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TEV = market value of equity + book value of debt + NCI - EXCESS cash.
Given that cash is typically subtracted from TEV, it's tempting to assume that your hypothetical would somehow increase TEV. In reality, only *excess cash -- extra cash and equivalents sitting on the balance sheet not in use by the business and available to pay down debt -- reduces TEV. In your hypothetical, this cash is being used in the business, so it would not be considered excess cash. Therefore TEV would not change.
Previous post is correct IF the cash is operating cash and not excess cash. If the firm is using excess cash to finance an acquisition, then TEV does change.
To try to help you with how to think about TEV. Suppose there are 2 identical $50,000 Teslas. Tesla A has nothing in the trunk. Tesla B has $30,000 cash in the trunk. Regardless of what is in the trunk, both Teslas operate identically. Someone would pay $80,000 for Tesla B, take the $30,000 cash out of the trunk, and it is as if they paid $50,000 for Tesla B. TEV is trying to get at the value of the firm's operating assets, which is the Tesla in this example.
In your example, if excess cash is used to finance the acquisition, the firm's operating assets increase while the excess cash in the trunk goes down. If the cash is operationally related cash, then as previous poster says, TEV doesn't change.
My take:
The TEV equation is simply the fundamental accounting identity (A=L+E or the value of the assets of the business equals the value of all contractual claims on these assets, so you can add pension liabilities, minority interests on the claims side) but with market values. Ignoring all complications on the claims side and breaking down assets into operating and non-operating we have (all in market value) Operating Assets (or Enterprise Value) + Non-Operating Assets = Equity + Debt. Non-Operating can be excess cash, short-term investment, associates, etc., again let's simpify and assume it is excess cash only. We thus have EV + C = Equity + Debt
Now if we purchase an asset with cash, it stands to reason we can only use excess cash, as operating cash (needed to keep business liquid) can not be taken without disrupting current operations (or would need to be replaced in some way, e.g. selling other assets somewhere, raising debt, etc.). Now three things can happen:
1. We purchase an asset for lower than its fair market value (financing a project with a positive NPV with present value of A and investment cost B, A > B), then C goes down by B, EV goes up by A. On the claims side for everything to balance, since all marginal value created flows to the equity directly E goes up by A-B > 0
2. We purchase an asset at its fair market value (financing a project with an NPV of 0, B=A) C goes down by B, EV goes up by A, E does not change
3. We purchase an asset for more than its fair market value (financing a project with a negative NPV with present value of A and investment cost B, A B), then C goes down by B, EV goes up by A. On the claims side for everything to balance, since all marginal value destroyed flows to the equity directly E goes down by A-B 0
In any case EV augments by A, the PV of the cashflows of the asset that was bought
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