Section 4.3: Saving for the Long Term - Building Your Nest Egg

Welcome to Professor Baker's Math Class! In this section, we'll dive into the exciting world of personal finance, specifically focusing on how to save for the future. Get ready to learn about calculating balances after regular deposits and determining the deposits needed to reach your financial dreams. Calculators are recommended for this lesson, so grab yours and let's get started!

Key Concepts

  • Regular Deposit Formula: This formula helps you calculate the future value of an account with regular deposits. The formula is: $$Balance = \frac{Deposit \times ((1 + r)^t - 1)}{r}$$, where $r$ is the interest rate per period and $t$ is the number of periods.
  • Annuity Payout Calculator: An annuity provides a stream of payments over time. Understanding how annuities work is crucial for retirement planning.
  • Annuity Yield Goal Calculator: This tool helps you determine the size of the nest egg you'll need to achieve a specific monthly income during retirement. Remember to use the APR divided by 12 for the interest rate ($r = APR/12$) and the number of years multiplied by 12 for the number of periods ($t = years * 12$).

Example 1: Calculating Future Balance with Regular Deposits

Let's say you deposit $20 into a savings account at the end of each month, and the account earns 7% APR. What will be the account balance after five years?

Solution:

  • Monthly interest rate: $r = \frac{0.07}{12}$
  • Number of deposits: $t = 5 \times 12 = 60$
  • Using the regular deposit formula: $$Balance = \frac{20 \times ((1 + \frac{0.07}{12})^{60} - 1)}{\frac{0.07}{12}} = $1431.86$$

So, the future value of your savings after five years would be approximately $1431.86!

Example 2: Determining the Savings Needed for a Goal

How much does your younger sibling need to deposit each month into a savings account that pays 7.2% APR to have $10,000 when they start college in five years?

We can use the Deposit needed formula : $$Needed Deposit = \frac{Goal \times r}{((1 + r)^t - 1)}$$

Solution:

  • Monthly interest rate: $r = \frac{0.072}{12} = 0.006$
  • Number of deposits: $t = 5 \times 12 = 60$
  • $$Needed Deposit = \frac{10000 \times 0.006}{((1 + 0.006)^{60} - 1)} = $138.96$$

Therefore, your sibling needs to deposit $138.96 each month to achieve their college savings goal.

Annuity and Retirement

Understanding annuities is essential for retirement planning. Remember these key terms:

  • Nest Egg: The balance of your retirement account when you retire.
  • Monthly Yield: The amount you can withdraw from your retirement account each month.
  • Annuity: An arrangement where you withdraw both principal and interest from your nest egg.

We hope this lesson helps you on your journey to financial literacy! Keep practicing, and you'll be saving like a pro in no time. Good luck!