september 18, 2007 risk training on economic capital counterparty credit risk: quantifying economic...

Risk Training on Economic CapitalSeptember 18, 2007

Counterparty Credit Risk:Quantifying Economic Capital

Martin [email protected]

September 18, 2007Counterparty Credit Risk

Risk Training onEconomic Capital

of 38Martin Goldberg

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.




Outline

¨ What is Economic Capital

¨ What is Counterparty Credit Risk

How does it differ from Issuer Credit Risk

How is it quantified

Default Only

Loan equivalent and Loss of Economic Value

How is it managed in ordinary times

¨ Economic Capital for Counterparty Risk




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.

Market Risk Regulatory Capital = VaR - 99% ten day

Worst 2 weeks of 4 years

Basel 2 Economic Capital - 99.9% one year

Worst year of the millennium - somewhat hard to backtest

Basel 2 PD Floor (~ AA rating) Economic Capital - 99.97% one year

Based on the .03% transition probability from AA-> D

Third worst year in 10,000

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?




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

-37.00-38.00-39.00-40.00-41.00-42.00-43.00-44.00-45.00-46.00-47.00-48.00-49.00-50.00-51.00-52.00-53.00-54.00-55.00-56.00-57.00-58.00-59.00-60.00-61.00-62.00-63.00-64.00-65.00-66.00

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%

0-20-40-60-80-100-120-140-160

Potential Credit Loss ($mm)

Pro

bab

ilit

y o

f C

red

it L

oss

Economic Capital

Economic capital

= Unexpected Loss

= Loss at very high CL – Expected loss.

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

Loss atvery high

CL Expected Loss




Extrapolating to EC level of risk

¨ Since we do not have 1000 years of data, let alone 10,000, what can we do instead?The 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 20th century but did not exist in the 19th.

Quick and dirty - pretend everything is normally or lognormally distributed and scale up accordingly.

Better - 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.

2

2

1)(

hZgZ

gXh eg

eBAZX




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.




Outline





Default Only







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.

The market value of the derivative does not explicitly depend on the counterparty

There 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.




Default-Only vs. Loss-of-Market-Value

¨ Counterparty risk can be measured in two incompatible ways

The banking book (accrual) view is that there is no loss until bankruptcy is declared or a permanent impairment is recognized.

The 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:

Bilateral - a legally enforceable netting arrangement exists.

Unilateral - 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.




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.




Outline





Default Only







Comparison to Debt

Debt

¨ Exposure to obligor depends on obligor’s actions

¨ Cannot issue negative debt

¨ Usually cross-default provisions, so exposures add

¨ Diversification across obligors reduces risk

¨ Mainly simple Discounted Cash Flows

Counterparty Exposures

¨ Exposure depends on market movements

¨ Can go long or short OTC derivatives

¨ 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




Outline





Default Only







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

Must include fat-tail effects - Basel requires it

Must backtest well and be calibrated to historical experience

Probably should include contagion

Interest rates are mean-reverting in the long run, so need to include that

Early attempts at this include the Alan Beilis / Jan Dash Macro-Micro Model [ 7 ]

There is no really good solution yet for this that I know of, but many are trying to solve it




Calculating Counterparty Risk 2

¨ Third Step - For each simulation path, calculate the risk-free market value of each contract at each time step

If the contract is too time-consuming to price, use a proxy that can be priced faster, preferably a closed-form solution.

This 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

Aggregation by counterparty

Include risk mitigants such as margining, and do unilateral differently from bilateral.

Assume no additional transactions

¨ Fifth - Collect into an Exposure Profile of potential exposure at different times and confidence levels.




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




Outline





Default Only







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 -

Simulate the market factors and derive exposure profiles

Calculate Expected Positive Exposure profiles as [Exposure Profile]+

Average to get a single number EPE for each counterparty

Simulate default and recovery




Outline





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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’)

where the effective remaining maturity M’ = Min(5 years,Max(1 year, tenor))

Probability of Default PD = max(.03%, counterparty’s default likelihood)

Loss Given Default LGD = 1 - Recovery Rate (not same as for loans or bonds)

K is the “default-only capital factor” given by a messy formula

MA is a maturity adjustment that is intended to capture migration risk, also a messy formula

Exposure 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.




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. “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].




Verbatim Quote from Basel 2




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).




Outline





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Credit Value Adjustments

¨ Bonds and loans are cash flows discounted at a credit spread above the risk-free 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 -

Whose spread? If we owe them money, is it our spread instead of theirs?

CVA applies to portfolio, not each exposure separately, for bilateral

Unilateral would as usual work differently

Need to introduce “maturity adjustments” to fit the Basel definition of a one-year horizon

Spread is another variable to simulate




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.




Outline





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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.




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-to-model 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.




What is Margined? Ref [5]




Cross-Product Netting

¨ Some margin accounts have a VaR-like limit, where the limit for a portfolio of many different exposures is calculated to be the (usually one-day, 99th 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 VaR engine to calculate CPM.




Contingent Credit Default Swaps

¨ Also called C-CDS

¨ Liquid market for some names

Only for a handful of vanilla derivatives underlyers

Some basis risk remains if you are actually protecting an exotic

Not 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




Outline





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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?




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.




References

1. [email protected], 732-742-8977

2. The Tukey gh 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_May07.pdf

4. http://www.quantlet.com/mdstat/scripts/stf/html/stfhtmlnode17.html

5. http://www.isda.org/whatsnew/pdf/2001Margin.pdf

6. http://www.bis.org/publ/bcbs116.pdf

7. Part VI of Quantitative Finance And Risk Management: A Physicist's Approach by Jan W. Dash (World Scientific Publishing Co © 2004 )

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