1. **State the problem:** Fez Fabulous Fabrics wants to acquire a $100,000 machine with an 8-year life and $20,000 residual value. They can either lease it with $16,000 annual payments in advance or buy it with a 12% loan. The tax rate is 35%, and depreciation is over 5 years. We need to find the present value (PV) of cash outflows for both options using the after-tax cost of debt as the discount rate and decide which is better. 2. **Key formulas and concepts:** - Present value of an annuity due (payments at beginning): $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ - After-tax cost of debt: $$r_{after-tax} = r \times (1 - tax\ rate) = 0.12 \times (1 - 0.35) = 0.078$$ - Depreciation tax shield reduces taxable income, so it lowers cash outflows. 3. **Calculate PV of lease payments:** - Lease payment $P = 16,000$ - Number of years $n = 8$ - Discount rate $r = 0.078$ $$PV_{lease} = 16,000 \times \frac{1 - (1 + 0.078)^{-8}}{0.078} \times (1 + 0.078)$$ Calculate: $$\frac{1 - (1.078)^{-8}}{0.078} = \frac{1 - 0.5403}{0.078} = \frac{0.4597}{0.078} = 5.895$$ So: $$PV_{lease} = 16,000 \times 5.895 \times 1.078 = 16,000 \times 6.352 = 101,632$$ 4. **Calculate PV of buying option:** - Initial cost = 100,000 - Residual value after 8 years = 20,000 (discounted back) - Loan payments have same schedule as lease, so loan PV = PV of lease payments = 101,632 - Depreciation is over 5 years (MACRS 5-year class), so depreciation tax shield must be calculated. 5. **Depreciation tax shield:** - Using straight-line for simplicity: annual depreciation = $\frac{100,000 - 20,000}{5} = 16,000$ - Tax shield per year = depreciation $\times$ tax rate = $16,000 \times 0.35 = 5,600$ - Depreciation lasts 5 years, so PV of tax shield: $$PV_{depr} = 5,600 \times \frac{1 - (1 + 0.078)^{-5}}{0.078}$$ Calculate: $$\frac{1 - (1.078)^{-5}}{0.078} = \frac{1 - 0.6806}{0.078} = \frac{0.3194}{0.078} = 4.095$$ So: $$PV_{depr} = 5,600 \times 4.095 = 22,932$$ 6. **PV of residual value:** $$PV_{residual} = 20,000 \times (1.078)^{-8} = 20,000 \times 0.5403 = 10,806$$ 7. **Total PV of buying option cash outflows:** $$PV_{buy} = Initial\ cost + PV_{loan\ payments} - PV_{depr} - PV_{residual}$$ $$PV_{buy} = 100,000 + 101,632 - 22,932 - 10,806 = 168,894$$ 8. **Compare alternatives:** - PV lease payments = 101,632 - PV buying cash outflows = 168,894 9. **Conclusion:** Leasing has a lower present value of cash outflows, so leasing is preferred because it costs less in present value terms when considering tax effects and financing costs.