A contract requires lease payments of $400 at the beginning of every month for 7 years.
a. What is the present value of the contract if the lease rate is 6.93% compounded annually?
Round to the nearest cent
b. What is the present value of the contract if the lease rate is 6.93% compounded daily?
Round to the nearest cent

To find the present value (PV) of the lease payments, we can use the formula for the present value of an annuity due since payments are made at the beginning of each period.

### a. Present Value with Annual Compounding

1. **Identify parameters:**
   - Payment (PMT) = $400
   - Number of payments (n) = 7 years × 12 months/year = 84 months
   - Annual interest rate (r) = 6.93% = 0.0693
   - Monthly interest rate (i) = r / 12 = 0.0693 / 12 ≈ 0.005775

2. **Formula for Present Value of an Annuity Due:**
   \[
   PV = PMT \times \left(1 + i\right) \times \frac{1 - (1 + i)^{-n}}{i}
   \]
   Where $i$ is the monthly interest rate and $n$ is the total number of payments.

3. **Calculation:**
   \[
   PV = 400 \times (1 + 0.005775) \times \frac{1 - (1 + 0.005775)^{-84}}{0.005775}
   \]
   \[
   PV = 400 \times 1.005775 \times \frac{1 - (1.005775)^{-84}}{0.005775}
   \]
   \[
   PV \approx 400 \times 1.005775 \times 59.43894 \approx 400 \times 59.77379 \approx 23909.52
   \]

So, the present value when compounded annually is approximately **$23,909.52**.

### b. Present Value with Daily Compounding

1. **Identify parameters:**
   - Daily interest rate (r_d) = (1 + 0.0693)^(1/365) - 1 ≈ 0.0001831 (approximately)
   - Total number of days (t) = 7 years × 365 days/year = 2555 days

2. **Formula for Present Value of an Annuity Due with daily compounding:**
   \[
   PV = PMT \times (1 + r_d) \times \frac{1 - (1 + r_d)^{-n}}{r_d}
   \]
   Here, $n$ should be the total number of daily payments, which is the same as the total number of days in this context.

3. **Calculation:**
   \[
   PV = 400 \times (1 + 0.0001831) \times \frac{1 - (1 + 0.0001831)^{-2555}}{0.0001831}
   \]
   \[
   PV \approx 400 \times 1.0001831 \times \frac{1 - (1.0001831)^{-2555}}{0.0001831}
   \]
   \[
   PV \approx 400 \times 1.0001831 \times 5462.52 \approx 400 \times 5465.62 \approx 2186248.70
   \]

So, the present value when compounded daily is approximately **$21,862.48**.

### Summary:
- a. Present Value (Annual Compounding): **$23,909.52**
- b. Present Value (Daily Compounding): **$21,862.48**

If you would like any further details or have questions, let me know! Here are some related questions you might consider:

1. How does the frequency of compounding affect the present value?
2. What would the present value be at a different interest rate?
3. How can I calculate the future value of these payments?
4. What are the tax implications of lease payments?
5. How do lease payments compare to purchasing an asset outright? 

**Tip:** Always consider the impact of compounding frequency on the present value calculations; more frequent compounding generally leads to a lower present value of future cash flows.

A contract requires lease payments of $400 at the beginning of every month for 7 years. What is the present value of the contract if the lease rate is 6.93% compounded annually and daily?

Explore the calculation of the present value of lease payments of $400 made monthly for 7 years at a lease rate of 6.93%. This includes solutions for both annual and daily compounding, with the present values being approximately $23,909.52 and $21,862.48 respectively.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation