Problem 22 - 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 $344.73.

Step 1: Understanding the Exercise

Mike wants to buy a ring and will be paying $30 each month for one year. The interest rate is 10% per year, compounded monthly. We need to find out the initial principal or the price of the ring given this information.

Step 2: Converting Annual Interest Rate to Monthly

Since the interest rate is compounded monthly, we convert the annual interest rate to a monthly rate. The annual interest rate is 10%, so the monthly interest rate is \( \frac{10\%}{12} = \frac{0.10}{12} \approx 0.0083333 \) per month.

Step 3: Using the Present Value Formula

The present value of the annuity formula is used here: \( PV = Pmt \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \). Substitute \( Pmt = 30 \), \( r = 0.0083333 \), and \( n = 12 \) into the formula.

Step 4: Calculating Present Value

Substitute the values into the formula: \( PV = 30 \times \left( \frac{1 - (1 + 0.0083333)^{-12}}{0.0083333} \right) \). Calculate \( (1 + 0.0083333)^{-12} \), subtract from 1, divide by 0.0083333, then multiply by 30.

Step 5: Final Price of the Ring

Evaluating the expression gives us \( PV \approx 344.73 \). This is the present value of all the payments and represents the price of the ring.

Key Concepts

Compound Interest

Compound interest is a powerful finance concept where interest is calculated not just on the initial principal, but also on the accumulated interest from previous periods. In simpler terms, you earn interest on your interest.

Here's how it works:

• Every period, the interest is added to the principal amount.
• The next period's interest is calculated on this new sum.

In Mike's ring financing case, the interest rate is 10% per year, compounded monthly. This means the interest is applied each month rather than waiting for the end of the year. By compounding monthly, the effective rate of growth on the original loan amount is slightly increased, meaning the actual cost of the ring will be higher than if the interest were compounded annually.

Monthly Payment Calculation

Monthly payment calculation is a frequent requirement when handling loans, annuities, or financing agreements. In Mike's situation, although he pays a fixed $ 30 a month, the real value of these payments changes because of the compound interest.

To calculate monthly payments:

• Determine the monthly interest rate (from annual rate).
• Multiply this rate by the outstanding loan balance or use it in a formula like the annuity formula.

In essence, while the dollar amount Mike pays is constant, the effective cost or what it represents in terms of the loan's principal changes as the interest compounds over time.

Present Value Formula

The present value (PV) formula is a fundamental part of understanding loans or any kind of time-value-of-money calculations. It helps to determine the total value today of a series of future payments, considering interest. The general form of the present value of an annuity formula is: \[ PV = Pmt \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]Where:

• \(Pmt\) = the payment amount per period
• \(r\) = the interest rate per period
• \(n\) = the number of periods

This formula helps convert future cash payments into a single lump sum amount in today's terms. In Mike's case, this formula is used to find how much he effectively owes today (the ring's price) by considering his future $30 monthly payments and the interest rate.

Interest Rate Conversion

Interest rate conversion is essential for calculations because financial products often quote rates annually but are compounded more frequently, like monthly or daily.

Here's how to convert an annual rate to a monthly rate:

• Divide the annual percentage rate (APR) by the number of compounding periods in a year. For monthly compounding, this number is 12.
• Thus, a 10% annual interest rate becomes approximately 0.833% per month, or formally \( \frac{10\%}{12} = 0.0083333 \) per month.

By ensuring the rate matches the payment period, calculations are kept accurate and reflective of real-life financial conditions. In the exercise, converting the rate was crucial to using the present value formula correctly.