* 2016 underlying profit 1.0 bln stg vs 1.3 bln in 2015
* Nationwide to stop selling car insurance
* Consumer confidence post-Brexit holding up (Adds CEO quote, details)
LONDON, May 23 (Reuters) - Britain's Nationwide Building Society said on Tuesday that its underlying profit fell by 23 percent in the financial year that ended in April, as costs increased and it did not pass on the full effect of interest rate cuts to savers.
Nationwide, Britain's second-biggest provider of mortgages, reported an annual profit of 1 billion pounds ($1.3 billion) compared with 1.3 billion a year earlier.
The lender also said it would no longer provide car insurance from June this year as it continues to pare back its business model to focus on its main product of home loans.
"Car insurance does not fit with our core purpose and strategy," Chief Executive Joe Garner told reporters in a phone interview.
Garner said the decision was not linked to broader concerns about rising risks in the auto finance sector in Britain, as surging use of complex leasing arrangements raises questions about customers' ability to pay if the economy deteriorates.
Nationwide said it would close other businesses no longer deemed central to its model, including international deposit taking and inheritance tax planning.
In November Nationwide said it would close its commercial real estate business after the outlook for the property sector darkened following Britain's vote to leave the European Union.
The lender said on Tuesday that consumer confidence had held up well since the vote, but that rising inflation was likely to squeeze household budgets in the coming months. ($1 = 0.7704 pounds) (Reporting By Lawrence White; Editing by Susan Fenton)