The Vasicek Single-Factor Loan Loss Distribution

A bank holding one loan either loses money or doesn't — a coin flip weighted by the borrower's credit quality. A bank holding ten thousand loans faces a genuinely different question: not "will this loan default" but "what fraction of the whole book will default, and how bad can that fraction get in a bad year." If loans defaulted completely independently of each other, the law of large numbers would flatten the portfolio's loss rate to something nearly certain, like the fraction of heads in ten thousand coin flips. Real loan books don't behave that tidily, because a recession pushes everyone's odds up together. The Vasicek model is the standard way to capture exactly that shared exposure in one formula simple enough to compute on a calculator, and it is the engine inside Basel bank capital rules.

Building the model, one piece at a time

Picture each borrower's financial health as an invisible score that drifts up in good times and down in bad times — call it the borrower's asset value, $A_i$. The borrower defaults if that score falls below a threshold $c$ representing their debts. Vasicek's trick is to split each borrower's asset value into two pieces: a systematic factor $Z$, one single number shared by everyone in the portfolio representing the state of the economy, and an idiosyncratic factor $\varepsilon_i$, noise unique to that borrower:

$$A_i = \sqrt{\rho}\, Z + \sqrt{1-\rho}\, \varepsilon_i.$$

In words: every borrower's fortune is a weighted blend of one shared economic wave $Z$ and their own private noise $\varepsilon_i$, both standard normal and independent of each other. The parameter $\rho$, the asset correlation, controls the blend — $\rho$ near 0 means borrowers are nearly independent, $\rho$ near 1 means they rise and fall as one.

Given a single portfolio-wide default probability $p$ (the average odds any one borrower defaults) and this correlation $\rho$, Vasicek showed that the fraction of the portfolio that defaults, conditional on the economy landing at a particular value of $Z$, has a closed form — and integrating over all possible values of $Z$ gives the full distribution of the portfolio's default rate:

$$P(X \le x) = \Phi\left(\frac{\sqrt{1-\rho}\,\Phi^{-1}(x) - \Phi^{-1}(p)}{\sqrt{\rho}}\right).$$

In words: this tells you the probability that the portfolio's default rate $X$ stays below any level $x$ you pick, using only two inputs — the average default probability $p$ and the correlation $\rho$ — and the standard normal cdf $\Phi$ and its inverse $\Phi^{-1}$.

Two worked examples

Example 1 — reading off a percentile. Take a portfolio with average default probability $p = 2\%$ and asset correlation $\rho = 0.15$, typical for a diversified corporate loan book. What default rate is exceeded only 0.1% of the time — the 99.9th percentile used in regulatory capital? Plug $x$ such that $\Phi^{-1}(x) = \Phi^{-1}(0.999) \approx 3.09$ into a rearranged form: $x = \Phi\left(\frac{\Phi^{-1}(p) + \sqrt{\rho}\cdot 3.09}{\sqrt{1-\rho}}\right)$. With $\Phi^{-1}(0.02) \approx -2.05$: numerator $= -2.05 + \sqrt{0.15}\times 3.09 \approx -2.05 + 1.20 = -0.85$, divided by $\sqrt{0.85} \approx 0.92$, giving $-0.92$. Then $x = \Phi(-0.92) \approx 17.9\%$. So even though the average default rate is only 2%, a genuinely bad year (one occurring roughly once every thousand years, in the model's own terms) could see nearly 18% of the book default — a direct consequence of correlation, not of any single loan being riskier than assumed.

Example 2 — the effect of correlation alone. Keep $p = 2\%$ fixed and drop $\rho$ to 0.05, a much more diversified book (unrelated industries, unrelated geographies). Repeating the same calculation: numerator $= -2.05 + \sqrt{0.05}\times 3.09 \approx -2.05 + 0.69 = -1.36$, divided by $\sqrt{0.95}\approx 0.97$, giving $-1.40$, so $x = \Phi(-1.40) \approx 8.1\%$. Halving correlation more than halved the tail loss rate, from 17.9% to 8.1%, with the average default probability never changing. Correlation, not average credit quality, is what drives the fat tail.

Where this shows up

Basel's Internal Ratings-Based capital formula is this model, with $\rho$ set by regulatory lookup tables (lower for retail mortgages, higher for large corporates) rather than estimated freely, so that capital scales directly with each portfolio's tail default rate rather than its average. Anyone building an internal credit-VaR model runs into the same formula, usually extended to multiple correlated systematic factors rather than one.