Problem 18 - Applied Calculus

Question

Each year, a family deposits $$\$ 5000$$ into an account paying \(8.12 \%\) interest per year, compounded annually. How much is in the account right after the \(20^{\text {th }}\) deposit?

Step-by-Step Solution

Answer

About $233,232.50 is in the account after 20 years.

Step 1: Understanding the Problem

The problem involves calculating the future value of a series of annual deposits in a bank account earning compound interest. The annual deposit is $5000, the interest rate is 8.12%, and the period is 20 years.

Step 2: Identify the Formula Needed

To find the amount in the account after 20 years with annual deposits and compound interest, we'll use the Future Value of a Series formula: \( FV = P \times \frac{(1 + r)^n - 1}{r} \), where \( P \) is the annual deposit, \( r \) is the interest rate per period, and \( n \) is the number of periods.

Step 3: Assign Values to Variables

Let's identify the values: \( P = 5000 \), \( r = \frac{8.12}{100} = 0.0812 \), and \( n = 20 \). These will be used in our formula to calculate the future value.

Step 4: Substitute Values Into the Formula

Now substitute the values into the formula: \[ FV = 5000 \times \frac{(1 + 0.0812)^{20} - 1}{0.0812} \]

Step 5: Calculate the Components

First, calculate \((1 + 0.0812)^{20}\):\[(1 + 0.0812)^{20} = 4.78638\]

Step 6: Compute the Future Value

Substitute back into the formula: \[ FV = 5000 \times \frac{4.78638 - 1}{0.0812} = 5000 \times \frac{3.78638}{0.0812} \]Simplify:\[ FV = 5000 \times 46.6465 = 233232.5 \]

Step 7: Final Result

The amount in the account right after the 20th deposit is approximately $233,232.50.

Key Concepts

Compound Interest

Compound interest is a powerful concept in personal finance. It is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
Unlike simple interest, where calculations are only on the principal amount, compound interest grows the account balance exponentially over time as interest is earned on interest.

• Essentially, this means your earning potential grows rapidly, especially as time extends.
• For example, with the 8.12% interest rate in our exercise, each year your deposit earns interest on the original deposit and on the interest accumulated up to that point.
• This compounding effect is where the real growth of your investments lies.

Understanding compound interest helps in realizing how effective it can be for long-term savings and why starting your savings early can be beneficial.

Annual Deposits

When saving money, annual deposits refer to the consistent addition of funds each year into an account. In our example exercise, the family deposits $5000 at the end of every year.
Regular annual deposits help build a habit of saving and compound growth amplifies your savings over time.

• Steady contributions like these make it easier to predict future savings as you can calculate how much money will grow through compound interest.
• Consistency is crucial; making deposits annually guarantees that each addition will benefit from the interest formula.

Annual deposits also benefit from the 'snowball' effect, where interest not only adds to the amount invested but also compounds, creating exponential growth.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to the current value, leading to the rapid increase of the quantity over time.
For savings, this is highly beneficial because it allows for a significant increase in funds through compounding and regular contributions.

• In our specific exercise, by continually adding $5000 annually, the growth is not linear but exponential due to compound interest.
• This means the account value increases more and more each year.
• The real power of exponential growth is seen over longer periods, such as our 20-year example, where the account value skyrockets compared to just linear growth.

Recognizing the exponential nature of investment growth is key to planning for financial goals over the long term.