Section 4-1: Saving Money - The Power of Compounding

Welcome to Section 4-1, where we dive into the power of saving money and understanding interest! Remember, calculators are your friend for today's lesson.

  • Key Concept: Simple Interest - Calculated only on the principal.
  • Key Concept: Compound Interest - Calculated on both the principal and accumulated interest. This is where the magic happens!

Simple Interest Formula:

$$Simple\,Interest = Principal \times Yearly\,Interest\,Rate \times Time$$

Let's look at an example: If you invest $2000 at a simple interest rate of 4% per year for 5 years, the interest earned would be:

$$Simple\,Interest = $2000 \times 0.04 \times 5 = $400$$

Now, let's consider compound interest. The Annual Percentage Rate (APR) is crucial. To find the periodic interest rate (r), we use the formula:

$$r = \frac{APR}{Number\,of\,Periods\,in\,a\,Year}$$

The Compound Interest Formula to calculate the balance after $t$ periods is:

$$Balance = Principal \times (1 + r)^t$$

Example: Suppose you invest $10,000 in a 5-year CD with an APR of 6%. Let's see how the compounding frequency affects the final value:

  1. Annually: $Balance = $10,000 \times (1 + 0.06)^5 = $13,382.26$
  2. Quarterly: $r = \frac{0.06}{4} = 0.015$, $t = 5 \times 4 = 20$. $Balance = $10,000 \times (1 + 0.015)^{20} = $13,468.55$
  3. Monthly: $r = \frac{0.06}{12} = 0.005$, $t = 5 \times 12 = 60$. $Balance = $10,000 \times (1 + 0.005)^{60} = $13,488.50$
  4. Daily: $r = \frac{0.06}{365} = 0.00016$, $t = 5 \times 365 = 1825$. $Balance = $10,000 \times (1 + 0.00016)^{1825} = $13,498.26$

Notice how compounding more frequently leads to a higher final balance! This is the power of compounding.

We also learned about the Annual Percentage Yield (APY), which represents the actual percentage return earned in a year, considering the effect of compounding. The formula is:

$$APY = (1 + \frac{APR}{n})^n - 1$$

Where $n$ is the number of compounding periods per year.

Present Value and Future Value:

  • Present Value: The initial amount invested.
  • Future Value: The value of the investment at a specific time in the future.

Doubling Time: Want to know how long it takes to double your investment? We can estimate it using the Rule of 72:

$$Doubling\,Time \approx \frac{72}{APR (as\,a\,percentage)}$$

Section 4-3: Saving for the Long Term - Building That Nest Egg

Section 4-3 focuses on long-term savings and annuities – essential for building your future nest egg! Here we will use the Regular Deposit Formula.

Regular Deposit Formula:

$$Balance\,after\,t\,deposits = \frac{Deposit \times ((1 + r)^t - 1)}{r}$$

Where:

  • $Deposit$ is the amount deposited each period
  • $r$ is the interest rate per period (APR/number of periods per year)
  • $t$ is the total number of deposits

Example: Saving for College. How much does your younger brother need to deposit each month into a savings account that pays 7.2% APR to have $10,000 in 5 years?

We can use the Needed Deposit formula:

$$Needed\,Deposit = \frac{Goal \times r}{((1 + r)^t - 1)}$$

First, calculate the monthly interest rate: $r = \frac{0.072}{12} = 0.006$. The number of deposits is $t = 5 \times 12 = 60$.

$$Needed\,Deposit = \frac{$10000 \times 0.006}{((1 + 0.006)^{60} - 1)} = $138.96$$

Thus, you need to deposit $138.96 each month.

Continue to explore these concepts and practice the formulas. You're on your way to becoming financially savvy!