Entry investment price in Private Equity (FIG)

I am trying to understand how PE firms determine the suitable entry valuation of a bank, assuming there is no leverage involved. Let's say a bank is valued at P/BV 1x, with cost of equity = RoE = 10%. If the PE firm invests at this value, bank performs accordingly and the firm exits at the projected terminal value, the PE firm would earn IRR = RoE = cost of equity, correct? If so, would the PE firm do this deal and why? There seems to be no alpha in the equation if the exit multiple is also going to be 1x. Or would the PE firm only invest if it can acquire shares at a discount?

Apologies if the line of questioning doesn't make sense or if i've messed up the logic.

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