Counterparty Credit Risk Quantifying Economic Capital Martin Goldberg

Counterparty Credit Risk: Quantifying Economic Capital Martin Goldberg martin@validationquant. com September 18, 2007 Risk Training on Economic Capital

The Usual Caveat ¨All opinions and views expressed in this presentation are my own personal opinion, and do not necessarily reflect the views of my employer, any of its clients, nor any other person or group. ¨Questions and comments are welcome. My e-mail address is on the cover page. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 2 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 3 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 4 of 38 Martin Goldberg

Regulatory and Economic Capital ¨Capital is a “Rainy Day Fund” to ensure your institution does not go bankrupt in a bad period ¨Potential unexpected loss, over a specified time horizon, at a specified confidence level. ØMarket Risk Regulatory Capital = Va. R - 99% ten day q Worst 2 weeks of 4 years ØBasel 2 Economic Capital - 99. 9% one year q Worst year of the millennium - somewhat hard to backtest ØBasel 2 PD Floor (~ AA rating) Economic Capital - 99. 97% one year q Based on the. 03% transition probability from AA-> D q Third worst year in 10, 000 q Very hard to calibrate or backtest with actual historical data Þ What was the effect on your bank of the invention of agriculture? Of writing? Þ How many of your swap counterparties defaulted when Rome was sacked? September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 5 of 38 Martin Goldberg

A Reminder: EC And Capital Horizon ¨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). ¨Insolvency risk is calculated by comparing EC to the Available Financial Resources (i. e. AFR = book capital – goodwill + fair market adjustment). ¨Here is an example of EC for Issuer Credit Risk of a large portfolio Economic capital = Unexpected Loss at very high CL September 18, 2007 Counterparty Credit Risk = Loss at very high CL – Expected loss. Economic Capital Expected Loss Risk Training on Economic Capital Expected loss should be covered by reserves and/or pricing. Page 6 of 38 Martin Goldberg

Extrapolating to EC level of risk ¨Since we do not have 1000 years of data, let alone 10, 000, what can we do instead? ØThe only derivative contract with a long history is the Japanese rice futures which has only a 400 year history, and most OTC derivatives trading started in the 20 th century but did not exist in the 19 th. ØQuick and dirty - pretend everything is normally or lognormally distributed and scale up accordingly. ØBetter - fit a fat-tailed distribution to the data you have and scale using that distribution. I recommend the Tukey g- -h distribution[2], where h controls the fatness of the tail, and the quantiles are known. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 7 of 38 Martin Goldberg

Extreme Value Theory and EC ¨Estimates of large percentiles, which correspond to rare events, are both difficult and possibly misleading. ¨Extreme value theory has some techniques for extrapolation but these are fraught with difficulties in practice[3]. ¨In a portfolio context, such as any of your firms, there is the added complication of contagion - correlations in normal times are usually irrelevant in a crisis. The relevant measure is tail dependence[4], which is hard to estimate since by definition you have very few data points, or more likely none. ¨The usual cheat scheme is to assume ergodicity - pretend that one year’s data on 1000 counterparties is a good proxy for any one firm followed for a millennium. Also one can assume stationarity - the nature of the market will be about the same in the near future as it has been in the recent past. This worked poorly for, say, Russian debt issued in 1915. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 8 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 9 of 38 Martin Goldberg

Counterparty Credit Risk Defined ¨Counterparty Credit Risk is the risk of loss to your firm due to a counterparty’s default on money owed you in a derivatives transaction. ¨The derivatives could be any OTC swap, forward, option, repo lending, or securities lending. ØThe market value of the derivative does not explicitly depend on the counterparty ØThere is no default-free exchange involved - the exchanges are assumed to have no risk of default and thus futures and listed options have no counterparty risk. ¨Basel 2 has a few ways of calculating the regulatory measure of counterparty risk - I will not address the more approximate measures but only the Advanced IRB approach. ¨The implementation of Basel 2 differs from one country to the next, and when in doubt ask your national regulator. For multinationals there may be a turf war or requirements for multiple incompatible calculations. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 10 of 38 Martin Goldberg

Default-Only vs. Loss-of-Market-Value ¨Counterparty risk can be measured in two incompatible ways ØThe banking book (accrual) view is that there is no loss until bankruptcy is declared or a permanent impairment is recognized. ØThe trading book (mark-to-market) view is that there is a loss in value if the counterparty becomes less creditworthy and a gain in value if the counterparty gets further from default. This value is measured as a Credit Valuation Adjustment (CVA) which ideally would move in tandem with the counterparty’s CDS spreads. ¨There is a legal distinction between two different kinds of counterparty: ØBilateral - a legally enforceable netting arrangement exists. ØUnilateral - No netting; each contract stands alone. If at default you owe them $100 on some swaps and they owe you $100 on others, you pay them $100 and go to court to try to collect some of the $100 they owe you. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 11 of 38 Martin Goldberg

Always adds a layer of optionality ¨If you owe them money when they default, you have not incurred any loss. ¨If they owe you money when they default, you are in line with everyone else they owe money to. ¨The payoff for counterparty risk on one contract is max(0, contract value) which is an option struck at zero. If the contract is linear, this is an option. If the contract is an option, this is a compound option. If the contract is an exotic, this is even more exotic. ¨For bilateral, this is an option on a portfolio. ¨The one exception is Extinguishers - derivatives where it says in the contract that the derivative vanishes with no payments in the event of default of either counterparty. These are not very common, but I have seen them, in FX especially. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 12 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 13 of 38 Martin Goldberg

Comparison to Debt Counterparty Exposures ¨Exposure to obligor depends on obligor’s actions ¨Exposure depends on market movements ¨Cannot issue negative debt ¨Can go long or short OTC derivatives ¨Usually cross-default provisions, so exposures add ¨Diversification across obligors reduces risk ¨Mainly simple Discounted Cash Flows September 18, 2007 Counterparty Credit Risk ¨Exposures net if bilateral, might be allowed to net OTC derivatives and repo’s ¨Diversification not as effective in a market meltdown ¨Option valuation - usually requires Monte Carlo simulation Risk Training on Economic Capital Page 14 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 15 of 38 Martin Goldberg

Calculating Counterparty Risk 1 ¨First Step - What are your exposures? Can you aggregate all OTC derivatives, repo, Sec Lending, etc. by counterparty? Often these are housed in incompatible systems that do not communicate. ¨Second - Real-world measure simulation of the entire market for many years ØMust include fat-tail effects - Basel requires it ØMust backtest well and be calibrated to historical experience ØProbably should include contagion ØInterest rates are mean-reverting in the long run, so need to include that q Early attempts at this include the Alan Beilis / Jan Dash Macro-Micro Model [ 7 ] ØThere is no really good solution yet for this that I know of, but many are trying to solve it September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 16 of 38 Martin Goldberg

Calculating Counterparty Risk 2 ¨Third Step - For each simulation path, calculate the risk-free market value of each contract at each time step ØIf the contract is too time-consuming to price, use a proxy that can be priced faster, preferably a closed-form solution. ØThis gets far messier if you want to assume market values are correlated with default likelihood. They are - many of the recent sub-prime defaults were arguably triggered by margin calls on repo exposures. ¨Fourth - Calculate the potential exposure for each path ØAggregation by counterparty ØInclude risk mitigants such as margining, and do unilateral differently from bilateral. ØAssume no additional transactions ¨Fifth - Collect into an Exposure Profile of potential exposure at different times and confidence levels. Page 17 of 38 September 18, 2007 Counterparty Credit Risk Training on Economic Capital Martin Goldberg

Exposure Profile of an Interest Rate Swap ¨This is a typical exposure profile of a typical 5 -year swap at a high confidence level. The sharp drops are coupon dates. Average Exposure September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 18 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 19 of 38 Martin Goldberg

Default-only calculation ¨Messiest technique - assume everything is correlated, and each simulated path includes a draw from the probability of default distribution, the recovery distribution, and also all the market factors. If you assume that default is not just correlated by a Pearson correlation with the market, but requires a more complicated copula to correctly describe the dependence, it gets even worse. This is closer to reality - firms can handle small changes in market conditions but there is often a trigger level past which they run out of alternatives and declare bankruptcy. If you are doing this whole exercise just for regulatory compliance, none of this is relevant. ¨Basel-compliant technique ØSimulate the market factors and derive exposure profiles ØCalculate Expected Positive Exposure profiles as [Exposure Profile]+ ØAverage to get a single number EPE for each counterparty ØSimulate default and recovery September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 20 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 21 of 38 Martin Goldberg

Basel 2 Rules for AIRB ¨In the Basel 2 AIRB framework, counterparty credit risk is treated as if it were a corporate exposure. Thus the regulatory capital charge C is ¨C = EAD K (PD, LGD) MA(PD, M’) Øwhere the effective remaining maturity M’ = Min(5 years, Max(1 year, tenor)) ØProbability of Default PD = max(. 03%, counterparty’s default likelihood) ØLoss Given Default LGD = 1 - Recovery Rate (not same as for loans or bonds) ØK is the “default-only capital factor” given by a messy formula ØMA is a maturity adjustment that is intended to capture migration risk, also a messy formula ØExposure At Default EAD is the loan-equivalent net expected positive exposure to a counterparty (“netting set”) if bilateral, and sum of expected positive exposures to each contract if unilateral. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 22 of 38 Martin Goldberg

Loan Equivalent and ¨Basel is a one-size-fits-all methodology, so naturally it wants to see all credit risk as loans. This requires a Loan Equivalent for counterparty risk to fit into the Basel 2 EAD slot. Ø“By using EPE, transactions with CCR are given the same standing as loans with the goal of reducing the capital treatment’s influence on a firm’s decision to extend an on-balance sheet loan rather than engage in an economically equivalent transaction that involves exposure to CCR. Consistent with this determination, the internal model and standardised methods to estimating EAD employ the concept of EPE. ” ¨Evan Picoult of Citigroup figured out how to do this, worked out the details with Eduardo Canabarro (then at GS), and Tom Wilde (then at CSFB) and Basel adopted this methodology [6]. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 23 of 38 Martin Goldberg

Verbatim Quote from Basel 2 September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 24 of 38 Martin Goldberg

The mysterious multiplier ¨ is defined as the ratio of Economic Capital computed with a full simulation of all defaults and all underlying factors in the real world measure divided by the EC computed with the fixed EPE required by Basel. Tom Wilde did some experiments and found a range of 1. 02 - 1. 43, with an average of about 1. 1. ¨Basel mandated = 1. 4 at first, but now allows internal calculations subject to the floor = Max( you calculate, 1. 2). September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 25 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 26 of 38 Martin Goldberg

Credit Value Adjustments ¨Bonds and loans are cash flows discounted at a credit spread above the riskfree rate. By analogy, counterparty exposures should have a credit spread baked in. ¨The CVA is the derivative value discounted at the risk-free rate minus the same derivative discounted at the relevant spread over risk-free. ¨Complications ØWhose spread? If we owe them money, is it our spread instead of theirs? ØCVA applies to portfolio, not each exposure separately, for bilateral ØUnilateral would as usual work differently ØNeed to introduce “maturity adjustments” to fit the Basel definition of a one-year horizon ØSpread is another variable to simulate September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 27 of 38 Martin Goldberg

Advantage of CVA over Default-Only ¨Since CVA varies with creditworthiness and market conditions, you can hedge counterparty risk as measured by CVA (not as much with unilateral). You can’t do that with a default-only view. ¨Hedge the loan-equivalent exposure to each counterparty, netted across all markets, with default swaps or with the counterparty’s debt securities. ¨Hedge the market exposure to each market factor, aggregating across all counterparties. ¨Still exposed to the cross-gamma of changes in market inducing changes in credit spread - not easy to hedge. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 28 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 29 of 38 Martin Goldberg

Ways to Manage Counterparty Risk There are several ways to manage counterparty credit risk, and most institutions use most or all of these. 1. Avoid it - since exchanges are assumed to be free from default risk, the counterparty credit risk for exchange-traded contracts is zero. 2. Ameliorate it - margining for OTC counterparties lowers the total amount at risk on any given day - most financial firms will offer cross-product margining agreements, but most OTC derivatives are not margined. 3. Diversify - this is an insurance-style hedge - assumes that defaults are independent or at least not 100% correlated - may be a bad assumption in the once-per-thousand-years systemic crisis where the whole banking system goes under. 4. Delta-hedge with debt or CDS of the counterparty - this may work quite differently if you do not have a good bilateral netting agreement, or the CDS seller is closely related to the counterparty - “wrong-way risk” such as buying protection on swaps with a foreign government from banks in that same country. 5. Sell it - Contingent Credit Default Swaps transfer counterparty risk and get it off your books. The regulators will need to agree that this is a true sale, but I expect that to happen. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 30 of 38 Martin Goldberg

Margining ¨A margin agreement reduces the exposure at default. A margin account is set up at one of the two counterparty’s institutions, or at a third-party margining manager. These work differently for bilateral than for unilateral counterparties. ¨Both counterparties (X and Y) agree to their threshold margin limit, which is usually based on credit quality. If the total exposure of X to Y (add up all positive exposures for unilateral, net all exposures of either sign for bilateral) exceeds the limit, based on the (usually daily) mark-to-market (mark-tomodel of the agreed calculation agent if necessary), then Y must deposit funds, either cash or acceptable securities, into the margin account by the end of a prespecified “cure period” to bring the exposure back to the limit. X’s total loss if Y defaults is at most the limit, plus the excess triggering the margin call, plus or minus the change in value during the cure period. Same for Y exposure to X. ¨Does default on a margin call trigger cross-default on all debts? Ask your lawyer. Page 31 of 38 September 18, 2007 Counterparty Credit Risk Training on Economic Capital Martin Goldberg

What is Margined? Ref [5] September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 32 of 38 Martin Goldberg

Cross-Product Netting ¨Some margin accounts have a Va. R-like limit, where the limit for a portfolio of many different exposures is calculated to be the (usually one-day, 99 th percentile) potential exposure tomorrow. This is a more objective way of setting limits, and responds appropriately to changes in the mix of exposure types. ¨You can buy off-the-shelf calculators for this - called Cross-Product Netting (CPN) or Cross-Product Margining (CPM). ¨One of my former employers used an exact copy of the corporate Va. R engine to calculate CPM. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 33 of 38 Martin Goldberg

Contingent Credit Default Swaps ¨Also called C-CDS ¨Liquid market for some names ØOnly for a handful of vanilla derivatives underlyers ØSome basis risk remains if you are actually protecting an exotic ØNot yet available on a portfolio of diverse underlying types, but the vanilla can have a complicated but non-stochastic amortization / accretion schedule ¨Bloomberg page ICVA ¨Bilateral flavor can be modeled as two unilaterals in opposite directions ¨Counterparty risk causes a correlation between credit and market risk ¨CCDS extinguish the credit risk part, and transform it into market risk, plus some counterparty risk to the issuer of the CCDS September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 34 of 38 Martin Goldberg

Outline ¨What is Economic Capital ¨What is Counterparty Credit Risk ØHow does it differ from Issuer Credit Risk ØHow is it quantified q. Default Only q. Loan equivalent and q. Loss of Economic Value ØHow is it managed in ordinary times ¨Economic Capital for Counterparty Risk September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 35 of 38 Martin Goldberg

Basel 2 Capital vs. Economic Capital ¨As mentioned before, Basel assumes no contagion or systemic risk in their AIRB calculation guidelines. In reality, the once-per-thousand-year worst case will certainly sink your firm. ¨Show of hands - How many people in the audience work for firms founded before the year 1007? September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 36 of 38 Martin Goldberg

Conclusion ¨Economic Capital for Counterparty risk, as defined in Basel 2 rules, is a useful measure of credit risk exposure separate from issuer credit risk. ¨If you want to hedge the exposures, it can be done using CVA as a dynamic measure. ¨There is lots of interesting work in this area still undone. ¨Don’t confuse the 99. 9% risk levels you measure with true economic risk in a true systemic credit event. September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 37 of 38 Martin Goldberg

References 1. Martin@validationquant. com, 732 -742 -8977 2. The Tukey g h distribution is getting quite good press. See http: //www. bos. frb. org/economic/wp/index. htm and the references there 3. http: //www. math. ethz. ch/~baltes/ftp/g-and-h_May 07. pdf 4. http: //www. quantlet. com/mdstat/scripts/stf/html/stfhtmlnode 17. html 5. http: //www. isda. org/whatsnew/pdf/2001 Margin. pdf 6. http: //www. bis. org/publ/bcbs 116. pdf 7. Part VI of Quantitative Finance And Risk Management: A Physicist's Approach by Jan W. Dash (World Scientific Publishing Co © 2004 ) September 18, 2007 Counterparty Credit Risk Training on Economic Capital Page 38 of 38 Martin Goldberg