U.S. home prices fell by the sharpest annual rate on record in the third quarter, a widely-watched index said on Tuesday. The Standard & Poor's/Case-Shiller U.S. National Home Price Index recorded a 16.6 percent drop during the third quarter over the same period a year ago. Prices are at levels not seen since the first quarter of 2004. The monthly indices also showed record declines. The 20-city index fell by 17.4 percent in September compared with a year ago, the largest drop since it began in 2000. The 10-city index plunged 18.6 percent, the biggest decline in its 21-year history. None of the cities in the Case-Shiller 20-city index saw annual price gains in September – the sixth straight month without an increase.