVALENT? IN WHAT CONTEXT, IF ANY, IS IT A GOOD APPROXIMATION?

WHY DOES QUESTION ARISE?• Internally, transactors need an easy method for evaluating return on risk – i.e. return on EC.

Therefore they need to have a loan equivalent of notional (and equivalent tenor) to plug into EC tables as a function of tenor and obligor risk rating.

• Basel II species risk weight functions as a function of risk parameters: PD, LGD, EAD and M. Therefore need to identify the “loan equivalent” that multiplies the Basel II risk weight.



EC USING EXPECTED POSITIVE EXPOSURE PROFILE AS A LOAN EQUIVALENT:

1) CALCULATE THE EXPECTED POSITIVE EXPOSURE PROFILE, EPE(t)K , OF EACH COUNTERPARTY K

a) Simulate a path, P, of market rates over time M(t)P

Loop

ove

r tho

usan

ds

of p

aths

P.

b) For simulated path, P, measure the simulated market value over time of each transaction with counterparty K.

c) For simulated path, P, derive counterparty exposure over time.i.e. For the counterparty K, for path M(t)P , derive Exposure(t)K,P

d) Derive each counterparty’s expected positive exposure profile


3) CALCULATE THE ECONOMIC CAPITAL FROM THE LOSS DISTRIBUTION DERIVED FROM EACH COUNTERPARTRY’S EXPECTED EXPOSURE PROFILE.

2) USING THE SET OF EXPECTED POSITIVE EXPOSURE PROFILES {EPExposure(t)K } FOR ALL COUNTERPARTIES

Calculate the potential loss distribution by simulating thousands of scenarios of default and recovery for the set of counterparties K.

Loop

ove

r tho

usan

ds o

f de

faul

t sce

nario

s.

EPExposure(t)K

Economic Loss - Counterparty Risk - Default Only AnalysisDEFINING THE “LOAN EQUIVALENT” FOR ECONOMIC CAPITAL

From a default only perspective.

Define α as the ratio of Economic Capital calculated in two different ways:

i) EC calculated with full simulation of both defaults and variable exposure.

ii) EC calculated with simulated defaults using a fixed EPE profile for each counterparty

T) CL, (P; Cap Econ

T) CL, (P; Cap Econ )(

ONLY DEFAULT SIM;FIXED_EPE_

ONLY DEFAULT FULL_SIM; =P; CL,Tα

Where: P = Particular portfolio of counterparties with transactions and assumptions about portfolio, e.g. PDs, Correlations, etc.

CL = Confidence Level that EC is measured at.T = Time Horizon over which EC measured.

Therefore: LOAN EQUIVALENTK = α * EPEK for counterparty K

Questions:- What factors does α depend on


? How much does α vary, firm by firm? - Is α stable over time i.e. how stable are characteristics of portfolios?

Economic Loss - Counterparty Risk - Default Only AnalysisISDA TESTS

• Create test portfolios and calculate α as a function of the characteristics of the portfolio:- Efective number of counterparties- Effective number of market factors- Probability of default of counterparties- Correlation of default.- Initial MTM of counterparties’ portfolios.- Other factors.

• Initial Proposal: Evan Picoult, Citigroup

• Simulations of Stylized Portfolios: Eduardo Canabarro, GS (now Lehman)

• Analytical Calculations: Tom Wilde, CSFB

• Measurement of α for Real Portfolios: Several Firms

• SUMMARY CONCLUSION: For Large Market Makers α ≈ 1.10

See: - ISDA web site. Papers on Counterparty Risk to Basel Committee, June 2003- Risk Magazine, September, 2003


Results from slide of Tom Wilde, CSFB


Asset corr'n

Spot value+/-

No factor

s

No. of cpties PD Conf

levelSystematic

riskActual

Portfolio AReference Portfolio B α = Α/Β

λ u K N p q Analytic M-Carlo Analytic M-Carlo Analytic M-CarloAnalyticBase case Percentile Percentile Percentile

22% 1.36 3 200 0.3% 99.9% 10.19 13.14 12.96 12.06 12.02 1.09 1.08Sensitivity to asset correlation

0% 1.36 3 200 0.3% 99.9% 0.51 6.09 NA 4.26 NA 1.43 1.4612% 1.36 3 200 0.3% 99.9% 5.31 8.99 8.91 7.43 7.73 1.21 1.1524% 1.36 3 200 0.3% 99.9% 11.30 14.08 13.96 13.04 13.05 1.08 1.0750% 1.36 3 200 0.3% 99.9% 30.69 32.70 32.50 32.06 31.82 1.02 1.02

Sensitivity to current market values22% 0 3 200 0.3% 99.9% 5.65 8.42 8.23 6.24 6.18 1.35 1.3322% 1 3 200 0.3% 99.9% 8.26 10.96 10.81 9.61 9.61 1.14 1.1222% 2 3 200 0.3% 99.9% 14.28 17.80 17.64 16.95 16.96 1.05 1.0422% 3 3 200 0.3% 99.9% 21.24 25.95 25.73 25.19 25.26 1.03 1.02

Sensitivity to the number of market risk factors22% 1.36 1 200 0.3% 99.9% 10.19 13.22 13.11 12.02 12.02 1.10 1.0922% 1.36 5 200 0.3% 99.9% 10.19 13.07 12.93 12.10 12.02 1.08 1.0822% 1.36 10 200 0.3% 99.9% 10.19 12.97 12.91 12.01 12.02 1.08 1.0722% 1.36 50 200 0.3% 99.9% 10.19 12.96 12.89 12.00 12.02 1.08 1.07

Sensitivity to number of counterparties22% 1.36 3 20 0.3% 99.9% 1.02 3.54 3.72 2.81 2.85 1.26 1.3122% 1.36 3 50 0.3% 99.9% 2.55 5.21 5.26 4.27 4.37 1.22 1.2022% 1.36 3 100 0.3% 99.9% 5.10 7.79 7.83 7.08 6.92 1.10 1.1322% 1.36 3 500 0.3% 99.9% 25.48 28.92 28.36 27.81 27.31 1.04 1.04

Sensitivity to probability of default22% 1.36 3 200 0.1% 99.9% 4.55 7.03 6.93 6.01 6.16 1.17 1.1222% 1.36 3 200 0.5% 99.9% 14.56 17.59 17.56 16.44 16.50 1.07 1.0622% 1.36 3 200 1.0% 99.9% 23.10 26.60 26.50 25.09 25.20 1.06 1.0522% 1.36 3 200 5.0% 99.9% 59.40 65.00 64.55 61.90 61.84 1.05 1.04

Sensitivity to confidence level22% 1.36 3 200 0.3% 99.0% 4.37 6.08 6.11 5.68 5.56 1.07 1.1022% 1.36 3 200 0.3% 99.5% 5.85 7.90 7.90 7.18 7.23 1.10 1.09

Actual portfolio A• This is the portfolio with full

stochastic exposures and correlations as per the settings described above.

Reference portfolio B• This portfolio is as A but with

each exposure fixed = EPE of the corresponding portfolio A counterparty.

α = Aq/Bq

• alpha measures the extra risk arising from the fact that exposures are variable and correlated.

Agreement • The analytic results agree well

to MC – fortunately!

2005 addendum:

α = 1.2 to take into account “general wrong way risk” –negative correlation of changes in general level of yield curve and credit spreads.

ECONOMIC CAPITAL FOR COUNTERPARTY RISK

ECONOMIC LOSS PERSPECTIVE


Economic Loss - Counterparty Risk - Full Economic Loss Analysis

FIRST QUESTION:

WHAT SHOULD BE THE EFFECT OF CREDIT SPREADS / RISK RATING ON DERIVATIVE VALUATION?

If all derivatives are (and should be) marked-to-market by discounting expected future cash flows at LIBOR Bid/offer midpoint, then valuation would be:

- Independent of counterparty risk rating and- Independent of counterparty credit spread.

In that case, changes in risk rating or spreads would not cause a change in economic value. Default only and economic loss analysis would be the same.


Economic Loss - Counterparty Risk - Full Economic Loss AnalysisHOW SHOULD RISKINESS OF OBLIGOR / MARKET SPREADS AFFECTTHE CURRENT MARKET VALUE OF FORWARD FX AND DERIVATIVES?

• BOND/LOAN VALUE = PV of cash flows discounted at Risk Free Rate – RISK PREMIUMLoan

= PV of cash flows discounted at (Risk Free Rate + SPREAD)

• RISK PREMIUM Loan ≅ PV Risk Free * Duration * Average SpreadLoan

• DERIVATIVE VALUE = PV of cash flows discounted at Risk Free Rate – RISK PREMIUM

CONTEXT FOR ASCERTAINING RISK PREMIUM OF DERIVATIVES. WHY THE CALCULATION IS NOT IDENTICAL TO THAT FOR LOANS:

1) PORTFOLIO ANALYSIS OF EXPOSURE

METHOD FOR CALCULATING RISK PREMIUM MAY DEPEND ON THE PURPOSE OF THE CALCULATION:

- Pricing

- Cost of credit

2) TIME VARYING, UNCERTAIN FUTURE EXPOSURE

3) FUTURE ASSET OR LIABILITY?– In general, at any future date

- Obligor could owe us (asset) or- We could owe obligor (liability)

– Therefore which party’s risk ratings should be taken into account?


LET US CALL THE CREDIT RISK PREMIUM OF THE COUNTERPARTY’S PORTFOLIO ITS CVA.CVA = Credit value adjustment for counterparty’s credit risk

THEREFORE, THE MARKET VALUE OF DERIVATIVE PORTFOLIO WITH A COUNTERPARTY IS:= Σ MARKET VALUE (discounted at risk free rate) - COUNTERPARTY RISK PREMIUM

= Σ MARKET VALUE (discounted at risk free rate) - CVA

SIMULATING LOSS DISTRIBUTION OF DERIVATIVE PORTFOLIO: ECONOMIC LOSS ANALYSIS

MEASURING THE CVA OF A COUNTERPARTY: Modification of a proposal by Bollier & Sorensen.

TWO PERSPECTIVES ON CVA: A UNILATERAL AND A BILATERAL PERSPECTIVE.

UNILATERAL CVA: CVACOUNTERPARTY K, UNILATERAL = CVA+CNTPY K

BILATERAL CVA: CVACOUNTERPARTY K, BILATERAL = CVA+CNTPY K - CVA–

CNTPY K

Credit premium of own firm’s expected asset from derivatives

Credit premium of CP’s expected asset from derivatives.

Calculated on a portfolio basis, taking into account potential future exposure





Market Value CP Portfolio = Σ MVCP Portfolio (risk free) - CVA CP Portfolio

CVA CP Portfolio_Unilateral = CVA+CNTPY

CVA CP Portfolio_Bilateral = CVA+CNTPY - CVA-

CNTPYExpected amount CP will owe to own firm.

A Counterparty's Expected Positive Exposure Profile

0

15

30

45

60

75

0 6 12 18 24 30 36 42 48 54 60

Time (Months)

Pote

ntia

l Exp

osur

e ($

MM

)

CVA+CNTPY K (Credit premium of own’s firm potential asset due to CP K)

)*** JJJ,KJ,K dftSpread CP ForwardExposure Expected ( J

∆∑ +=

Calculate and sum over each forward period J, for CP K.

Expected amount own firm will owe to CP.


A Counterparty's Expected Negative Exposure Profile

0

15

30

45

60

75

0 6 12 18 24 30 36 42 48 54 60

Time (Months)

Pote

ntia

l Exp

osur

e ($

MM

) CVA-CNTPY K (Credit premium of CP K’s potential asset due to own firm)

Calculate and sum over each forward period J, for CP K

)*** JJJJ,K dftSpreads Firm' Own ForwardExposure Expected( J

∆∑ −=



Market Value CP Portfolio = Σ MVCP Portfolio (risk free) - CVA CP Portfolio

CVA CP Portfolio_Bilateral = CVA+CNTPY - CVA-

CNTPY (bilateral perspective)

CVA CP Portfolio_Unilateral = CVA+CNTPY (unilateral perspective)

EXAMPLES

Example 1:

• ASSUME:– Only One Swap With Counterparty– Counterparty And Own Firm Have Same Risk Rating.– Potential Change in Value Has Symmetric Shape For Pay or Receive Fixed Swaps.

(e.g. flat yield curve).

• CONSEQUENCE FOR CVA unilateral and bilateral

Example 2:

• ASSUME:– Only One Swap With Counterparty– Counterparty Is BBB And Own Firm Is AA.

• CONSEQUENCE FOR CVA unilateral and bilateral


Counterparty Risk Issues – Maturity and CVA

Deriving M for Unilateral CVA:

• For simplicity, let us define Spread = average spread for counterpartyEPE1 yr = Average EPE over one year horizon.

Therefore:

∑=

=portfolio oflife Full

1kkkk df∆tEPE * Spread CVA

∑=


1kkkk df∆tEPE * Spread CVA ∆∆

• In analogy to the relationship between the change in the credit spread of a bond and its duration, we can define the effective M by the equation:

M * EPE * Spread CVA year1∆∆ ≡

• We therefore have the definition of M, from a unilateral perspective:

∑

∑

=

== Year 1

1 kkkk

Maturity

1 kkkk

dft

dft

EPE

EPE M

∆

∆ M is simply the ratio of:

The area under the full-lifetime EPE curve divided by the area under the 1-year discounted EPE curve.


For fuller discussion including M for bilateral CVA, see Evan Picoult and David Lamb (2004) Economic Capital for Counterparty Credit Risk, Chapter in Economic Capital: A Practioner Guide, London, Risk Books

Counterparty Risk Issues – Maturity and CVA

A Counterparty's Exposure Profile

0

25

50

75

100

125

150

0 6 12 18 24 30 36 42 48 54 60

Time (Months) ==>

Pot

entia

l Exp

osur

e ($

MM

)

Lifetime area under discounted curve

140.7 EPEPortfolio ofLife Full

1 kkkk dft =∑

=∆

A Counterparty's Exposure Profile

0

25

50

75

100

125

150

0 6 12 18 24 30 36 42 48 54 60

Time (Months) ==>

Pote

ntia

l Exp

osur

e ($

MM

)

EPE 75.9 EPE Year 1Year 1

1 kkkk dft ==∑

=∆

One year area under discounted curve

Therefore M = (140.7/75.9) yrs= 1.85 yrs= 22.2 months



CONSEQUENCE FOR PSE ECONOMIC CAPITAL.

TO SIMULATE ECONOMIC LOSS DISTRIBUTION:

• SIMULATION NEEDED TO INCORPORATE POTENTIAL DEFAULT AND RECOVERY

- Simulate potential exposure

- Simulate potential defaults

- Simulate potential recovery, given default

• SIMULATION NEEDED TO INCORPORATE POTENTIAL LOSS DUE TO CHANGES IN RISK RATINGS AND/OR CREDIT SPREADS:

- Need to simulate how CVA could change over time per obligor due to:

- Potential changes in exposure profile at a future date- Due to changes in market rates and volatilities/correlations.

- Potential changes in obligor’s risk rating / credit spread at a future date.

- A double level of simulation



Hedging Counterparty Risk

Market Value CP Portfolio = Σ MVCP Portfolio (risk free) - CVACP Portfolio

Gives rise to market risk Gives rise to counterparty credit risk

∑=



• For a specific portfolio of U.S. Dollar LIBOR interest rate swaps, the discounted area under the EPE curve, Σ(EPEk ∆tk dfk), will be a function of the terms and conditions of all the swaps and the structure and volatility of the U.S. Dollar LIBOR yield curve.

• Therefore for such a portfolio:

)()(

}){}{(

x contracts;f * Spread x contracts;f * Spread

σr(t) contracts;f * Spread

df∆tEPE * Spread CVA

k

kkkportfolio oflife Full

1k

==

=

= ∑=

kk ,

∆x*∆Spread*xSpread

f ∆x*xf * Spread df∆tEPE * ∆Spread ∆CVA

2portfolio oflife Full

1kkkk ∂∂

+∂∂

+= ∂∑=


Hedging Counterparty Risk

∑=



∆x*∆Spread*xSpread

f ∆x*xf * Spread df∆tEPE * ∆Spread ∆CVA

2portfolio oflife Full

1kkkk ∂∂

+∂∂

+= ∂∑=

Hedge potential changes in market rates in proportion to CP spread.

Determines notional value of interest rate derivatives need to buy to hedge interest rate risk (or more general hedges needed to hedge more general market risk).

Potential risk of correlation of change in spread and changes in market rates.

Determines degree of correlation risk, which may not be possible to hedge.

Hedge spread risk in proportional to discounted EPE curve.

Determines notional value of credit default swaps to buy.

Effectively have transformed counterparty credit risk into market risk.

SUMMARY

• PORTFOLIO SIMULATION OF COUNTERPARTY EXPOSURE PROFILE.– Described method.

• ECONOMIC CAPITAL FOR LOAN CREDIT RISK– Described default only simulation.– Described loss of economic value simulation.

• ECONOMIC CAPITAL FOR COUNTERPARTY RISK - DEFAULT ONLY:– Described full coherent simulation of potential exposure and default.– Described approximation using incoherent simulation with expected

exposure profile scaled up by factor α .

• ECONOMIC CAPITAL FOR COUNTERPARTY RISK - ECONOMIC LOSS:– Need to first define Credit Value Adjustment (CVA) for credit risk of

counterparty’s portfolio. – Need to simulate default, recovery and changes in CVA over time.

• DYNAMICALLY HEDGING COUNTERPARTY RISK


Documents

Measuring and Dynamically Hedging Counterparty Credit Exposure …ieor.columbia.edu/files/seasdepts/industrial-engineering... · Measuring and Dynamically Hedging Counterparty Credit