# Roll Rates

## Contents

## Definition

**Roll Rates** help quantify the Delinquency and Default behaviour of credit portfolios with large number of borrowers. The name suggests the *rolling* (transfer) of borrowers from one *state* of delinquency to another.

Mathematically the computation of roll rates is related to the estimation of a Transition Matrix, more specifically the transition rates between the various states of the adopted State Space. The size n of this square (n x n) matrix corresponds to the number of distinct Past Due days that are selected as informative (each on of them considered to be one distinct state).

## Usage

*Roll Rate Analysis* is used in a variety of Credit Portfolio Management tasks, in particular in Consumer Finance:

- Providing relevant Credit Portfolio Information
- Estimating a roll rate matrix constitutes a simple type of a credit Risk Model in that it allows projecting likely outcomes over the future periods. Combined with further metrics (Loss given default), roll rates can form the basis for Expected Credit Loss calculations as required under IFRS 9 and CECL accounting
- Helping with the development of credit scorecards and Credit Scoring Models (by identifying which the type of delinquency that is adequately correlated with eventual default]]
- In the context of managing seriously delinquent accounts (Non-Performing Loan) the roll rates of delinquent accounts into performing states are denoted as cure rates

## The Roll Rate Matrix

The collection of all relevant roll rates is called a *Roll Rate Matrix*. The matrix is defined by

- the period (observation time internal) that is used
- an enumeration of states that capture all possible states at the beginning and the end of the time interval
- a set of transition probabilities (roll rates) from any state to any other state during that time interval

### The Period

A 30-Day (monthly) observation period is typical. It reflects primarily the fact that Contractual Cash Flows for retail financial products typically have monthly repayments schedules (hence this is the period over which new information is produced)

### The List of States

The collection of states will include (depending on the nature of the borrowers and credit products) the union of the following sets:

- An enumeration of distinct Current states (if there are more than one)
- An ordered list of Delinquent states. While any granularity is possible in principle, it is quite typical that the shortest period is 30 days past due, whereas the longest can be 90 days past due or 180 days past due.
- An enumeration of distinct Default states (including such possibilities as Forbearance, Foreclosure etc)

### The Graph of Possible Transitions

The graph of possible transitions lays out possible paths between states. An Absorbing Default State will not have paths leading back to performing status

Dividing the current month's delinquency bucket by the prior delinquency bucket, calculates the month's roll rates in the previous month.

## Issues and Challenges

- When managing portfolios with significant Non-Performing Loan segments the Roll Rate Matrix must capture accurately the NPL Life Cycle
- The underlying assumption of Roll Rate Analysis is that future accounts will continue to flow through delinquent buckets as they have in the past. Changes in the economic environment or other possible risk factors affecting a given portfolio may introduce significant dynamics (variability) in the roll rates. Various forms of Stress Testing and the use of Conditional Transition Matrix are useful to mitigate this weakness (and hidden Model Assumptions)
- Roll Rate Analysis constructs a
*summary*Risk Metric. It must be complemented with further tools e.g. vintage analysis to reveal other dimensions of the evolution of a credit system

## See Also

- For credit portfolios that are managed using a Credit Rating System (more common for Corporate Loan or Bond portfolios) one can also estimate the Rating Migration Matrix. See Roll Rate versus Rating Migration Matrices for a comparison.
- How to Calculate a Roll Rate Matrix