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On the appendix for Chapter 19 of Principles of Corporate Finance (BMA), it discusses the topic of "Discounting Safe, Nominal Cash Flows", in which case they argue that

However, suppose we ask what depreciation tax shields are worth by themselves. For a firm that’s sure to pay taxes, depreciation tax shields are a safe, nominal flow. Therefore, they should be discounted at the firm’s after-tax borrowing rate

However, I remember that in the APV method, V_L = V_U + PV(Tax Shield). When we discount the tax shield here, we use the pre-tax borrowing rate (suppose the company keep the debt amount fixed).

I was wondering why is there a difference between the discount rate of tax shields?

Reference: Brealey, Myers and Allen, Principles of Corporate Finance

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Assuming the company does not default, the equity holders receive the benefit of the tax shield as an annuity. That means, as you suggested, one should discount the tax shield at the cost of debt. But, equity holders receive profits after tax, which implies using the after-tax borrowing rate. See the Holthausen and Zmijewski textbook for a more complete discussion.

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