Welcome to a lesson on Effective Yield.
Also known as Effective Rate of Interest,
as well as Annual Equivalent Rate.
In this lesson, we'll determine the effective yield formula
for compounded interest.
And also use the formula to determine effective yield.
The effective yield for compounded and continuous interest
is the equivalent annual simple interest rate
that would yield the same return after one year.
Here's the simple interest formula,
the compounded interest formula,
as well as the continuous interest formula.
And below, we have the formulas for effective yield
for compounded interest as well as continuous interest.
This lesson focuses on effective yield
for compounded interest
so let's begin by determining where this formula comes from.
Remember the effective yield for compounded interest
is going to be the equivalent annual simple interest rate
that would yield the same return after one year.
So starting with the simple interest formula,
and the compounded interest formula,
we'll let t equal one for one year.
And then the simple interest rate r equal to r sub e,
which is the effective yield.
And therefore the amount A in both accounts
would have to be the same.
The amount in the account paying simple interest
would be P times the quantity one plus r sub e.
And the amount in the account paying compounded interest
would be P
times the quantity one plus r divided by n
raised to power of n.
Again we let t equal one in both formulas.
To represent one year.
And we let the simple interest rate r be equal to r sub e.
Because it would be the effective yield.
So now if we solve this equation for r sub e,
we'll have our effective yield formula
for compounded interest.
So let's first divide both sides by P.
Simplifying this would be one and so would this.
So we have one plus r sub e
equals the quantity one plus r divided by n,
raised to the power of n.
And now I subtract one on both sides.
So we have r sub e is equal to the quantity
one plus r divided by n, raised to power of n minus one.
Which is our effective yield for compounded interest.
Let's look at a couple other examples.
We want to determine the effective yield for an account
paying four percent interest compounded quarterly.
So we know r is equal to four percent as a decimal,
should be zero point zero four.
Interest is compounded quarterly,
there's four quarters in a year.
So n equals four.
This is all we need in order to determine
the effective yield.
R sub e
is equal to the quantity one plus r divided by n is
zero point zero four, divided by four
raised to the power of n,
so it's raised to the fourth power minus one.
For now we go to the calculator.
So we have open parentheses one plus point zero four
divided by four, close parentheses.
Raise the power of four, right arrow,
minus one, enter.
As a decimal we have approximately
zero point zero four zero six
We do want to convert this to a percentage
by multiplying by 100 and adding a percent sign.
Times 100.
Notice how I typed in a 1000 so,
left arrow, delete.
Now we have 100, enter.
So the effective yield is approximately 4.06%.
To multiply by 100, we could just move the decimal point
to the right two places.
So this tells us, simple interest rate of 4.06%
will yield the same return as an account paying
four percent interest compounded quarterly
after one year.
Let's look at a second example.
We want to determine the effective yield
for an account paying seven percent interest
compounded monthly.
So now r is equal to seven percent.
Which equals zero point zero seven.
Interest is compounded monthly, so n equals 12.
Which means the effective yield r sub e is equal
to the quantity one plus
r divided by n would be zero point seven divided by 12.
Okay, want zero point zero seven.
Raised to power of 12 minus one.
Now we'll go back to the calculator.
Open parentheses, one plus
point zero seven
divided by 12, close parentheses.
Raised power of 12,
right arrow, minus one, enter.
Four decimal places we have approximately
point zero seven two three.
If we're into a percentage, this would be
approximately seven point two three percent.
So the effective yield is seven point two three percent,
which means, an account paying
seven point two three percent simple interest
will yield the same return as an account paying
seven percent interest compounded monthly after one year.
I hope you found this helpful.