Etienne KoehlerBarclays CapitalEtienne.Koehler@univ-paris1.fr

CVA - VaR

January 2012

Shahram AlavianRoyal Bank of ScotlandShahram_Alavian@yahoo.com

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Preliminaries Credit Value Adjustment (CVA ) Value at Risk (VaR) VaR for CVA

The Proposed Approach VaR properties VaR and Monte Carlo simulation VaR for CVA

Examples A non-linear payoff Application to Regulatory VaR

Concluding Remarks Limits of the approach Challenges

In Nutshell

Statement Calculate VaR on CVA using a single batch run instead of multiple runs; once for each sensitivity.

Benefit Faster execution of VaR

Approach Insert a 1-day time horizon in the simulation timeline. Model the distribution of the change in value, at 1-day horizon, as a function of the underlying factors. Use this function to reprice the CVA changes in the same way a trade’s pricing formula is used to calculate its VaR.

What we say … How we go about it ….

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Credit Value Adjustment (CVA ) Value at Risk (VaR) VaR for CVA

Preliminaries

4

Definition It is a Fair Value Adjustment to an OTC Trade. It is equivalent to the risk that a dealer has taken on its counterparty’s credit by entering into an OTC trade

Objective Accounting, risk mitigation, and regulatory Unilateral versus bilateral

Management Accounting – Managing its volatility in the balance sheet. Risk Mitigation – Selling protection to other Proposed business functions (desks) Regulatory Capital – Minimizing the cost of regulatory capital

Preliminaries - CVA

Dealer Counterparty

=Dealer Counterparty

+Dealer Counterparty

Defaultable Cash flow

Default Free Cash flow

Option to Default

5

Main Components of CVA Market component Credit component (including default) Cross gamma component

Hedging CVA Inconsistencies in hedging from different objectives Hedging for Default? Hedging for Capital Hedging for volatility

The hedged CVA book, П is

Preliminaries – CVA Continued …

HCVACVA

6

Value at Risk (VaR) Is A measure of unexpected loss due to market move. tool for calculating capital qualitative representation of the volatility of the balance-sheet. risk-limit measure for trading desks

Sensitivity Based Sensitivities are sent to market risk department. VaR and its relevant break-down numbers are then returned. Not all sensitivities are included

Historical Based Recent (3 year) daily time series “stressed period “

Preliminaries – VaR

7

CVA Volatility leads to: CVA Capital

CVA as a fair value adjustment brings volatility to the balance sheet For firms using unilateral CVA (no DVA), the unexpected loss due to a rise in credit spread of the market can be very large. Newly introduced regulatory CVA capital (BASEL III)

Limit on CVA-VaR Level - how much? Bucket – Region, industry, currency, …

Preliminaries – VaR for CVA

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VaR properties VaR and Monte Carlo simulation VaR for CVA

Proposed Approach

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1. Instantaneous: It excludes the duration of the portfolio, omits any cash flow during the time horizon and limits the change of value to the instantaneous change in the underlying risk factors, only.

2. Conditional: It is conditional on its initial value at current time.

3. Functional: Similar to a pricing function, it is a function of the risk factors driving its value. This feature provides the means for generating VaR from any arbitrary distribution.

Proposed Approach – VaR Properties

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Monte Carlo Approach N risk factors are simulated under a joint process. For each path j and time t, we have (dependence on t is implied)

Proposed Approach – VaR and Monte Carlo

j

Njjjj RFRFRF )()2()1( ,,, RF

tt jj ,RF

Valuation of an asset π conditional on each time t and path j will result

Therefore, we have a distribution of π, as a function of its underlying risk factors, for each time t.

Defining a 1-day Implied VaR. Calculate the conditional values of the asset at δ = 1-day time horizon for each path j.

j

In future slides, we replace π with expected exposure (EE) and CVA itself.

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1. Create an Instantaneous Change

Proposed Approach – VaR and Monte Carlo

00, jj

0'0,' jj

'

0,'0,',

jj

jjjj

This is not an instantaneous change. Therefore, we pick up another path j’.

and create an instantaneous change by

Calculate the change in values of the asset at 1-day time horizon for each path j.

2. Create a Conditional Change

0' j

So we have a conditional change from j’ which given the above assumption, we a have conditional on spot.

Having simulated a large number of paths, one can find a path for which

<= This change is from any j’ to any j

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3. Create a Functional Change

Proposed Approach – VaR and Monte Carlo

',', jjjj RF M

Assuming there exists a model, M, representing the conditional changes as a function of the changes in the risk factors

with

jjjj RFRFRF '',

0,22

0,120,110',', ; RFRFRFβRF k

x

kkkjjjj M

Using a linear regression model, for example,

Putting It All Together ][%99,1 jday RFVaR M

with

)()()1()1(11 ,, N

tN

tttj jjjj RFRFRFRF RF

simulated. represents and historical represents that Note jj RFRF

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Proposed Approach – VaR for CVA

Change of Value of a Hedged CVA Book

Re-writing the above in the same notation as previous section

HCVACVA

000 Hjjj CVACVA

Following our prescription, we need to calculate

jCVA

which we can do using a rollback method like Least Squared MC, under two different approaches.

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Proposed Approach – Method I

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CVA

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RF

RF

βRFβRF

βRF

RFRF

RF

M

M

Joint Simulation of the Credit and Market

Step-by-Step

βRF ;00 ,, jjCVAjjCVA M

CVA

15

Proposed Approach – Method II

1

,,,,

k

kckkdk EExEExCVA

Simulation of the market, only CVA (bilateral case)

EE

SimulatedNot

discounted and simulation From

1

1,

1,

x

EE

PQ

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kc

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16

Proposed Approach – Method II

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Terms Cross

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ttttEEEE

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CVA

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xEExEE

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kkkk

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jkEE

kj

k

dkj

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j

kj

ckj

kj

dkj

ckj

kdkj

k

k

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RF

βRFM

Simulation of the market, only Change in CVA (bilateral case)

17

Proposed Approach – Method II

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k

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βRF

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M

M

Step-by-Step

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A Power Option An Interest Rate Swap and Its Application to Regulatory VaR

Examples

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Examples – A Power Option

Motivation For This Example To show the effectiveness of linear regression when modelling a non-linear trade

Motivation For This Trade The power option is a highly non-linear trade with an analytical pricing formula.

750 elements historical ofNumber

5,000Paths ofNumber

0.5. and both for ies volatilitHistorical

1

005.0

02.0)0(3.0

100)0(25.0

0,maxpayoff 2

HS

LGD

CVAr

dwdwE

HdwdtH

dH

SdwdtrS

dS

SK

c

H

HS

H

S

T

Setup Assumption Rates and Vols are not

stochastic. Counterparty has sold

us a put on its stock

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Examples – A Power Option [ ρ = 0 ]

55

2210

82

765

52

210

;

II-Method

;,,

I-Method

SSSS

SHHHSSSSH

kkkk

EE

kkkkkkkCVA

k

β

β

M

M

For every strike, with ρ = 0, both S and H were simulated. First, the VaR using benchmark(BM) was produced by pricing the CVA with current market data and then pricing the CVA with each of the 750 scenarios. The VaR of this distribution, for this strike, produces one point under BM label. Separately, Method-I, and Method-II were used to calculate their corresponding VaR numbers. Each VaR makes a data point under the Method-I and Method-II labels, respectively. This process was then repeated for strikes ranging from 4,000 to 40,000. The longest part of the exercise was the generation of the VaR using BM, since it had to reprice the CVA 751 times for each strike value.

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Examples – A Power Option [ ρ = -0.95 ]

SHHHSSSSH kkkkkkkCVA 8

276

55

2210;,,

I-Method

βM

The same set of simulations was repeated, for the same range of strikes,using ρ =-0.95, incorporating a large WWR. In this case, however, only Method-I was calculated. This figure illustrates the effect of WWR to VaRusing Method-I.

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Examples – An Interest Rate Swap

Motivation For This Example Sensitivity of the approach to various volatility values To compare the BASEL –III CVA regulatory capital using method-I.

Motivation For This Trade Trade’s value can be negative creating the exposure non-linear and making

the rollback process challenging.

750 elements historical ofNumber

8,000Paths ofNumber

1. to0 from varies

1.1. to0 from aries , , of ies volatilitHistorical

1

.2 and ofy volatilitImplied

05.0)0(02.0)0(

vH

LGD

dwdwE

Hr

rH

H

c

HS

Setup Assumption Current period is also

the “stress Period” There are no hedging

instruments.

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Examples – Effect of Volatility on Approach

rHHHrrrrH kkkkkkkCVA 8

276

55

2210;,,

I-Method

βM

Since the regulatory VaR does not take any contribution from the un-hedged exposure variations, the first test would be to compare the credit component of the VaR for various historical spread volatilities. This is done by generating a time series for H while keeping the historical volatility of the rates to zero. The results are then compared with the regulatory VaR. Using the regulatory VaR as the benchmark, one can also observe that the VaR methodology proposed here can generate, from a single implied volatility of 20%, the correct VaR for different historical volatilities generated from various time series.

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Examples – Regulatory VaR and WWR

rHHHrrrrH kkkkkkkCVA 8

276

55

2210;,,

I-Method

βM

The next objective is to compare the regulatory and the proposed VaR for a given historical volatility of market (both, hazard rate and the rates) and for various correlation. This is done by performing a Monte Carlo simulation for each level of ρ, generating the time series matching the corresponding correlation and calculating the VaR as prescribed in method-I. Since the regulatory VaR depends only on the historical volatility of the hazard spread, it produces a flat line. Note that for ρ=0, method-I includes variations in the market component and regulatory VaR does not.

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Concluding Remarks

Limits of the Approach Implied Volatility – In cases where rollback methods were used to obtain the conditional prices of option, there will be no volatility to regress against.

Challenges A Robust rollback algorithm – Obviously, there is no analytical formula to calculate the conditional CVA or exposures. This means we need a robust rollback method in order to obtain convergent conditional values . Nonlinearity of the exposures – Even when using a rollback one still needs a substantial number of paths for the exposures and CVA values to converge as they are highly nonlinear. A different VaR platform – Almost in all cases, all desks use the same VaR platform. This method now requires a different one.

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Questions and Comments

Thank you