It’s been nearly two months since Dunkin’ Brands (DNKN) went public, in one of the year’s most successful IPOs. The coffee and ice-cream shop operator has seen its shares rise by more than 40% — including a 7.5% bump since its first public trade — compared to an 11.5% decrease in the S&P 500 over the same period of time.

So Fortune stopped by corporate headquarters in Canton, Mass., to have a conversation with CEO Nigel Travis — who formerly held senior executive roles with companies like Papa John’s, Blockbuster and Burger King. What follows is an edited transcript:

Fortune: You ran Dunkin for over two years as a private equity-owned company. Are you worried about the added responsibilities, and reduced flexibility, that many “taken-public” CEOs complain about?

Travis: I may be slightly different from many of the people you talk to. I actually enjoy being in the public environment. I think it’s the competitive spirit. I enjoy the cut-and-thrust. I like the fact that we deal with the journalists and the analysts and the investors. It makes you sharper. Every day at the end of the roadshow, I sat down with [CFO] Neil Moses and said, ‘That was a great point that one guy made.’ You get a wide diversity of opinions, which helps you manage better. Even just reading analyst reports provides an education.

But the private equity firms have been great to work with and still own a majority of the company. They’ve taken a very strong, long-term approach and that’s not going to change.

Is there a plan for how to bleed out the PE firms?

There is not a concrete plan that I can articulate to you today. My very strong hope is that they stay in for some time because they’ve contributed significantly to building this business and can continue to do so.

Why was this past summer the right time to take Dunkin public?

It was about a year ago that I first identified to our private equity people that I thought the business would be in a good situation sometime this year to go public. We could see the momentum building, particularly on the Dunkin’ U.S. side, which is 73% of the revenue. We also had a very good refinancing last year that was followed by a re-pricing of the debt, which told us that our story had some resonance.

We also felt that it was a moment in which a lot of the work we needed to do was behind us, but there was still some to be done. That gave investors the opportunity to share in some of the upside.

And I think we hit the market just about perfectly. The market has had a rough time since our IPO, but our stock has done pretty well.

Indeed, you have traded strong. But there have been critics, Barron’s for example, which argue you’re overpriced compared to competitors like Starbucks (SBUX), Tim Hortons (THI) and McDonald’s (MCD).

I’m not going to comment on where we should or shouldn’t be, because that’s a dangerous thing to do. Plus, I’m biased. But there been all kinds of analysts coming out with their thoughts today – five of them were buys, five were holds and one was a sell. So there seems to be more positive than not.

About comparing us to those other companies, I’d argue that it’s tough to do because we have this incredibly strong franchise model – an asset-light model, as is the buzzword – that investors seem to find very attractive. If you go look at some of the companies we get compared with – Panera (PNRA), Starbucks, Chipotle (CMG) – they’ve got a large number of company-owned stores. Tim Hortons is very focused on Canada, so that’s a difference. It’s not like comparing Burger King to McDonalds. We’ve in a different league.

You mentioned Tim Hortons, which had a tough time expanding outside of Canada. How will Dunkin keep expanding successfully outside of the U.S. market?

First thing I say is we have two brands, and Baskin Robins has tremendous international traction. It’s over twice the size of either Dunkin’ international or Baskin domestic. So that’s a real proven international powerhouse that we think can take us anywhere. Dunkin’ is a different animal, but it’s a highly-flexible animal.

Part of that dichotomy seems to be because ice cream is ice cream most anywhere, while beverages – particularly coffee and tea – are so culture-specific. Does that mean doughnuts are a bigger part of your international Dunkin’ brand than they are domestically?

Yes, I think that’s so. I was in Berlin the other week, which we’re using as the hub for our German expansion of Dunkin’, and was told that doughnuts were doing very well there. But I’d challenge them about how doughnuts don’t have the frequency coffee does. Well, they told me people came in every day for doughnuts, which is a big difference from the U.S. And we see similar things in Southeast Asia, which in many ways makes the beverage equation less important in certain international markets.

What is the benefit of Dunkin’ to Baskin, and vice versa?

I think the benefit is probably demonstrated by a series of planning meetings we had last week. The gentleman who runs Baskin U.S. said to me: I wrote down 17 pages of notes related to international. In many ways it’s the benefit of having like businesses and like problems and giving you different solutions to the same problem.

We’ve got a new store design, for instance, that’s been very successful in Baskin international and another new store design for Baskin U.S. and can now put two next to each other and say what works well here and there and pick the best parts. I think the whole issue of beverages and donuts is a benefit. We also have 1,150 stores in the U.S. that have combos, Baskin and Dunkin together – for historical reasons, but we have them – and we want those franchisees to be successful because the whole mantra we operate on is how can we improve franchisee profitability.

Could you envision a third brand, either organically or via acquisition?

I’ve got no desire to own another brand. We’re busy. As I said very clearly on road show: The chances of us picking up another brand are very, very low.

When PE firms bought Dunkin in 2006, they talked about trying to refresh the customer experience, including by making the stores more comfortable to hang out in. Has that worked?

We’re slowly evolving. One of the things we think our customers have missed is the opportunity to have tasty snacks in the afternoon, and our afternoon business now is actually stronger than many people realize. For example, the Goldman Sachs analyst report – the one downgrade – noted that we now have a stronger afternoon snacking business than McDonalds.

Yes, people have to sit there. They have to have WiFi. I think our newer stores we’re probably putting in more “soft” seating and one of the things that is a real success here is the number of remodels that we’ve done….

You’ve now lived in the Boston suburbs for just under three years. Have you been surprised by how deeply Dunkin’ permeates local culture?

You hear about it. You read about it. To live it is another thing. I always talk in speeches about how the Boston Duckboat tour talks about us four times and that’s an example of it. It’s probably a magic that’s tough to replicate.

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