To directly answer your question with the equation from before, EqV + Debt - Cash = EV. EqV up and Cash down (spent on repurchasing shares). The “-“ in front of the cash and the actual negative cash creates a positive that offsets the drop in equity value.

However, taking a more theoretical approach, I’d recommend reading about The Modigliani–Miller theorem. It discusses how in a vacuum, financing decisions (share repurchases / raising debt etc) do not affect enterprise value

 

You're assuming P/E is held constant -  in that scenario, your total equity value doesn't change at all, because the market is valuing your earnings at the same level as pre-repurchase. In that scenario, your EV would decrease because you're expending cash to repurchase shares, but your equity value is not changing. It is not a realistic scenario.

 

Equity value decreases since cash is leaving the business and thus not available to investors after the buybacks. EV is unchanged.

 

You are using cash to buy equity

Repurchased shares go into a contra equity account

Using cash to repurchase shares

Cash is down. Your contra equity goes up, which means your equity goes down

Equity value decreases. Cash decreases same amount. Enterprise value unchanged (since EV has EQv MINUS cash, so EQv is neg and cash is neg, but it it is a double neg since you subtract cash in EV, and thus EV is the same

Source: ECM Village Idiot

 

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