The question ‘Why does the universe exist?’ can be answered in two steps:
1) First of all we will have to know as to whether there is any God or not;
2) If we can somehow come to know that there is a God, then we can further ask the question as to why he created the universe. When we will have the answer to this question, we will also come to know as to why the universe exists, why we exist.
The above two have already been answered here1 and here2 respectively.
But even if we somehow come to know the reason as to why the universe exists, yet this will not answer all the questions. This is because we can still ask the question: Why does God exist? If there is a God, then what is the reason that there will have to be a God at all? If we think that there is actually such a reason, then we must keep in mind that we will have to find this reason within God’s existence itself and not outside of it, as otherwise there will be an infinite regress. That means God must have to be a necessary being, not contingent. But what is the reason due to which God will have to exist at all? I think I have already answered this question here3, where I have shown that the existence of nothing is self-contradictory and that therefore only something can exist, and not nothing. I have also shown that simply by default this something will always be spaceless, timeless, changeless, immortal, all-pervading, one, unborn, uncreated, without any beginning, without an end, everlasting and non-composite.
So I think I have answered all the questions that can possibly be asked: Why does God exist? Why does the universe exist? Why do we exist? God will have to be there because existence of nothing is self-contradictory. Being there he will have to create the universe in order to overcome his utter loneliness.