American Security Corp., citing sliding interest rates, yesterday announced the largest single profits jump since it began releasing quarterly figures five years ago, reporting a 30 percent rise in profits for the 1980 second quarter.

The parent company of the American Security Bank, the city's second-largest, reported second-quarter profits of $6.37 million ($1.75 a share), compared with $4.91 million ($1.35) for the same quarter in 1979.

For the first six months of the year, the holding company reported profits of $11.5 million ($3.17), compared with 1979 six-month figures of $9.87 million ($2.70). American Security's first-quarter profits rose by 10.5 percent.

"Our asset-liability management positioned us well to capitalize on the favorable spreads created by the rapidly declining interest rate environment of the past two months," said W. Jarvis Moody, chairman of American Security.

Moody predicted that "barring abnormal loan losses," the company's third- and fourth-quarter profits "should be on target" but said that as iterest rates bottom out American Security's profits would stabilize at lower, more normal levels.

Release of the holding company's second-quarter results comes less than three months after Moody, who has run American Security since March, warned stockholders that 1980 "may be one of the most difficult years for the banking industry in recent memory."

Another bank, Madison National Bank of Washington, also reported a significant profits increase for the period, reporting a 25 percent rise in 1980 first-half profits over the same period last year.

After-tax profits at Madison National rose to $1.12 million ($2.75 a share) for the first six months of the year, up from $901,214 ($2.20) for the first half of 1979.

American Furniture Co. yesterday reported a steep drop in profits for the quarter ended May 31. The Martinsville company said profits for the quarter fell from $926,000 in 1979 to $402,042 for the second quarter of the fiscal year. Sales rose slightly to $19.18 million for the quarter, compared with $19.04 million for the same period last year.

For the six-month period, the company reported profits of $804,000 (29 cents), compared with profits of $1.41 million (50 cents) for the six-month period ended May 31, 1979.