Problem 37 - Calculus

Question

Assume \(10 \%\) interest (so \(a=1+i=1.1)\). With annual compounding of deposits \(s=\$ 1000\), what is the balance in 20 years?

Step-by-Step Solution

Answer

The balance after 20 years is $57,275.

Step 1: Understanding the Variables and Formula

To find the balance in 20 years, we need to use the formula for the future value of an annuity. The formula is:\[ A = s \times \frac{(1+i)^n - 1}{i} \]where:- \( A \) is the future value of the annuity.- \( s \) is the amount of each deposit, which is $1000.- \( i \) is the interest rate per period, which is 0.10.- \( n \) is the number of periods, which is 20 years.

Step 2: Plugging in the Values

Now substitute the given values into the formula:\[ A = 1000 \times \frac{(1.1)^{20} - 1}{0.1} \]This will allow us to calculate the future value after 20 years.

Step 3: Calculating the Power of (1.1) to 20

Calculate \((1.1)^{20}\). Using a calculator:\[ (1.1)^{20} = 6.7275 \]This is the factor by which the investment grows after 20 years.

Step 4: Simplifying the Future Value Formula

Substitute \((1.1)^{20} = 6.7275\) back into the future value formula:\[ A = 1000 \times \frac{6.7275 - 1}{0.1} \]So now, subtract 1 from 6.7275, and divide by 0.1.

Step 5: Final Calculation of Future Value

Simplify the expression:\[ A = 1000 \times \frac{5.7275}{0.1} \]\[ A = 1000 \times 57.275 \]\[ A = 57275 \]This means the future value of the annuity, or the balance after 20 years, is $57,275.

Key Concepts

Future Value of an Annuity Formula

The future value of an annuity formula is a fundamental tool in financial mathematics. It helps you calculate how much money you will gain from regular deposits over a certain period, considering a fixed interest rate. The formula is expressed as:\[A = s \times \frac{(1+i)^n - 1}{i}\]Where:

• \(A\) represents the future value of all annuity payments combined.
• \(s\) is the amount of each regular deposit.
• \(i\) refers to the interest rate per period.
• \(n\) denotes the total number of periods.

By applying this formula, investors can estimate the accumulated value of their deposits, making it easier to plan for financial goals. Understanding each part of the formula is crucial, as you can see how changes in deposit amounts, interest rates, or time periods affect the overall future value.

Interest Rate

Interest rate is the percentage at which money grows over time. It is a key factor in annuity calculations. In our exercise, the interest rate is 10% annually.
Here’s why it matters:

• A higher interest rate means your investment grows faster, increasing the future value of your annuity.
• Conversely, a lower interest rate slows down growth, resulting in a smaller future value.

Different financial products may offer various interest rates. It's essential to compare them to determine the best option for maximizing returns. When working with annuities, the interest rate is crucial, as it exponentially impacts the growth over time.

Compound Interest

Compound interest allows your investments to grow at an increasing rate. With compound interest, you earn interest on both the initial principal and the accumulated interest from previous periods.
In our exercise, deposits are compounded annually. This means the interest earned each year is added to the principal, and the next interest calculation will include this accumulated amount. Consequently:

• Your investment continuously grows, leading to a larger future value over long periods compared to simple interest, which only calculates interest on the principal amount.
• The more frequently interest compounds, the higher the future value of the investment.

Understanding compound interest is essential in financial mathematics, as it reflects real-world scenarios where interest earns on itself, maximizing returns over time.

Financial Mathematics

Financial mathematics involves applying mathematical techniques to solve financial problems. It’s pivotal in making informed decisions about investments and understanding various financial instruments.
Key components include:

• Calculating present and future values of investments, helping in evaluating annuities and other financial products.
• Understanding interest rates and their impact on investments, aiming to optimize returns.
• Using mathematical formulas to predict and analyze the behavior of investments over time, providing strategic insights.

Financial mathematics is invaluable for anyone aspiring to make sound financial decisions and increase wealth efficiently. By mastering these concepts, you can better navigate the complexities of the financial world.