Problem 17 - Algebra and Trigonometry

Question

Financing a Ring Mike buys a ring for his fiancee by paying \(\$ 30\) a month for one year. If the interest rate is 10\(\%\) per year, compounded monthly, what is the price of the ring?

Step-by-Step Solution

Answer

The price of the ring is approximately $354.65.

Step 1: Understand the Problem

Mike makes monthly payments of $30 for one year at an interest rate of 10% per year, compounded monthly. We want to find the present value, or the actual price, of the ring.

Step 2: Identify Formula

The problem involves an annuity due, where payments are made at the start of each period. The present value of an annuity due can be found using the formula: \[ PV = Pmt \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r) \]where \(Pmt\) is the payment amount, \(r\) is the monthly interest rate, and \(n\) is the number of payments.

Step 3: Calculate Monthly Interest Rate

The annual interest rate is 10%, so the monthly interest rate \(r\) is given by dividing the annual rate by 12 months:\[ r = \frac{0.10}{12} = 0.0083333 \]

Step 4: Calculate Number of Payments

Since Mike pays monthly for one year, the number of payments \(n\) is equal to 12.

Step 5: Substitute Values into Formula

Substitute \(Pmt = 30\), \(r = 0.0083333\), and \(n = 12\) into the formula:\[ PV = 30 \times \left( \frac{1 - (1 + 0.0083333)^{-12}}{0.0083333} \right) \times (1 + 0.0083333) \]

Step 6: Simplify and Solve

Perform the calculations:1. Calculate \((1 + 0.0083333)^{-12}\): \[ (1 + 0.0083333)^{-12} \approx 0.902095 \]2. Calculate \(\frac{1 - 0.902095}{0.0083333}\): \[ \frac{1 - 0.902095}{0.0083333} \approx 11.7288 \]3. Calculate \(PV\): \[ PV \approx 30 \times 11.7288 \times 1.0083333 \] \[ PV \approx 354.65 \]

Step 7: Interpret the Result

The present value, or the price of the ring, calculated from the annuity formula, is approximately $354.65.

Key Concepts

Present Value

The concept of present value is crucial when evaluating the worth of future cash flows in today's terms. In Mike's ring financing scenario, the present value represents the current cost of the ring, relative to his future monthly payments. Essentially, it tells Mike what the total value of his payments is in today's dollars, accounting for the fact that money now is more valuable than money later due to the potential earning of interest.
To calculate the present value of an annuity, you need to know:

• The periodic payment amount (in Mike's case, $30 monthly)
• The interest rate (measured per period)
• The total number of periods

The formula reflects that future payments are reduced by the interest rate to show what they are worth now. This insight helps individuals or businesses plan and evaluate financial decisions where future payments can't just be added up without considering the time value of money.

Compounded Interest

Compounded interest arises when interest is calculated on the initial principal and also on the accumulated interest of previous periods. For Mike's situation, the interest compounds monthly, which means each month's interest calculation is added to the next month's principal.
The monthly interest rate is found by dividing the annual rate by the number of months. Here, Mike's annual interest rate of 10% becomes a monthly rate by dividing by 12, yielding approximately 0.0083333 (or 0.83333%). This small percentage compounding every month slightly increases the total cost over time, compared to simple interest which would only apply to the initial principal.
This compounding effect makes understanding the actual interest costs more important, as compounded interest can significantly impact the overall cost of financing if not managed carefully.

Monthly Payments

Monthly payments in annuity calculations reflect consistent amounts paid over time, like Mike's $30 monthly payment for the ring. These payments are characterized by their regularity and are usually designed to cover both principal and interest components of a financial commitment.
When calculating these payments' effects, you often use them to determine either the present value, as seen here, or the future value, if you are looking to save or invest. In Mike's case, knowing his monthly payments allows him to establish a predictable budget and manage his cash flows effectively.
Utilizing monthly payments with an annuity formula is especially common in calculating loan repayments, saving plans, or evaluating financial products designed around regular contributions.