Anthem Inc. shares jumped by the most in more than 10 years after the health insurer said it expects its plan to move up the launch of its pharmacy-benefits business by nine months to help drive strong profit growth.

The company forecast adjusted earnings per share of $19 for 2019, exceeding analysts’ estimates. The forecast reflects growth across Anthem’s businesses as well as some benefit from switching existing customers to its new pharmacy-benefits manager, called IngenioRx.

Anthem executives have said they expect to save $4 billion a year on drug costs from the new venture, which will replace its existing...