Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Lowell Corporation wishes to establish a sinking fund to retire a $$\$ 200,000$$ debt that is due in 10 yr. If the investment will earn interest at the rate of \(9 \% /\) year compounded quarterly, find the amount of the quarterly deposit that must be made in order to accumulate the required sum.

Step-by-Step Solution

Answer

The required quarterly deposit for Lowell Corporation to retire the $200,000 debt in 10 years with a 9% annual interest rate compounded quarterly is approximately $1,902.93.

Step 1: Convert annual interest rate to quarterly interest rate

To convert the 9% annual interest rate to a quarterly interest rate, we divide it by 4. \(r = \frac{9\%}{4} = \frac{0.09}{4} = 0.0225\) The quarterly interest rate (r) is 0.0225 or 2.25%.

Step 2: Calculate the number of payments

There will be 4 payments per year for 10 years, so the total number of payments (n) is: \(n = 10 \cdot 4 = 40\)

Step 3: Rearrange the formula to solve for the quarterly deposit (P)

We need to rearrange the future value formula to find P: \[P = FV \cdot \frac{r}{(1 + r)^n - 1}\]

Step 4: Plug in the values and calculate the quarterly deposit (P)

Now, we can plug in the values for FV, r, and n into the formula: \(P = 200,000 \cdot \frac{0.0225}{(1 + 0.0225)^{40} - 1}\) First, calculate the term inside the brackets: \((1 + 0.0225)^{40} - 1 \approx 2.3653\) Now, calculate the quarterly deposit: \(P = 200,000 \cdot \frac{0.0225}{2.3653} \approx 1902.93\) The required quarterly deposit for Lowell Corporation is approximately $1,902.93.

Key Concepts

Understanding Compound Interest

Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. It's a fundamental concept in finance that allows investments to grow over time.

Consider it as 'interest on interest.' It can be thought of as a snowball effect because, as the principal grows, the amount of interest accrued increases, leading to even more interest being earned. This is the power of compounding in action. It's an incentive to save early and often, as the effects of compounding become more pronounced over time.

For those managing loans or investments, understanding how compound interest works is crucial. It not only affects savings but also the overall cost of borrowing money. Compound interest can significantly increase the future value of a sum of money, making it a vital component for anyone looking to save for future goals, such as education or retirement.

Quarterly Compound Interest Explained

When discussing quarterly compound interest, we're breaking down the compounding period into quarters, or a four-part year. Instead of interest being added to the principal just once per year, it happens every quarter which means four times a year.

By increasing the frequency of compounding, the investment grows faster than with annual compounding, because interest gets the opportunity to earn interest in itself within the year. Even though the interest might seem small, compounding quarterly quietly amplifies the earnings over time. That's why when computing future values or required savings, like in the Lowell Corporation exercise, it's important to convert annual rates to quarterly rates to get a precise calculation.

The user must remember that the more frequently interest is compounded, the greater the effect on the future value of your investments. Therefore, if given a choice, and all else being equal, a quarterly compounding option can result in higher returns than an annual compounding one, due to the more frequent application of the interest rate.

Future Value of Annuities and its Significance

An annuity is a series of equal payments made at regular intervals over a period of time. The future value of an annuity calculates the value of this series of payments at a specific point in the future, taking into account compound interest.

The concept is incredibly important when planning for scenarios where regular payments are involved, such as retirement contributions, loan repayments, or, just like in Lowell Corporation's case, setting aside money in a sinking fund. An annuity can be a powerful tool to ensure that a large sum of money will be available when you need it.

To determine the future value of an annuity, you must understand the timing of the payments (e.g., quarterly, annually) and the compounding frequency. Having this clear can help someone plan their finances effectively, ensuring that enough money is saved today to meet future financial obligations. This future value annuity equation is versatile and can be tailored to match any saving or investment strategy, which helps individuals and businesses like Lowell Corporation to accurately forecast their financial situations.