103: The number of months it would take to sell off all the foreclosed homes in banks’ possession, plus all the homes likely to end up there over the next couple years, at the current rate of sales.

How much should we worry about a new leg down in the housing market? If the number of foreclosed homes piling up at banks is any indication, there’s ample reason for concern.

As of March, banks had an inventory of about 1.1 million foreclosed homes, up 20% from a year earlier, according to estimates from LPS Applied Analytics. Another 4.8 million mortgage holders were at least 60 days behind on their payments or in the foreclosure process, meaning their homes were well on their way to the inventory pile. That “shadow inventory” was up 30% from a year earlier.

Based on the rate at which banks have been selling those foreclosed homes over the past few months, all that inventory, real and shadow, would take 103 months to unload. That’s nearly nine years. Of course, banks could pick up the pace of sales, but the added supply of distressed homes would weigh heavily on prices — and thus boost their losses.

The government is understandably worried about the situation, and its Home Affordable Modification Program has made an impact by helping people stay in their homes and avoid foreclosure. As people who enter the program catch up on their payments, the number of homeowners 60 or more days delinquent has fallen 9% over the past two months.

Now, though, the effect of modifications could be on the wane. According to Goldman Sachs, HAMP started less than 80,000 trial modifications in March, less than half the number in the peak month of October 2009. At the same time, a growing number of modifications are being canceled as borrowers prove unable to pay. By Goldman’s count, about 68,000 were canceled in March.

All this means that little can stop banks’ inventory of distressed homes from growing. Too many people owe too much more on their homes than they can afford. For the housing market, that could mean a long-lasting hangover.