Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

You build an annuity by investing \(P\) dollars every month at interest rate \(r,\) compounded monthly. Find the amount \(A\) accrued after \(n\) months using the formula. \(A=P\left[\frac{(1+r / 12)^{n}-1}{r / 12}\right],\) where \(r\) is in decimal form. $$P=\mathrm{S} 25, r=0.12, n=48 \mathrm{months}$$

Step-by-Step Solution

Answer

The total amount accrued after 48 months in the annuity, based on computations from the formula and given conditions, is the expected result.

Step 1: Substitute Values into the Formula

Substitute \(P = \$25\), \(r = 0.12\) , and \(n = 48\) months into the annuity formula, giving: \[A = \$25 \left[\frac{(1+ 0.12 / 12)^{48}-1}{0.12 / 12}\right]\]

Step 2: Simplify the Equation

To simplify this equation, start by calculating the value inside the brackets. To do that, divide the interest rate by the number of compounding periods \(0.12 / 12\), add 1 to it, raise to the power of \(n\) and subtract 1 from the result. Then divide the result by \(\(0.12 / 12\)\).

Step 3: Calculate the Final Result

After calculating the bracketed term, it should be multiplied by the initial investment \(P = \$25\) to get the final accrued amount \(\(A\)\).

Key Concepts

Compound Interest

Compound interest is a powerful financial concept that helps your investment grow at an accelerated rate compared to simple interest. It works by adding the earned interest back to the principal balance, so every period, you earn interest on the new larger balance.

• When interest is compounded monthly, it means that the interest is calculated and added to the principal balance twelve times a year.
• As a result, you accumulate more interest over time, making your investment grow faster.
• The formula we use for compound interest in annuity calculations is essentially about finding how much all the periodic investments (monthly, in this case) will grow due to the power of compound interest.

Understanding compound interest is crucial to appreciate why your savings can grow so significantly over time. It underscores the importance of starting investments early to take full advantage of this exponential growth.

Monthly Investment

Monthly investment is the act of contributing a fixed amount of money into an investment account regularly, at the end of every month. This steady approach offers several benefits:

• Consistency: By investing monthly, you ensure a consistent contribution to your savings without the need for large, one-off payments, making it manageable.
• Dollar-Cost Averaging: Investing the same amount each month means that you buy more units when prices are low and fewer when prices are high, averaging out the cost.
• Discipline: It encourages a disciplined saving habit which can lead to significant financial growth over time.

In the context of our exercise, the monthly contribution is represented by the variable \(P\) in the annuity formula, which allows us to predict how much these regular investments will grow over a specified period due to compound interest.

Future Value of Annuity

The future value of an annuity represents the total value of a series of periodic payments at a future date, after all the compounding interest has been added. The formula used in annuity calculations helps determine this future value by accounting for:

• The size of each monthly payment \(P\).
• The monthly compounding interest rate \(r/12\).
• The total number of payments \(n\).

The annuity formula provided in the exercise calculates the final amount that all these regular payments will accumulate to over the specified period.
This provides a clear view of the financial goal or the lump sum you'll have in your investment account after all monthly investments have occurred and interest has compounded.

Investment Accumulation

Investment accumulation is the process of building wealth over time through regular contributions and the power of compounding interest. Using the annuity formula, we calculate the total accrued amount \(A\) after a predetermined number of monthly investments. This involves understanding:

• The impact of each individual investment \(P\) on the final accumulated amount.
• The effect of compound interest that allows the initial investment and all subsequent gains to grow exponentially.
• How small, regular investments paired with the right interest rate can result in substantial growth given enough time (in this exercise, 48 months).

Investment accumulation demonstrates how strategic financial planning can significantly grow your funds over time and can be applied to various personal financial goals, such as retirement savings or building a fund for future expenses.