The maker of cardboard boxes leases a warehouse and pays $5,000 at the beginning of each month for 5 years. If interest rates are 4.25% compounded monthly, what is the present value (in dollars) of the payments? (Round your answer to the nearest cent.)

To determine the present value of the lease payments, we can treat it as an annuity problem, where the payments are made at the beginning of each period (each month). This is an **annuity due** situation, as payments occur at the start of each month.

### Formula for Present Value of an Annuity Due:
\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
where:
- $ PV $ = Present Value
- $ P $ = Payment per period ($5,000)
- $ r $ = Monthly interest rate (annual rate divided by 12)
- $ n $ = Total number of payments (number of months)

### Given Data:
- Monthly payment $ P = 5,000 $
- Interest rate $ 4.25\% $ annually, so monthly interest rate $ r = \frac{4.25}{12 \times 100} = 0.00354167 $
- Number of months $ n = 5 \text{ years} \times 12 = 60 $

### Step-by-step Calculation:
1. **Monthly interest rate**: 
   \[
   r = 0.0425 / 12 = 0.00354167
   \]

2. **Number of payments (months)**: 
   \[
   n = 5 \times 12 = 60
   \]

3. **Present Value using the formula**:
   \[
   PV = 5,000 \times \left( \frac{1 - (1 + 0.00354167)^{-60}}{0.00354167} \right) \times (1 + 0.00354167)
   \]

Now, calculating this step-by-step:

1. Calculate $ (1 + 0.00354167)^{-60} $:
   \[
   (1.00354167)^{-60} \approx 0.812996
   \]

2. Subtract from 1:
   \[
   1 - 0.812996 = 0.187004
   \]

3. Divide by $ r $:
   \[
   \frac{0.187004}{0.00354167} \approx 52.8012
   \]

4. Multiply by $ (1 + 0.00354167) $:
   \[
   PV = 5,000 \times 52.8012 \times 1.00354167 \approx 265,278.88
   \]

### Conclusion:
The present value of the payments is approximately **$265,278.88**.

Would you like further details on this calculation or have any questions?

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Here are 5 related questions:
1. How would the present value change if the interest rate were higher or lower?
2. What is the difference between an annuity due and an ordinary annuity?
3. How can this formula be applied to other types of loans or lease agreements?
4. How does the compounding frequency affect the present value in annuity problems?
5. How would inflation affect the future value of money in such lease agreements?

**Tip:** When dealing with annuity problems, it's crucial to distinguish between an **annuity due** (payments at the beginning of the period) and an **ordinary annuity** (payments at the end of the period).

Calculate the present value of $5,000 monthly lease payments over 5 years with a 4.25% interest rate compounded monthly. This detailed solution covers key financial math concepts like annuities due and compound interest, suitable for college-level financial mathematics.

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