It would have been fun.

[keannu]

Is the underlined possible? I think it means a past presumption, but it seems impossible according to others' opinion. They say a past can't be a certainty by using "would", but I thought "would" can mean a presumption with "have+p.p" as it means a presumption for the present or the future like "The party would be fun tonight."

ex)My friend went to Bon Jovi's concert yesterday. It would have been fun.

[philo2009]

What the speaker clearly means here is

It would have been fun for me to go to the concert last night.

(I.e. If I had gone, it would have been fun.)

[keannu]

What the speaker clearly means here is

It would have been fun for me to go to the concert last night.

(I.e. If I had gone, it would have been fun.)

What about "the party itself would have been fun" without relating me to it? Isn't it possible?

[bhaisahab]

What about "the party itself would have been fun" without relating me to it? Isn't it possible?

No.

[keannu]

No.

What is this explanation from a material? When you presume about the past, can you only use "could" or "might" not "would" which seems stronger in certainty than the first two?

would: Presumption or expectation

*That would be Jo calling. I'll answer it.

* We saw a police helicopter overhead yesterday morning. | Really? They would have been looking for those bank robbers

[5jj]

We are in a rather grey area here.

If we are dealing with an assumption of certainty, then the modal is normally 'will':

My friend went to Bon Jovi's concert yesterday. It will have been fun.
That will be Jo calling. I'll answer it.

A. We saw a police helicopter overhead yesterday morning.
B. Really? They will have been looking for those bank robbers.

In some contexts, it is possible to 'distance' the certainty by backshifting the modal:

A.That's a distinguished-looking person sitting next to your father in this photo.
B. Hmm, The college rowing team. Ah, that would be my cousin James.

This is probably the explanation for your second example: We saw a police helicopter overhead yesterday morning. | Really? They would have been looking for those bank robbers

It is probably also the explanation for the first; That would be Jo calling. I'll answer it. My personal feeling is that that utterance is unlikely. The immediacy of the confirmation of the thought makes 'will' far more likely.

[philo2009]

What about "the party itself would have been fun" without relating me to it? Isn't it possible?

The speaker is hardly likely to be making such an absurd，not to say egocentric, claim. Common sense plainly rules out that particular interpretation!

[Raymott]

What about "the party itself would have been fun" without relating me to it? Isn't it possible?

I have to disagree with my colleagues. But that's fine, because we are making different assumptions about the context.
The other interpretation is anything but egocentric. It shows concern for your friend.

My friend went to Bon Jovi's concert yesterday. It would have been fun for her. She's been depressed lately and she needed something like that to cheer her up.

[keannu]

I have to disagree with my colleagues. But that's fine, because we are making different assumptions about the context.
The other interpretation is anything but egocentric. It shows concern for your friend.

My friend went to Bon Jovi's concert yesterday. It would have been fun for her. She's been depressed lately and she needed something like that to cheer her up.

This is what I wanted, Actually I listed this as a question in my test material for new students and I meant it for "it could (might) have been fun for her"(but would stronger than could or might). If you hadn't confirmed it, I would become a fraud teacher faking students.