I'm reading Rosenbaum, and they note that ROIC uses a pre-interest earnings numerator while ROE and ROA use an earnings metric net of interest expense.

Why is this? The book didn't really explain the reasoning behind this.

You want to account for the effects of leverage when looking at ROE and ROA. Net income is the amount of income available to equity holders, so it makes sense to consider that as their return. In this case, you want a levered numerator.

Invested capital is comprised of both equity and debt capital, so you want to use a numerator that is unlevered, which in this case would be EBIT (as you're not accounting for the interest expense).

It's similar to how when you're doing a valuation based off FCF, using a levered free cash flow would give you an equity value while using an unlevered free cash flow would give you an enterprise value.

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If net income is for equity holder why do u use assets which comprise of both equity and liability then?

I think you're thinking that Assets = Liabilities + Equity in a literal sense. That's not the case. That equation just means that, in order for the balance sheet to balance, the size of your assets must be equal to the size of your liabilities plus your equity (to remain solvent, liquid, and operational).

May be wrong here (just a sophomore) but ROE and ROA naturally are equity metrics so it makes sense to incorporate interest expense, since that will not go towards the company's equity holders. ROA may be a little more iffy but I think it is still valid because the assets don't incorporate the liabilities of the company.

Think about it this way: if a company levers up and another does not would their ROE metrics make sense if it were EBIT/Equity? The levered company is going to have much higher EBIT and appear to have a greater ROE. Likewise, if you use Net Income/Invested Capital the company without leverage is going to seem much more attractive because of the interest expense, even though the invested capital is generating a much greater rate of return to all investors.

For ROIC you are looking at ALL invested capital, so you should use EBIT which is cash flow that is received by both equity and debt holders.

ROA should use NOPAT.