Etienne Koehler Barclays Capital Etienne.Koehler@univ-paris1.fr

1. CVA - VaR Shahram Alavian • Royal Bank of Scotland • Shahram_Alavian@yahoo.com Etienne Koehler Barclays Capital Etienne.Koehler@univ-paris1.fr January 2012

2. In Nutshell • What we say … • How we go about it …. • Statement • CalculateVaR 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. 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

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

4. Preliminaries - CVA Dealer Counterparty Defaultable Cash flow = Dealer Counterparty Default Free Cash flow + 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 Dealer Counterparty Option to Default

5. Preliminaries – CVA Continued … 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

6. Preliminaries – VaR 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 “

7. Preliminaries – VaR for CVA 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, …

8. Proposed Approach VaR properties VaR and Monte Carlo simulation VaR for CVA

9. Proposed Approach – VaR Properties 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. Conditional: It is conditional on its initial value at current time. 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.

10. Proposed Approach – VaR and Monte Carlo • Valuation of an asset πconditional 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. 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) • Defining a 1-day Implied VaR. • Calculate the conditional values of the asset at δ = 1-day time horizon for each path j. • In future slides, we replace πwith expected exposure (EE) and CVA itself.

11. Proposed Approach – VaR and Monte Carlo • Calculate the change in values of the asset at 1-day time horizon for each path j. This is not an instantaneous change. Therefore, we pick up another path j’. and create an instantaneous change by Create an Instantaneous Change <= This change is from any j’ to any j • Create a Conditional Change • Having simulated a large number of paths, one can find a path for which So we have a conditional change from j’ which given the above assumption, we a have conditional on spot.

12. Proposed Approach – VaR and Monte Carlo Assuming there exists a model, M, representing the conditional changes as a function of the changes in the risk factors with Using a linear regression model, for example, Create a Functional Change • Putting It All Together with

13. Proposed Approach – VaR for CVA • Change of Value of a Hedged CVA Book Re-writing the above in the same notation as previous section Following our prescription, we need to calculate which we can do using a rollback method like Least Squared MC, under two different approaches.

14. Proposed Approach – Method I • Step-by-Step Joint Simulation of the Credit and Market

15. Proposed Approach – Method II Simulation of the market, only CVA (bilateral case)

16. Proposed Approach – Method II Simulation of the market, only Change in CVA (bilateral case)

17. Proposed Approach – Method II • Step-by-Step

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

19. Examples – A Power Option • Motivation For This Trade • The power option is a highly non-linear trade with an analytical pricing formula. • Setup • Assumption • Rates and Vols are not stochastic. • Counterparty has sold us a put on its stock Motivation For This Example To show the effectiveness of linear regression when modelling a non-linear trade

20. Examples – A Power Option [ ρ = 0 ] 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.

21. Examples – A Power Option [ ρ = -0.95 ] 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 VaR using Method-I.

22. Examples – An Interest Rate Swap • Motivation For This Trade • Trade’s value can be negative creating the exposure non-linear and making the rollback process challenging. • Setup • Assumption • Current period is also the “stress Period” • There are no hedging instruments. Motivation For This Example Sensitivity of the approach to various volatility values To compare the BASEL –III CVA regulatory capital using method-I.

23. Examples – Effect of Volatility on Approach 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.

24. Examples – Regulatory VaR and WWR 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.

25. 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.

26. Questions and Comments Thank you