CR 21 – The Evolution of Stress Testing Counterparty Exposures

Counterparty credit risk (CCR) is the danger that the other side of a derivative contract defaults while the contract is worth something to you. Few corners of risk management changed as fast as this one from the late 1990s onward, and the change was concentrated in the statistics used to measure the risk. Stress testing, by contrast, moved slowly, and the gap between sophisticated statistical measures and fairly basic stress tests is the theme that runs through this lesson. The natural fix is to build stress tests on top of the statistical measures.

From add-ons to simulated exposure

Through the 1990s, managing a derivatives counterparty meant judging its creditworthiness and tracking what it currently owed. After the near collapse of Long-Term Capital Management in 1998, the Counterparty Risk Management Policy Group (CRMPG, 1999) criticised banks on both counts and asked for better measures. Capital rules were crude: the 1988 Basel Accord added a fixed percentage of gross notional to current exposure, ignoring how a trade’s value might drift over its life.

Cheaper computing let banks simulate future paths of market variables and revalue trades along each path, showing how exposure could evolve. The statistical toolkit then developed in two steps. Potential exposure models came first and were used to measure and limit risk to each counterparty. Those models were then extended into expected positive exposure (EPE) models, which allowed a derivative position to sit inside a portfolio credit risk model in much the same way as a loan (Canabarro, Picoult and Wilde, 2003). Both steps belong to the view of CCR as a credit risk, and that view dominated practice from 2000 to 2006, becoming the basis of the Basel II capital rules (BCBS, 2005). Netting agreements and margining were built into the models over the same period.

Wrong-way risk and the market risk view

Researchers also identified a problem peculiar to derivatives, called wrong-way risk (Levin and Levy, 1999; Finger, 2000). It arises when the counterparty’s credit quality is correlated with the exposure, so that the amount at risk tends to grow just as default becomes more likely. A loan has a fixed exposure, so this interaction cannot happen there, and tools borrowed from loan risk management do not handle it easily.

Alongside the credit view, a market risk view was forming. Pricing the possibility of counterparty default as a credit valuation adjustment (CVA) was first described for swaps (Sorensen and Bollier, 1994; Duffie and Huang, 1996) and spread widely once the fair value accounting standard FAS 157 (FASB, 2006) required it. Before 2007, bank credit spreads were calm and CVA was a small slice of portfolio value, so few treated it as a risk to manage until the crisis of 2007 to 2008. Credit spreads and exposure amounts both swung violently, and their combined effect produced large losses at some banks and large, unusual gains at others. Banks have since built frameworks to manage CVA risk, and Basel III introduced a CVA charge in regulatory capital (BCBS, 2011).

The four exposure measures used throughout this topic follow the Basel definitions (BCBS, 2005). All of them are defined at the level of a netting set, which is the group of trades with one counterparty that can be offset against each other in a bankruptcy. The first looks only at today, while the other three describe a distribution of exposures that a simulation produces for future dates.

The four definitions

Current exposure is the greater of zero and today’s market value of the trades in the netting set. It is the amount that would be lost if the counterparty defaulted right now and nothing were recovered in bankruptcy. Because it is what the bank would pay to replace the trades, current exposure is often called the replacement cost.

Peak exposure is a high percentile of the exposure distribution at some future date before the longest trade in the netting set matures. The percentile is typically 95% or 99%, and the calculation is repeated for many future dates, which produces a profile running out to the final maturity. Peak exposure is the measure banks use for potential future exposure (PFE) limits.

Expected exposure (EE) is the mean of the same distribution at a given future date. Like peak exposure, it is generated for a grid of dates up to the longest maturity.

Expected positive exposure (EPE) collapses the EE profile into one number. It is a time-weighted average of the expected exposures, where each EE is weighted by the share of the total time interval that its period represents. For minimum capital, the averaging runs over the first year, or over the life of the longest contract if that is shorter.

A bank can manage CCR as a credit exposure that it limits and mitigates, or as a price (the CVA) that it hedges in the market. Both views are legitimate ways to run a derivatives book, and the choice shapes how the trading desk and the risk function are organised (Picoult, 2005; Canabarro, 2009). The difficulty is that adopting either view on its own leaves the bank blind to what the other view sees.

What each view misses

Suppose a bank treats CCR purely as a credit risk. It sets PFE limits on each counterparty and contains default losses through collateral and netting, yet accounting still forces it to carry CVA inside the fair value of its derivatives. If nobody watches how that CVA moves with spreads and exposures, the balance sheet can deliver nasty surprises. Now suppose the bank treats CCR purely as a market risk and dynamically hedges its CVA to keep mark-to-market losses small. It is still exposed to a sudden collapse in a counterparty’s creditworthiness, or to an outright default that the hedges did not anticipate. A derivatives dealer faces both problems at once and cannot pick only one view.

How the risk is managed under each view

The two views also lead to different management habits.

Under the credit view, little can be done once trades are on the books, so the emphasis falls on credit evaluation and on mitigants arranged up front. When the counterparty defaults, the bank must rebalance its book by replacing every trade with that name in a single burst. Under the market view, the bank replaces trades gradually and before any default, buying positions in proportion to the counterparty’s probability of default. A name with a low default probability has only a small share of its trades replaced in advance, but as its credit quality deteriorates, a growing share is moved to other counterparties. By the time default arrives, the replacement has already happened.

Because CCR has a credit face and a market face, it generates a long list of measures. The credit side supplies current, peak and expected exposure, while on the market side there is the CVA itself and the risk that it changes, which might be measured by the value-at-risk (VaR) of CVA. Each of these can be reported for the whole portfolio and for every counterparty, so even the plain question of how much counterparty risk the bank holds is hard to answer. Deciding which of these measures to stress is harder still.

A comparison with market risk shows the scale of the problem. When a market risk manager runs an equity crash scenario, the output is one or two figures: the immediate loss on today’s portfolio and perhaps a stressed VaR. Those are easy to interpret. The CCR manager running the same scenario needs a result for each counterparty and another for the portfolio, and needs each under both the credit and the market view. So the count of results is at least twice the number of counterparties, plus one. If the manager also stresses risk measures, rather than only applying instantaneous shocks to values, the count at least doubles again, and it grows further because several CCR measures matter at once. Sub-portfolios by region or industry multiply it again.

None of this lets the bank skip the exercise. Supervisory guidance such as SR 11-10 (Federal Reserve Board, 2011) expects counterparty exposures to be stress tested, and there is no other route to a full picture of derivatives risk.

Stressing current exposure is the most widely used CCR stress test. The bank chooses a scenario of risk factor moves, revalues each counterparty’s portfolio under it, and reads off the stressed current exposure. Senior management typically sees one table ranking the top counterparties by current exposure, followed by a separate table for each scenario ranking them by stressed current exposure. The rankings usually differ by scenario, since the counterparty most hurt by an equity crash is rarely the one most hurt by an interest rate shock.

The table below reproduces a report for an equity market fall of 25%. The derived final column shows how much each exposure jumps, and stressed exposure is measured after the shock but before any extra collateral is called.

Values as reported in Lynch’s illustrative equity crash report; the increase column is derived.

Three weaknesses of stressed current exposure

The report shows which names would worry the bank in a crash and how much each would owe, but it has three weaknesses.

First, the figures cannot be aggregated sensibly. Summing the stressed exposures gives the loss if every counterparty defaulted in the scenario, a result that only an apocalypse would justify. Feeding stressed current exposures into a portfolio credit model is also wrong, because such models are built to take expected exposures rather than current ones (Canabarro, Picoult and Wilde, 2003). That is why the report shows no total.

Second, the measure ignores credit quality. It values the trades but says nothing about whether the counterparty is able or willing to pay, and a US$200 million exposure to a newly launched hedge fund is a very different risk from a US$200 million exposure to a triple-A corporate. Rating-dependent limits on stressed exposure are possible in principle, but setting them for every scenario and rating bucket is daunting, and peak exposure already offers a more consistent basis for limits. So ambiguity remains. Counterparty C has the larger stressed exposure (US$119m against US$76m), but it is rated AA while D is rated BBB, and nothing in the table says which matters more.

Third, because the measure carries no credit information, it cannot reveal whether exposure and credit quality move together, so it offers almost no insight into wrong-way risk. Stressed current exposure is a good monitoring tool for individual names, but it gives neither a portfolio view nor any reading of credit quality.

To bring CCR into a credit framework, the starting point is the way banks already stress their loans. For provisioning, a bank estimates the unconditional expected loss on a pool of borrowers. Each borrower’s expected loss is the product of its probability of default p_i, its exposure at default ead_i and its loss given default lgd_i, where the probability may itself depend on other variables. Summing over the N borrowers gives the pool’s expected loss.

A common stress test treats EAD and LGD as fixed and puts all the stress into the default probability. The probability is modelled as a function of drivers such as an important exchange rate or the unemployment rate. Setting some of those drivers to stressed values gives a conditional, stressed probability of default, written p_i^s, and the stressed expected loss follows.

The distinction matters. Stressed expected loss is the full expected loss under the stressed scenario, while the stress loss is only the increment over the unstressed expected loss. A bank can produce these numbers easily, either by raising default probabilities directly or by stressing the macroeconomic variables and balance-sheet items on which those probabilities depend. Results are available by borrower and for the pool.

The loan framework carries over to derivatives once the right exposure is chosen. PD and LGD are handled as before, but exposure at default is now random because it depends on market variables. The quantity that lets a derivative position enter a portfolio credit model alongside loans, giving a high-percentile loss for the combined book, is EPE multiplied by an alpha factor (Picoult, 2005; Wilde, 2005). This alpha-scaled EPE is often called the loan equivalent of the derivatives exposure. Alpha depends on the quantile at which capital is measured; here it is the alpha calibrated at expected loss, so it need not equal the capital alpha.

Stress loss is again EL^s minus EL. Default probabilities can be stressed directly or through their drivers, which now include balance-sheet values, macroeconomic indicators and prices of financial instruments, and the resulting derivatives stress loss can simply be added to the loan stress loss.

Stressing default probabilities

The table below gives a PD stress modelled on the jump in default rates by industry after the dotcom crash of 2001 to 2002. EPE and LGD are held constant, and expected losses, stressed expected losses and stress losses can all be totalled and combined with the loan book.

EPE, LGD, PD and EL figures as printed in Lynch’s dotcom PD stress table; the PD multiple column is derived.

Derivatives add a set of variables that loans do not have. EPE depends on market variables such as equity prices and swap rates, so the bank can shock those and watch exposure respond. Unlike a PD stress, a market shock does not have a predictable direction in aggregate. Whether expected losses rise or fall depends on many things, including which way the bank’s book leans, which counterparties post margin and which have posted excess margin. Stresses to the drivers of PD, by contrast, usually push every counterparty the same way.

A market shock to EPE also does not need to be combined with the loan book, because loan exposures do not respond to market variables. There are two qualifications. Expected losses on loans can still rise if default probabilities are correlated with market variables, and loan commitments and some other loan products do have stochastic exposure.

How the shock is applied

Banks normally apply an instantaneous shock, often the one used for stressed current exposure; shocks at a future date or in sequence over time are possible but rare. In practice the bank shocks today’s value of the trades and then reruns the exposure simulation from that shocked start. How much EPE responds depends on, among other things, how well the portfolio is collateralised and how far in or out of the money it is.

The table below recasts the equity crash of Table 2 as an expected loss stress, holding each counterparty’s PD constant and letting only exposure move. Expected losses are small and quoted in thousands of dollars because the default probabilities involved are small.

PD, CE, EPE and EL figures as printed in Lynch’s credit framework equity crash table; MtM and collateral are as in Table 2, and the EPE change column is derived.

Joint stresses and wrong-way risk

Stressing credit quality and market variables together is simple in concept but hard in practice, because little ties macroeconomic or balance-sheet drivers of PD to the market variables that drive exposure. Equity-based default models (Merton, 1974; Kealhofer, 2003) come closest to a link. Even there, exposure reacts to a shock at once, while it is unclear whether an equity-based PD should. The same gap makes wrong-way risk hard to capture, and most attempts are ad hoc. The best method available in the credit framework is to stress current exposure, find the counterparties most exposed to the scenario, and then examine each of them for wrong-way risk.

Even so, the credit framework is a clear step beyond stressed current exposure: losses can be aggregated with the loan book, and counterparty quality enters the result. Treating CCR as a market risk goes further, because PD is then inferred from market variables and joint stresses of credit quality and exposure become easier.

In the market risk view, the bank cares about the market value of counterparty risk and about how that value changes when market variables, including the counterparty’s own credit spread, move. Most often the bank stresses its unilateral CVA, which reflects only the possibility that its counterparties default. If it also allows for its own default to its counterparties, it is working with bilateral CVA, which comes later in this lesson.

A common simplified formula for the CVA to counterparty n, which ignores wrong-way risk, comes from Gregory (2010):

Here EE_n*(t_i) is the discounted expected exposure in period i, calculated under the risk-neutral measure, q_n*(t_i-1, t_i) is the risk-neutral marginal default probability of counterparty n between t_i-1 and t_i, T is the final maturity, and LGD_n* is the risk-neutral loss given default. Summing over N counterparties gives portfolio CVA, and a stressed CVA uses the same structure with stressed inputs.

Every component depends on market variables. The marginal default probabilities come from the counterparty’s credit spreads, LGD is set by convention or taken from market spreads, and EE depends on the values of the trades. A stressed CVA applies an instantaneous shock to some of these variables, affecting EE, the default probabilities or both. LGD could be stressed too, but CVA is linear in LGD, so the effect is obvious. A shock to current values works like the current exposure stress earlier, except that it changes the expected exposures in every later period, so the whole EE profile must be recalculated. Shocks to marginal default probabilities usually come from credit spread shocks.

CVA stress tests compared with expected loss stress tests

Both frameworks rest on the same product of LGD, exposure and default probability, yet the numbers they produce differ sharply. The biggest reason is that CVA uses risk-neutral inputs where expected loss uses physical ones. CVA also covers losses over the whole life of the trades, whereas expected loss uses a set horizon, and CVA takes its default probability from market prices.

A market-based default probability brings a real benefit: correlation between default probability and exposure can be built in. Hull and White (2012) show how, and they highlight a stress test that follows naturally, namely a stress of that correlation itself. The correlation can move measured CVA considerably, and because it is highly uncertain, a bank should test what happens to its profit and loss if its assumed correlation is wrong.

A complete picture of CVA profit and loss under a scenario must also include the liability side. That component of bilateral CVA, usually called the debit valuation adjustment (DVA), is the value to the bank of its own option to default on its counterparties. Its formula mirrors CVA with two changes. First, expected exposure is replaced by negative expected exposure (NEE), which is the expected exposure seen from the counterparty’s side. Second, the bank can only profit from its own default if the counterparty is still alive, so the counterparty’s survival probability S_n* enters the sum. The same adjustment is made to the CVA term, since a loss from the counterparty’s default now requires that the bank has not defaulted first. Gregory (2010) gives the bilateral formula, with subscript I denoting the bank:

Survival probabilities, like default probabilities, now depend on CDS spreads, and the result depends on the bank’s own credit spread. That produces counterintuitive outcomes: an improvement in the bank’s own credit quality shrinks the DVA and so registers as a loss. In a bilateral stress test, the bank must also consider how its own spread is correlated with its counterparties’ spreads. Stress loss is computed as before, as stressed BCVA minus current BCVA.

Aggregating BCVA stress with market risk stress tests

Because BCVA treats CCR as a market risk, it can be folded into the bank’s market risk stress testing in a consistent way. The gain or loss on BCVA under a scenario is simply added to the stress result for the trading book, provided one identical set of market shocks drives both calculations.

Banks have started to go beyond stressing current exposure, but the methods are still being developed, and few banks combine CCR stress results with loan portfolio stress tests or trading book stress tests in a consistent way. Better models of exposure and CVA make that combination possible, provided two traps are avoided.

Using current exposure where expected exposure belongs

Since nearly every bank already stresses current exposure, it is tempting to reuse those numbers when combining CCR with loans or trading positions. The earlier sections show that EE or EPE is the right exposure amount, and substituting current exposure can cause large errors. A normal approximation to expected exposure (Gregory, 2010), accurate for linear derivatives with no intermediate payments, shows why. Current exposure responds to a shock all or nothing: an in-the-money position passes the full shock through, and a deeply out-of-the-money one passes none. Expected exposure responds smoothly, because it averages over future values on both sides of zero.

The figure shows the pattern for a US$1m rise in market value. Deep out of the money, current exposure does not move at all, yet expected exposure still rises a little, because some future paths end in positive territory. Deep in the money, both pass it through almost fully. The danger zone is the at-the-money portfolio, where current exposure and expected exposure differ by almost half the size of the shock.

Relying on delta approximations

The second trap is to estimate exposure changes from delta sensitivities rather than full revaluation. Exposure is a highly nonlinear function of market values, because of the floor at zero and the effect of collateral, so a linear approximation can be badly misleading even though it saves computing time, and the errors are not obvious from the output. Portfolios near the money that are hit with large price moves are the most prone to these errors.

Bringing the views together

Too many stress tests can obscure the risk, yet a reasonable number is needed, and both views are required because fair value losses and default losses occur however the bank manages CCR. The credit view combines with the loan book and the market view with the trading book, but merging default stresses and fair value stresses into a single comprehensive test remains unsolved. There is also an irony in the timing: regulators began pushing derivatives into central clearing to reduce counterparty risk at the very moment the tools for managing that risk were improving fastest.