First American Bank of Virginia reported fourth-quarter profits of $3.2 million ($1.73 per share) up 14 percent from the $2.8 million net income for the final three months of 1981.

For the year, the bank had net income of $12.1 million ($6.48), up less than 1 percent from the $12 million ($6.42) in profits the bank had in 1981.

Milton L. Drewer Jr., president and chief executive officer, attributed the small year-to-year increase to the rising costs of attracting deposits because of deregulation, which "coincided with a local economy weakened by a national recession, characterized by weak demand for loans and other credit services."

He said that banks will have to charge more for loans and service charges to offset the higher costs of attracting and maintaining deposits.

According to the bank's report, however, its net interest spread--essentially the difference between what it pays for deposits and takes in from loans and other similar activities--grew 8.2 percent to $47.4 million from $43.8 million. Its non-interest income, such as fees and service charges, grew more than 20 percent to $10.1 million from $8.2 million.

But the bank's non-interest expenses, such as wages and equipment, rose 16 percent from $35.9 million to $41.7 million, enough to swallow nearly all the increases in interest income and non-interest income.

The Virginia bank had assets of $1.1 billion on Dec. 31, a 15 percent increase over 1981. Deposits rose 14.2 percent to $938 million, while loans outstanding increased to $589 million from $575 million.