What are the benefits of staying calm? What are the disadvantages of staying calm? There are a few parts to this question because it's not as simple as it reads.
My quick answer is that I think you should stay real, there are times you will do so calmly and times less calmly.
When you are a CEO hopefully you share the vision of the company you are working for not just intellectually but viscerally. Hopefully you get emotional at times. Hopefully your team gets emotional at times. Im not talking about unproductive emotion i'm talking about authenticity and care.
Hopefully you've hired great people who share the vision and are just as passionate. Hopefully you genuinely respect your team and exercise strong leadership skills that keep them motivated to do great work. But if you hide your emotions, you miss opportunities. At the same token if you treat people badly, they won't want to work for you. You need to find balance but most importantly you need to be real because that's how you create culture. When you're CEO, you shouldn't play it safe because in order to create the culture that's going to help you achieve the most success, you need to show what success looks like and that takes emotion. If something important were to go wrong and you feel like there is a misalignment, communicate and show your frustration, but also show confidence in your teams ability - don't make them feel like a failure... That's the key.
So, calmness in the sense of respecting your team is very important and something a great CEO will never stray away from. Calmness in the sense of lack of real communication and feeling can limit your potential. Be real!