CR 10 – Credit Value at Risk

Credit VaR is the credit loss over a stated time period that a portfolio will not exceed at a chosen confidence level. The wording mirrors market risk VaR, but the two measures differ in what counts as a loss, how far ahead they look and how the loss distribution is built. Some credit VaR models count only default losses, while others also count the value lost when a borrower is downgraded or its spread widens.

Banks compute credit VaR for two separate purposes. Regulatory capital is the amount supervisors require the bank to hold against credit risk, and its rules prescribe the model and its parameters. Economic capital is the bank's own view of the capital its risks demand, and it feeds return on capital measures for individual business units. A bank may therefore run one model for each.

Real-world probabilities for scenarios, risk-neutral ones for valuation

VaR is a scenario analysis measure, so the likelihood of each scenario should come from real-world default probabilities estimated from default history. Once a scenario is generated, the portfolio is priced at the horizon date, and pricing uses risk-neutral default probabilities implied by market prices. Since real-world default probabilities sit below the risk-neutral ones, swapping their roles would distort both the scenario weights and the horizon values.

Horizon and tools

Market risk VaR is usually a one-day measure produced by historical simulation. Credit VaR for positions not held for trading typically uses a one-year time horizon, and because it must generate defaults, and sometimes rating changes, for many obligors at once, it needs a more elaborate model.

Why credit correlation drives the answer

In a recession most firms are hurt at once and their default probabilities rise together; in a strong economy they fall together. That shared sensitivity to the cycle is the main source of credit correlation. A lender with many loans faces a fatter right tail when correlation is high, because bad years bring many defaults at once, and correlation tends to climb exactly when conditions are stressed.

A ratings transition matrix lists, for each starting rating, the probability of ending a given period in every rating category, default included. Matrices are estimated from history, on a bank's internal scale or an agency scale (Moody's, S&P or Fitch). The one-year matrix below covers every company S&P rated from 1981 to 2020, adjusted for moves to not rated. Its last column, added here, sums the cells right of the diagonal: the chance of ending lower than the starting grade.

Source: S&P Global, as reproduced in Hull. Final column derived here.

Each row describes one starting grade. An A-rated company keeps its A rating over the year with probability 92.61%, rises to AA with probability 1.67% and slips to BBB with probability 5.23%. Its one-year default probability is just 0.05%, or 5 in 10,000. Default is absorbing: once in D, a company stays there.

Longer horizons by matrix multiplication

If we are willing to assume that a rating change in one year tells us nothing about the change in the next, the one-year matrix can be chained. Multiplying the matrix by itself gives the two-year matrix, and raising it to the fifth power gives the five-year matrix shown next.

The diagonal shrinks, so keeping a grade for five years is far less likely (58.92% for AAA against 89.85% over one year), and default probabilities climb sharply, to 24.59% for B and 70.18% for CCC/C.

Horizons shorter than a year are harder. A six-month matrix is the square root of the one-year matrix and a three-month matrix its fourth root; such a matrix root must be found numerically, not cell by cell. The one-month matrix, the twelfth root of Table 1, follows.

Over one month almost all probability sits on the diagonal. A BBB company keeps its rating with probability 99.28%, falls to BB with probability 0.35% and defaults with probability 0.01%, figures used later for a portfolio rebalanced monthly.

The same root technique produces a matrix for any horizon. Applied with an exponent of one half, it gives a six-month AAA row in which the probability of remaining AAA is about 94.78% and the probability of moving to AA about 4.92%. Fractional powers above one also work, so a 1.25-year matrix is simply Table 1 raised to the power 1.25.

The independence assumption and ratings momentum

Every one of these calculations treats rating changes in different periods as independent, and that is not quite true. A company that has just been downgraded is more likely than its peers to be downgraded again soon afterwards, a pattern known as ratings momentum (studied by Altman and Kao, and by Lando and Skodeberg). Comparing Table 2 with an observed five-year matrix confirms it: companies in the top three grades actually default or suffer large downgrades more often than the chained matrix predicts. For many purposes, though, the independence assumption is an acceptable approximation.

Vasicek's model uses a one-factor Gaussian copula to find a high percentile of the default rate in a loan portfolio. Define WCDR(T, X), the worst case default rate, as the Xth percentile of the distribution of the default rate over a period of length T. The model links it to two inputs: PD, the probability of default over time T, and ρ, a copula correlation parameter that captures how strongly defaults move together.

For one loan with exposure at default EAD and loss given default LGD, the Xth percentile of its loss is the worst case default rate scaled by the amount at stake.

Gordy (2003) showed that in a one-factor world this result can be extended to a portfolio. When there are n loans and each one is only a small part of the whole, the Xth percentile of the portfolio loss is approximately the sum of the loan-level figures, each loan using its own PD and therefore its own WCDR.

Use under the Basel II internal ratings-based approach

Under the Basel II internal ratings-based (IRB) approach, banking book regulatory capital is set by Equation 2 with T fixed at one year and X at 99.9%, a loss exceeded once in a thousand years, while banks supply PD, EAD and LGD. Each term may also carry a maturity adjustment, because a loan lasting beyond one year can lose value through deterioration without defaulting.

Choosing the correlation parameter

Structural models imply that ρ for two companies is roughly the correlation between their asset returns, for which equity return correlation is a workable proxy. So ρ for a portfolio can be estimated as the average equity correlation among the borrowers, or among comparable listed firms when borrowers are private.

The Gaussian copula builds in very little tail correlation, a known weakness; a copula with stronger tail dependence can replace it.

Credit Risk Plus, proposed by Credit Suisse Financial Products in 1997, borrows the analytic approximations insurers use for claim counts. Take n similar loans, each with one-year default probability q, so qn defaults are expected. If defaults are independent, the number of defaults m follows the binomial distribution.

With q small and n large, the binomial is well approximated by the Poisson distribution, which needs only the expected number of defaults.

Equation 3 still holds approximately when default probabilities differ across loans, as long as all of them are small and q is taken as the average one-year default probability in the portfolio.

Making the default rate uncertain: the negative binomial

Equation 3 treats next year's default rate as known, yet default rates swing widely from year to year. A convenient fix is to let qn, the expected number of defaults, be random with a gamma distribution of mean μ and standard deviation σ. Mixing the Poisson over that gamma turns it into a negative binomial distribution.

Here Γ(x) is the gamma function, equal to (x − 1)! for integer x. Table 4 sets μ = 4 and lets σ grow from zero. The first column is Equation 3; the others come from Equation 4, with the implied α and p shown at the top.

Source: Hull. The α and p rows are derived here from μ = 4.

At σ = 0.1 the column is almost the Poisson column, so the negative binomial collapses to the Poisson as uncertainty vanishes. As σ rises, probability drains from the middle into zero defaults and the far right tail. If every default costs the same amount, the 99.9% VaR is the loss from the 99.9 percentile number of defaults: 11 defaults when the default rate is certain, but 98 defaults when σ = 10.

Monte Carlo simulation is a more flexible way to build in default rate uncertainty, especially when several exposure categories have different but correlated default rates. One recipe runs as follows.

1. Draw an overall default rate for the year, for example by sampling from a history of annual default rates.
2. Translate it into a default rate for each category of obligor, using a model such as a regression of each category's historical default rate on the overall rate.
3. Draw the number of defaults in each category.
4. Draw a loss given default for every default.
5. Add up the total loss.
6. Run steps 1 to 5 thousands of times to obtain the distribution of total loss.
7. Read the required VaR from that distribution.

Step 1 can be refined. Annual default rates are autocorrelated, so next year's rate is better modelled from this year's rate or other observed economic variables than drawn at random from history.

Where the default correlation comes from

Default rate uncertainty is what creates default correlation here: all companies share one default rate that may turn out high or low, so their defaults rise and fall together. Without that uncertainty, defaults are independent, losses are fairly symmetric and many defaults are very unlikely (the first column of Table 4); with it, the distribution becomes positively skewed.

Vasicek's model and Credit Risk Plus produce a distribution of losses from defaults only; a downgrade costs nothing in either until it ends in default. CreditMetrics, proposed by JPMorgan in 1997, is built to capture losses from downgrades as well. Its foundation is a ratings transition matrix such as Table 1, using either a bank's internal ratings or agency ratings.

One simulation trial

For many counterparties the one-year credit VaR comes from Monte Carlo simulation. Each trial draws an end-of-year rating for every counterparty, using the correlation model in the next section. If a counterparty survives, its transactions are revalued at the one-year point using the credit spreads for its new rating, and the change in value is the credit loss. If it defaults, the loss is the exposure at default multiplied by one minus the recovery rate, with the recovery rate itself usually drawn from an empirical distribution. Total credit losses across trials form the loss distribution from which VaR is read.

Revaluation needs a term structure of credit spreads for each rating category at the horizon: either today's observed curves, or curves that depend linearly on a credit spread index with its own distribution.

Derivatives and the link to CVA

When the transactions are derivatives, the revaluation extends the credit valuation adjustment (CVA) calculation. Split the time from today to the end of the longest transaction into n intervals; let q_i be the default probability in interval i, v_i the present value of expected net exposure, after collateral, at the midpoint of interval i, and R the recovery rate. If interval j is the first one starting at year one, the expected cost of defaults from year one onward, viewed from today, is the following.

On a trial with no default in the first year, the simulated one-year term structure of spreads gives a new set of interval default probabilities q_i* for i ≥ j. The credit loss on that trial is the change in expected default cost.

If the counterparty is upgraded, q_i* falls below q_i and the credit loss is usually negative, a gain. On a trial with a first-year default, the default time is sampled, the exposure at that time is determined, and the loss is that exposure times one minus the recovery rate.

CreditMetrics joins counterparties' rating moves with a Gaussian copula, and the copula correlation for a pair of companies is normally taken from the correlation of their equity returns. A one-factor equity model such as the capital asset pricing model keeps a large correlation matrix internally consistent.

From a normal draw to a rating

Suppose equity returns of one A-rated firm and one B-rated firm have correlation 0.2. On each trial we draw two standard normal variables, x_A and x_B, with correlation 0.2. Each variable is mapped to a rating by slicing the standard normal distribution at the inverse normal of the cumulative transition probabilities in the company's row of Table 1, starting from the best grade.

For the A-rated company the probabilities of ending at AAA, AA and A are 0.0003, 0.0167 and 0.9261 (the original text prints the second value as 0.167, a slip for 0.0167). The thresholds follow from the cumulative sums.

So the company moves to AAA below the first threshold, to AA between the first two, and stays A for x_A between -2.1201 and 1.5813. It defaults above N^-1(0.9995) = 3.2905, reflecting its 0.05% default probability. One caution: the inverse normal of 0.0003 is -3.4316, not the printed -3.3416 (transposed digits); the printed boundary would imply an AAA probability near 0.04% instead of 0.03%.

The B-rated company works the same way, with probabilities 0.0002, 0.0008 and 0.0017 of moving to AA, A and BBB.

The B-rated company defaults when x_B exceeds 1.7744. This is labelled N^-1(0.9619), one minus the 3.81% default probability, but N^-1(0.9619) is 1.7732; 1.7744 is N^-1(0.9620), the B row's non-default total, which reaches 96.20% through rounding. The gap is immaterial. Table 5 gathers every zone.

Thresholds as printed in Hull, based on Table 1.

Same total loss, different timing

Built on identical assumptions, CreditMetrics and Credit Risk Plus should agree on long-run losses; they differ in when losses are recognised. Picture a borrower that slides from A to BBB in year 1, from BBB to B in year 2, and defaults in year 3. Credit Risk Plus records nothing in years 1 and 2 and the whole loss in year 3. CreditMetrics books a revaluation loss in each of the three years, and in theory those three amounts add up to the single Credit Risk Plus loss.

Credit spread risk is the risk that a position loses value because credit spreads widen, with no default. Trading book credit instruments are priced off spreads, so their VaR or expected shortfall (ES) must allow for spread moves.

Historical simulation and its problems

The direct route is historical simulation of a one-day 99% VaR, scaled by √10 to obtain a 10-day 99% VaR. The bank collects daily spread changes for every exposure over a window such as 500 days, and scenario k applies day k's changes to all companies together. Two problems arise. Any company still alive today did not default in the historical window, so the scenarios carry no possibility of default; one patch is to treat a spread above some level as a default, and regulators instead add a separate jump-to-default charge. In addition, spreads for many companies are not updated daily, so the data can be poor.

A CreditMetrics-style alternative

The second route builds a 10-day transition matrix by the root method and estimates, from history, a 10-day spread change distribution for each rating category. Each Monte Carlo trial samples every company's rating outcome and the category spreads, then revalues the portfolio. Correlation enters twice: a Gaussian copula links rating changes, and spread moves across categories are treated as perfectly, or almost perfectly, correlated, so when A spreads widen, other grades widen too.

Values, probabilities and losses from Hull; yield and cumulative probability columns derived here.

A constant level of risk assumption means the portfolio is treated as if it were rebalanced periodically to bring its risk back to the starting level. Compare two ways of holding BBB bonds for a year: buy and hold keeps them whatever happens, while the constant level of risk strategy sells any bond that is no longer BBB at a month end and buys a BBB bond in its place. The buy-and-hold outcomes come from the BBB row of Table 1; the rebalancing outcomes come from the BBB row of Table 3, applied twelve times.

Defaults

Under buy and hold, the one-year default probability is 0.0017. With monthly rebalancing, each month carries a default probability of 0.0001, so the probability of exactly one default in twelve months is binomial.

That is lower than 0.0017, and two or more defaults are negligible. Rebalancing removes the bonds that have drifted towards default before they get there.

Downgrades to BB

Buy and hold gives a 0.0378 probability of ending the year at BB. Under rebalancing, the monthly probability of a fall to BB is 0.0035, so the chance of exactly one such downgrade is as follows.

The printed figure of 0.04404 does not reconcile; the product is 0.0404, but the conclusion is unaffected because 0.0404 still exceeds the buy-and-hold 0.0378. The probability of more than one downgrade is about 0.001.

So buy and hold suffers more defaults and big downgrades, while rebalancing brings more frequent small rating changes that each cost little. VaR and ES, driven by severe outcomes, are generally smaller under the constant level of risk assumption.

The three models differ in what counts as a loss and where default correlation comes from. Basel rules require credit VaR with a one-year horizon and 99.9% confidence, while economic capital may use a different confidence level.

For credit spread risk in the trading book, the choice is between historical simulation of spread changes and a model of rating transitions combined with movements in average spreads for each rating category.