National Bank of Washington, troubled throughout most of last year by federal and internal probes of questionable loan and credit policies, has reported one of the sharpest declines in annual profitability ever posted by a major area financial institution.

In a very brief statement that did not provide complete financial data and offered no explanations for the earnings performance, the city's third-largest bank said operating profits fell about 45 percent to $2.77 a share in 1980 from $5 the previous year. Total operating earnings data were not listed in the bank's press release.

The fourth-quarter performance was worse: Per-share earnings were down to 54 cents from $1.51 a year earlier, at a time when most area banks were reporting substantial increases because of higher interest income on loans. Using figures provided earlier for the first nine months of 1980, indicated NBW new income for the fourth quarter was $741,000 vs. $2.07 million a year ago.

NBW's second quarter actually was the weakest period of the year, as the bank earned just 37 cents a share after boosting reserves for possible loan losses and accounting for unusual one-time expenses associated with the loan probes.

Net income for the full year, which includes any gains or losses from selling securities, declined to $4 million ($2.91 a share) from the record $6.9 million ($5) in 1979 and far below the $6 million earned in 1978. It was the lowest annual new income figure for NBW since 1975, when the bank earned $3.7 million.

Deposits, assets and loan business at the bank all declined in 1980, according to the bank statement. Assets as of Dec. 31 fell to $863 million from $923 million a year earlier. Deposits declined to $671 million (the lowest level since 1978 and a decline of $20 million in the fourth quarter alone) from $745 million, and loan volume decreased to $504 million from $536 million.

NBW's new chairman, Luther Hodges, Jr., could not be reached yesterday for elaboration on the press release. In a separate statement, however, Hodges said three new persons have been named to the NBW board of directors. They are:

Phillips Peter, vice president for corporate government relations of General Electric Co., in Washington.

Robert Washington Jr., a partner in the law firm of Danzansky, Dickey, Tydings, Qunit & Gordon and former D.C. Democratic chairman.

Sanford Greenberg, Chairman of American Metals and Alloys Inc. and a technology adviser to the White House under President Johnson.

Also sworn in at a board meeting last week were two directors whose elections were announced earlier: former congressman Robert Giaimo and retired Air Force general Brent Scowcroft, a former national security adviser.