Portfolio Credit Risk: Correlation, Vasicek, and the Basel Formula

Everything up to now has priced one borrower at a time. But nobody holds one borrower — banks hold portfolios, and a portfolio’s risk is not the sum of its parts. Ten thousand loans, each with a 2% PD, can produce a placid book that loses about 1% a year forever, or a time bomb that loses 15% in one bad year — with identical single-loan statistics. The difference is one number: how much the borrowers move together. This module builds the machinery for that number, ending at the formula that decides how much capital nearly every bank on earth must hold.

Concentration: the risk you can see

The visible portfolio sin is concentration, and it comes in three flavors: single-name (one borrower large enough that its idiosyncratic failure hurts — the reason regulators cap large exposures at a fraction of capital), sector (a hundred borrowers who are really one bet on oil prices or commercial real estate), and geographic (the regional bank’s original sin — every borrower shares one local economy). Concentration is measurable with nothing fancier than a Herfindahl index, and the cure — granularity, diversification across names and sectors — is obvious even when it’s commercially painful.

The deeper problem is that you can build a perfectly granular, sector-balanced portfolio and still get destroyed, because diversification only removes idiosyncratic risk. What’s left is the risk that borrowers share by living in the same economy — and that brings us to correlation.

Correlation: the input that runs the show

Default correlation between two typical investment-grade borrowers is tiny — the joint default probability might be a few basis points. Intuition says something that small can’t matter. Intuition is wrong, for a reason worth internalizing: with thousands of borrowers there are millions of pairs, and the portfolio’s tail is driven by the sum of all that pairwise co-movement, not by any single pair. Small correlation × enormous number of pairs = the dominant driver of extreme losses. In the simulator below, moving ρ from 5% to 30% — with PD and LGD frozen — multiplies the 99.9% loss several times over. No other input in credit does that.

It is also the input we measure worst. Defaults are rare; joint defaults are rare². Nobody has enough direct data, so everyone estimates correlation indirectly: from equity co-movement (borrowing Merton’s logic from Module 6 — if defaults are driven by asset values, asset correlation ≈ what equity correlation lets you infer), from rating co-migration (Module 8’s transition data), or from the cycle-dependence of realized default rates. The estimates disagree, and the portfolio model swallows whichever one you feed it. Remember this when a model reports a 99.9th percentile to four significant figures.

CreditRisk+: the actuarial route

One respectable school of thought — CreditRisk+, published by Credit Suisse in 1997 — refuses to model why defaults correlate. Borrow the fire-insurance toolkit: each borrower defaults with a small probability, defaults in a sector are Poisson-ish, and correlation enters by making the Poisson intensities themselves random (a gamma-mixed Poisson — bad years are years the whole sector’s intensity drew high). The magic is analytic: no Monte Carlo, closed-form loss distribution via probability generating functions. It’s the reduced-form philosophy of Module 7 scaled to a portfolio, and it remains popular where speed and auditability beat structural storytelling.

The Vasicek model: Merton, applied to everybody at once

The structural route won the regulatory argument, and it deserves the space. Oldrich Vasicek’s 1987 result starts from Merton’s picture and adds one ingredient: a single common factor. Borrower $i$‘s (standardized) asset return is

where $M$ is the state of the economy (one draw per year, shared by everyone), $\varepsilon_i$ is borrower-specific noise, and $\rho$ is the asset correlation. Borrower $i$ defaults if $X_i < \Phi^{-1}(\text{PD})$ — exactly Merton’s “assets below the barrier,” with the barrier calibrated to the right PD.

Now condition on the year. Given the economy drew $M = m$, defaults become independent (the only thing borrowers shared was $M$), and each happens with probability

This is the engine of the whole model: in a good year ($m$ high), everyone’s conditional PD is low; in a bad year, everyone’s rises together. Correlation has been converted into a common intensity — note the family resemblance to CreditRisk+‘s random intensity; the two schools meet in the middle.

The asymptotic step. Let the portfolio become infinitely granular — thousands of loans, none material. Conditional on $m$, the law of large numbers kills all idiosyncratic noise: the realized default fraction equals $p(m)$ exactly. The only randomness left is the economy itself. One random variable, pushed through a monotone function — so the loss distribution comes out in closed form:

That is the Vasicek distribution — the entire loss distribution of a granular portfolio, from two parameters. Play with it:

Three experiments worth running. Drop ρ toward 1%: the density collapses into a spike at EL — this is the diversification dream, losses as predictable as an actuarial table. Push ρ to 40%: the density smears across the axis and grows a long right tail — same EL, radically different risk. Then hold everything and switch the confidence from 95% to 99.9%: watch how much further out the VaR marker jumps at high ρ than at low ρ. Correlation doesn’t change what you expect to lose; it changes what you can lose.

ASRF and the Basel IRB formula

Basel II needed a formula with a specific, unusual property: portfolio invariance. The capital charge for a loan had to depend only on that loan’s own characteristics — not on what else is in the portfolio — or the rule would be unworkable across thousands of banks. Gordy (2003) proved essentially only one model has that property: the Asymptotic Single Risk Factor framework — Vasicek’s model, taken as regulation. One factor, infinite granularity; then each loan’s contribution to the portfolio’s 99.9% loss is just its own conditional expected loss in the 99.9th-percentile bad year:

Read it left to right: the loss rate in a 1-in-1,000 year, minus expected loss. EL is subtracted because — Module 1’s oldest lesson — expected loss is priced into the loan; capital exists for the unexpected part. Multiply $K$ by 12.5 (the reciprocal of the 8% minimum) and by EAD, and you have the risk-weighted assets from Module 9. The IRB inputs — PD, LGD, EAD — are the bank’s own (Module 2 scorecards, Module 3 ratings); the formula and the correlation are the regulator’s.

Where does the regulator get ρ? Prescribed by asset class: corporates get a PD-dependent 12–24% (lower-PD firms are assumed more systematic — big safe firms fail with the economy, small risky ones fail on their own), residential mortgages a flat 15%, qualifying revolving retail 4%, with a size adjustment for SMEs. A maturity adjustment scales corporate charges up for longer loans (more time for downgrade risk — a Module 8 effect smuggled into a one-period model). The downloadable workbook implements every piece with live formulas, including the exact regulatory ρ(PD) and maturity functions, on a worked loan portfolio.

Where this connects

• The latent variable $X_i$ is Module 6’s Merton model with the balance sheet abstracted away; the threshold trick for migration (not just default) is Module 8’s CreditMetrics.
• The conditional-independence architecture — correlate through a common factor, then defaults are independent given the factor — is the single most reused idea in portfolio credit. You will see it again immediately in Module 11, where the Gaussian factor becomes a Gaussian copula and its thin tails become the crisis.
• The 99.9% confidence level, the RWA machinery, and the output floor politics all live in Module 9.

Try it yourself

The Excel workbook below builds the whole module in auditable formulas: a Vasicek sheet (conditional PD given the economy, the closed-form loss CDF, VaR at any confidence), and a Basel IRB sheet with the exact corporate correlation function ρ(PD), the maturity adjustment b(PD), per-loan capital K, and RWA for a worked eight-loan portfolio. Change one loan’s PD and watch its capital charge move; change the confidence from 99.9% to 99% and see how much of the requirement is pure tail.

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