#### PD Methods

When it comes to CECL, segmenting your portfolio is an integral step in properly assessing risk. When calculating loss rates and allowances, how you segment your portfolio can greatly affect the outcomes. One situation where the outcomes are not necessarily affected by segmentation is when using probability of default (PD). Before you take that statement as an absolute, know that this, of course, comes with caveats. Essentially, we’re saying segmentation doesn’t matter with PD, and while that’s mostly true, it depends on how you’re looking at your portfolio. For example, let’s say you’re calculating your loss rates and allowances for your entire auto portfolio. When implementing VE’s PD model, the model returns an expected loss for each loan. If all loans have the required data to produce an expected loss, we simply sum the expected loss across loans to arrive at an unadjusted allowance. If a loss rate is desired, we can divide this sum by the remaining balance of the class to obtain the total loss rate. No matter how we segment the loans, the total expected loss and total loss rates will remain the same as long as we are taking all loans within the class into consideration. Individual segment expected losses and loss rates of course are completely dependent on the loans that define the segment, but when taking a holistic view the results do not change.

Let’s explore some examples to clear up any confusion. We’ll simplify our first example and say we currently only have five auto loans, which can be found in the table below.

In this table, each row represents a single loan. We have the direct/indirect status, the lifetime PD, the expected loss, and the remaining balance. All together, our expected losses sum to $72.50, and our loss rate is the total expected loss divided by the total remaining balance, or rather $72.50/$8,000 = 0.91%.

Let’s now segment based on status. With our direct autos (loans A and B), our expected losses sum to $21, and our loss rate is $21/$3,000 = 0.70%. With our indirect autos (loans C, D, and E), our expected losses sum to $51.50, and our loss rate is $51.50/$5,000 = 1.03%. When we segment our portfolios, our respective expected losses and loss rates change, which reflect the risk associated with each segment. However, our total expected loss ultimately ends up being $21 + $51.5 = $72.5. Regardless of how you segment, the final overall outcome will be the same. Nevertheless, this doesn’t discount the value in segmenting to assess risk across segments. The key is that segmentation doesn’t change our total expected losses or loss rates for PD as long as we’re accounting for all loans within the class.

Let’s now explore an example where segmentation can affect our outcomes for PD. In this example, we are concerned with missing data. The table below is a copy of the previous table, but now loan B is missing some key data such that PD (and subsequently expected loss) cannot be calculated.

In this situation, the total loss rate becomes $56.5/$6,000 = 0.94%; without data from loan B, our loss rate has increased by about three basis points. Because we do not have an expected loss derived from the PD calculation for loan B, we multiply loan B’s remaining balance by the total loss rate to obtain the expected loss, which is 0.0094*$2,000 = $18.83. This results in a total expected loss of $75.33, and the total loss rate will remain unchanged.

Because we are missing loan B’s PD-derived expected loss, our total loss rate is much more heavily weighted toward the indirect loans. When we apply the loss rate to loan B’s remaining balance, we are treating loan B’s expected loss behavior more similarly to indirect autos than direct autos. This is a situation where segmenting can help. If we segment based on status again, our new loss rate for direct autos is now $5/$1,000 = 0.50%. Theoretically, this loss rate should be more representative of loan B’s loss behavior. We now multiply the direct auto loss rate of 0.50% by loan B’s remaining balance to obtain an expected loss of $10. This results in a total expected loss of $66.50 and a total loss rate of $66.50/$8,000 = 0.83%, a decrease of seven basis points.

These examples illustrate that segmentation only impacts the total expected loss in the presence of missing data. If we have the proper information for all loans within the class, then segmentation will not affect our outcomes for the PD methodology. Most institutions will have issues with missing data, and this example shows how important it can be for an institution to have as much clean data as possible for CECL PD calculations. However, understanding that a “perfect” dataset is more myth than reality, we emphasize the importance of making sure the loans with missing data are properly represented within their respective classes.

#### Loss Rate Methods

Let’s now explore a non-PD example. Let’s work with static pool. Recall that static pool is a loss rate method that is calculating the portion of the current balance we do not expect to recover using a historical loss rate from a similar pool of loans. Let’s continue to work with our direct and indirect autos in the table below.

This table is a little different from the previous tables. The rows of this new table do not represent single loans. Instead, we are looking at segmentations, namely direct autos, indirect autos, and all autos, or rather, the entire class. For each segment, we also have the historical loss; the historical balance; the loss rate, which is the historical loss divided by the historical balance; the current remaining balance; and the expected loss, which is the product of the remaining balance and the loss rate.

If we do not segment, then our total expected loss is $148.8 (found in the bottom right cell), and our base loss rate is 1.24%. If we do segment, then our total expected loss is $30+$126 = $156, and our base loss rate is $156/$12,000 = 1.3%. We’ve increased our total loss rate by six basis points simply by segmenting. Thus, it’s clearly important to take care when setting up segments.

Does segmentation always result in a change in your base loss rate for non-PD methods? No, not always. If the percentage breakdown of the current remaining balances of the segments is exactly equal to the percentage breakdown of the historical balances of the segments, then the total loss rate will be unchanged. Since $1,000/$2,500 is not equal to $3,000/$12,000, and $1,500/$2,500 is not equal to $9,000/$12,000 in this example, there was a change in the total loss rate when using segmentation.

To summarize, segmenting your portfolio is an excellent way of assessing underlying portfolio risk. Defining segments requires great care, especially depending on the method you use. In the case of PD, when taking all loans into account, how we segment will not affect the total expected losses and loss rates; yet, missing data can throw a wrench into this. With other methods, such as static pool and vintage, how we segment will almost always change our total expected losses and loss rates.