Are taxes necessary for the cost of debt financing to be less than the cost of equity financing?
Not really.
While it is true that interest on debt is tax deductible which makes debt cheaper but the actual reason of cost of debt being lower than cost of equity is the fact that equity holders are the last stakeholders to get their money back in case of bankrupcy. Hence they demand higher rate of return. Equity holders will never accept a return on investment that is lower than debt holders. This is because equity holders are always subordinate to debtors. In case of bankruptcy, debtors will be paid first, so they are at low risk and so the return of debt will always be lower than return of equity.
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