Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you are saving to buy some cattle. You plan to put \(\$ 200\) into an account every month for the next three years ( 36 deposits) to pay for the cows. You put your money into an account paying interest of \(4.5 \%\) per year compounded monthly. Immediately after the 36 th deposit, how much money will you have in your cattle fund?

Step-by-Step Solution

Answer

After 36 deposits, you will have approximately \$8165.69 in your cattle fund.

Step 1: Understanding Annuity Formula

The future value of an ordinary annuity formula is \( FV = P * [(1 + r/n)^(nt) - 1] / (r/n) \) where P is the periodic payment, r is the annual interest rate, n is the number of compounding periods in a year, and t is the time in years.

Step 2: Substituting the Values

Substitute the known values into the formula. P = 200, r = 4.5/100 = 0.045, n = 12 (monthly compounding), and t = 3.

Step 3: Calculating Future Value

Perform mathematical operations in the formula: \( FV = 200 * [(1 + 0.045/12)^(12*3) - 1] / (0.045/12) \)

Step 4: Solve for FV

On calculating, we find that FV is approximately \$8165.69, which will be the total money saved for the cattle fund after 36 deposits.

Key Concepts

Compound Interest Explained

Compound interest is a fascinating concept in financial mathematics that describes how your initial investment or savings grow over time. It works by calculating interest not only on the original amount of money but also on the accumulated interest from previous periods. This means your money grows at an increasing rate, allowing your investments to benefit from the magic of compounding.

For someone saving money, like in the case of saving to buy cattle, compounding usually occurs periodically, such as monthly or annually. The formula that incorporates compound interest for an annuity looks a bit complex, but the idea is quite simple. You multiply the fixed periodic payments into a formula that accounts for both growth due to compounding and the number of deposits you make.

Key points to remember about compound interest include:

• It can significantly impact the future value of your savings or investments.
• Frequent compounding generally increases the amount of interest earned.
• It's essential to understand the compounding period, as this dictates how often the interest is added to the principal.

Understanding Financial Mathematics

Financial mathematics provides us with the tools and equations needed to make informed financial decisions. In the context of savings and investments, it helps predict future values, understand payment schedules, and calculate returns based on interest rates.

An important formula used in financial mathematics is related to finding the future value of an annuity. This formula, as mentioned in the solution, helps determine how much a series of regular payments will grow over time, considering a specific interest rate and compounding frequency.

To use these mathematical principles efficiently, consider the following steps:

• Identify the parameters involved, such as payment amount, interest rate, and compounding period.
• Use the correct formula that applies to your specific financial scenario.
• Input the values correctly and perform calculations to find the desired outcome, such as the future value of your savings.

Understanding and applying financial mathematics can help you make better financial decisions, enabling you to achieve your savings goals effectively.

Planning a Savings Plan

A savings plan is crucial for achieving financial goals, like buying cattle as in our example. It involves setting aside a portion of your income regularly to build up a fund for future use. To create an effective savings plan, you should consider the following elements:

Firstly, decide on a specific goal, such as the amount needed for your future purchase and the timeline within which you wish to achieve it.

Next, determine how much you can realistically save each month. This might involve assessing your current financial situation and making any necessary adjustments to your budget.

Once you have your goal and savings amount set, select a financial account that offers a competitive interest rate. The choice of account is important as a better interest rate can help your savings grow faster through compound interest.

Finally, stick to your savings plan by making regular deposits, as scheduled. This discipline, combined with the power of compound interest, will help you reach your financial target in the planned timeline.