βasset = βdebtWd + βequityWe

Wd, We: the proportions of debt and equity

The above equation is provided in Cost of Capital (Corporate Finance) pp no. 19.

Further, it says

βasset = βdebt(D/D+E) + βequity(E/D+E)

Then it says,

But interest on the debt is deducted by the company to arrive at taxable income, so the claim that creditors have on the company’s assets does not cost the company the full amount but, rather, the after-tax claim; the burden of debt financing is actually less due to interest deductibility. We can represent the asset beta of a company as the weighted average of the betas of debt and equity after considering the effects of the tax-deductibility of interest:

So,

βasset = βdebt((1-t) x D/D+E) + βequity(E/(1-t) x D+E)

My question is rather simple.

I understand interest arising out of debt provides a tax shield. Hence not all the interest costs are ‘wasted’. Some go to reduce the tax-outflow of the company. However, in the calculation of Unlevered Beta (Asset Beta) why does one recognise the tax shield on the principal portion of the debt (D, being the market value of debt). Even considering D is the present value of all future cash flows of the debt, the future cash flows include the redemption of face value, which amount is not a tax shield, but just a payment of principal. Is the formula right in taking the after-tax cost of D?

Thanks in advance for your inputs.