The maker of cardboard boxes leases a warehouse and pays $8,000 at the beginning of each month for 5 years. If interest rates are 2.75% compounded monthly, what is the present value (in dollars) of the payments? (Round your answer to the nearest cent.)

We need to calculate the present value of a series of monthly payments of $ 8,000 $ made at the beginning of each month for 5 years, with an interest rate of 2.75% compounded monthly.

### Step 1: Define the Variables
- **Monthly payment**: $ P = 8,000 $
- **Interest rate per month**: 
  \[
  i = \frac{2.75\%}{12} = \frac{0.0275}{12} = 0.00229167
  \]
- **Number of payments (months)**: 
  \[
  n = 5 \times 12 = 60
  \]

Since the payments are made at the beginning of each month, this is an **annuity due**.

### Step 2: Formula for Present Value of an Annuity Due
The present value $ PV $ of an annuity due can be calculated using the formula:
\[
PV = P \times \left( \frac{1 - (1 + i)^{-n}}{i} \right) \times (1 + i)
\]
Where:
- $ P $ is the monthly payment,
- $ i $ is the interest rate per month,
- $ n $ is the number of payments.

### Step 3: Calculation
Let's substitute the known values into the formula.

\[
PV = 8,000 \times \left( \frac{1 - (1 + 0.00229167)^{-60}}{0.00229167} \right) \times (1 + 0.00229167)
\]

Now, let's compute the value of the present value.The present value of the payments is approximately $ 449,009.84 $ dollars, rounded to the nearest cent.

Would you like more details on the calculation, or do you have any questions?

Here are five related questions you might explore:
1. How does increasing the interest rate affect the present value of the payments?
2. What would the present value be if the payments were made at the end of each month?
3. How would a different lease term (e.g., 10 years) impact the present value?
4. How does compounding frequency (e.g., daily vs. monthly) influence the result?
5. What if the monthly payment amount changes over time?

**Tip:** The present value of an annuity due is always higher than that of an ordinary annuity, because payments are made at the beginning of each period.

Calculate the present value of monthly $8,000 lease payments made at the beginning of each month for 5 years, with a 2.75% interest rate compounded monthly. Learn about the annuity due formula and how interest rates affect the value of long-term payments. Suitable for high school and college students.

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