Lease or buy the piece of equipment

  To determine whether to lease or buy the piece of equipment, we need to analyze the total costs associated with each option and make a comparison based on the present value (PV) of those costs. Let’s break down the calculations for both options. Option 1: Buying the Equipment Purchase Details: - Cost of Equipment: $250,000 - Down Payment (10%): [ \text{Down Payment} = 0.10 \times 250,000 = 25,000 ] - Financed Amount: [ \text{Financed Amount} = 250,000 - 25,000 = 225,000 ] - Interest Rate: 8% - Loan Term: 5 years Annual Installments Calculation To find the annual payment for the financed amount, we use the formula for an annuity: [ PMT = P \frac{r(1+r)^n}{(1+r)^n - 1} ] Where: - (P) = principal (loan amount) = $225,000 - (r) = interest rate per period = 8% or 0.08 - (n) = number of periods (5) Plugging in the values: [ PMT = 225,000 \frac{0.08(1+0.08)^5}{(1+0.08)^5 - 1} ] Calculating further: [ PMT = 225,000 \frac{0.08 \times 1.46933}{1.46933 - 1} = 225,000 \frac{0.1175464}{0.46933} \approx 225,000 \times 0.25057 \approx 56,786.25 ] So, the annual payment is approximately $56,786.25. Total Payments Over Loan Term Total payments over 5 years: [ \text{Total Payments} = PMT \times n = 56,786.25 \times 5 \approx 283,931.25 ] Salvage Value After 8 years, the scrap value of the equipment is $25,000. Present Value of Costs When Buying 1. Present Value of Down Payment: [ PV_{\text{down}} = 25,000 ] 2. Present Value of Annual Payments (Years 1-5): The present value of an annuity can be calculated using the formula: [ PV_{\text{annuity}} = PMT \times \frac{1 - (1 + r)^{-n}}{r} ] Where (PMT = 56,786.25), (r = 0.08), and (n = 5): [ PV_{\text{annuity}} = 56,786.25 \times \frac{1 - (1 + 0.08)^{-5}}{0.08} ] Calculating this gives: [ PV_{\text{annuity}} = 56,786.25 \times \frac{1 - (1.36049)^{-1}}{0.08} \approx 56,786.25 \times \frac{1 - 0.73503}{0.08} ] [ PV_{\text{annuity}} \approx 56,786.25 \times 3.30374 \approx 187,698.29 ] 3. Present Value of Scrap Value: The present value of receiving $25,000 at year 8 is calculated as: [ PV_{\text{salvage}} = \frac{FV}{(1+r)^n} = \frac{25,000}{(1+0.08)^8} \approx \frac{25,000}{1.85093} \approx 13,513.21 ] Total Present Value When Buying Now we can sum all the present values: [ PV_{\text{total buy}} = PV_{\text{down}} + PV_{\text{annuity}} - PV_{\text{salvage}} ] [ PV_{\text{total buy}} = 25,000 + 187,698.29 - 13,513.21 \approx 199,185.08 ] Option 2: Leasing the Equipment Lease Details - Annual Lease Payment: $100,000 for 8 years - Maintenance Costs: Not specified but assumed to be included in total lease cost for this analysis. - Discount Rate: 8% Present Value of Lease Payments The present value of lease payments can be calculated using the annuity formula: [ PV_{\text{lease}} = PMT \times \frac{1 - (1 + r)^{-n}}{r} ] Where (PMT = 100,000), (r = 0.08), and (n = 8): [ PV_{\text{lease}} = 100,000 \times \frac{1 - (1 + 0.08)^{-8}}{0.08} ] Calculating this gives: [ PV_{\text{lease}} = 100,000 \times \frac{1 - (1.85093)^{-1}}{0.08} ] [ PV_{\text{lease}} \approx 100,000 \times \frac{1 - 0.54027}{0.08} ] [ PV_{\text{lease}} \approx 100,000 \times \frac{0.45973}{0.08} ] [ PV_{\text{lease}} \approx 100,000 \times 5.746625 = 574,662.52 ] Conclusion: Lease vs Buy - Total Present Value when Buying: $199,185.08 - Total Present Value when Leasing: $574,662.52 Decision Given that the present value of buying the equipment ($199,185.08) is significantly lower than that of leasing it ($574,662.52), it is financially more advantageous to buy the equipment. Pros and Cons of Each Option Buying Pros: - Ownership of the asset after payments are complete. - Potential for salvage value after use. - Depreciation tax benefits. Buying Cons: - Larger initial cash outflow due to down payment. - Responsibility for maintenance and repairs. Leasing Pros: - Lower initial cash outflow (no down payment). - Predictable expenses without worrying about asset depreciation. Leasing Cons: - No ownership at the end of the lease. - Total payments may exceed purchase costs in the long term. - Responsibility for maintenance costs which can add up. In summary, based on financial analysis and consideration of ownership benefits, purchasing the equipment is the optimal choice in this scenario.    

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