A closely watched index released today showed home prices tumbling by the sharpest annual rate ever in July, but the rate of monthly declines is slowing. The Standard and Poor’s/Case-Shiller 20-city housing index fell a record 16.3 percent in July from the year-ago month, the largest drop since its inception in 2000. The 10-city index plunged 17.5 percent, its biggest decline in its 21-year history. Prices in the 20-city index have plummeted nearly 20 percent since peaking in July 2006. The 10-city index has fallen more than 21 percent since its peak in June 2006. No city in the Case-Shiller 20-city index saw annual price gains in July, the fourth straight month that’s happened. However, the pace of monthly declines is slowing, a possible silver lining. Between May and July, for example, home prices fell at a cumulative rate of 2.2 percent — less than half the cumulative rate experienced between February and April. But there’s “no evidence of a bottom,” said David M. Blitzer, chairman of the index committee at S and P.