Through the cycle vs point in time PD

Ask two credit teams for the probability that the same company defaults next year and you can get two different answers, both correct. One is point in time, the other through the cycle. The difference is not a disagreement. It is a different question.

Two questions, one obligor

Point in time (PIT) PD answers: given today’s economy, how likely is this obligor to default over the next twelve months? It moves with the cycle, rising into recessions and falling in recoveries.

Through the cycle (TTC) PD answers: averaged over a full credit cycle, good years and bad, how likely is a default in a year? It describes the obligor’s underlying quality and barely moves when the economy does.

Each has its home. IFRS 9 and CECL provisioning, short-tenor pricing and early warning want PIT. Basel IRB capital, rating-agency style grades and long-term risk appetite want TTC, because capital should not collapse in good years and spike in bad ones. Most real rating systems are hybrids somewhere in between, which is exactly why the bridge between the two matters.

The bridge: a one-factor model

The standard bridge is the Vasicek one-factor model, the same model behind the Basel IRB formula. Each obligor’s asset return mixes a systematic factor shared by everyone, the state of the economy, with an idiosyncratic shock of its own:

A_i = \sqrt{\rho}\,Z + \sqrt{1-\rho}\,\varepsilon_i, \qquad Z,\ \varepsilon_i \sim N(0,1)\ \text{independent}

The obligor defaults when $A_i$ falls below a threshold set by its TTC PD, $\Phi^{-1}(\text{PD}_{\text{TTC}})$ , where $\Phi$ is the standard normal distribution function. Condition on the state of the economy $Z$ and you get the PIT PD:

\text{PD}_{\text{PIT}}(Z) = \Phi\!\left(\frac{\Phi^{-1}(\text{PD}_{\text{TTC}}) - \sqrt{\rho}\,Z}{\sqrt{1-\rho}}\right)

Positive $Z$ is a good year and pushes the PIT PD down; negative $Z$ is a bad year and pushes it up. Average over every possible state of the economy and you recover the TTC PD exactly:

\mathbb{E}_Z\big[\text{PD}_{\text{PIT}}(Z)\big] = \text{PD}_{\text{TTC}}

The asset correlation $\rho$ controls how cyclical the segment is. Basel sets it between 12% and 24% for corporates; trade credit books of small, cyclical buyers can sit at the top of that range or above.

What it looks like in numbers

Take an obligor with a TTC PD of 1% and $\rho = 15\%$ :

State of the economy	$Z$	PIT PD
Strong year	+1.0	0.16%
Median year	0.0	0.58%
Mild downturn	−1.0	1.77%
Recession, about 1 year in 15	−1.5	2.92%
Severe, 1 year in 100	−2.33	6.12%

Two things stand out. The PIT PD ranges across more than an order of magnitude over the cycle. And in a median year it sits well below the TTC average, at 0.58%, because the distribution is skewed: a few bad years carry most of the defaults. A book priced on PIT PDs from a run of median years is underpriced for the cycle, even though every individual year looked accurate.

Calibrating it, step by step

1. Anchor the TTC PD. Use the long-run average default rate per grade, over a window that contains at least one full cycle including a genuine downturn:

\text{PD}_{\text{TTC}} \approx \overline{\text{DR}} = \frac{1}{T}\sum_{t=1}^{T} \text{DR}_t

A window of only benign years is the most common way to end up with a TTC PD that is quietly too low.

2. Estimate the asset correlation. Default rates that swing a lot from year to year imply a high $\rho$ . Fit it by maximum likelihood on the default rate series, or by matching moments: under the model, the variance of the annual default rate is

\operatorname{Var}(\text{DR}_t) = \Phi_2\!\left(\Phi^{-1}(p),\, \Phi^{-1}(p);\, \rho\right) - p^2

where $\Phi_2$ is the bivariate normal distribution function and $p$ is the TTC PD. Short histories make $\rho$ noisy, so sense-check it against regulatory and industry values.

3. Back out the cycle, year by year. Invert the PIT formula to read the state of the economy from each year’s observed default rate:

\hat{Z}_t = \frac{\Phi^{-1}(\text{PD}_{\text{TTC}}) - \sqrt{1-\rho}\;\Phi^{-1}(\text{DR}_t)}{\sqrt{\rho}}

With a 1% TTC PD and $\rho = 15\%$ , an observed default rate of 0.6% maps to $\hat{Z} \approx 0$ , a median year, while 2.5% maps to $\hat{Z} \approx -1.34$ , a recession.

4. Forecast the next $Z$ . Regress $\hat{Z}_t$ on leading indicators, such as GDP growth, unemployment, insolvency trends and bank lending standards, and project it forward:

Z_{t+1} = \beta_0 + \boldsymbol{\beta}^{\top}\mathbf{x}_t + e_{t+1}

For IFRS 9, run this under several probability-weighted macro scenarios rather than a single forecast.

5. Apply it to every grade. Push each grade’s TTC PD through the PIT formula with the forecast $Z$ . One number moves the whole book consistently.

A pragmatic alternative: the logit shift

When there is too little default history to trust $\rho$ , a common shortcut is to shift every grade by the same amount in log-odds space:

\operatorname{logit}\big(\text{PD}^{g}_{\text{PIT}}\big) = \operatorname{logit}\big(\text{PD}^{g}_{\text{TTC}}\big) + \delta_t, \qquad \operatorname{logit}(p) = \ln\frac{p}{1-p}

Solve for $\delta_t$ so that the exposure-weighted PIT PD of the portfolio matches your central tendency forecast. It preserves rank ordering and is easy to explain. Its weakness is that it moves every grade by roughly the same proportion, whereas the one-factor model, correctly, makes strong grades more sensitive to the cycle in relative terms than weak ones.

How to tell if you got it right

A good PIT PD tracks realised default rates year by year: back-test each grade with binomial or Jeffreys tests. A good TTC PD stays stable: grade migration should be modest, and default rates should average out to it over the cycle. If your “TTC” grades migrate heavily in a downturn, the system is more point in time than you think.

And avoid the classic double count: feeding PIT PDs into a capital model that then applies a stressed $Z$ on top. Stress a TTC input, or condition a PIT input, but never both.

Banu Munnangi is SVP, Credit Reinsurance AI at Marsh Re, where he leads data and AI for the Credit, Bond and Political Risk specialty. More writing · Expertise

Views are my own and do not represent my employer. Examples use illustrative figures, not client or employer data.

Through the cycle vs point in time PD: the maths, the calibration, and when to use which