## Calculating Annual Percentage Yield (APY) 141-32

This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I’m going to show how to calculate
annual percentage yield, or APY. Our problem says find the annual percentage yield for a
stated, or nominal, interest rate of 4.5%, compounded quarterly and
compounded continuously. Here’s what this means: you know that if you put money into a bank and the bank told you they would pay
4.5% interest and the interest was compounded quarterly
or maybe compounded continuously, at the end of the year you would have ended up earning a little
bit more than that 4.5%. The amount you actually earned at the end
of the year is the APY, and what we want to do is
figure out what the APY would be for 4.5% — let’s start
with it compounded quarterly. So to make this an easy problem,
since it doesn’t say how much money you’re investing, let’s assume that the amount you’re
putting into the bank, the principal, is just one dollar, and let’s assume you’re putting into the
bank for one year. So now let’s work with the formula we have for interest which is
compounded. The formula says that ‘A’, this is the amount you[re going to get back equals P times (1 + r/n) aised to the nt power. Now remember, the principle is one dollar, so multiplying this by one doesn’t
make any sense. We can just get rid of that P altogether. And ‘t’ is 1, so multiplying n times t is not going to help, so I’ll just get rid of the 1. Now we have ‘A’ equals (1 + r/n) raised to the n. So remember, ‘r’ is the interest rate, the annual interest
rate. If we state that as a decimal, it’s .045, and ‘n’ is the number of compounding periods
per year. It’s compounded quarterly, so ‘n’ is 4. So we just want to divide .045 by 4, and we get point .01125. That means that (1 + r/n) is the same as 1 + .01125. So we can just say ‘A’ equals 1.01125 — I just added those numbers
together — and we want to raise it to the n power. Well, n is 4, because its quarterly. So now it’s a very simple problem. It’s just 1.01125 raised to the 4th power. We can put this
into the calculator and what you’ll find is you if put that
in the calculator you get 1.045765….. I’ll just round that to ‘A’ equals approximately 1.0458. Now remember, ‘A’ is the amount of money you get back from the bank. ‘A’ includes both the amount
of money you invested and the interest. Well in our problem you
invested one dollar, so if you want to find the interest we
have to subtract one dollar from the ‘A’. That means the interest is going to equal
approximately .0458, that’s just 1.0458 minus 1. And now we want to turn that into a percentage. So we just multiply that by
100 and add a percent sign. It’s going to be 4.58%. That’s the answer to the compounded quarterly part, 4.58%, which kinda makes sense — it’s a little
more than the 4.5% we started with. Let me review the steps for this and
then we’ll go on to the compounded continuously part. All we did was take the formula for compound
interest, A=P + (1 + r/n) raised to the nt and got rid of the P and the t, because we said those numbers were 1.
Then we had a much simpler formula. We put in the numbers that we had —
the percentage rate and the number of compounded periods. Then we calculated what ‘A’ equaled. We remembered that ‘A’ also includes the
amount of money we invested, which was 1. We subtracted 1 from
that and we ended up with the interest rate. We turned the
interest rate into a percentage and that’s our answer. Now let’s do the compounded continuously part. The formula for compounded
continuously is ‘A’ equals P times e to the rt. Once again, let’s assume that P is \$1
and the time is 1 year. So we don’t need that P or the t, so
now we just have ‘A’ equals e to the r, and 4.5%. We want that as a decimal. So that’s ‘A’ equals e to the .045. I’ll use acalculator to figure out what that
is, and what we get is the
‘A’ equals approximately 1.046027… and once again this number keeps
going. I’ll round this and make it ‘A’ equals approximately 1.046, and then we just want to subtract the
original one dollar we invested from this 1.046. So the interest is .046. We turn that into a percentage by multiplying by 100 and adding a percent
sign. So it’s just went 4.6%, and that’s the APY for the compounded
continuously part of the problem. You might want to
do this yourself a couple of times just to practice it. It’s a pretty logical procedure and I think
you’ll be fine with it. Take care. I’ll see you next time.

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1. Raul Torres says:

This is exactly what I needed. Thank you.

2. animeisbae says:

thanks so much. I really needed this.

3. Justin Rocha says:

You are the real MVP

4. Jeremiah Moyes says:

What is E?

5. Al May says:

how would you find the answer if you were to make deposits monthly in your bank account?

6. Alric McDermott says:

What's the "e" stand for?

7. Snav x says:

And e is?

8. Rex Light says:

What if compounded monthly?

9. Selam Hagere says:

What's the e value?

10. Rebecca Jones says:

Thanks!!!