SunTrust Banks has agreed to pay nearly $2 million in fines and buy back millions of dollars in auction-rate securities as part of an agreement with industry regulators.

Two SunTrust entities were involved in the agreement announced Thursday. SunTrust Robinson Humphrey will pay a $1.65 million fine, and SunTrust Investment Services will pay a $300,000 fine.

The Financial Industry Regulatory Authority, an industry self-regulatory body, said it had reached agreements in principle with SunTrust and several other companies across the country. The companies will offer to buy back a total of more than $1.8 billion in the securities.

FINRA did not break out how much each company may buy back. Several weeks ago, SunTrust announced the buyback could cost the company about $500 million.

A FINRA spokesman said an investigation revealed that the companies’ sales and marketing material underplayed the risks involved.

Auction-rate securities are a complex type of debt banks have typically sold to corporate clients and wealthy investors as a safe alternative to a cash or money-market fund. Banks set the interest rate at weekly and monthly auctions, hence the name.

However, the market for auction-rate securities collapsed in February amid the global credit crunch.

FINRA said it has about 50 more investigations open.