Could the Mortgage Companies do this? 
Instead of foreclosing on a property, what if they re-write the loan in the following way:
1) keep the principle amount the same.
2) lower the rate to today's lower rate.
3) and amortize for 40 years.

Why would they?
1)A) principle of the note stays the same for the future payoff.
1)B) Keeping the principle the same means that the borrower needs to stay in the property until the market re-appreciates to the value of the principle to avoid selling at a loss. 
This means more people will stay put, and at the same time, be able to pay for their house.
2) and 3) will make the payments easier to afford.

More Why would they?
Most loans are paid off by sale or refinance within the first 7 years of the loan. 
By making the payments affordable, but keeping the principle amount as is, the loans will perform longer and mitigate much of the loss.

Good for the owner of the property (of course):
$100,000 @ 10% = $877/month on a 30 year amort. 
$100,000 @ 6% = $550/month on a 40 year amort.
This allows a reduction of $327/month in payments, but keeps the principal to be paid, and extends the life of the debt.

Better than foreclosure for the Investor?
The future values are dramatically lower for the second loan, but ONLY if they go to final payment. 
In reality, they will both be paid off by the seventh year based on average ownership statistics.
That would mean a "loss" of  about $45,000* on paper versus an actual loss in the event of an actual foreclosure.  *(the 7 year FV of the 10% note - FV 6% note = $194,871.71-$150,363.03 = +-$45,000)

A foreclosure loss may include:
- 12-24 months of no payment ($877 x 24 = -$21,000)
- legal fees (-$4000)
- broker fees (up to -$6000)
- loss of value of this property and other properties in the neighborhood (-$15,000 min)
- REO staff increases (-$10,000)
$21 + 4 + 6 + 15 + 10 = more than $56,000 in actual loss over 2 years

By re-writing the note, the Investor reduces their loss:
The "new note" performs over the next 84 months instead of costing actual money over the next 24 months.
84 x $550 = $46,200 that the note returns over the next 24 months as it performs versus $0 during 24 month foreclosure.
$46,200 - 45,000 = $1,200 positive  OR  Straight $56,000 loss over the next 2 years.  

Other useful benefits include:
Property values do not decline as dramatically as when many foreclosures take over a market.
The loss numbers get much larger much faster when the face loan value increases. 

Could they do this if they wanted to? Legally?  Some have said there is no incentive for a bank to do this. I thought banks used math?  What do you think? Would it work? Is it good for them? Is it good for real estate?