Apollo Funds will pay $42.75 per share, representing a 6.6 percent premium to Aspen stock's closing price on Monday, the companies said.
The deal, which is expected to close in the first half of 2019, has been approved by Aspen's board after a strategic review, the companies said.
"We aim to leverage Apollo's resources and deep expertise in financial services to support the company (Aspen) as it embarks on its next chapter," said Alex Humphreys, Partner at Apollo.
Upon completion, Aspen will be a privately held portfolio company of the Apollo Funds.
The Bermuda-based insurer, which reported a loss of about $15 million in the second quarter, has been exploring a potential sale since early this year.
The company suffered losses in three out of last four quarters, hit by hurricanes, the California wildfires and the Mexican earthquakes.
Goldman Sachs and J.P. Morgan Securities acted as financial advisers to Aspen, while Apollo was advised by Willis Towers Watson and Libero Ventures.