October 24, 2021

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Consumer installment loans appear in many forms, including residential mortgages, automobile loans, student loans, and online unsecured consumer loans. These loans are aggregated and pooled by lenders, who finance their lending activity by selling the loans either directly to a strategic or financial counterparty via a whole loan sale, or by selling the loan pool to a bankruptcy remote trust, who funds the loan purchase by selling debt securities backed by the loan cashflows (mortgage backed securities “MBS” or asset backed securities “ABS”). Unlike corporate debt securities, where principal and interest payments typically represent direct obligations of the corporate issuer, MBS or ABS principal and interest payments generally depend solely on the repayment activity and cashflows from the underlying loan pool. As a result, investors in either whole loan or security formats must carefully forecast the scheduled payments, prepayments, delinquencies, and defaults for the loan pool, over time.

Typically, investors may specify a conditional repayment rate (CRR) and conditional default rate (CDR), which measure the percentage of a pool’s remaining principal balance that is prepaid or defaults in a given year. These annual measures are de-compounded to calculate single month mortality (SMM), which is the analogous prepay or default percentage for a given month (as a consequence of consumer loans typically being monthly-pay obligations). These are generally top-down measures, meaning that historical CRR and CDR performance data is used to inform the CRR and CDR values used to forecast future performance. This approach is relatively quick and easy to implement based on readily reported CRR and CDR data, and is easily adapted to potential differences in go-forward performance. However, this approach generally overlooks nuances in the fundamental drivers or ultimate performance, namely new delinquencies (“DQs”) and the “roll” of those new DQs through progressively more overdue buckets until default.

A potential alternative forecasting method is to utilize a Markov Chain to simulate the network of payment and delinquency statuses over a loan’s life (or a pool of loans). A Markov Chain is used to represent a network of states (nodes) connected by probabilistic transitions. The network’s connections (edges) specify the probability of the transition from one state to another for each network cycle, and by iterating through some representative number of cycles, we can obtain the network state (loan status distribution) through the pool’s life.

Table 1: State transition probabilities

In this simplified model, each node on the network graph represents a discrete loan payment status. Each connection between nodes (an edge) represents a possible transition between states during a monthly payment period. During each monthly cycle, a dollar of loan principal balance may:

• if performing:
  • remain performing if principal was not due
  • make a scheduled principal payment when due
  • make an early principal payment (prepayment)
  • become delinquent and enter the lowest bucket of the DQ process
• if delinquent:
  • move one node toward performing by making a late payment
  • remain at the same DQ node by making only the currently due payment
  • move one node toward default by not making the currently due payment
  • default, possible only for loans at the 120+ days late node

Table 1: State transition probabilities

Note that in this simplified view, only these transitions are defined. Certain nuances like significantly DQ loans (60+ days past due) may not become performing in one cycle like they may in reality, however unlikely. Instead, the loan must transition to progressively less severe DQ buckets before becoming current or performing again. Similarly, a DQ loan must progress through the full DQ pipeline before defaulting. Implementing the full measure of nuance and complexity is certainly feasible; this would just require adding additional edges between nodes and defining additional transition probabilities.

The transition table above must also take into account the shifting proportion of aggregate performing principal balance that is paid on schedule each month. Consumer loans are typically structured with a fixed monthly payment that fully amortizes the loan principal balance over the specified payment term. As a result, a relatively higher proportion of the monthly payment is allocated to interest early in the loan’s paydown. As the loan balance decreases, there is a lower dollar amount of monthly interest and more of the fixed payment amount is allocable to pay down principal. Given an initial balance, $b$, annual simple interest rate (APR), $r$, and loan maturity term in months, $n$, the fixed monthly payment, $p$, can be calculated as follows:

$$ p=b\frac{\frac{r}{12}(1+\frac{r}{12})^n}{(1+\frac{r}{12})^{n-1}} $$

Then for each period within some range (generally advisable to take the loan term and add a cushion for a late payment and delinquency workout tail) the scheduled interest paid is just the previous loan balance times the monthly interest rate, $\frac{r}{12}$, the scheduled principal paid is the fixed payment, $p$, minus the interest payment. You can then calculate the percentage of the previous loan balance that makes this transition in the $k^{th}$ month, $s_k$, by dividing the scheduled principal payment in month $k$ by the principal balance in month $k−1$. Sample payment curves are included below, each for a hypothetical $100 loan with a stated maturity term of 60 months, and either 5% APR or 20% APR. The y-axis can be interpreted as the percentage of the initial principal balance that is allocated to a given month’s principal, interest, or total payment.

Sample principal and interest allocations for 5% and 20% APR 60-month loans

Sample loan balance paydown path for 5% and 20% APR 60-month loans