Market cap = Equity Value ? - Help

Hey guys, hope you all are fine. Please help me with this question:

Company A does an IPO and issues 100,000 out of 400,000 shares and the share price is $20. So then shares outstanding is 100,000 and market cap is $2 million.

In my view the actual value of the firm is $8 million, since 1/4 is public and this stake is worth $2 million. But I heard it is wrong because market cap values the entire equity, and not only the public stake. So, the entire equity value is worth $2 million.

It doesn't make sense to me since if you buy all the public shares you are going to pay $2 million and have only 25% of the company, so market cap is not valuing the entire equity.

I appreciate if you guys help me to understand what is wrong with my view.

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