> ## Documentation Index
> Fetch the complete documentation index at: https://docs.creditbenchmark.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Aggregate Calculation Methodology

> Portfolio PD aggregates are calculated using a back-calculation approach anchored to the most recent month and validated by the Opinion Change Indicator.

The Aggregate PD is a portfolio-level metric reflecting average credit risk across a defined entity set.

The goal is to produce an index on a **moving pool of entities** that captures **genuine bank opinion changes** — filtering out moves caused purely by entities entering or leaving the universe.

It is built using a **back-calculation approach** — anchoring to the most recent month as a baseline, then working backwards through time using only month-on-month PD changes validated by the [Opinion Change Indicator (OCI)](/methodology/consensus-calculations/opinion-change-indicator).

This ensures the time series stays consistent with the current portfolio view and filters out noise from contributor population shifts.

<Tip>
  The back-calculation approach means the most recent month always reflects the full current entity universe. Historical values are derived from that baseline — so the series never distorts when entities join or leave.
</Tip>

## Mathematical Process

<Steps>
  <Step title="Logarithmic Transformation">
    Transform the PD for each entity $i$ at time $t$ into log space:

    $$
    \operatorname{LogPD}_{i,t}=\log\!\bigl(\operatorname{PD}_{i,t}\bigr)
    $$
  </Step>

  <Step title="Difference in Log PD">
    Compute the month-on-month change in log PD for each entity:

    $$
    \Delta\operatorname{LogPD}_{i,t} = \operatorname{LogPD}_{i,t}-\operatorname{LogPD}_{i,t-1}
    $$
  </Step>

  <Step title="OCI Adjustment">
    Retain only changes where the OCI confirms a genuine shift in bank opinion. If the sign of the OCI does not match the sign of the log PD change, set the difference to zero:

    $$
    \operatorname{Adj}\Delta\operatorname{LogPD}_{i,t}=
    \begin{cases}
      \Delta\operatorname{LogPD}_{i,t},
        & \text{if }\mathrm{sign}(\mathrm{OCI}_{i,t})
          = \mathrm{sign}\!\bigl(\Delta\operatorname{LogPD}_{i,t}\bigr)\\[4pt]
      0, & \text{otherwise}
    \end{cases}
    $$
  </Step>

  <Step title="Average Log Difference">
    Average the adjusted log differences across all $n$ entities for month $t$:

    $$
    \overline{\Delta\operatorname{LogPD}}_{t} = \frac{1}{n}\sum_{i=1}^{n}\operatorname{Adj}\Delta\operatorname{LogPD}_{i,t}
    $$
  </Step>

  <Step title="Baseline — Most Recent Month">
    Establish the baseline by averaging raw log PDs across all entities at $t_{\text{latest}}$:

    $$
    \operatorname{AggLogPD}_{t_{\text{latest}}} = \frac{1}{n}\sum_{i=1}^{n}\operatorname{LogPD}_{i,t_{\text{latest}}}
    $$

    This is the starting point for all back-calculation.
  </Step>

  <Step title="Back-calculate Previous Months">
    Derive each prior month by subtracting the average log difference:

    $$
    \operatorname{AggLogPD}_{t-1} = \operatorname{AggLogPD}_{t}-\overline{\Delta\operatorname{LogPD}}_{t}
    $$

    Repeat iteratively for all months prior to $t_{\text{latest}}$.
  </Step>

  <Step title="Convert Back to PD">
    Exponentiate to return to the probability scale:

    $$
    \operatorname{AggPD}_{t}=\exp\!\bigl(\operatorname{AggLogPD}_{t}\bigr)
    $$
  </Step>
</Steps>

## Output Views

The Aggregate PD can be expressed three ways:

<AccordionGroup>
  <Accordion title="Segment PD Average">
    The absolute probability of default for the segment — the direct output of the back-calculation:

    $$
    \operatorname{AggPD}_{t} = \exp\!\bigl(\operatorname{AggLogPD}_{t}\bigr)
    $$
  </Accordion>

  <Accordion title="Segment PD Rebased">
    Relative change from a chosen base date $t_0$, showing directional risk movement:

    $$
    \text{RelChange}_t(\%) = \frac{\operatorname{AggPD}_{t}-\operatorname{AggPD}_{t_0}}{\operatorname{AggPD}_{t_0}} \times 100
    $$
  </Accordion>

  <Accordion title="Segment Average Rating">
    The aggregate PD mapped to the CB rating scale:

    $$
    \text{Rating} = f(\operatorname{AggPD}_{t})
    $$

    Where $f$ is the CB PD-to-rating mapping function; see the [Rating Scale](/methodology/rating-scales).
  </Accordion>
</AccordionGroup>