A lease valued at ​$30000 requires payments of ​$4500 every three months. If the first payment is due two years after the lease was signed and interest is 12​% compounded​ quarterly, what is the term of the​ lease?

To determine the term of the lease, we need to understand the situation clearly. We have:

- The lease value (present value, $ PV $) is $30,000.
- Payments ($ PMT $) of $4,500 are made every three months (quarterly).
- The first payment is due two years after the lease was signed.
- The interest rate is 12% compounded quarterly.

First, let’s convert the annual interest rate to a quarterly interest rate:
\[ r = \frac{12\%}{4} = 3\% = 0.03 \]

Given that the first payment is made two years after the lease was signed, this means the payments are deferred for 8 quarters (since there are 4 quarters in a year).

We need to find the number of payments, $ n $, such that the present value of these payments equals the lease value of $30,000. The present value formula for an annuity with deferred payments is:

\[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)^{-d} \]

where:
- $ PV $ is the present value ($30,000),
- $ PMT $ is the payment amount ($4,500),
- $ r $ is the quarterly interest rate (0.03),
- $ n $ is the number of payments,
- $ d $ is the number of quarters the payments are deferred (8).

Rewriting the equation to solve for $ n $:
\[ 30000 = 4500 \times \frac{1 - (1 + 0.03)^{-n}}{0.03} \times (1 + 0.03)^{-8} \]

First, calculate $ (1 + 0.03)^{-8} $:
\[ (1 + 0.03)^{-8} = (1.03)^{-8} \approx 0.7894 \]

Now substitute this back into the equation:
\[ 30000 = 4500 \times \frac{1 - (1 + 0.03)^{-n}}{0.03} \times 0.7894 \]

Next, simplify the equation:
\[ 30000 = 4500 \times \frac{1 - (1.03)^{-n}}{0.03} \times 0.7894 \]
\[ 30000 = 4500 \times 0.7894 \times \frac{1 - (1.03)^{-n}}{0.03} \]
\[ 30000 = 3552.3 \times \frac{1 - (1.03)^{-n}}{0.03} \]
\[ 30000 = 118410 \times (1 - (1.03)^{-n}) \]
\[ \frac{30000}{118410} = 1 - (1.03)^{-n} \]
\[ 0.2534 = 1 - (1.03)^{-n} \]
\[ (1.03)^{-n} = 1 - 0.2534 \]
\[ (1.03)^{-n} = 0.7466 \]

Now, solve for $ n $ by taking the natural logarithm:
\[ -n \ln(1.03) = \ln(0.7466) \]
\[ -n \times 0.0295588 = -0.29219 \]
\[ n = \frac{0.29219}{0.0295588} \]
\[ n \approx 9.88 \]

Since the number of payments must be an integer, we round up to 10. Therefore, the term of the lease is approximately 10 quarters (2.5 years) after the first payment is made.

### Summary
The term of the lease is 10 quarters (or 2.5 years) after the first payment, which starts 2 years after the lease was signed.

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How is the present value of an annuity calculated?
2. What is the significance of deferring payments in lease agreements?
3. How does the interest rate affect the total cost of a lease?
4. What are the differences between compounded interest and simple interest?
5. How do you convert annual interest rates to quarterly interest rates?
6. What other financial applications use the concept of present value?
7. How do lease agreements impact business accounting?
8. What are the benefits and drawbacks of leasing vs. buying equipment?

**Tip:** Always ensure to read the fine print of lease agreements to understand the terms, interest rates, and payment schedules accurately.

Learn how to calculate the term of a lease valued at $30,000 with quarterly payments of $4,500, starting two years after signing. Includes step-by-step solution using compound interest formulas. Suitable for financial professionals and analysts.

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