CR 11 – Portfolio Credit Risk

A lender or investor rarely holds a single credit exposure. A typical book mixes bonds and commercial paper with off-balance-sheet items such as guarantees, and it may also carry credit default swaps (CDS) and other credit derivatives. The same book can hold many different obligors, and it can also hold several claims on one obligor at different levels of its capital structure, for example senior debt alongside preferred shares. These details matter for measuring risk, though the models here set most of them aside.

When a single credit-risky position is analysed on its own, five sources of risk and return come into view:

• the probability that the obligor defaults;
• the loss given default (LGD), which is one minus the fraction recovered after default;
• rating migration, meaning the chance and size of a downgrade that falls short of default;
• spread risk, meaning a change in the market spread demanded for a given rating;
• for distressed debt, restructuring, where a negotiated deal among the owners and creditors or a ruling in bankruptcy court can impose losses on particular classes of debt.

Moving from one position to a portfolio adds one ingredient that the single-name view cannot supply: default correlation, which governs how likely it is that several obligors default together. To keep that ingredient in clear view, this lesson takes each default probability and each recovery rate as given and sets the other sources of return aside.

The modelling approach used here

The tool developed in the later sections is a factor model whose drivers are latent factors with normally distributed returns. Once the factor values are known, the defaults of different firms become independent of one another, and the whole analysis looks at one future horizon that is common to every credit in the portfolio. The version used here is narrowed twice: it tracks only default events, not migration between ratings, and it uses only one factor. The CreditMetrics framework applies the same structure to the full distribution of rating migrations as well as defaults, so the version in this lesson can fairly be called default-mode CreditMetrics. Commercial models such as Moody’s KMV and CreditMetrics normally run in migration mode, where both upgrades and downgrades and outright defaults can occur.

The structure has two practical strengths: the factor can be tied to observable data such as equity returns, and the model is tractable, so many results come out in closed form.

Take two firms (or two sovereigns, if the positions are government debt) with default probabilities π₁ and π₂ over a horizon τ, and let π₁₂ be the probability that both default before τ. Each firm’s fate can be coded as a Bernoulli variable x_i that equals 1 on default and 0 on survival. The horizon must be fixed in advance, because a Bernoulli trial only makes sense for a defined interval, and in a portfolio model every credit shares that same interval.

The joint default probability π₁₂ is a genuinely new parameter. It cannot be worked out from π₁ and π₂ unless an extra assumption is imposed, the obvious one being that the two defaults are independent, in which case π₁₂ = π₁π₂.

The four outcomes

The product x₁x₂ equals 1 only when both firms default and 0 in the other three cases. The table lists the outcomes, starting from the joint default and working back, together with the number of defaults in each.

The four outcomes are mutually exclusive and their probabilities sum to one. So, the probability of one or more defaults can be read either as one minus the no-default probability or as the sum of the other three rows.

Moments and the correlation formula

The Bernoulli variables have simple moments. The mean of each is its default probability, the mean of the product is the joint default probability, and the variance of a Bernoulli variable is p(1 – p).

Dividing the covariance by the product of the two standard deviations gives the default correlation.

The numerator is the excess of the actual joint default probability over its value under independence. When the defaults are independent that excess is zero and so is ρ₁₂. Otherwise the correlation rises in a straight line with the excess, because the denominator depends only on the single-name probabilities.

The relationship can be turned around so that the correlation, rather than the joint probability, is the primitive input. Most portfolio models, including the one in this lesson, take that route and specify a default correlation.

Correlations that look tiny can multiply the joint default probability many times over, because single-name default probabilities are themselves small. The two worked examples below show the effect with a pair of investment-grade names and then with original numbers.

Because π₁π₂ is a product of two small numbers, even a modest covariance term is large relative to it, so the joint default probability is very sensitive to the correlation input.

With two credits the default probabilities, one pairwise correlation and the requirement that probabilities sum to one pin down all four outcomes. That neat closure breaks down as soon as a third credit is added. Three credits produce eight joint outcomes (no default, three single defaults, three double defaults and one triple default), yet single-name probabilities, pairwise correlations and the adding-up condition supply only seven restrictions. In general n credits create 2ⁿ distinct default events, while the correlation framework supplies only n + 1 + n(n – 1)/2 conditions.

So, a model built only on default probabilities and pairwise correlations cannot determine the full joint distribution of defaults once there are more than two names. It is nonetheless a pragmatic compromise, since the alternative for a book of just 10 credits would be to estimate or simply assume 1,024 separate event probabilities.

Too many parameters, and the sign restriction

Even the reduced framework is heavy. A portfolio with N obligors needs N default probabilities, N recovery rates and a full matrix of pairwise default correlations, which the source counts as N(N – 1) entries; because the matrix is symmetric, only N(N – 1)/2 of those values are distinct. A common shortcut is to set every pairwise correlation equal to one parameter. That parameter then has to be non-negative, since a matrix in which every firm is negatively correlated with every other cannot be positive-definite and produces nonsensical results.

Positions that sit awkwardly in the framework

Several common holdings were left out of that example because they do not behave like plain default exposures. Contingent liabilities such as guarantees and revolving credit agreements act much like options on credit quality. CDS basis trades depend on market and credit risk, but their profit is driven mostly by technical factors, meaning temporary dislocations in how market participants are positioned; during the subprime crisis, for instance, the CDS basis widened sharply because funding liquidity dried up. Convertible bonds carry equity and equity volatility (vega) risk that can matter as much as their credit risk.

Most companies that issue debt default rarely, and that rarity has two consequences for correlation. First, there are very few joint defaults in the historical record, which makes default correlation hard to estimate. Most studies put one-year default correlations at around 0.05, but the estimates swing a lot across time periods, industries and countries of domicile, and they often come out negative. Second, default correlations are small numbers in absolute terms.

That second point can mislead anyone used to reading correlations in other settings. In a regression, a correlation of 0.05 would usually be dismissed as trivial and 0.5 regarded as strong. Default probabilities, however, are typically around 1 percent, except for the small group of firms rated CCC and below, so the probability that any particular pair defaults together is also very small. Against such a small base, an optically small default correlation can raise the joint default probability many times over, which is exactly what Example 1 showed when a correlation of 5 percent lifted the joint probability from about 0.31 basis points to about 3 basis points.

So, a default correlation should be judged by its effect on joint defaults and the loss tail, and treating 0.05 as negligible understates the chance of clustered defaults.

Portfolio Credit VaR is built in the same way as the VaR of a single credit: take a quantile of the credit loss distribution at the chosen confidence level and deduct the portfolio’s expected loss.

Default correlation has a powerful effect on portfolio risk, but it works on the volatility and the extreme quantiles of loss, not on the expected loss. With a default correlation of 1 the portfolio behaves like one credit and gains nothing from diversification. With a correlation of 0 the number of defaults follows a binomial distribution and diversification can be substantial.

A stylised portfolio

Consider a portfolio worth $1,000,000,000 split equally among n credits. Each credit has default probability π and a recovery rate of zero, so a defaulted position is wiped out. All positions sit at the same level of the capital structure and each belongs to a different obligor, since two claims on the same obligor would simply act as one bigger position. Time value is ignored, or equivalently all amounts are treated as future values. With n = 50, every position is worth $20,000,000.

Correlation of 1: the all-or-nothing case

If the default correlation is 1, either every credit defaults (probability π) or none does, whatever the value of n, so the portfolio behaves as though n = 1. The expected loss is π × 1,000,000,000 and the only two outcomes are a loss of $1,000,000,000 and a loss of zero, the latter with probability 1 – π.

When π is larger than one minus the confidence level, the loss quantile is the full $1,000,000,000 and the Credit VaR equals that amount less the expected loss. When π is smaller than one minus the confidence level, the quantile is zero and Credit VaR is negative, because the expected loss is always subtracted. Take π = 0.02 at 95 percent confidence: there is a 98 percent chance of no loss, so the 95th percentile loss is zero, and subtracting the expected loss of 0.02 × 1,000,000,000 = 20,000,000 gives a Credit VaR of -$20,000,000, in effect a gain. The number is well defined but not very informative.

Correlation of 0: the binomial case

With zero correlation the number of defaults is binomial with parameters n and π, which fills in many outcomes between the two extremes. For n = 50 and π = 0.02 the expected loss is still $20,000,000. The figure plots the cumulative probability of the number of defaults.

Because the probability of two or fewer defaults is 0.92 and of three or fewer is 0.98, the 95th percentile is three defaults. Three defaults cost $60,000,000, and subtracting the $20,000,000 expected loss leaves a Credit VaR of $40,000,000. Raising the count to n = 1,000 while holding the portfolio value fixed shrinks the dispersion further: the 95th percentile becomes 28 defaults, or a loss of $28,000,000, and the Credit VaR falls to $8,000,000.

* The source prints 1 default in these two cells, but the printed share of 0.000 and the negative Credit VaR both correspond to 0 defaults, which is the correct binomial quantile.

Reading the pattern

Each row of the table is an application of the same rule: defaults at the quantile, divided by n and multiplied by $1,000,000,000, minus the expected loss. For a given portfolio size, a higher default probability means a higher Credit VaR. For a given default probability, Credit VaR falls as the portfolio becomes more granular, meaning it holds more independent credits, each a smaller share of the whole. The fall is steepest when π is high, and the converse is just as important: when default probabilities are low, adding granularity buys much less reduction in VaR.

In the limit of a very large number of tiny independent positions, the realised loss converges with probability 1 to the expected loss. The single credit loses nothing with probability 1 – π and everything with probability π, whereas the fully granular portfolio loses 100π percent almost certainly. Its credit loss then has no volatility and its Credit VaR is zero.

The extreme cases above allowed default correlation to be only 0 or 1 and left no room for firm-specific risk. The single-factor model lets default correlation take any value between those extremes by giving each credit a sensitivity, or beta, to a common market factor, while an idiosyncratic shock captures everything specific to the firm.

For firm i, the return on assets is split into a market part and a firm-specific part. The market factor m and the idiosyncratic shock ε_i are standard normal, and β_i scales the firm’s exposure to the market.

The model assumes that the market factor and every idiosyncratic shock are uncorrelated, and that the idiosyncratic shocks of different firms are also uncorrelated with one another.

Moments of the asset returns

Under these assumptions each a_i is itself a standard normal variable: its mean is zero and its variance is β_i² + (1 – β_i²) = 1. Because both building blocks have unit variance, β_i is also the beta of firm i to the market factor. The only route by which two firms’ returns are linked is their shared exposure to m, so the covariance, which equals the correlation here, is the product of the two betas.

The default trigger

As in the single-credit structural model, firm i defaults when its asset return falls to or below a threshold k_i. That threshold measures, in standard deviations, how far the log asset value sits above the asset level at which default occurs. Since a_i is standard normal, the unconditional default probability is simply the standard normal distribution function at the threshold.

Once the market factor has taken a particular value, the asset returns of different firms, and therefore their default events, are independent of one another. This property, conditional independence, follows directly from the assumption that firms are linked only through the market factor, and it is what makes the single-factor model so convenient for portfolio credit risk.

Fix the market factor at a realised value m̄. The asset return is then driven only by the idiosyncratic shock, around a shifted mean.

For any positive beta, the conditional mean moves to β_im̄ and the conditional standard deviation drops from 1 to √(1 – β_i²), even though the default threshold k_i stays exactly where it was. A bad market draw pulls the whole distribution towards the threshold, so more of it lies in the default region.

Three effects of fixing the market factor

First, the conditional default probability differs from the unconditional one unless either m̄ = 0 or β_i = 0, meaning the market shock is exactly zero or the firm has no exposure to the economy. Unconditionally, endless pairs of market and firm shocks can combine to cause default, but that is no longer true once the factor is fixed. Given m̄, default occurs exactly when the idiosyncratic shock is at or below a cut-off, and that cut-off falls in a straight line as m̄ rises; in a weak economy a smaller negative shock is enough.

Second, the conditional variance of the asset return is 1 – β_i², below the unconditional variance of 1. Third, the conditional returns β_im̄ + √(1 – β_i²)ε_i of two firms depend on independent shocks, so their default outcomes are independent.

Putting these together, the conditional distribution is normal with mean β_im̄ and standard deviation √(1 – β_i²), and the conditional cumulative default probability is a standard normal distribution function evaluated at the new, standardised distance to default.

The intuition is that, for a known state of the economy, the chance of default is set by how far, in standard deviations, the idiosyncratic shock has to fall below zero. Plotted against m, p(m) slopes downwards: near 1 in deep recessions and near 0 in booms, with a steeper transition when beta is high.

Asset return correlation and default correlation are related but very different numbers. In the single-factor model, the asset returns of any two credits i and j follow a bivariate standard normal distribution with correlation β_iβ_j.

A joint default occurs when both returns fall into the region below their thresholds, so the joint default probability is the bivariate normal distribution function, written here as Φ₂, evaluated at the two thresholds.

Substituting this joint probability into Equation 11.1 turns the asset correlation into a default correlation.

The homogeneous case

From here on every firm is given the same parameters, β_i = β, k_i = k and π_i = π. Any pair then has asset correlation β², and the default correlation collapses to a single expression.

Going the other way, from a target default correlation to the beta that produces it, has no closed form and needs a numerical root search over β.

An asset correlation of 0.315 maps to a default correlation of only 0.05, and 0.25 maps to 0.034. Default correlations sit far below the asset correlations that generate them, which fits the small values seen in historical data.

Now apply the homogeneous model to a granular portfolio of n credits, where n is large, each credit has a loss given default of $1, and each is such a small slice of the whole that idiosyncratic risk is negligible.

Conditional loss level

For a given market factor value, the credits’ asset returns are independent standard normals, so the law of large numbers applies across the portfolio. The loss level x(m), the defaulted share of the portfolio, tends to the conditional default probability of any single credit.

So, once the market factor is known, the portfolio loss is known. Given the two parameters, the default probability and the correlation, the market factor alone drives portfolio returns.

From conditional to unconditional loss distribution

What a risk manager needs is the unconditional distribution of the loss level X. Each loss level is generated by one market factor value, so the two share the same probability. The procedure works through every possible loss level analytically rather than by drawing random scenarios:

1. Treat the loss level X as a random variable and consider every value x from 0, meaning nothing is lost, up to 1, meaning everything is lost.
2. For each x, find the market factor value m̄ at which a single credit’s conditional default probability equals x, by inverting Equation 11.2.
3. Read off the probability of that market factor value from the standard normal distribution.
4. Repeat across loss levels to build the whole distribution of X.

Because the loss level rises as the market factor falls, a loss of x or more happens exactly when the market factor ends at m̄ or below. The probability of that event is Φ(m̄), which gives the upper tail of the loss distribution; the cumulative distribution function is its complement.

The homogeneous model has only two levers. The default probability π fixes the unconditional expected level of defaults. The market correlation β² decides how widely those defaults are spread across states of the economy. When the correlation is high, defaults pile up quickly as business conditions worsen, and when it is low, only an extremely bad economic scenario can push the default rate high.

The two extremes

As β approaches 1 (perfect correlation), and remembering that idiosyncratic risk has already been removed by granularity, only two outcomes survive. If the market factor ends at or below k, nearly every credit defaults and the loss rate is 1; if it ends above k, hardly any credit defaults and the loss rate is 0. The portfolio then behaves like the single credit of the earlier stylised example.

As β approaches 0 (zero correlation), defaults have no link to the market factor and idiosyncratic risk is diversified away, so the loss rate is very likely to sit very close to the default probability π. That matches the fully granular uncorrelated portfolio, whose Credit VaR tends to zero.

Between the extremes

For intermediate values, a higher correlation raises the probability of both very few and very many defaults and lowers the probability of middling outcomes. The expected loss does not move, but the upper quantiles of loss rise, so Credit VaR increases with correlation. Figure 3 shows exactly this: the beta 0.8 curve starts much higher at very small loss rates and approaches 1 much more slowly at large ones.