Say a company wants a $100MM term loan so they can issue a $100MM dividend to its owners.
On the pro forma BS, debt obviously increases by $100MM, but what is the offset to this increase? A decrease in equity, but what specifically do you deduct from?
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Think about it in two steps, raising the TL, then paying the dividend. In the first step, cash is up by $100, debt also up by $100. Second step, cash is down by $100, equity also down by $100.
You can put the two steps together and say it's cash neutral. Then debt and equity offset by $100 in either direction, still balance.
Here's a good div recap overview: https://multipleexpansion.com/2018/01/31/Dividend-Recap/
Retained earnings...
Break the transaction down into two smaller items; the capital acquisition (term loan), and the dividend payment to shareholders. To help with this sort of stuff I used to break down the B/S to "Cash Asset + Noncash Assets = Liab. + Contr. Capital + Earned Capital" where the Earned Capital section would contain both the Retained Earnings (what you'll use for the second part of the transaction) and Accumulated other comprehensive loss items. So, looking specifically at the question you asked about the effect on the equity section, if a cash dividend of $10million is paid out then Cash Asset and Earned Capital should both equal -10m. Since dividend payments directly influence the Retained Earnings item, they bypass the income statement so you're done after adjusting these accounts.
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