Interest-bearing Bank Accounts & Inflation Part I-Math w/Business Apps, Compound Interest Chapter


Interest-bearing bank accounts dealing
with compound interest. The first objective will be looking at the
different types of interest bearing accounts. We have regular savings
accounts this is generally the type of savings accounts were introduced to as a
child. They are easy to use, the positive is that they’re
liquid which means you deposit money today you can take that money out
tomorrow. There’s no time restriction on it. Because they’re found at a bank most banks are FDIC which stands for
Federal Deposit Insurance Corporation. In the event that the bank should go out of
business the FDIC insurance kicks in and will
give you your money up to $250,000. The downside is they are very low interest
rates. And here’s a table from a bank showing what a regular savings account is
requiring there be $100 in the savings account. And they’re earning
one tenth of one percent interest rate on it. So not very high at all but again
some of those other pluses outweigh the interest being paid. We also have
checking accounts that will pay interest which is definitely a plus provided you
have a balance in your account. And this may or may not be a positive but to earn
the interest you need to carry a minimum balance. When that minimum balance
isn’t maintained and there could be a fee assessed. So be sure when you are looking
at opening up a new checking account that you check into the options and what
seems to fit your financial situation. So let’s look at how we compute daily
interest. Sometimes we will have interest savings accounts that pay
compound interest daily. We’ll see them in savings accounts, passbook savings, or
checking accounts that pay interest as well. We can use the formula that we use
for compound interest that we saw previously. And here is an example if we
have $500 at 4% compounded daily for 3 years this is what the
formula would look like. Here’s our principal times the quanity one plus our
interest rate as a decimal divided by the compounding period which is daily.
So we use 365 days raised to the “n” the number of compounding periods 3 years at 365
per year which results in the amount of $563.74. Or we can use a table, on the
table we would look at something that is set up for compounded daily and then the
interest rate that the account is paying. Similar to compound interest tables that
we’ve seen in previous sections we’ll pull that value from the table which
replaces the quantity that’s shaded here in our compound interest formula. But
leaves us with multiplying that value by the principal amount to give us our
answer. Here’s an example of a compound interest table for daily frequency and
it’s set at a 3.5% interest rate. Very similar to the table that
we’ve seen before the columns in blue are the number of days and adjacent to
it is the value that we will use to replace this portion of our compound
interest formula. So here we have an example set aside
$5,500 in his savings account earning 3.5% interest compounded daily
for 50 days. How much interest will be earned? We could use our formula or using this
table we need to make sure that we’re using the appropriate table. This is set
for 3.5% interest and compounded daily which is the case for this problem.
So our next step will be to find the number that corresponds with 50 days.
Here on the third column of ‘Number of Days’ we find 50 the number adjacent to it
will replace this portion of our compound interest formula. Which leaves
us only to take the principal times that value to give us the total amount
in the account after 50 days. Let’s look at another one, $3200 dollars is invested in a
savings account earning 3.5% interest compounded daily for 85
days. How much interest will be earned? We’re still dealing with compounded
daily at 3.5%, so we use this table we’re going to locate where 85 is with our multiplier.
Multiplying our deposit by that value from the table gives us our total amount in
the account after 85 days. But the problem is asking how much interest is
earned. We will take our balance at 85 days and subtract the initial
deposit to determine the amount of interest that’s earned in the account.
Let’s take a look at maybe a more practical problem in the sense that
deposits are added periodically and we’re interested in what amount would be
after a certain amount of time or maybe we have withdraws taken. So in this first sample
problem we’re going to use the 3.5% interest compounded daily. On January 1, this bank was opened and the amount of $8,756 was deposited, in
February another amount and in March a third amount was deposited. And the question
is asking, what is the balance on March 31 and ultimately what
is the interest earned? So we’re going to take this as though it were three separate
problems. Here we have our timeline we need to know how many days this first deposit is
going to be in there. If nothing else happened in this account on January 1
this amount was deposited and we’re interested in how much money that
accrued due to compound interest being paid daily on March 31. Well if we do
the calendar math there are 90 days between March 31 and January 1. So
we’ll use our table with the 3.5% interest for compounded
daily, we’ll find our multiplier that is associated with the
90 days and multiply that initial deposit. So this would be the balance
after the 90 days for just that first deposit. Then we’ll go to the
February 11th deposit this money is sitting in here from February 11 until
March 31 it too will be accumulating interest but only from February 11
until March 31, which is 48 days. To calculate the interest earned on that
amount we will look up 48 days on our table, which this is our multiplier times
it by that deposit that’s only sitting in the account
for 48 days and we have the ending balance. The last step here is
looking at how long is the March 21st deposit of $650 in the account and
earning interest. Well March 21 to March 31 is 10 days. The
interest earned on that account and the ending balance for that deposit is this
multiplier times 650. Now if we add these 3 values together it will give us
the balance from these 3 separate deposits on these different dates. That
answers the first question. To determine the amount of interest we need to take
this balance and subtract off those principal deposits that were made
through the course of this 90 period. So here’s what that would look
like we take our balance after the 90 days subtract each of those three. And here
we’re showing that they added those 3 principal deposits and then subtracted
that quantity from the balance as of March 31. The result is an earning of
$80.82. Here’s another example where we have some withdraws that are occurring,
so let’s take a look at this one. On April 1 MVP Sports opened a savings
account with a deposit of $17,500. A withdrawal of $5000 was made 21 days later, and another
withdrawal of $980 was made 12 days before July 1. Find the balance on July 1.
To solve a problem with a withdrawal we kind of reverse our thinking a
little bit to calculate this. On 21 days past this initial deposit $5000 was
withdrawn. Which means that $5,000 sat there for the first 21 days before it
was moved or withdrawn and it was earning interest. We have another
withdrawn made 12 days before July 1st. Doing some calculation that means that
this $980 sat in there from April 1 until 12 days before July 1.
Which would apparently if we do that calculation July 1 is 182nd day of the
year, 12 days before that is 170th day. April 1 is the
91st in the difference between the 170th and 91st is 79 days. So this $980 before it was withdrawn was
compounding interest every day for 79 days. What stayed in the account the
entire period from April 1 to July 1 is the difference between the initial
deposit and these two withdrawals. So if you take that initial deposit and
subtract the two withdraws that were made this means
there was this amount of money that was left in the account for that time period
from April 1 to July 1. We then need to calculate what the balance is for
each one of these three values: $5,000 times we’ll look up the number from our
table associated with 3.5% interest compounded daily for 21
days this is what this would have grown to. For $980 sitting in their 79
days we find the multiplier and the multiplier associated
with 91 days, which is actually below the table in a little note there’s where
you’ll find your multiplier for that. These then are what the balances are for
these accounts with their interest. We’re almost done, what we need to do is add the 3 up
but recognize we did withdraw the $5,000 and the $980 which gives us an
ending amount of $11,638.49. You could have subtracted your $5000 off
from here to get a $10.08 interest earning on that amount sitting for 21
days before the withdrawal. And if we take our $980 withdrawal away from here
it means it earned $7.45. We could have added the interest from these two onto here and
still arrived at that same value.

Paul Whisler

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