How an Auto-Loan Portfolio Moves Through Risk States
The Cleveland Fed auto-loan model treats repayment as movement through several states, not as a switch that flips directly from “safe” to “default.” The important question here is how those states connect and how a model turns the connections into a multi-period forecast. Once that structure is clear, a changing unemployment coefficient has a visible place in the system: it changes some transition probabilities, which changes the population exposed to later transitions, which changes the cumulative forecast.
Four states, with different meanings and consequences
A loan is current when its payment is less than 30 days overdue. It is delinquent when a payment is 30 to 89 days late. That middle state is not a cosmetic category. A delinquent borrower may cure and return to current, remain delinquent, worsen into default, or leave the observed sample without a recorded default. Each possibility carries a different implication for the next period’s risk.
The model defines default broadly: at least 90 days delinquent, repossessed, bankrupt, or charged off. Non-default exit covers loans that leave the observed portfolio without being classified as default. The paper gives examples including full payment at term, prepayment or refinancing, borrower death, and sale by the lender. The reason for an exit is usually not observed in the data. A loan that disappears from the sample is therefore not automatically evidence of a successful payoff, even though it is treated as a non-default exit when no default is recorded.
Default and non-default exit are absorbing states in the model. Once a loan enters either state, it no longer returns to current or delinquent in the transition system. That is a modeling choice with an intuitive forecasting purpose: a charged-off or otherwise defaulted loan is no longer part of the performing portfolio, and a loan that has left without observed default is no longer available for later observations. The two absorbing states should still be kept conceptually separate. Default represents the loss-related event the model is trying to forecast; non-default exit can represent several outcomes with different economic meanings.
The resulting system is easier to understand as a set of possible paths than as a single default rate. A current loan can stay current, become delinquent, default, or exit. A delinquent loan can return to current, remain delinquent, default, or exit. A cure matters because it moves the loan back into a different risk environment. A prepayment matters because it removes the loan from future default exposure. A direct default matters because it removes the loan through the loss-bearing absorbing state. The model needs all of these routes to keep the future at-risk population aligned with the observed state history.
The diagram also shows why the state definition creates a measurement boundary. The data are observed at points in time, not continuously. If a loan becomes delinquent and then leaves between observations, the record may not reveal whether it cured, paid off, was sold, or defaulted before disappearing. The paper follows a procedure used in Federal Reserve stress testing to make assumptions about some such exits, using the prior loan state and other observed characteristics. That procedure makes forecasting possible, but it does not turn an unobserved event into a directly measured one.
The state machine also says what information persists and what is lost. A loan that remains current retains its place in the performing population and can face future transitions. A loan that becomes delinquent carries forward the fact that it is in a more exposed starting state; if it cures, the next transition is evaluated from current rather than delinquent. A defaulted loan no longer contributes to future performing-state transitions. An exited loan no longer contributes either, even though the reason for exit may be economically important but unavailable. These are not merely bookkeeping decisions. They determine which loans receive a future probability and which are removed from the calculation.
From relative risk to a cumulative forecast
The model estimates the transition probabilities from current and delinquent starting states with separate multinomial logistic regressions. For each starting state, the possible destinations are modeled together. The probability of moving to a destination is compared with the probability of staying in the same starting state, while borrower characteristics, loan features, lender type, and macroeconomic variables are held at their specified values.
That comparison is called a relative risk in the paper. If a coefficient’s exponentiated value is greater than one, the destination becomes more likely relative to staying in the starting state when the associated input increases, all else equal. A value below one means the destination becomes less likely relative to remaining in place. The comparison is not itself a probability. It is a ratio whose translation into a probability depends on the other possible destinations and on the starting state’s baseline probabilities.
This distinction matters for reading the unemployment result. Saying that a one-unit change in an unemployment variable raises the current-to-default relative risk by a given percentage does not mean that every current loan’s default probability rises by that many percentage points. A current loan may also remain current, become delinquent, or exit. The multinomial model allocates probability across those destinations, so changing one relative risk can alter the final probability only after the competing transitions are considered. The same relative-risk movement can therefore have different probability effects for portfolios with different starting states, borrower mixes, or baseline transition rates.
The model’s joint structure imposes two useful constraints. Every predicted transition probability must lie between zero and one, and the probabilities from a given starting state must sum to one. This prevents the forecast from assigning independent probabilities that are individually plausible but collectively impossible. It also forces a change in one route to be interpreted alongside the alternatives. If the estimated tendency to move from current to default changes, the forecast still has to decide how the full set of current-state destinations is redistributed.
A simple thought experiment shows why that matters. Suppose two otherwise similar loans begin current, but one model assigns separate risks to delinquency, default, and exit without accounting for the fact that those outcomes compete for the same loan in the same period. The three numbers might look reasonable in isolation, yet their total could leave too little or too much probability for remaining current. The multinomial setup treats the destinations as a common allocation. It does not tell us that the destinations have the same causes; it ensures that the forecast describes one loan’s next observed state rather than several incompatible events occurring independently.
The same allocation principle clarifies what it means for an unemployment coefficient to weaken. A smaller relative risk for current-to-default does not automatically mean that the missing probability becomes “no risk.” It may be reflected in staying current, moving first into delinquency, or exiting without default, depending on the other estimated relationships. The economic interpretation must follow the entire row of the transition matrix. This is especially important when the data show unusual movement in a competing route: a model can miss default partly because it overstates another destination, or it can miss the future at-risk population by leaving too many exited loans in the calculation.
For a one-period forecast, the process is straightforward in principle. Start with the observed state of each loan, apply the relevant transition probabilities, and aggregate the predicted destinations. A two-year forecast repeats the operation. The first transition produces a new distribution across current, delinquent, default, and exit. The second transition applies the current-state probabilities to the loans still current and the delinquent-state probabilities to those now delinquent. Loans already in either absorbing state stop participating in later performing-state transitions.
The portfolio forecast is therefore a moving distribution, not a fixed list of loans to which the same default probability is applied twice. If the first period sends more loans into delinquency, the second period inherits a larger group governed by delinquent-state relationships. If the first period sends more loans into exit, the second period has fewer loans available to default. If a stress scenario changes both unemployment and another input, the effect is not necessarily additive in the final cumulative rate because the transition probabilities are recomputed jointly. The matrix preserves the accounting of the paths, while the parameter estimates determine how much probability flows along each path.
This recursive logic is why a one-period coefficient can matter beyond one period. Suppose an unemployment shock increases the modeled chance that current loans become delinquent. The immediate consequence is not only a possible increase in current-to-delinquent movement. It also creates a larger delinquent population for the next forecast step. That population may cure, remain delinquent, default, or exit according to a different row of the transition system. Conversely, if more current loans exit without default, fewer loans remain exposed to future default transitions. The cumulative result depends on the whole path, not on a single coefficient read in isolation.
The formal forecast combines an initial distribution of states with a sequence of transition matrices. Each matrix depends on loan and borrower characteristics and on the macroeconomic conditions assigned to that period. Some characteristics change deterministically: loan age and borrower age advance as time passes. Other characteristics, such as outstanding balance and mortgage status in the paper’s forecast exercise, are held constant at their latest observed values because their future paths are not modeled in the same way. The macroeconomic path is treated as known for the diagnostic comparison.
That last assumption is useful for isolating model behavior, but it is not how an ordinary ex ante forecast operates. If the future unemployment path itself must be forecast, uncertainty about that path is added to uncertainty about the transition relationship. The paper’s cumulative exercise asks a narrower question: given the realized path and stated assumptions about other inputs, what would different parameter vintages have predicted? It is a way to expose parameter instability, not a claim that a lender could have known the future macro path in advance.
The Markov label adds another boundary. The model conditions the next transition on the loan’s current state and observed covariates, rather than on the full sequence of earlier states. That conditional simplification is useful because it keeps the transition system estimable and interpretable. But it does not demonstrate that the full history is irrelevant. A borrower who has recently cured, a loan that has experienced repeated delinquency, or a loan with unobserved servicing changes may carry information not captured by the current state alone. The model’s results are therefore statements about the relationships represented by this conditional specification.
The same caution applies to non-default exits. Treating exit as absorbing prevents a vanished loan from continuing to accumulate predicted default risk, which is preferable to pretending that an unobserved loan remains in the portfolio. Yet if some exits are actually defaults that occurred between observations, the state transition is misclassified. The authors specifically flag this possibility when interpreting unusual delinquent-to-exit estimates. A model can be more realistic than a default-only calculation while still inheriting uncertainty from the way the data record events.
This architecture gives the unemployment relationship a precise role. It is one input into particular transitions, not a direct command that the portfolio’s final loss rate must rise by a fixed amount. When that input is estimated from a different economic regime, the state machine can propagate the mismatch through delinquency, cures, exits, and absorbing defaults. Understanding that propagation is essential before asking whether the coefficient itself stayed stable. The next issue is the portfolio feeding the machine: borrowers, lender channels, and loan characteristics do not enter as decoration. They determine the baseline against which macroeconomic sensitivity is measured.