Aggregating Loan-Level SMM into Cohort Prepayment Rates

Summary

The document shows how to calculate a cohort-level single monthly mortality rate (SMM) from loan-level data. The cohort SMM is the total prepaid monthly balance divided by the total scheduled balance across the loans in that cohort. Equivalently, it is a weighted average of individual loan SMMs, with each loan’s scheduled balance providing its weight. This gives larger loans proportionate influence on the aggregate rate.

The answer demonstrates the equivalence algebraically by expressing the cohort ratio as a sum of loan-level SMMs multiplied by scheduled-balance weights. The question also gives the standard conversion from monthly SMM to conditional prepayment rate (CPR), but the explanation focuses on aggregation of SMM rather than deriving that annualization formula. The method assumes the supplied loan balances and prepayment amounts are defined consistently for the period; it does not discuss data-quality issues, cohort formation, or adjustments for servicing and loan-level conventions.

Key ideas

• Cohort SMM equals aggregate prepaid balance divided by aggregate scheduled balance.
• The cohort SMM is a weighted average of individual loan SMMs.
• Scheduled loan balance is the weight applied to each loan’s SMM.
• The annual CPR conversion is separate from the balance-weighted aggregation step.

Tags

# How to calculate aggregated conditional prepayment rates


# How to calculate aggregated conditional prepayment rates












I understand that the formula for loan-level CPR is:

$$ CPR = 1 - ( 1 - SMM)^{12} $$ where $$ SMM = \frac{Prepaid monthly balance}{Scheduled Blance} $$

But Im not too sure how the formula changes when aggregating the loan-level data to a cohort-level. Does the formula for $SMM$ then become as follows, where the sum is calculated across all the loans that fall into that cohort:

$$ SMM = \frac{sum_{Prepaid monthly balance}}{sum_{Scheduled Blance}} $$

And if this second equation for $SMM$ is correct, how does the $SMM$ equation go from 1 to 2?

## Answer by Sharad (score 1, accepted)

https://quant.stackexchange.com/a/58655

The aggregation formula for SMM is correct. The SMM for the cohort can be defined to the weighted-average SMMs of the individual loans (in the cohort), where the weights are the scheduled balances of the individual loans. A little bit of algebra shows that this is the same as your expression for the SMM of the cohort:

\begin{align*} \mbox{SMM}_{\mbox{Port}} &= \frac{\sum_i (\mbox{sched}_i - \mbox{curr}_i)}{\sum_i \mbox{sched}_i} \\ &= \sum_i \frac{1}{\sum_i \mbox{sched}_i} \frac{\mbox{sched}_i}{\mbox{sched}_i}(\mbox{sched}_i - \mbox{curr}_i) \\ &= \sum_i \frac{\mbox{sched}_i}{\sum_i \mbox{sched}_i} \mbox{SMM}_i \end{align*}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.