What is the effective interest rate on the same truck and loan terms except no buy-out option at the end of the lease and $40,000 truck FMV after 5 years?

To find the effective interest rate for the loan given the truck's final market value (FMV) and other terms, we need to gather some information and use the following approach. Here's the general setup of the problem based on your description:

1. **Truck Cost**: Let's denote the initial cost of the truck.
2. **FMV at End of Lease**: $40,000.
3. **Loan Duration**: 5 years.
4. **No Buy-out Option**: Indicates that the truck's residual value is taken into account in the payments.
5. **Effective Interest Rate**: This is the annual rate that results in the equivalent periodic payments over the lease term.

The problem can be interpreted as a calculation involving the present value of an annuity with a residual value at the end of the term. This involves setting up an equation for the annuity payment using the loan's structure.

### **Step-by-step Approach**:

1. **Loan Amount**: Let's denote the loan principal as $ P $. This is the truck's initial value.

2. **Residual Value (FMV)**: The final market value of the truck is $40,000.

3. **Annuity Payments**: We need to find the effective interest rate $ r $ that would equate the present value of the annuity (monthly payments) plus the present value of the residual value to the initial loan amount.

4. **Effective Interest Rate Calculation**:
   - The relationship can be expressed using the formula for the present value of an annuity with a future residual value:
     \[
     P = \frac{PMT \times (1 - (1 + r)^{-n})}{r} + \frac{FV}{(1 + r)^n}
     \]
     Where:
     - $ P $ = Initial cost of the truck
     - $ PMT $ = Monthly payment
     - $ r $ = Monthly interest rate (the effective interest rate is the annualized version of this)
     - $ n $ = Total number of payments (months)
     - $ FV $ = Residual value at the end of the lease ($40,000)
   
5. **Solve for $ r $**: The equation requires iterative or numerical methods to solve for the monthly rate $ r $. Once $ r $ is found, the annual effective interest rate can be calculated as:
   \[
   \text{Effective Annual Rate} = (1 + r)^{12} - 1
   \]

Could you please provide additional details such as the truck's initial cost and the monthly payment amount? These are needed to proceed with the specific numerical calculation.

Learn how to calculate the effective interest rate for a truck lease with a $40,000 final market value after 5 years and no buy-out option. This guide covers the use of present value of annuity formulas, loan amortization concepts, and effective interest rate calculations.

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