CR 12 – Credit Risk

A derivative that is cleared bilaterally, rather than through a central counterparty, leaves each side exposed to the possibility that the other will fail. Trades of this kind between two companies are usually documented under an ISDA Master Agreement, the standard contract published by the International Swaps and Derivatives Association. Where both sides are financial institutions, regulation now goes further: variation margin must change hands as values move, and initial margin must be lodged with third parties. The initial margin is sized so that, with 99% confidence, it absorbs whatever losses could accumulate across a 10-day window.

The Master Agreement spells out what counts as an event of default. Missing a payment that is due on the outstanding derivatives, failing to post collateral (margin) when it is called, or entering bankruptcy all qualify. Once such an event happens, the side that has not defaulted is entitled to terminate every outstanding transaction under the agreement at once.

When termination produces a loss

Termination hurts the nondefaulting party in two distinct situations, and it helps to keep them apart because the second is easy to overlook.

Either way the survivor is an unsecured creditor of a failed company and recovers only part of its claim. The bid-ask cost of replacing the terminated trades is ignored here.

The credit valuation adjustment (CVA) that a bank calculates for one counterparty is the present value of the cost the bank expects to bear if that counterparty defaults. The debit valuation adjustment (DVA, also called the debt valuation adjustment) is the mirror image: the present value of the cost the counterparty expects to bear if the bank itself defaults.

DVA can look odd at first, because it treats the bank’s own possible failure as a source of value. The logic is that a bank in default might not make all the payments its derivatives require, and payments it never makes are a gain to the bank and a loss to the counterparty. So, DVA is added to the value of the portfolio from the bank’s point of view, while CVA is subtracted.

Adjusting the no-default value

Standard pricing models such as Black-Scholes-Merton produce a no-default value, meaning the value that would hold if neither side could ever fail. Write that value, measured from the bank’s side, as f_nd. The value once both sides’ default risk is recognised is:

To estimate the two adjustments, let T be the life of the longest derivative outstanding with the counterparty, and cut the period from today to T into N intervals. For each interval the bank needs a default probability and a loss given that default, and it adds up the products:

The bank carries one CVA and one DVA for every counterparty, and both move whenever market variables, the counterparty’s credit spreads or the bank’s own credit spreads move. That behaviour makes them derivatives in their own right, and banks hedge the risk in CVA (and sometimes DVA) with the same tools they use for any other derivative book: Greek letter sensitivities, scenario analysis and similar measures.

The q_i in the CVA sum must be risk-neutral default probabilities. CVA is a valuation of future cash flows, and the exposures that feed it are generated with risk-neutral valuation, so the probabilities have to come from the same world. Market credit spreads supply them directly.

The starting point is a set of credit spreads for the counterparty at several maturities. Interpolating between them gives s(t_i), an estimated spread for each interval end point t_i, where i runs from 1 to N. If R is the recovery rate expected when the counterparty defaults, the average hazard rate (the default intensity) between today and t_i is approximately:

A constant average hazard rate turns into a survival probability through the exponential function, so the chance that the counterparty is still solvent at time t_i is exp[−s(t_i)t_i/(1 − R)]. The probability of defaulting inside interval i is then the drop in survival probability from the start of the interval to its end:

The bank’s own q_i* values come out of exactly the same calculation, using the bank’s credit spreads in place of the counterparty’s.

Estimating each v_i is the expensive part of CVA. With no collateral in place, the bank runs a Monte Carlo simulation of every market variable that drives the no-default value of its trades with the counterparty, moving them forward in a risk-neutral world from today to T. On each trial it revalues the whole portfolio at the midpoint of every interval and records the exposure:

A portfolio that is a liability to the bank creates no exposure. Averaging the exposures at a midpoint across all trials, discounting the average to today and multiplying by one minus the recovery rate gives v_i. The counterparty’s exposure to the bank is treated the same way to obtain v_i*.

The cure period when collateral is posted

Collateral complicates matters because the bank must know, on each trial, how much collateral each side would be holding at the midpoint if a default struck then. The usual assumption is that a counterparty heading into default stops posting collateral and stops handing back any surplus it holds c days before the default. The parameter c is typically 10 or 20 days and goes by two names: the cure period, or the margin period of risk. So, to find the collateral held at a midpoint, the simulation also has to value the portfolio c days earlier. Once the collateral-adjusted exposures are found, v_i is computed from their average just as in the uncollateralised case.

Peak exposure and incremental CVA

Alongside CVA, banks usually report peak exposure at each midpoint, defined as a high percentile of the simulated exposure distribution. With a 97.5% percentile and 10,000 trials, the peak exposure at a midpoint is the 250th highest of the 10,000 simulated exposures there. The largest of these figures across all midpoints is the maximum peak exposure. Strictly, as a scenario measure it should use real-world rather than risk-neutral default estimates, a point usually ignored.

Because the simulation is costly, banks keep every sampled path of market variables together with every portfolio valuation. When a new trade is proposed, only that trade needs to be valued along the stored paths to see its incremental effect. A new trade whose value moves with the existing portfolio tends to raise both CVA and DVA, while one that moves against it, for example a trade that wholly or partly unwinds earlier positions, tends to reduce them.

Wrong-way risk and right-way risk

The method above treats the counterparty’s default probability as independent of the bank’s exposure, which is often reasonable. When the two are positively correlated, so that the counterparty is more likely to fail exactly when the bank is owed the most, traders call it wrong-way risk. A negative correlation is right-way risk. More elaborate models exist to capture this dependence.

Netting

Netting treats every transaction under the Master Agreement as one combined contract when a default happens. Take a bank with three uncollateralised trades against one counterparty, whose values are +$10 million, +$30 million and −$25 million. Treated separately, a negative trade cannot offset a positive one, so the exposures are $10 million, $30 million and $0, totalling $40 million. With netting, the three trades are a single position worth $15 million, and the exposure falls from $40 million to $15 million.

Collateral

Collateral may take the form of cash, normally interest-bearing, or of marketable securities. Securities are often subject to a haircut, meaning only a percentage of their market value counts towards the collateral requirement. Between financial institutions the collateral terms are now set by regulation. Derivatives also enjoy favourable treatment in bankruptcy: the nondefaulting party simply keeps the collateral posted by the other side, without the long and costly legal proceedings that a lender would normally face.

Downgrade triggers

A downgrade trigger is written into the Master Agreement when a bank deals with a nonfinancial counterparty. If the counterparty’s credit rating drops below a stated level, say BBB, the bank gains the right to demand collateral or to close out all outstanding derivatives at market value. The protection has two weaknesses. It does nothing when the rating falls in one large jump, for instance from A straight to default. It also works only when few such triggers exist: a counterparty that has granted triggers to many dealers will face collateral calls from all of them at once, and it may be unable to meet any of them.

Downgrade triggers at AIG

AIG shows the second weakness clearly. By 2008 it had written many derivatives guaranteeing the AAA-rated tranches of ABS CDOs, and many of those contracts let it avoid posting collateral as long as its rating stayed above AA. On September 15, 2008, Moody’s, S&P and Fitch all cut AIG below AA. The guaranteed tranches were performing badly, collateral calls arrived from many counterparties at the same moment, AIG could not meet them, and only a massive government bailout prevented bankruptcy.

Two simple portfolios allow CVA to be calculated without Monte Carlo simulation. The first is a single uncollateralised derivative that can only pay the bank, at a single time T. A bought European option with remaining life T, sold to the bank by the counterparty, is the standard case.

The exposure at any future date is the no-default value of the option at that date, since the value can never turn negative. The present value of a future no-default value is just today’s no-default value, f_nd. So, every interval has the same expected loss:

Substituting into the CVA sum, and noting that DVA is zero because the bank owes nothing on a bought option, gives the value after credit risk:

Valuing the derivative like the counterparty’s bond

A zero-coupon bond maturing at T and issued by the counterparty is itself a derivative of this kind. If the bond and the derivative share the same recovery rate, the bond satisfies an identical equation with B in place of f, where B_nd is the bond’s no-default value. Dividing one equation by the other shows that f/f_nd = B/B_nd. Writing the risky bond price as e^−yT and the riskless price as e^−y_ndT, where y and y_nd are the two yields, produces:

In words, a one-sided derivative can be valued by raising the discount rate applied to its expected risk-neutral payoff by the T-year credit spread of the counterparty.

The second special case is a bank that has agreed, under an uncollateralised forward contract, to buy an asset from a counterparty for price K at time T, with no other trades between them. Let F_t be the forward price at time t for delivery at T, and let r be a constant risk-free rate. The long forward position is worth the following at time t:

The bank is exposed only when this is positive, so its exposure at t is max[(F_t − K)e^−r(T−t), 0], which equals e^−r(T−t) max[F_t − K, 0]. That is the payoff of a call on the forward price. In a risk-neutral world F_t has expected value F₀, and the standard deviation of ln F_t is σ√t, where σ is the volatility of the forward price. The expected exposure therefore takes a familiar Black-style form:

Discounting the expected exposure at a midpoint t_i back to today and allowing for recovery gives v_i. The two discount factors combine into a single factor for the full life T:

Using more default dates would refine the estimate, and DVA could be found the same way from the bank’s side.

Default correlation describes how far defaults by two companies tend to arrive close together in time. Firms in the same industry or region are hit by the same external shocks and can run into trouble together. The economy itself pushes default rates up in some years and down in others. A default can also directly cause another, which is called credit contagion.

The practical significance is large. Because defaults cluster, credit risk cannot be fully diversified away in a portfolio, and the loss distribution of a portfolio of exposures to many counterparties depends heavily on how correlated the defaults are. Default correlation is also the main reason that risk-neutral default probabilities exceed real-world ones: investors demand compensation for a risk they cannot diversify.

Reduced form models

A reduced form model lets each company’s hazard rate follow a stochastic process and links those processes to macroeconomic variables. When company A’s hazard rate is high, company B’s tends to be high too, and that shared movement creates default correlation. These models are mathematically tractable and naturally capture the way economic cycles generate correlated defaults. Their main weakness is that the default correlation they can produce is limited. Even if two hazard rates move in perfect lockstep, the chance that both companies default within the same short window is usually very small. Where two firms share an industry and country, or one depends heavily on the other, a high correlation may be warranted, and letting the hazard rate jump sharply is one remedy.

Structural models

A structural model builds on Merton’s model: a company defaults when the value of its assets falls below some level. Correlating the asset value processes of companies A and B produces default correlation. The advantage over reduced form models is that any desired level of correlation can be reached. The drawback is that structural models are liable to be slow to compute.

Practitioners lean heavily on the Gaussian copula model for time to default, and it can be shown to behave much like Merton’s structural model. It assumes every company defaults eventually and focuses on how the probability distributions of default times for different companies are correlated. It works with either real-world default probabilities, whose left tail can come from rating agency data, or risk-neutral ones implied from bond prices.

Let t₁ and t₂ be the default times of two companies. If they were normally distributed, a bivariate normal would describe them jointly. Default times are nowhere near normal, so the model maps each one onto a standard normal variable first:

This is a percentile-to-percentile transformation. The point 5% of the way through the distribution of t₁ maps to x₁ = -1.645, the 5th percentile of the standard normal, and the 10th percentile of t₁ maps to x₁ = -1.282. By construction each x_i is standard normal. The modelling assumption is that the x_i are jointly multivariate normal, which is what using a Gaussian copula means.

The appeal is that the joint distribution of default times is pinned down by the individual cumulative distributions plus one correlation per pair, called the copula correlation. The correlation structure is estimated separately from the marginal distributions, and the approach scales to n companies. In practice the copula correlation between two firms is often approximated by the correlation of their equity returns.

Specifying a separate correlation for every pair of companies becomes unmanageable as the portfolio grows. A one-factor model replaces all those pairwise numbers with one loading per company:

F is a common factor that affects defaults at every company, and Z_i is an idiosyncratic factor for company i alone. F and every Z_i are independent standard normals, and each a_i is a constant between −1 and +1. The copula correlation between x_i and x_j is then simply a_ia_j. The loading a_i is sometimes estimated from how company i’s equity returns move with a broad, well-diversified market index.

Default probability conditional on the factor

Let Q_i(T) be the probability that company i defaults by time T. In the copula, default by T means N(x_i) is below Q_i(T), which is the same as x_i lying below N⁻¹[Q_i(T)]. Substituting the one-factor equation and isolating Z_i gives the default condition:

Once F is fixed, Z_i is the only random element, so the conditional default probability is the normal probability of that inequality:

In the special case where every company has the same default distribution Q(T) and every pair the same correlation ρ, each loading is a_i = √ρ, and the expression simplifies:

Credit VaR mirrors market risk VaR: a 1-year credit VaR at 99.9% confidence is the credit loss that will not be exceeded over 1 year with 99.9% certainty.

Take a bank holding a very large book of similar loans, and assume, as an approximation, that every loan has the same default probability and every pair the same correlation. Under the one-factor Gaussian copula, the conditional probability Q(T | F) is then a good approximation to the proportion of the portfolio that defaults by T, because with many loans the idiosyncratic factors average out and only F matters. F is standard normal, so we can be X% certain that F will exceed N⁻¹(1 − X) = −N⁻¹(X). Putting that worst-case value of F into the conditional formula gives the default rate that will not be exceeded with X% confidence over T years:

Here Q(T) is the probability of default by T and ρ is the copula correlation between any two loans. The result is due to Vasicek, and it underlies some of the formulas regulators use for credit risk capital. A rough credit VaR follows by scaling by the portfolio size L and the loss rate:

A loan of size L_i contributes L_i(1 − R)V(X, T) to this total.

CreditMetrics builds the whole distribution of credit losses by simulation. For a 1-year horizon, each Monte Carlo trial samples a rating change, including possible default, for every counterparty, then revalues all outstanding contracts to find the year’s total credit loss. Repeating this many times produces a loss distribution from which credit VaR is read.

The method is computationally heavy but has two advantages. Losses include the fall in value from downgrades as well as from defaults, with upgrades counted as negative losses. It can also roughly incorporate credit mitigation clauses such as netting and collateral.

The rating transition matrix

Historical rating migration data from the agencies drives the simulation. The matrix below gives 1-year transition probabilities in percent for 1981 to 2019, computed from S&P data, with moves to the withdrawn rating (WR) category redistributed proportionally across the other categories. Columns run from default back up to AAA, with a row total as a check.

Reading across the A row, a company rated A has a 92.49% chance of keeping that rating for a year, a 5.27% chance of slipping to BBB and a 0.05% chance of defaulting.

Correlating rating changes with a Gaussian copula

Rating changes of different counterparties are not independent, so a Gaussian copula links them in the same way it linked default times. The copula correlation for two companies is usually set equal to the correlation of their equity returns, estimated with a factor model. Each company gets a standard normal draw, and the cumulative transition probabilities in its row are turned into cut points with N⁻¹.