Problem 1
With simple interest, we take a percentage of the loan amount (or deposit depending how you frame things) to determine the interest paid or earned.
The simple interest formula is
i = P*r*t
Let's look at an example: You deposit $100 into an account earning 5% simple interest. You want to know how much interest you earn over 2 years.
i = P*r*t
i = 100*0.05*2
i = 10
You've earned $10 in simple interest. Notice how this is 10% of the original deposit ($100). You can think of the 5% interest rate doubling to 10% because we're working over a 2 year period.
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Compound interest is where we can't take a simple percentage of the deposit. This is because we must use this formula
A = P*(1+r/n)^(n*t)
variables are:
- A = final amount after t years
- P = deposit
- r = interest rate in decimal form
- n = compounding frequency
- t = number of years
An example: You deposit $100 at an interest rate of 5%. The interest is compounded quarterly (4 times a year). How much interest have you made in 2 years?
The input values will be
- P = 100
- r = 0.05
- n = 4
- t = 2
So,
A = P*(1+r/n)^(n*t)
A = 100*(1+0.05/4)^(4*2)
A = 110.448610118141
A = 110.45
i = interest
i = A - P
i = 110.45 - 100
i = 10.45
You've made $10.45 in interest over the two year period.
As you can see, we used an exponential function to determine compound interest. Whereas with simple interest, the function was linear. Exponential functions grow much faster compared to linear functions.
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Problem 2
Savings accounts offer higher interest rates, which encourages people to save their money (hence the name). Checking accounts on the other hand are used to write checks and are there for immediate withdrawals of cash. Checking accounts are also used for debit cards, such as for online shopping.
Both types of accounts are useful in their own different ways. Linking them together is handy to quickly move money without having to deal with tons of paperwork and/or lots of time spent.
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Problem 3
APR = annual percentage rate
APY = annual percentage yield
The APR value is often the stated item on a loan, since it is the smaller of the two values. When it comes to savings accounts in banks, they tend to state the APY value since it is larger.
Let's look at an example:
The APR is 5% and the money is compounded monthly (12 times a year). Compute the APY.
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Solution:
The formula we use is
y = (1+r/n)^n
where y is the APY and r is the APR
In this case we have
Giving us the following
y = (1+r/n)^n - 1
y = (1+0.05/12)^12 - 1
y = 1.05116189788173 - 1
y = 0.0512
Then multiply that by 100, aka move the decimal point two spots to the right, to arrive at an APY of approximately 5.12%
This example demonstrates how compound interest slightly bumps up the amount of return. We've gone from 5.00% to 5.12%
This slight increase is because the money is compounded 12 times a year. Each time we compound, we're adding to the pile ever so slightly. There's a limit to this process (see the continuous compounding interest formula); i.e. we can't grow the APY forever.
If we were using simple interest, then the APY formula shown above would not apply. For simple interest situations, the APY and APR are the same value.