 Hello and welcome to this session. This is Professor Farhad in which we would look at conventions for annualizing rate of return. Simply put, we're going to look at the APR versus the EAR, the annual percentage rate versus the effective annual rate. These topics are usually covered on the CFA and the CPA exam, essentials or principles of investment course graduate or undergraduate. As always, I would like to remind you to connect with me on LinkedIn if you haven't done so. YouTube is where you would need to subscribe. I have 1700 plus accounting, auditing, tax, finance as well as Excel tutorial. If you like my lectures, please like them, share them, put them in playlists. If they benefit you, it means they might benefit other people. Connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources, the complement and supplement, this course as well as your other accounting and finance courses. I strongly check out, I strongly encourage you to check out my website. So the best way to illustrate this APR versus EAR is to show you an example in numbers to kind of explain the difference. Let's assume you have $10,000 and you decided to invest this $10,000 in a bank. And you have Bank A and Bank B. Bank A is offering you 12% interest and Bank B also offering you 12% interest. Bank A told you the interest will be a simple, 12% simple interest. Bank B said the interest is 12% compounded monthly. So let's see what's going to happen after a year or 12 months after you keep that money in your bank account. Let's start with Bank A. In Bank A, after 12 months, you're going to have the, you're going to have your original amount, $10,000 plus the principal. So you're going to take $10,000 times 1 plus B38, which is 12%. Therefore, you're going to have $11,200. So what's going to happen in Bank B? Bank B said, I'm going to give you 12%, but that 12% will be compounded every month. What does that mean? It's mean every month. So rather than giving you 12% annually, what I'm going to do, I'm going to take the 12% and compound it on a monthly basis. So it's going to be 12% divided by 12 months. So every month, I'm going to, you're going to earn 1%. What does that mean? It means a month after you deposit this money, I'm going to take your, I'm going to take your $10,000 multiplied by 1.01, which is 1%, 1.01, which is 1%. And you're going to have in your bank account, $10,100. This is for January. For February, I'm going to take the $10,100 and I'm going to multiply it by 1.01 and you're going to have $10,201. For March, I'm going to take your $10,201 multiplied by 1.01. So simply put, after you go through all 12 months for this, you are going to have in your bank account, $11,268. Hold on a second. That's more than the simple rate. That's correct, because what was happening is this. Every month, the bank was taking your $10,000, simply put, your bank was taking your $10,000 multiplied for the first month, 1.01. Then for the second month, it was taking the $10,000, multiplying it by 1.01, another 1.01. For the third month, it was taking the $10,000, multiplying it by 1.01 for the first month, 1.01 for the second month, and 1.01 for the third month. And this will repeat itself until by the 12 month, you're going to take the $10,000, simply put, and you're going to multiply it by 1.01 raised to the 12th power, because remember, 1.01 times 1.01, it's the same thing as 1.01 raised to the second power. 1.01 times 1.01 times 1.01, it's the same as 1.01 raised to the third power. So what's happening, your interest rate is being compounded on a monthly basis. Now we could do this on a daily basis. I can take 12% divided by 365 and find the rate, then take the $10,000 times that rate. That's going to be a small rate, but let me just show you what will be, just kind of so you understand. So if I take 12 divided by 365, the rate will be 0.12%, yes, 12%, the rate will be 0.00329. But you'll have to do this over, every day you'll have to take the $10,000 times 1.00329, 365 times, or you can compound it twice a year, or you can compound it four times a year. I did 12 times because it's easy to see 12% and 1%. So this is what happened here. So what does that mean? It means over a period of a year, you earned 1,200 in interest. Over a period of a year here, you earned 1,268. So notice here, because you chose Bank B, which compounded monthly, you made more interest. Now that will work also the same if you borrow money from Bank B. If you borrow money from Bank B and the loan is compounded monthly, you'll have to pay 1,268. If you borrow from Bank A, you only pay 1,200. And here we're talking about if you deposit the money, but it works the same way if you have a loan. So what is your effective rate really? What is your effective rate? How much did you get versus your original amount? Well, you invested 10,000 and you made 1,200. If I take 1,200 divided by 10,000, I'll get 12%. What is your effective rate here? Well, your effective rate is 12.68. Notice there is a difference between the two. Now, so if we want to find the effective rate, how do we find the effective rate? Because what you are quoted for Bank A is APR of 12% compounded annually. So the effective rate is 12%. But how do we go from 12% to 12.68? Simply put, all what we have to do is take 1.01 raised to the second, 12 to the nth power, whatever that power is. So simply put what I can do is this. To find out what's my effective rate, I can take my 1 plus the rate per period raised to the n. So what is 1 plus the rate per period? 1 plus the rate per period is 01, 1% because I'm doing it monthly raised to the 12th power. And if I do that, I will get 1.1268. All I have to do is subtract 1 from this formula. And if I subtract 1 from this formula, I will get 12.68, which is, I want to turn it into a percentage, 12.68%. So to go from APR to EAR, you will take 1 plus the rate over the number of periods. Now let's assume, let's assume I'm doing this on a quarterly basis. Quarterly basis means if I'm going to take 12%, so I'm going to take 12% divided by 4, and it's going to give me 3%. So now it's going to be compounded on a quarterly basis. So how do I know if the bank says compounded change this to quarterly? So what's going to happen is this, if this is compounded quarterly, first I have to find my quarterly rate and I did find my quarterly rate is 3%. Now what I would do is I will take 1 plus 0.03 raised to the fourth power because I have four quarters in a year minus one. So let's do this. So I'm going to take 1 plus 0.03 raised to the fourth power. It's going to give me 1.1255. I'm going to take, let me just subtract the one from the formula here because I can do that. Then I will subtract 1 minus 1. So the rate, if it's quarterly, it is 12.55, which is lower than, lower the quarterly rate is lower than the monthly rate because the more it compound, you know, monthly is more. So this is monthly and this is quarterly. Now if I want to do it semi-annually, what happened to this? Okay, 12.55. Now if I want to do it semi-annually, semi-annually the rate is 6%. So what I would do is 1.06 and I raise it two times a year raised to the second power and that's going to give me 12.36. So this is semi-annually. Now again, I know if I do it daily, if I do it daily, it's going to be way higher. We said the daily rate is 0.00329. So let's do this. So 1 plus this rate raised to the 365, all of this minus 1 to find if it's compounded daily minus 1 and that's going to give me a rate of 12.75. This is the daily, which is higher than everything else. The daily is higher. So the daily is 12.75, monthly 12.65, quarterly 12.55, semi-annually 12.36 and obviously annually is 12%. Why 12%? Let's use the formula. Let me use the formula here and I'm going to take 1.12 raised to the 1, raised to the 1th power. Okay, I don't have to do this, but I'll show it to you so this way. You see it's 12%, minus 1, minus 1, it's 12%. Therefore, if it's compounded annually, it's 12%. So hopefully, now we know how to compute, how to convert from APR to AAR. When you get a loan, they usually show you the APR and in the fine print, they'll tell you how often it's compounded. So you have to be careful, you have to be careful, not the APR, it's what matters is the EAR. So make sure you know how to compute the EAR. So if you're depositing money, if you're saving, if you're making an investment, the more compounded, the more compounds, the better off you are. If you are taken alone, the last compound, the better off you are. Usually credit card rates, credit card rate, they compound on a daily basis because they want to get as much money out of you as possible. So let's work this example to see how we can compute this. Suppose you buy a 10,000 face value treasury bill maturing at six months for 9,900. On the bills, on the bills maturity date, you collect the face value. The face value is obviously $10,000. You collect the face value, $10,000. Because there are no other interest payment, the holding period for six months is how much? Well, I made $100 return on a $9,900 investment. Simply put, I made 1.01% for six months, for six months. So this is my rate of return for holding period for six months. Now, if I want to find out what is my annual return, obviously I'm going to do this twice. So if this is six months, then if I do the same thing again, because you want to find your annual return, you're going to take 0.0101 times 2, and the APR will be 2.02. This is the APR, 2.02. This is the APR. So what is the EAR for this investment? Well, the EAR, remember, you're going to take 1 plus the compounding 0.0101, and you're going to raise this, you're going to do this twice minus 1. And that's going to give us a little bit more than 2.202. It's going to give us 2.03, because it's compounded semi-annually. Therefore, the EAR on this investment is 2.03. So what we did first is we find out what's the rate for six months, $100, that's how much you made, divided by $9,100, 1.01. Well, this is for six months. You multiply this by 2 to find your annual percentage rate. Then you're going to convert the annual into an EAR. You will take the period raised to the second power, because it's compounded semi-annually, minus 1, and you will find that 1.203, minus 1 will give you 2.03. So this is how you convert this. In the next session, what we'll look at is inflation and real interest rates. Simply put, how does the inflation affect the real interest rate? Now we're getting more and more into how to measure actually investments. As always, I'm going to invite you to like this recording and visit my website, farhatlectures.com, for additional resources for this course, as well as your accounting and finance courses, your professional certification. Good luck, study hard, and stay safe.