Interesting question. I will address this more like a real project, which you can convert in an interview case.
1. Clarify the problem.
"How will you reduce the operational cost for a credit card company?"
To me, this lacks clarity.
1.1 What do you mean by operational costs? Do you mean the cost of goods sold, operating expenses as well as overhead expenses? Let's assume so.
1.2 What do you mean by a credit card company? The network (VISA, Master card, ...) OR do you mean the issuer of the credit card (American Express, Bank of America, Barclays, ...). Let's assume the latter.
1.3 Do you mean the operating costs for the entire issuer (e.g., Barclays) OR just the costs associated with this credit card issuing business? Let's say this is in the CC business.
1.4 Are they looking across all sectors within their CC business or just in the consumer segment? Let's assume Consumer segment
1.5 What is the client trying to achieve, and why? (obviously, the client wants to improve the bottom line, but why is he looking into this area in particular?)
THIS PROCESS IS WHAT IS CALLED BY STRATEGIC SCOPING. You are trying not only to define clearly your objective function but give you the basic knowledge of what is going on.
2. Refine the question the team is trying to answer
Now through our probing questions, we learn about the real problem the client was trying to solve; in fact, "can we reduce Barclay's operating costs in the consumer segment?" is the correct question.
3. Develop the issue tree
We know need to break operating costs into its drivers:
3.2 Transaction processing
3.4 Risk monitoring
3.5 Customer service
3.6 Interchange fees
4. Prioritize drivers
Now, we can go forward to doing three things.
4.1 Starting by putting numbers on each lever, so we know how significant each component is in the overall cost structure.
4.2 We look historically to see how these numbers have been growing in the past three years.
4.3 We do a quick benchmark to the competitors to see how their cost structure differs from our client.
From here, we can already prioritize the drivers that will cause the most impact to the client.
5. Top-down approach/hypothesis generation
Seeing how well (or bad) the client has been doing in each driver, we can do a quick back of the envelope calculation to see how impactful it could be if the client were to adopt best practices across the industry. This approach is just an 80/20 rule to make sure we are not spending too much time in opportunities that will not generate enough impact.
It's also good to generate hypotheses for the drivers that were prioritized in step 4.
6. Data gathering / bottom-up analysis
Now that you know which levers to focus and what hypothesis you want to test to make the case to the client, you need to start making data requests and building a proper financial model.
In this case, you want to understand if there is a reason for the client to have this particular cost structure and if he could even adopt the best practices that you are thinking of recommending.
From your analysis, you will probably be able to create a 2x2 matrix. On one side, you can benchmark how well the client is performing against its peers from mediocre to high, and on the other side, you can have if these activities are non-core and core or if they are adding value or not.
From here, most likely, you will be able to recommend a set of actions such as outsource, divest, invest or stay the same.
In summary, this would be 80% of the rationale it would be going into something like what you asked in a "real" engagement. For a case interview, the rationale is the same but boiled down to oblige to the scope of the situation
My 2 cents!