CVAFVA Challenge

Sep 25, 2013

Information Security Identification: Confidential

Practical CVA/FVA Calculations: Complexity and Challenges

Presented by Dongsheng Lu, Managing Director

Sep 25, 2013

Consistent Accounting

Terms Explanation

Fair Valuation of financial Instrument

FVACVAVV FCFair

FairV


Valuation under fully-collateral assumption

Credit Value Adjustment

One way CVA

Funding Value Adjustment

Funding cost and benefits

Fair

FCV

CVA

FVA

Credit and Funding Concepts

CVA/FVA: both coming from borrowing/lending. Default vs funding. Base: Perfect credit, fully collateralized cash CVA: credit default only FVA: anything related to funding, value of collateral, cost of posting collateral,

Bond spread vs CDS: funded vs non-funded default spread

u’: cpty default risk


LX=u+L

u’: cpty default risk

u

u’

L

Funding: Balance sheet and Borrowing

Businesses

Capital

Debt Issues

Equity Financial institutions rely heavily on

borrowing and are highly leveraged. The better capitalized, the higher the credit

rating.

Firm’s funding level represents market/investor’s view on firm’s credit as well as market liquidity


and

Investments

Securities

Asset Liability

as market liquidity

Firm’s Funding activities:

Deposits Issue debt Issue equity Convertible bond Funding centers Practical difficulty to compare different

funding activities

Funding

DeskCpty

Lend $1

LOBMarket

Lender

Lend $1 Lend $1

LOB Lending

LOB Borrowing

CPTY Borrowing

FundDeskBorrowing

FundDeskLending

Market Lending

Unsecured Lending/Borrowing


-$1

$1.2

$1

-$1.1

$1

-$1.2

$1

-$1.1

-$1

$1.1

-$1

$1.1

t=0

t=T

Internal Funding Transfer

Derivative Activity (CVA/DVA/FVA)

External Borrowing

Assumptions

Applies to financial institutions with high leverage

At any given time, there is an equilibrium funding cost for the firm. Assuming the existence of a funding curve.

Funding policy and operations varies from firm to firm

Given a policy and funding curve


Firm should be marking the future cash flow exactly the same way, whether it is issued debt or swaps or complex derivatives

Debt issuance spread would give the funding level across maturities

Funding spread include: Firm’s credit, Market’s liquidity

The same discounting should be applied for derivatives

Assumptions

There is one true economic value for each market participant for a given trade

Every market participant’s economic value could be different for the same trade

Different valuation reflects the competitiveness of the market participants

“Market Price” Market’s average exit value


same way as LIBOR as average funding rate

Valued based on average funding rate

Funding value adjustment (FVA) depends on how one treats all other adjustments

A lot of confusion as to what is FVA

It is a relative value

Base: Fully collateralized. Base + Credit + Funding

Funding value adjustment Collateral value Part of discounting spread

FVA Debate: FVA in Fair Valuations?

LOB1

LOB2

LOB3

Borrow(-)/Lend(+)

+ + +

+ +

+

* *

Total Funding Requirement: Sum of funding cost/benefit

IndividualBusinesses


Given funding spread X at firm level, all LOBs mark book based on X The sum of funding/cost reflect the total funding requirements at the firm level Funding from outside at firm level balanced with sum of all LOBs funding needs Individual business level funding X marking Equivalent Market value for LOBs +

apply funding X at the firm level Proper incentive for the individual businesses

- - -LOBn

**

**

FVA Debate: FVA in Fair Valuations?

Aggregated Short/Medium/Long term funding needs

Ideally matched funding needs and actual outside funding

Reality Short/Medium term funding liability/borrowing Longer term asset/lending


Regulations more balanced asset/liability matching in terms

Funding cost should be charged at the individual LOB level to reflect the proper incentive.

Market change: for example, S&P 500 index total return (LIBOR+30/40bps from close to LIBOR flat)

Typical Client Quoting with CVA/FVA

CSA / CCP?

LCH Pricing

CME Pricing

LCH /CME ?

Secured Pricing

CVA ~ 0 Collateral asset

Differential Discounting

Pricing

CCP

YES

LCH

CME

Margin rule + Incrementl Risk


Fully Collateralized?

Collateral asset Collateral CCY

CSA Pricing

CollateralThrshld

Rating Trigger Break Clause …

Collateralized Portion

Un-collateralized Portion

CVA Funding

Cost/Benefit

CSA

YES

NO

SCEN

SCEN

Other Pricing Terms

Market Hedging Cost Credit Hedging Cost Capital/Balance sheet Charge Profit Margin


Classify all funding situations, construct funding curves

Collateral Currency Xccy spread curve Optimal xccy funding

Basic Funding Spread Curves

Base discounting: e.g. USD Collateral Different Currency Different Asset Construct Funding spread curve



Collateral Asset Cash/Treasury Agency GSE Corporates Munis

Break Clause

Dynamic Funding Curve Generation

Max(FND1,FND2, … FNDN)

A Typical Example: JPY Swaption Trading

Differentiate Collateral: JPY or USD or CSA, differential discounting

Differentiate Underlying swap: LCH/CCP or CSA

- LCH Swap: central cleared swap- CSA: different assumptions- Complex situation: mixture discounting CSA+LCH


- Complex situation: mixture discounting CSA+LCH

Forward vs Spot premiums

Physical vs Cash settlement

Customer trading: CVA, FVA, Capital

t=0 Option Swap

CSA Discounting

LCH Swap: OIS Discounting

Example: A CSA governed Swaption settle into LCH Swap


t=0 Option Expiry

Swap Maturity

Mixture Discounting Example

Before option expiry, economics follow CSA discounting

After option expiry, settle into LCH swap, native OIS discounting

t=0Break Date 1

Deal Maturity

Credit Risk Exposure

No Credit risk

Example: Break/Trigger/Exit Clause/Mutual Put/Replacement


Mutual Put, Break/Trigger Clause

CVA: No Credit risk after break date, limited tenor risk paid at day one

FVA: Discounting for cashflow after break date is ambiguous

FVA: Average market funding is LIBOR?

Replacement event ?

Scenario Based CVA/FVA Calculation

Scenario Based Pricing

Rating based threshold scenario rating scenario valuation scenario threshold

Trigger/Exit

No Credit risk after trigger Exiting value calc (LIBOR) Replacement cost

Triggered?

V < 0

Collateralized: Max(-V-H,0)

Collateral discounting

Unsecured: -Min(-V,H)


scenario threshold

Rating Trigger scenario triggered?

Break Clause

Unsecured: Min(V,H’) Credit Exposure: Min(V,H’) CVA: Min(V,H’)*u’*dt Funding cost: Min(V,H’)*X*dt

Collateralized: Max(V-H’,0) No CVA Collateral Discounting

V < 0

V > 0

No DVA Fund Benefit: Min(-V,H)*X*dt

Scenario Based CVA/FVA Calculation

Market Simulations (M)(e.g. 10,000 paths)

CVAFVA

Aggregation

CSA

ExposureExposure

CreditCredit

CreditCredit

FundingFunding


Credit Simulations (C)(e.g. 100,000 paths)

Capital Requirement:

Counterparty Credit risk (Basel II) CVA VaR based (Basel III) Advanced/Standardized approach

Challenge: Complex CSAs

Rating Migration

Rating Based ThresholdDowngrade eventsReplacement cost

Break/Default Clause

Mutual put Termination event (ATE)Legal/Netting

Collateral Definition

Collateral AssetCollateral CurrencyHair Cut Initial marginsCollateral Damage


Need scenario based credit

Credit simulation with migration Apply CSA in scenarios

Additional Exit Event

No credit risk after exit Funding cost calculation Exit event vs market liquidity

Collateral Modeling

Collateral choice: static funding curve. Optionality impractical.

Hair cuts/Margins

Challenge: Massive Computations

Market RiskAny market risk within the normal

derivatives pricing/risk management are risks for CVA/FVA, including rates, FX, stock price, volatility/skew and correlations etc .

Pricing,

Credit Risk

Specific Name riskGeneric credit risk

Live TradingLive Pricing/Quoting Incremental CVA/FVA Calc


Pricing,Risk Management,

Hedging and Perf Measurement

Generic credit riskDowngrade/Default riskHedging strategy

Wrong Way Risk

Specific counterparty WWR WWR hedge

Capital Requirement

CVA VaR BASEL III Capital Standardize vs Advanced methodology


Market RiskPrice Levels, delta/gammaRates/delta buckets, gamma ladderBasis (Libor basis, xccy basis etc) bucketsVega distribution/Skew exposureCorrelation levels

Credit Risk

Specific Name credit bucketsGeneric Index credit buckets, hedge ratio

Each ScenarioMarket, Credit

Simulations, and

Scenarios Generation


Generic Index credit buckets, hedge ratio Proxy hedges, hedge ratio

Wrong Way Risk

Different Correlation levels Factor based WWR

Regularoty Requirement

Credit sensitivities for CVA VaR Stress testing: market and credit

Simulations, and Aggregations

Sensitivities Buckets

Hedge Ratio etc


Each ScenarioMarket, Credit

Scenarios Generation

Efficient Computing

Efficient CVA/FVA computation design

Adjoint Algorithmic Differentiation


Distributed Computing

GRID Computing GPU Computing

Market, Credit Simulations, and

Aggregations

Sensitivities Buckets

Hedge Ratio etc

Node Node Node Node Node

Backward Pricing: credit/funded discounting

Credit Discounting: (similarly for funded discounting)

drttur PRee )1(1)(

)/()/(/ unfundriskfreeVfundedriskyVFVACVA

CVA/FVA Calculations

Forward Simulation: collecting defaults and funding cost/benefits


E(t): Exposure at risk at time tD(t): discountingX(t): funding spread

T

T

d

dttDtXtEFVA

dttDtPRtECVA

0

0

)()()(

)()()1)((

Forward Simulation: collecting defaults and funding cost/benefits

Backward Discounting:

Pros: Easy to incorporate spread impact Can incorporate credit impact on exercise boundary

Cons:

Difficult to net portfolio of deals with same CSA Difficult to apply more complex CSA terms Difficult to calculate wrong way risk, incremental CVA

Forward Simulation:

Pros: Can net portfolio deals across same CSA easily Can deal with very complex CSA terms easily Fast computation of wrong way risk and incremental CVA

Cons: Need to develop a global simulation model and a complex correlation model Approximation: exercise boundary not affected by credit/funding

Forward Simulations Steps

Consistent multicurrency/asset simulations (IR, FX, Equity … )

Along with credit simulations + ratings transition

Valuation of all instruments once market factor exposure

CVA/FVA Calculations

Valuation of all instruments once market factor exposure

Aggregation of market factor exposure with credit/ratings

Application of netting and CSA

Collect default loss and funding cost/benefits

Credit Rating Collateral Threshold

AAA 50M

AA 20M

A 5M

BBB 0M

Credit Rating Event

BBB Termination

Eligible Currencies

USD

EUR

CSA Complexity

Rating dependency of collateral posting Downgrade provisions Collateral asset/currencies Automatic/Discretionary terminations Legal/Netting opinion

Asset Haircut

Cash 100%

Treasury < 2y 101%

Treasury (2y-5y) 103%

Treasury(5y-10y) 105%

Treasury(>10y) 108%

GSE Passthroughs

115%

Corprts/Munis 120%

GBP

CSA Complexity Modeling

Needs to model ratings in the future Market risk simulation and credit

simulations at the same time Correlations could become important

Challenge: CSA Netting with Complex Portfolio

Generic Time Grid

Generic Time Grid

Specific Trade Pricing:

Trade specific Cashflows, Resets, Notifications, Exercise etc Requires a trade specific time grid Different grid for different deals Different grid for different deals

Specifc Time grid For Deal j

Specifc Time grid For Deal k

Methodology 1:

Market factor calculations based on trade specific grid Regression/interpolation of scenario valuations:

V = V(R,t), R: regression variables Use regressed/interpolated values on generic grid/scenarios Exposures Aggregation of market factor exposures with credit/ratings

Methodology 2:Methodology 2:

Direct valuations based on trade specific grid: simulation based (such as BGM) Exposures are obtained for trade specific grid Interpolate market factor exposures from trade specific grids to generic grid

Methodology 1:

Pros: Can use any trading quality pricing model regression

Cons:

Accuracy of regression/interpolation: explanatory variables and power

Methodology 2:Methodology 2:

Pros: Consistency can be built among all market risk factors

Cons: Need to develop a global simulation model Need consistent correlations among risk factors and for all trades.

Random number Mapping

Random number for generic time grid => derive trade specific grid RN Conserve correlation: equivalent random number generation

Black: Generic Time GridOrange: Trade Specific Grid

dt dt’

dz

dz’’

Known dz’

Need

Black: Generic Time GridOrange: Trade Specific Grid

dt dt’

dz

dz’’

Known dz’

Need

Brownian Bridging Random Number

Orange: Trade Specific Griddz’’dz’’NeedNeed

Adding random numbers:

Correlation Conserved:

Credit simulation:

Aggregation vs market valuations Requires correlations among Market and Credit risk factors Use generic time grid CVA = Credit_Aggregation(CSA, Credit Ratings, Market Values) FVA = Funding_Agg(CSA, Credit Ratings, Market Values, Funding Spread)

Challenge: Credit Simulation and Ratings Migration

Generic Time Grid

FVA = Funding_Agg(CSA, Credit Ratings, Market Values, Funding Spread)

Methodology 1:

Simulate credit spreads in scenarios Map credit spreads to ratings credit migrations

Credit Simulation

Methodology 2:

Structural model simulation using ratings transition matrix Risk neutralization of transition matrix Defaults from ratings migration calibrated to market traded CDS

Credit Simulation: Finite Markov Process + Ratings Transition

Rating k

Default

AAA

A

B

AA

C Dynamic simulation

Rating i Rating j

Rating m

Structural asset model Based on a N-rating system. Probability of defaults calibrated step-wise

1...00

............

...

...

~ 22221

11211

n

n

ppp

ppp

p

Credit Simulation: Transition Matrix Propagation

1...00

1...00

............

...

...

1...00

............

...

...

~ 22221

11211

2

1

22221

11211

Tn

TT

Tn

TT

Tn

T

T

n

n

ppp

ppp

P

P

P

ppp

ppp

p

Credit Simulation: Transition Matrix Propagation

n

t = 1 t = T

Credit Default: Jump to default and Transition to Default

AAA AA A BBB DC

Credit Calibration: Generic Stepwise Transition Matrix Calibration

1

1

2

1

1

1

12

11

......

~

i

dn

i

d

i

d

inn

in

in

t

P

P

P

p

p

p

Pi

Risk Neutralizing Markovian Transition Matrix

P(Default) A B C

1 0.31% 1.72% 6.28%

2 0.72% 4.27% 11.80%

5 2.60% 13.00% 25.60%

10 7.00% 30.00% 48.00%10 7.00% 30.00% 48.00%

TMatrix(Annnual) A B C D

A 96.0% 2.50% 1.19% 0.31%

B 0.40% 83.0% 14.87% 1.73%

C 0.41% 1.00% 92.3% 6.29%

D 0.00% 0.00% 0.00% 100%

Credit Simulation: Structural Asset Model and Calibration

1

2

1

1

12

11

......

~i

d

i

d

in

in

t

P

P

p

p

Pi

Exact Calibration to specific name CDS

Specific name calibration after generic TM calibration From generic TM calibration, probability of default not calibrated exactly In this calibration process, default probabilities are adjusted to match CDS

exactly One can assign partial default to scenarios to match default exactly

11

......i

dn

inn Pp

Market Factors

Base ScenarioMarket/Credit

Correlation Stress

Wrong Way Risk: Market/Credit Correlation Stress)

Correlations

Market Factors

Credit Factors

Keeping same market factor exposure

Solving credit factors incrementally

Credit Risk Hedging with Rating Trigger

AAA AA A BBB DC

Credit Rating Collateral Threshold

AAA 50M

AA 20M

A 5M

BBB 0M

Credit Rating Event

BBB Termination

Incremental CVA Calculation for a New Deal

Given exposure V(ijk) for portfolio Calculate single deal exposure U(ijk) Portfolio CVA = Agg(credit, V) New Portfolio CVA’ = Agg(Credit, U+V) Incremental CVA = CVA’-CVA

Incremental CVA Calculations

Incremental CVA = CVA’-CVA Credit aggregation: fast process

Incremental CVA Calculation for a Existing Deal

Given exposure V(ijk) for portfolio Calculate single deal exposure U(ijk) Portfolio CVA = Agg(credit, V) Portfolio without deal CVA’ = Agg(credit,V-U) Incremental CVA = CVA-CVA’

CVAFVA Challenge

Documents

funding level

funding policy

default vs funding

maturities funding spread

equilibrium funding

funding concepts cvafva

total funding requirement

value of collateral