Lesson 5.5 Exponential Functions and Investing

Key Objectives

• Learn how to figure out how much money you’ll have in the future if you invest it and earn interest.

• Understand how to calculate the value of an investment that grows a long time.

• Discover how to find out how much money you need to invest today to reach a goals in the future.

• Use charts and graphs to understand how investments work.

Understanding Investments

• Future Value (FV): This is how much your money grows over time if you invest it.

  • Example: If you invest $1,000 at a 6% interest rate, it means your money will grow.

How to Calculate Future Value

• There’s a special formula we can use: ( y = a(1 + r)^x )

  • Where:

    • ( a ) is the starting amount (initial value).

    • ( r ) is the interest rate.

    • ( x ) is the number of years.

  • Example: For $1,000 at 6% interest for a certain number of years:

    • For Year 0: ( S = 1000.00 )

    • For Year 1: ( S = 1060.00 )

    • For Year 2: ( S = 1123.60 )

    • And so on…

Visual Aids

• You can make a table with future values to see how money grows over the years.

• You can also draw a graph to see the growth clearly compared to other ways of earning interest.

Compounding Interest

• When we say the interest is compounded, it means that you earn interest not just on your initial investment but also on the interest that accumulates.

  • For example: If you have $1,000 at 6% interest compounded monthly, the formula looks like this: ( S = P (1 + {r}/{k})^{kt} )

    • Where ( P ) is your starting amount, ( r ) is the interest rate, ( k ) is how many times interest is applied each year, and ( t ) is the number of years.

    • For $1,000 at 6% compounded monthly for 5 years:

    • ( S \approx 1348.85 )

Continuous Compounding

• When you keep adding interest all the time, it’s called continuous compounding. The formula is:

  • ( S = Pe^{rt} )

• As you increase how often interest is applied, the total amount of money you’ll have grows significantly.

  • Example: If you invest $2,650 at 12% interest for 8 years, you'd end up with about $6,921.00!

Finding Present Value

• Present Value (PV): This is how much money you’d need to invest today to reach a certain amount in the future.

  • Formula: ( P = \frac{S}{(1+i)^n} )

  • Example: To know how much to invest now in order to have $15,000 in 7 years at a certain interest rate, plug the numbers into the formula!

Investment Models

• We can also study real-life data to see how investments work. We can use tools to graph these changes and view them easily.