Let's break down the problem step by step: ### **Given Data** - **Equipment cost:** $250,000 - **Stamp duty & registration expenses:** $5,000 - **Total amount to be financed:** $250,000 + $5,000 = **$255,000** - **Interest rate (fixed):** 8% p.a. - **Lease term:** 3 years (36 months) - **Residual value:** $100,000 (at end of term) - **Repayment frequency:** Monthly ### **Step 1: Calculate the Amount Financed** The full amount financed is **$255,000**. ### **Step 2: Lease Repayment Formula** For a financial lease with a **residual value**, the monthly repayment (PMT) is calculated as: \[ PMT = \frac{PV - \frac{FV}{(1 + r)^n}}{\sum_{i=1}^{n} \frac{1}{(1 + r)^i}} \] But it's easier to use the **Present Value of an Ordinary Annuity with a future value (balloon payment)** formula: \[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} + \frac{FV}{(1 + r)^n} \] Where: - \( PV \) = Present Value (amount financed) = $255,000 - \( FV \) = Residual Value = $100,000 - \( n \) = Number of monthly payments = 36 - \( r \) = Monthly interest rate = 8% / 12 = 0.6667% = 0.006667 We need to solve for PMT. #### Rearranged for PMT: \[ PMT = \frac{PV - \frac{FV}{(1 + r)^n}}{\frac{1 - (1 + r)^{-n}}{r}} \] ### **Step 3: Plug in the values** #### Calculate \((1 + r)^n\): \[ (1 + r)^n = (1 + 0.006667)^{36} \] First, calculate \(1 + r\): \[ 1 + 0.006667 = 1.006667 \] Now, raise to the 36th power: \[ 1.006667^{36} \approx e^{36 \times \ln(1.006667)} \approx e^{36 \times 0.006645} \approx e^{0.2392} \approx 1.2701 \] #### Calculate \((1 + r)^{-n}\): \[ (1 + r)^{-n} = 1 / 1.2701 \approx 0.7873 \] #### Calculate denominator: \[ \frac{1 - (1 + r)^{-n}}{r} = \frac{1 - 0.7873}{0.006667} = \frac{0.2127}{0.006667} \approx 31.91 \] #### Calculate \(FV/(1 + r)^n\): \[ \frac{FV}{(1 + r)^n} = \frac{100,000}{1.2701} \approx 78,740 \] #### Calculate numerator: \[ PV - \frac{FV}{(1 + r)^n} = 255,000 - 78,740 = 176,260 \] #### Now, PMT: \[ PMT = \frac{176,260}{31.91} \approx 5,524.30 \] ### **Final Answer** #### **Monthly Repayments:** \[ \boxed{\$5,524.30} \] --- **Summary Table:** | Item | Amount | |-------------------------------------- |---------------| | Amount financed (purchase + fees) | $255,000 | | Interest rate (monthly) | 0.6667% | | Lease term | 36 months | | Residual value (end of lease) | $100,000 | | **Monthly lease repayment** | **$5,524.30** | --- **If you need an Excel formula for this:** ```excel =PMT(8%/12, 36, -255000, 100000) ``` This will give you the same answer.

**1. Introduction** Understanding the financial aspects of equipment acquisition involves several key concepts, including the valuation of assets, financing mechanisms, and the calculation of repayment schedules. When a business seeks to purchase equipment through financing, it must consider the total purchase cost, additional expenses such as stamp duty and registration, and the terms of the financing arrangement. A financial lease is a common method of financing where the lessee pays periodic lease payments over an agreed period, often with a residual value at the end of the lease term. To determine the monthly repayments under such a lease, it is essential to understand the principles of present value calculations, interest rate application, and amortization with a residual value. The interest rate (cost of borrowing) impacts the repayment amount, and the residual value affects the amount financed after accounting for the expected residual worth of the asset. This introduction lays the foundation for understanding how to approach the calculation of lease repayments, emphasizing the importance of the total amount financed, interest rate, lease period, and residual value. It also underscores the necessity of translating these concepts into mathematical formulas to arrive at precise repayment figures. **Explanation:** This conceptual background is vital as it connects the physical purchase and associated costs with the financial models required to determine repayment schedules. Recognizing the roles of interest rates, residual values, and present value calculations enables a clear understanding of how lease payments are structured and how they relate to the initial investment. This comprehension ensures that subsequent calculations are grounded in fundamental financial principles, facilitating accurate and meaningful results. --- **2. Presentation of Relevant Formulas Required To Solve The Question** **a) Present Value of an Ordinary Annuity (Loan Repayments):** \[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \] - *Description:* This formula calculates the current value of a series of fixed periodic payments (PMT) over \(n\) periods at a periodic interest rate \(r\). It is applicable when a series of payments are made to amortize a loan. **b) Present Value of a Future Sum (Residual Value):** \[ PV_{residual} = \frac{FV}{(1 + r)^n} \] - *Description:* This formula discounts the residual value \(FV\) to its present worth at the start of the lease, considering the interest rate \(r\) over \(n\) periods. **c) Calculating the Periodic Payment (PMT) with Residual Value:** \[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} + \frac{FV}{(1 + r)^n} \] - *Rearranged to solve for PMT:* \[ PMT = \frac{PV - \frac{FV}{(1 + r)^n}}{\frac{1 - (1 + r)^{-n}}{r}} \] - *Description:* This formula determines the periodic payment needed to amortize the full amount \(PV\), considering the residual value \(FV\) at the end. **d) Summary of Variables:** | Variable | Description | Units | |------------|--------------|--------| | \(PV\) | Present value of the financed amount | Dollars ($) | | \(PMT\) | Periodic (monthly) payment | Dollars ($) | | \(FV\) | Residual value at lease end | Dollars ($) | | \(r\) | Monthly interest rate | Decimal (e.g., 0.006667) | | \(n\) | Total number of periods (months) | Number (36) | **Explanation:** These formulas are fundamental in calculating lease payments that account for interest and residual value. The present value of an annuity captures the total value of periodic payments, while discounting the residual value aligns future expectations with current valuation. Together, these formulas facilitate precise computation of monthly lease installments, which are essential for financial planning and decision-making. --- **3. A Detailed Step-by-Step Solution** **Step 1: Calculate the Total Amount to be Financed** - **Purchase Price:** \$250,000 - **Stamp Duty & Registration Expenses:** \$5,000 - **Total Cost (PV):** \$250,000 + \$5,000 = **\$255,000** **Explanation:** The total amount to be financed includes the purchase price plus any additional costs directly attributable to acquiring the asset. These expenses are incorporated into the financed amount to reflect the true initial investment. --- **Step 2: Determine the Monthly Interest Rate and Total Number of Payments** - **Annual Fixed Interest Rate:** 8% - **Monthly Interest Rate (\(r\)):** 8% / 12 = 0.6667% = 0.006667 - **Lease Term:** 3 years = 36 months (\(n\)) **Explanation:** Converting the annual interest rate into a monthly rate aligns with the monthly payment schedule. The total number of payments is calculated based on the lease duration in months, which influences the amortization schedule. --- **Step 3: Calculate the Present Value of the Residual Value** - **Residual Value at Lease End:** \$100,000 \[ PV_{residual} = \frac{100,000}{(1 + 0.006667)^{36}} \] Calculating \((1 + 0.006667)^{36}\): \[ (1.006667)^{36} \approx e^{36 \times \ln(1.006667)} \approx e^{36 \times 0.006645} = e^{0.2392} \approx 1.2701 \] Then, \[ PV_{residual} = \frac{100,000}{1.2701} \approx 78,740 \] **Explanation:** Discounting the residual value to present terms accounts for the time value of money, reflecting how much the residual worth is worth today. --- **Step 4: Calculate the Annuity Factor** \[ \frac{1 - (1 + r)^{-n}}{r} \] Calculate \((1 + r)^{-n}\): \[ (1.006667)^{-36} \approx 1 / 1.2701 \approx 0.7873 \] Then, \[ 1 - 0.7873 = 0.2127 \] and dividing by \(r\): \[ \frac{0.2127}{0.006667} \approx 31.91 \] **Explanation:** This factor converts a periodic payment into its present value, considering the interest rate and the number of periods. It is essential for calculating the amount of each installment. --- **Step 5: Compute the Monthly Lease Payment (PMT)** Using the rearranged formula: \[ PMT = \frac{PV - PV_{residual}}{\text{Annuity Factor}} \] Where: \[ PV = 255,000,\quad PV_{residual} = 78,740,\quad \text{Annuity Factor} = 31.91 \] Calculate numerator: \[ 255,000 - 78,740 = 176,260 \] Calculate PMT: \[ PMT = \frac{176,260}{31.91} \approx 5,524.30 \] **Explanation:** The monthly payment is derived by amortizing the financed amount minus the discounted residual value over the lease period, considering the applicable interest rate. --- **Final Result:** \[ \boxed{\$5,524.30 \text{ per month}} \] This amount ensures that all payments, including interest and residual value, are covered over the 3-year lease period. --- **Summary:** The equipment purchase, including stamp duty and registration costs, is financed through a 3-year financial lease with a residual value of \$100,000. Considering the interest rate of 8% per annum, the monthly repayment amount is approximately **\$5,524.30**. This structured repayment plan balances the initial costs, interest obligations, and residual value to facilitate the acquisition of the equipment in a financially sustainable manner.