U.S. home prices have fully recovered from their steep plunge during the housing bust and Great Recession, according to a private measure.

The Standard & Poor's CoreLogic Case-Shiller national home price index is slightly above the peak it set in July 2006, after rising 5.5 percent in September from a year earlier. The milestone comes after more than four years of steady gains.

Still, prices have not fully recovered in many cities and other gauges show that home prices remain below their peaks.

Steady job gains and low mortgage rates have encouraged more Americans to buy homes. Yet the supply of available homes has dwindled, setting off bidding wars and pushing up home prices.

Seattle, Portland and Denver reported the largest annual gains in September for the eighth straight month.