Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Mike's Sporting Goods sells elliptical trainers under two payment plans: cash or installment. Under the installment plan, the customer pays $$\$ 22 /$$ month over 3 yr with interest charged on the balance at a rate of \(18 \% /\) year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan.

Step-by-Step Solution

Answer

The cash price for an elliptical trainer is approximately $645.00, as it is equivalent to the present value of all the installment payments.

Step 1: Determine the installment term and interest information

The installment plan has the following information: - Monthly payment: $22 - Term of installment: 3 years (36 months) - Interest rate: 18% per year compounded monthly To find the cash price, we need to calculate the present value of all the monthly payments for 36 months.

Step 2: Calculate the present value of the installment payments

The formula for calculating the present value (PV) of a stream of cash flows with constant payments and interest rate is: PV = \(\frac{PMT \times (1 - (1 + i)^{-n})}{i}\), where - PMT = monthly payment - i = monthly interest rate (annual interest rate divided by 12) - n = number of monthly payments In our case, PMT = $22, the annual interest rate is 18%, so the monthly interest rate i = (18% / 12) = 0.18 / 12. There are 3 years of monthly payments, which means n = 3 × 12 = 36 monthly payments. Now let's substitute these values into the formula and calculate the present value of the installment payments.

Step 3: Substitute the values into the formula and solve for PV

PV = \(\frac{22 \times (1 - (1 + 0.18/12)^{-36})}{0.18/12}\) Now we compute the present value: PV ≈ $645.00 The present value of all the installment payments is approximately $645.00.

Step 4: Cash price

Since the cash price is equivalent to the price paid by a customer using the installment plan, the cash price for the Elliptical trainer is the present value of all the installment payments: Cash price ≈ Present value = $645.00 Therefore, the cash price for an elliptical trainer is approximately $645.00.

Key Concepts

Understanding the Time Value of Money

The concept of the time value of money is fundamental in understanding why a dollar today is worth more than a dollar in the future. It's based on the idea that money available now can be invested to earn interest, thereby growing over time. As a result, when we are comparing or calculating the value of financial transactions that span over various periods, this principle must be considered to ensure an accurate valuation.

For instance, when someone opts for installment payments over a lump sum cash option, they're essentially deferring payment into the future. The cash price is determined by calculating the present value of these future installment payments, effectively 'discounting' them back to their value in today's dollars. Without taking the time value of money into account, one could not correctly evaluate the true cost of such deferred payment plans.

Installment Payment Plans

An installment payment plan allows customers to pay for goods or services over time rather than in a single lump sum. These plans often include interest to compensate the seller for the delayed receipt of funds. The interest is critical because it recognizes the time value of money — that is, the seller could have invested the full amount immediately if paid in cash.

From the consumer's perspective, an installment plan may be more manageable because it spreads out the financial burden. However, the tradeoff comes in the form of total price paid due to the added interest. For anyone considering an installment plan, a key factor to weigh is whether the benefits of spreading out the payments outvalue the additional cost due to interest charges.

Interest Rate Compounded Monthly

When dealing with loans or installment payment plans, the interest rate compounded monthly drastically affects the total amount of money paid over time. Compounding is the process where interest is earned on both the initial principal and the accumulated interest from previous periods. Compounding monthly means that the interest is calculated and added to the balance once every month.

Even if the annual interest rate remains the same, more frequent compounding results in a higher effective annual rate and, consequently, a higher amount of interest over the life of the loan. This concept emphasizes the importance of understanding the terms of financial agreements; the same advertised annual interest rate can lead to very different costs when compounded differently.

Financial Mathematics: A Tool for Informed Decisions

In the realm of financial mathematics, we use mathematical formulas and calculations to assess the value and viability of financial products or services. This includes evaluating investments, determining loan payments, and calculating present and future values of cash flows. In the example of the elliptical trainer, we used a present value calculation to determine the cash price equivalent to an installment payment plan. By computing present value, one can compare different payment options on a common ground.

Understanding the principles of financial mathematics is not just academically beneficial but also crucial for making informed personal finance decisions. The ability to accurately evaluate and compare different payment and investment options can lead to significant savings and more beneficial financial outcomes over time.