A bank features a savings account
that has an annual percentage rate of 4.7%
with interest compounded quarterly.
Marie deposits $4,500 into the account.
The first question is how much money
will Marie have in the account in one year?
And then second, what is the annual percentage yield
for the savings account?
Remember, the annual percentage yield
is the simple interest rate that would produce
the same return at the end of one year,
as an account that paid compounded interest.
So, for the first step,
we'll determine the balance after one year
using the compounded interest formula, shown here below.
Where A, the amount after T years,
is equal to P sub zero, the initial deposit of $4,500,
times the quantity one plus R divided by N,
where R is the annual interest rate expressed as a decimal
and N is the number of compounds per year.
So, because the percentage rate is 4.7%,
4.7% as a decimal is 0.047,
which means R is 0.047.
And then N, the number of compounds per year, is four,
because the interest is compounded quarterly
and there are four quarters in a year.
This is raised to the power of N times T where, again,
N is four and T is time in years,
and because the time is one year, T is one.
And then I'm going to go to the calculator
and round to the nearest cent.
We enter 4,500 times the quantity one plus 0.047
divided by four, this is raised to the power
of four times one, or four,
so we press the exponent key, four, and then enter.
To the nearest cent,
the account balance after one year is $4,715 and 26 cents.
And now to determine the annual percentage yield, or APY,
and there are two possible formulas, let's show both.
This first formula requires that we have
the balance after one year,
the second formula does not require
the balance after one year.
Using the first formula, the APY is equal to
the balance after one year minus the starting balance,
and this difference is then divided by the starting balance.
Which means the APY is equal to the balance after one year,
which is $4,715 and 26 cents,
minus the starting balance of $4,500,
and then this difference is divided by
the starting balance of $4,500.
Because we're asked to round the percentage
to the hundredths place value, or two decimal places,
then this will give us the decimal,
we'll have to round this quotient
to the fourth decimal place.
So, we have, in the numerator, 4,715.26 minus 4,500,
so this is in parentheses, and then divided by 4,500.
If we round to four decimal places,
because we have a three in the fifth decimal place,
we round down to 0.0478.
Which means the APY is 4.78%.
So, again, this means after one year an account that pays
4.78% simple interest would have the same return
as the given account that pays
4.7% interest compounded quarterly.
But let's also find the APY using the second formula,
where the APY is also equal to the quantity
one plus R divided by N, raised to the power of N minus one,
and, of course, we should get the same result.
So, we have the quantity one plus R divided by N
is going to be the annual interest rate as a decimal,
which is 0.047 divided by N,
which is four because we have interest compounded quarterly,
this is raised to the power of N,
which is four and then minus one.
So, going back to the calculator,
we have open parentheses one plus 0.047
divided by four, closed parentheses,
this raised to the fourth power.
To get out of the exponent position we press the right arrow
and then minus one, enter.
Notice there's a slight difference between these two values,
and that's because, remember, the 4,715.26 was rounded,
and this is exact.
But, rounding the the fourth decimal place,
we still have 0.0478.
Which gives us the same APY of 4.78%.
I hope you found this helpful.