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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.