When a household headed by an elderly person supports someone in a nursing home, the cost averages about $20,500 a year and about half of that comes directly from the household's private funds, not from insurance or government aid, according to a study released yesterday by the American Association of Retired Persons.

The study, conducted by David Kennell of ICF Inc., a consulting firm on health and other public-policy issues, suggests that by far the biggest gap in health-care coverage for the aged is in protection against the high costs of long-term nursing-home care.

According to the study, health spending in 1986 for all elderly persons will total $150 billion, excluding insurance premiums. Medicare and other public programs plus private insurance will pay for most of this. But the elderly will pay $37.3 billion out of their own pockets, and by far the biggest single chunk of this -- $16 billion -- will be for nursing-home care.

The study shows that while more than nine-tenths of the cost of hospital care for the elderly is covered by government programs or insurance (Medicare alone covers two-thirds of hospital costs) and more than four-fifths of the cost of doctor care, only half the $33 billion spent annually for nursing-home care of the aged comes from government programs or insurance, while half is out-of-pocket.

Robert J. Rubin, former assistant secretary of Health and Human Services who is executive vice president of ICF, said, "Most people are unaware . . . that they don't have coverage" for long-term nursing-home care under either Medicare or private insurance policies. Only Medicaid pays for long-term nursing home care, but the patient must be indigent to qualify. The survey estimates that overall, the average elderly household in 1986 will incur health expenditures, including insurance premiums, of $8,340. About $2,670 of this will come from the family. Medicare will cover about $3,300.

The average elderly household will pay about 11.6 percent of its gross income on direct out-of-pocket payments and health insurance premiums. But the average masks substantial differences between various income and demographic groups, the study said. For example, for those 85 and over, out-of-pocket expenditures and premiums represent 42 percent of income, reflecting that many of these people are in nursing homes with huge out-of-pocket outlays. (ICF estimated 1.6 million elderly in nursing homes in 1986, out of about 29 million persons 65 and over.)

Low-income households -- below $10,000 a year -- are estimated to spend 56.6 percent of their income on out-of-pocket medical outlays and insurance premiums. This, too, reflects the high costs of nursing-home care. When an elderly person of low income enters a nursing home, the expense usually eats up all or most of his or her income.

The higher the income, the lower the percent that goes for out-of-pocket medical costs: Households with income above $75,000 spend 2.2 percent on out-of-pocket medical costs and insurance.

One recent federal revenue-raising proposal is that the government should count as income to the elderly -- and thus subject to the income tax -- the portion of the Medicare doctor-insurance program that is paid for by general revenues from the Treasury -- about $576 a person. The study found that if this were done it would barely increase -- from 11.6 percent to 11.9 percent -- the share of the income the elderly paid out-of-pocket for medical care. Because low-income households pay little tax anyhow, it would affect them little, if at all.