The document included estimates of the cost from lives and jobs lost as well as damage to property and infrastructure.
In the 1980s, it said, the annual cost was about $50 billion, quadrupling to $200 billion per year in the last decade.
"Weather-related economic impacts are especially high in fast-growing, middle-income countries due to increasingly exposed, valuable assets," said the report.
In these economies, "the average impact of disasters equaled one percent of GDP (gross domestic product) over the six years from 2001 to 2006, 10 times higher than the average for high-income countries."
Those further down the ladder of development experienced a correspondingly greater loss of GDP.
Hurricane Tomas wiped out the equivalent of 43 percent of the GDP of St Lucia in 2010.
The 2008-11 drought in the Horn of Africa, which at its peak saw 13.3 million people without enough food, caused estimated losses of $12.1 billion in Kenya alone.
The World Bank said more must be done to help vulnerable countries adapt to climate change and prepare for weather disasters.