Exponential Mulah
Are you ready to spend some mulah? Or are you ready to save some dough? This section is all about using the equations for loans, investments, and present value which you can use when you are shopping for that car or house.
These formulas can arm you with ways to calculate how much money you are actually spending if you leave a balance on your credit card. We will learn how to plug values into these equations and quickly see how much you will pay over the life of the loan. Just won the lottery? You are in luck. You will be able to calculate how much money you can make over a period of time on your investments. Young mulah, baby.
Exponential functions really come into play when we are talking about loans or savings accounts.
The general equation for investments is:
where A is the amount of the investment or loan after a certain number of years t. P is the principal, or amount you started with. The r is the interest rate and the n is the number of times that the investment or loan is compounded every year. For your reference, here is a compounding table to help you decide what to use for n.
Sample Problem
So you won the lottery, $1 million dollars it is! And because you are such a savvy saver, you have decided to invest this into an account that will get 5% annual interest rate which is compounded monthly. Let's calculate how much you will have in this account after 20 years if you didn't spend any of it.
Here is what we know:
P = 1000000
r = 5% = 5/100 = 0.05
n = 12
t = 20
Substituting those values in:
You can almost triple your initial investment thanks to exponential growth and compounding! The way compounding works is that every month, the monthly percentage rate is applied to the cumulative total of your savings. You get more money the more you save because your balance is always on the rise.
Time is money, and money is time. We want to know how much a loan or annuity is worth to you right now. That is called present value. In simple terms, present value means: "What is the future value worth today?"
For our purposes, we will look at a loan for a car since you already might be doing this or will be in the near future. The general form for a loan or annuity is the present value equation:
where PV stands for present value of the loan, R is the monthly payment, r is the annual percentage rate (APR), t is the total years for the loan, and n is the number of times per year the loan will be compounded. (Pssss BIG HINT: If you are asked finance questions involving a monthly payment, this is the equation to use.)
Sample Problem
Calculate the present value of a loan if your monthly payments are $350 for 3 years, monthly compounding and the APR is 3.8%.
Here is what we know:
R = 350
r = 3.8% = 0.038
n = 12
t = 3
Substituting those values in:
The amount you will have paid on this car will be almost $12000 (in present value) after 3 years of payments.
Example 1
Calculate the amount of a savings account after 10 years of savings that starts with $1500 and is compounded monthly at a rate of 5%. |
Example 2
How much will you pay for a car (in present value), if your payments are $225 for 36 months at an APR of 6%? |
Example 3
How much mulah will you accumulate in a savings account if you start with $10000, the interest rate is 3%, the bank compounds daily, and you save for a lifetime of fifty years? |
Exercise 1
Calculate the total amount of money you will pay on a 30 year loan if the principle was $100,000 which is compounded monthly, and the interest rate is 4%.
Exercise 2
If a savings account starts at $500, the interest rate the bank pays you is 2%, and the savings is compounded daily, how much will you have in 5 years?
Exercise 3
How much will a $27,500 car loan cost in total if there is semi-annual compounding at 6% interest for 36 months?
Exercise 4
What is the present value of a car loan with 3 years of payments of $500, monthly compounding, and an interest rate of 5%?
Exercise 5
You find out you have a rich great aunt and when you turn 18, you get an allowance! Or rather, an annuity. Calculate the present value of an annuity with monthly payments to you of $200, semi-annual compounding, for the next 5 years at an interest rate of 3%.