CRL, I completely understand your question, and you are on the right path by asking it. I generally view these situations as a denial, and would recommend sending an AA notice as the conservative approach. Now, the way you worded your original question, I could see where an AA notice may not technically be required (given more specifics we don't have), but I personally wouldn't want to have this discussion/argument with an examiner and would just send the AA notice to be done with it. Here is what I mean: If you hadn't told the applicant that they needed a certain LTV (which we don't know what was said to the applicant) and they had no way of knowing you might/were going to deny the loan, then I could see it be considered withdrawn and not require an AA notice. HOWEVER, what usually happens is the loan officer tells the applicant they need a certain value and when the value comes in low, the applicant assumes the loan can't be done so they back out or "withdraw", but this isn't a really a "withdraw" for Reg B purposes as your loan officer has essentially told them you won't do the loan (you are denying them) because the value came in low.
So for your specific question, I would do one of two things. First, I would just send the AA notice and be done with it. Alternatively, (and you may be auditing this after the fact), I would want clarification from the loan officer as to what was actually said. If they borrower was lead in any way to believe the bank wouldn't do the loan due to the low value, then I would consider it a denial. If they weren't lead to believe the bank wouldn't do the loan, then you might be okay - but I would caution that this should be an exception to the rule and not a regular occurrence or you could find your risk of examiner/auditor scrutiny increasing.
Again, I think you are on the right track with your question. I've personally seen instances in the past where LOs were steering applicants to "withdraw" when an appraisal comes in low, but examiners ended up citing a systemic violation saying that by steering the applicant, the loan was actually denied (since the LO was basically telling the applicant the loan can't be done as applied for).
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Adam Witmer, CRCM
All statements are my opinion, not those of my employer, and should not be taken as legal advice.
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