/How come when the tax rate on cost of debt increases, this lowers the cost of equity requirment? you would expect from an intuitive basis, that more taxes on debt would increase the risk of equity since more operating earnings from the company will be eating away of higher taxes on these earnings.
I am pretty sure that, in general, an increasing tax rate t does not lead to a lower required return on equity. I assume you are getting at the modigliani miller formula, which is:
re = r0 + (r0 - rd)*(D/E)*(1-t)
Note that in this case r0 is the required return in an unlevered firm with otherwise similar characteristics (especially similar tax rate). So if you change the tax rate in your formula above you also need to change r0, which is a function of t.
Think of it practically. Tax rate on debt increase is not what an investor is concerned as that tax is paid by debt holders/lenders. As an investor, what would drive my required return is:
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total leverage in the business
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asset risk within the company
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corp tax rates
In short, anything that would have a dampening or elevating effect on the cash flow available to me will encourage me demand a higher/lower risk premium for investing in your business.
Also, higher tax rate on debt lowers cost of debt not equity. MM proposition 2 with taxes - value is max at 100% debt(assuming all else constant).