Mr Comyn has declared Australia's largest bank has not had the right leaders - until now.

Seven months into his stint in the top job, Mr Comyn has presented himself at the banking royal commission as the right person to change the culture at CBA after a series of scandals.

Has CBA had the right leaders in the past?

"No," was Mr Comyn's frank response on Tuesday.

Does it have the right leaders now?

"We will see," he said.

"I hope so. Yes."

Mr Comyn told the inquiry he repeatedly lobbied to scrap consumer credit insurance products during his time as CBA's head of retail banking.

He had a "robust discussion" with Mr Narev a month after an April 2015 audit revealed 64,000 CBA customers were sold credit card plus insurance when they were unemployed and therefore ineligible to make a claim.

After a fascinating and intriguing chat about chess grandmaster Garry Kasparov's attendance at a CBA client function, Mr Comyn "enthusiastically" tried to convince Mr Narev to suspend sales of consumer credit insurance products.

He was overruled and told "temper your sense of justice".

Slightly irritated by the comment, Mr Comyn said he took that to mean "calm down".

"He was referring to I was enthusiastically pursuing my argument which I thought was right. And he was telling me to temper it."

Mr Comyn raised the issue again with Mr Comyn in 2016.

"I was insufficiently persuasive," he said.

Mr Comyn said he was willing to forgo the $150 million the bank made annually from the "poor value" consumer credit insurance products.

Mr Comyn said Mr Narev believed the product was still relevant for customers and commercial considerations were not his primary focus.

CBA may end up having to compensate more than half a million customers over the add-on insurance products.

The bank is already paying $46 million to 154,000 customers over the mis-selling of credit card and loan protection insurance.

An independent review in September raised concerns about the sales process or value of the consumer credit insurance products for another 374,000 customers.

Mr Comyn said CBA will extend its remediation program to cover those people if necessary, agreeing it would likely have to compensate a significant number of additional customers.

Mr Comyn admitted issues such as the insurance mis-selling and the charging of fees for no service, for which CBA is paying $116 million in remediation, revealed the bank prioritised short-term profits ahead of customers.

"We didn't sufficiently prioritise customers.

"We didn't understand the potential for harm."

Mr Comyn said the culture of the bank must change and strike the right balance between delivering good outcomes for customers and delivering sustainable, long-term returns to shareholders.

Australian Associated Press