MM - Compound Interest Lesson

Compound Interest

You have studied simple interest in previous courses, and we explored compound interest in the previous unit - exponential and logarithmic functions. Now, we will review and explore simple and compound interest again so that we can apply these concepts to model various real-world problems.

Simple interest is something of the form of P = I*R*T, where P=Principal, I=Interest, R=Rate, and T=Time.

The following is an example of compounding interest.

Mary knows she needs to plan for the future. She wants to invest in a savings account and is researching the best choice. All the banks use formulas with compounded interest. When an investment has compounded interest, the interest due is added to the principal. Therefore, the next payment or new principal contains the old payment and the new interest. The formula is: equation image indicator $LaTeX: A=P\left(1+\frac{r}{n}\right)^{nt}$ $A=P\left(1+\frac{r}{n}\right)^{nt}$ .

A=amount, t= time in years, P=principal, n= # times compounded per year, and r= annual interest

When using the formula above you have to use the following values for "n".

Compounded	N =
Annually	1
Semiannually	2
Quarterly	4
Monthly	12
Weekly	52
Daily	365

If you let the number of times the principal is compounded, or "n", increase without a limit, then it is continuously compounded. The formula is given below -

equation image indicator $LaTeX: A=Pe^{rt}$ $A=Pe^{rt}$

A=amount, t= time in years, P=principal, r= annual interest, and e is the natural base found on your calculator.

Mary will use the information from each bank to see how much money she would have after 20 years assuming she invests $2000 to start with (principal).

Watch the following videos that will explore the modeling of real-world problems.

Let's explore the following example.

Anthony loves his grandmother and gladly accepts her $200 gift, but he doesn't want her to open up a checking account and put extra money in it, as she's done too much for him already. He believes he has a better idea, anyway. Anthony remembered that he took some notes in a finance course he was enrolled in at the local community college regarding interest-bearing accounts:

Compound Interest (2 types) notes...

1. n compounding periods: equation image indicator $LaTeX: A=P\left(1+\frac{r}{n}\right)^{nt}$ $A=P\left(1+\frac{r}{n}\right)^{nt}$

2. Continuous compounding: equation image indicator $LaTeX: A=Pe^{rt}$ $A=Pe^{rt}$

Remember: A = Final Amount, P = Principal (starting or initial amount), r = interest rate (decimal form), n = compounding periods per year, t = time in years

Values of n: Annually = once per year, Semiannually = twice per year, Quarterly = four times per year, Monthly = twelve times a year, Weekly = fifty - two times per year

There are two banks that Anthony will decide between to open a savings account (one type of interest-bearing account): Bernoulli Bank or Euler Federal Bank. Bernoulli Bank offers a 6.75% quarterly-compounded interest rate, and Euler Federal offers a 6.75% continuously-compounded interest rate.

Anthony plans on investing his $200 gift - plus an additional $50 bill that he found in the sofa cushion when he was looking for change - in a savings account for one year.

Anthony gives the simplified form of the equation representing the amount earned at any given month at Bernoulli Bank by the following equation, which involves a transformation of an exponential function.

equation image indicator $LaTeX: A=250\left(1+\frac{0.0675}{4}\right)^{4t}so\:A=250\left(1.016875\right)^{4t}$ $A=250\left(1+\frac{0.0675}{4}\right)^{4t}so\:A=250\left(1.016875\right)^{4t}$

When Anthony graphs the function, he finds it looks like the following:

graph with a curve going up

Next, Anthony makes a table to show himself which bank would truly give him the best return on his investment.

table comparing Bournouli Bank versus Euler Federal Bank for 6 years

Anthony's grandmother had a second idea to give him a sum of $200.00, and then give him $15.00 per month for 6 months. This would have yielded Anthony a total of $290.00 after the six months. If Anthony chooses the savings account with Euler Federal, how long will it take for Anthony to save the same amount that he would have had in six months with his grandmother's second idea?

Using the formula associated with Anthony's savings account at Euler Federal, he found the following results:

NOTE: Anthony needs to use the inverse of an exponential function (logarithm) to solve this exponential function.

equation image indicator $LaTeX: 290=250e^{0.0675t} \\ 1.16=e^{0.0675t} \\ \ln \:1.16=\ln e^{0.0675t} \\ \frac{\ln 1.16}{0.0675}=t\\ so\:t\approx2.2\:years$ $290=250e^{0.0675t} \\ 1.16=e^{0.0675t} \\ \ln \:1.16=\ln e^{0.0675t} \\ \frac{\ln 1.16}{0.0675}=t\\ so\:t\approx2.2\:years$

Anthony could have also used his graphing calculator in the following way:

Let $LaTeX: y_1=250e^{0.0675x}\&\:y_2=290$ $y_1=250e^{0.0675x}\&\:y_2=290$
Choose an appropriate viewing window
Next, Press 2nd CALC (#5 intersect)
This will yield the solution of x = 2.1988

Therefore, even though Anthony starts with a $50 higher initial amount with his savings account, it would still take him over two years to earn the same amount that he would in six months using his grandmother's second idea.

It's now time for us to explore and practice working with Compound Interest.

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