A bank features a savings account
that has an annual percentage yield of 5.9%
with interest compounded monthly.
Cody deposits $5,500 into the account.
We first want to determine how much money
will Cody have in the account in one year
and then what is the annual percentage yield
or APY for the savings account.
For a quick review, the annual percentage yield
or APY is also referred to as the effective
annual yield, the effective annual rate,
or annual equivalent rate.
All these are the simple interest rate
that would produce the same return at the end
of one year as an account that paid compounded interest.
And this can be helpful when comparing different
investment accounts that pay different compounded interest.
We can determine the APY using either of these
two formulas, where this first formula requires
the balance after one year and the second formula doesn't.
In this video, we will take a look at both formulas.
Well, for the first step we need to determine
the balance in the account after one year
that pays 5.9% interest compounded monthly.
And therefore we use the compounded interest
formula shown here below to determine the balance
after one year.
So the balance after one year A is equal
to P sub zero, the starting amount of $5,500,
times the quantity one plus R divided by N
where R is the annual interest rate as a decimal
and N is the number of compounds per year.
5.9% as a decimal is 0.059 and the number
of compounds per year is 12 because the interest
is compounded monthly and there are 12 months a year.
And this is raised to the power of N times T
where again N is 12 and T, the time in years, is one.
And now we go to the calculator and round this
to the nearest cent or nearest penny.
We enter 5,500 times the quantity one plus 0.059
divided by 12, and this is raised to the power
of 12 times one or 12.
We press the exponent key, 12, and then enter.
To the nearest cent, we have $5,833.42.
And now we'll determine the APY using both formulas.
For this first formula we have the beginning
balance after one year minus the starting balance
all divided by the starting balance.
So notice how this first formula does require
the balance after one year, which we already have.
So the APY is equal to the balance after one year
which is $5,833.42 minus the starting balance
of $5,500 and this difference is divided
by the starting balance of $5,500.
Because we're told to round the percentage
to the hundredths place value or two
decimal places, and this will give us the decimal
form of the percentage, we'll have to round
the decimal to four decimal places.
So we have in parentheses 5,833.42 minus 5,500
and this difference is divided by 5,500
to four decimal places is approximately 0.0606
which as a percentage is 6.06%.
So because the APY is 6.06% for an account
that pays 6.06% simple interest after one year
would have the same return as the given account
that pays 5.9% interest compounded monthly.
But let's also determine the APY using
the second formula where the APY is equal
to the quantity one plus R divided by N
raised to the power of N minus one.
Notice how for this formula we don't have to use
the ending balance after one year.
We would have the quantity one
plus R divided by N is 0.059 divided by 12 again.
And it's 12 because we have interest compounded monthly.
This is raised to the power of N, which is 12
and then minus one.
Going back to the calculator, we have
in parentheses one plus 0.059 divided by 12
and this is raised to the power of 12.
We now need to press the right arrow to get out
from the exponent position.
And then minus one and enter.
Notice how there's a slight difference
between these two values because remember
the 5,833.42 is rounded to the hundredths
place value or to the nearest cent.
But to four decimal places we still
have approximately 0.0606, giving us the same
percentage of 6.06 for the APY.
I hope you found this helpful.