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