Home prices have sunk to 2002 levels, effectively wiping out almost a decade’s worth of home equity across the U.S. and imperiling the fragile economic recovery as Americans confront the falling value of their biggest investment. A closely watched home-price index released Tuesday, the S&P/Case-Shiller National Index, showed that prices nationwide fell 4.2% in the first quarter after declining 3.6% in the fourth quarter of 2010. The index had seen increases in 2009 and early 2010.