What is the effective interest rate on a lease for a $170,000 truck with $0 down, $4,000 monthly payment over 5 years, and no buy-out option at the end of the lease and $40,000 truck FMV after 5 years?

To calculate the effective interest rate (also known as the internal rate of return or IRR) on a lease, we need to find the rate at which the present value of the lease payments equals the net cost of the truck. Here's how to approach the problem:

### Step-by-Step Solution:

1. **Initial Information:**
   - Truck cost (Principal): $170,000
   - Monthly lease payment: $4,000
   - Lease term: 5 years (60 months)
   - Residual value (FMV after 5 years): $40,000
   - No down payment, no buy-out option.

2. **Cash Flow Calculation:**
   - Monthly payment of $4,000 is made for 60 months.
   - There is no down payment or upfront payment, so the cash flow starts from $0 at time zero.
   - At the end of the lease, the truck is returned with an estimated fair market value (FMV) of $40,000.

3. **Net Cash Outflow:**
   The lessee's total cash outflow consists of the monthly payments:
   \[
   \text{Total Payments} = 4,000 \times 60 = 240,000
   \]

4. **Net Cost:**
   The effective cost of the lease is calculated by subtracting the residual value from the total payments made:
   \[
   \text{Net Cost} = 240,000 - 40,000 = 200,000
   \]
   This net cost represents what the lessee paid above the residual value.

5. **Effective Interest Rate Calculation:**
   To find the effective interest rate, we need to solve for $ r $ in the equation:
   \[
   \sum_{n=1}^{60} \frac{4,000}{(1 + r/12)^n} = 170,000 - \frac{40,000}{(1 + r/12)^{60}}
   \]
   where:
   - $ r $ is the annual effective interest rate.
   - $ \frac{r}{12} $ represents the monthly interest rate.

6. **Solving for the Interest Rate:**
   The effective interest rate is calculated using financial tools or software like Excel, using the IRR function with the given cash flows. I’ll calculate this for you.

Let me compute the effective interest rate.The effective annual interest rate on the lease is approximately **18.89%**.

This means that, considering the lease payments and the residual value, the lessee effectively pays an annual interest rate of 18.89% on the truck.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the effective interest rate change with a different monthly payment?
2. What if there were an initial down payment? How would that affect the interest rate?
3. How is the effective interest rate different from the nominal rate in leasing contracts?
4. Can the IRR method be used for other financial products, like loans or mortgages?
5. What would be the impact of a different residual value on the interest rate?

**Tip:** The effective interest rate (IRR) is sensitive to the timing and amount of cash flows, making it a useful tool for comparing different financing options.

Learn how to calculate the effective interest rate (IRR) for a lease on a $170,000 truck with no down payment and a $4,000 monthly payment over 5 years. Includes detailed steps using IRR and present value formulas, considering a $40,000 residual value at the end of the lease term.

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