 Hello and welcome to the session. This is Professor Farhad in which we would discuss rates of return using arithmetic geometric and dollar weighted average. These topics are covered on the CPA as well as the CFA exam, also an essential or principles of investments graduate or undergraduate course. In this session, I will explain those methods and I will also work them using Excel sheet and show you the advantages and disadvantages for each method. 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 1,700 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 and connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources, the supplement that complement this course as well as other courses, the CPA exam, the CFA exam, CMA exam. If you're looking to add more resources to your education, check out my website. We're going to start by talking about the holding period return or basically your rate of return for a particular period. I'm going to work a simple example with you first. Assume, let's assume you purchased a home and this is, you know, this is how I'm going to, this is my home, this is my drawing and you paid for this home, $100,000. That's your purchase price. And what you did is you rented this place, you rented it. In the end of the year, you had $5,000 net rental income. So you made $5,000 from the rent. At the end of the year, you sold the house for $110,000. You sold it. If I ask you, what is your rate of return or what's your holding period return on this house? Well, let's, let's think about it. Your rate of return is composed of two things. First of all, from the house itself, you made $10,000. So $110,000 minus $100,000. From the house itself, you made $10,000 profit. That's called capital gain because the home appreciated in value plus from the rental you made $5,000. So overall, you made capital gain of $10,000 from the income $5,000. And you'll divide this by your original investment. You invested $100,000. Simply put, your rate of return is 15%, which is $15,000 divided by $15,000 divided by $100,000. Your rate of return is 15%. And basically, this is how we compute the rate of return. Part of it is capital gain. And part of it is basically income. Here it's income. We might call it dividend. We might call it interest. So it's capital gain plus what you received in income from that investment. So holding period on a share of stock, basically the same, reflect either the increase or a decrease. This home could also go down in value on the price of the share over the investment period, as well as any dividend income. For the home, you have rental income. It is defined as dollar earned in a price appreciation plus dividend. For example, to take a look at the formula, the ending price is 110, whether it's a stock or a home. The beginning price is 100,000. The rental income was 5,000 divided by 100,000. This is what I did. So the definition assumed that the dividend is spate at the end of the holding period. Otherwise, we have to take care of the reinvestment process, the reinvestment income. So what we assume, we assume the dividend is received at the end of the period. So the percentage return from dividend, which is cash dividend divided by the beginning price is the dividend yield. So if I ask you, what is the rental yield? Rental yield by itself is $5,000 divided by $100,000, which is 5%. So this is the rent, my rent yield. But if we're talking about a stock, this is the dividend and this is the stock price. So it will be called the dividend yield. Same thing. So and we add the capital gain appreciation is 10,000 divided by 100,000 equal to 10%. 10% plus 5% equal to 15%. So dividend yield plus capital gain equal to the holding period return. Let's take a look at a quick example. The price of a share of a stock is currently $100 and your time horizon is one year. You don't expect to receive cash dividend of $4, which is expected yield of 4%. And by the end of the period, the stock will be sold at 110. So basically, 110 minus 100, you're going to make $10 from the stock itself plus $4 from the dividend. In total, you're going to make $14 by investing $100. The total return is 14%. Your dividend yield is 4%. Your capital gain yield is 10. So your capital yield is 10. Your dividend yield is 4. Together, they'll give you 14%. Now, this concept also applies for whatever investment you are working with. For example, if we're dealing with bonds later, well, the interest is the interest on the bond is the income. Therefore, you'll have an interest yield and capital gain appreciation. Now, measuring investment return over a multiple period is a little bit more involved than a single period. Here, we looked at a single period. When we have more than one period, then we're going to have to use a different measurement because think about it. I know how much I made in one year, 10%. Now, the following year, I might make 10% or something else. So I have to compute the period, which is now, it's two years or it could be three years or four years. So over several periods, how do we do that? So you might want to measure how well a mutual fund has performed over the preceding five years. In this case, you might have more than one measurement. It becomes a little bit more ambiguous. And this is what we need to illustrate. We're going to look at an example to illustrate how we compute this. So consider a fund that started with a million dollars. This is what the fund is starting with with money. It receives additional funds from new and existing shareholders and also redeemed shares of existing shares. So the net cash inflow can be positive or negative. So in addition to the million dollars throughout the year, some people cash out, more people and add money to the fund. And this is what the fund looks like. So it's very important as we read this graph properly in order to solve what we need to solve. Okay. So I'm going to read it very, very closely. So this is the first quarter. We started with a million dollar, one million. For that quarter, we made a return of 10%. Well, guess what? A million dollar multiplied by 10%, which is to be more specific, one million multiplied by 1.1 will bring us to one million, 100,000. This is the total asset before inflows. Then the investors added $100,000 to the fund. So they added $100,000. Now we have at the end of the quarter, $1.2 million. So this is how we went from a million to 1.2. Part of it was 100,000 was part of the 10% and another 100,000, the investors put money. The second quarter, we started with 1.2, which is the ending period. We start the beginning period. Then we're going to take 1.2 million times 1.25. The return was 25%. Now we're up to 1.5 million. Then the investors added half a million because we made a lot of profit. They liked it. So they added half a million. We end up the second quarter with $2 million. I just made this up that they like the fund. It doesn't mean that's why, but I'm just kind of trying to make it a little bit more interesting. So we have 2 million by the second quarter. We're starting the third quarter. The third quarter, we started with 2 million and the fund experienced a 20% decrease in value. So we have 2 million and we're going to multiply the 2 million now by 0.8 because we lost 20%. By 0.8, we're going to be left with 1.6 million. So we went down to 1.6 million and the investors kind of panicked and they withdrew $800,000. They took out $800,000 because they panicked. Now what happened? We end up the fund, end up the third quarter with half a million, with $800,000. Then in the fourth quarter, guess what? In the fourth quarter, we made a 20% return. So we're going to take 0.8 times 1.2. Now we're back to 960,000. The investors put in another $600,000. We end up the fund at the fourth quarter, 1.56 million. So this is what we have. So it's very important that you understand what you are giving because we're going to be performing some computation to find out what is the rate of return. What is the rate of return of this portfolio? So hopefully you were able to follow. So there are several alternative measures of performance with each its own advantages and shortcomings. We have the arithmetic average, which is the easiest one, geometric and dollar weighted average. These measures vary considerably. So it's very important to understand the differences between them because it's very important when someone quotes your rate of return. So what I want to know is this arithmetic, geometric, or dollar weighted return? Because you will see we're going to have three different returns. Starting with the most simple and arithmetic simply put, you made 10%, 25%, negative 20, and 20. So you'll add them up. 10, 25, minus 20, plus 20 divided by 4 period. Your arithmetic return is 8.75. That's pretty straightforward. That's it. So this is the arithmetic and this is most likely what you are familiar with. Now let's take a look at the geometric method. The geometric method is also called the compound rate of return or the time weighted series. So it's time weighted, time weighted, cannot dollar weighted. So not dollar weighted, time weighted. So we don't take into account the dollar amount. We only take the time over the period. So the geometric average of the quarterly return is equal to the single period return that would give the same cumulative performance as a sequence of the actual return. So basically what we're going to do, we're going to take each period return and multiply it by the next period as a compounding effect. And you will see how we place in the formula. So we calculate the geometric method by compounding the actual period by period return and then finding the single per period rate that will compound to the same final value. So what we'll do, we're going to take all the return, compound them together and find out what is kind of the net return. So in this example, let's take a look to see how we work this example. Here's what's going to happen. We're going to take the return of the first period. This is the formula. You will take 1 plus 0.1, 1 plus 0.1 times, you multiply it by 1 plus 0.25. The second period. You multiply it by the negative 2, it means 0.8. So 1 minus, I'm going to put it 1, 1 minus 0.2, which is 0.8. Then you multiply this by 1.2, 1.2, which is 1 plus 0.2. Let's just be consistent, 1 plus 0.2. Then what you do is you raise those 1 divided by n. The number of period, the number of period is 4. Then you subtract 1. So this is the formula. Now I'm going to show you in an Excel sheet how you compute this and what does that actually mean. Because I know you read this. You may not understand what does that mean, but I'm going to show you exactly mathematically what does that mean. So if we perform this computation, if you perform this computation, you're going to get a return of 7.19. Now the question is, do you understand what does that mean, 7.19? That's the question. So first, let me show you the formula. So this is the formula. And it gives us 7.19%. And by the way, mutual fund, they have to show you the geometric average. Just be careful. They may call it geometric average. They may call it the compound rate of return. Or they might call it the time-weighted series. But the point is, they cannot use the arithmetic. They use the geometric. And just FYI, the geometric from a mathematical perspective would always be lower than the arithmetic, equal or lower. It can never be higher. So if you perform the geometric and you find out it's higher than the arithmetic, you made a problem. You made a mistake. So let me first show you how to compute this in an Excel sheet, then show you what does that mean, 7.19 on this fund. So first, what I'm going to do, I'm going to show you what does the 7.19 mean. I'm going to show you how we compute this in an Excel sheet. So we can take any amount. We can take $1,000. We can take $1,000. We can take $1,000. It doesn't matter. I'm going to use $1,000 because the geometric number, it doesn't care what amount you invested in. And what happened is, remember, our answer was 7.19. It means over the four period, we earned 7.19%. Let me show you. So if we take $1,000, the year one, we invested at 7.1 at the end of year one. So at the end of year one, we had $1,071.90. Again, we could use millions. We could use 10,000 of millions. It doesn't matter any dollar amount. We start period two. We start period two. So this is period two with this amount, $1,071. Again, this amount, we said we're going to be earning 7.19. And that's going to give us, at the end of period two, at the end of period two, $1,148.97. We're going to start period three. Period three, we have $1,148.97. It's going to grow at 7.19. And that's going to give us, at the end of year three, $1,231.58. Now we're going to start period four. And period four, we are starting with $1,231.58. It's going to grow at 7.19. And that's going to give us $1,320. So this is year four. So this is the number that we end up with. This is the number that we end up with. So what does that mean? Why did we do all of this? It means. It means. So what does the geometric mean? And why is it kind of accurate from a percentage perspective? It's accurate. So how much did we earn? We did earn, we did earn 7.9, 7.19 over a period of four years. Because if you invest any amount, $1,000, a million, 10 million, it's going to assume if you put $1,000 in this fund, you would receive at the end of four years $1,320, which is your actual rate of return. Your actual rate of return is 7.19. This is what it means. This is what it means. So remember, the average arithmetic, it gives us 7.85%, which is not true. What you did. In other words, if you put that $1,000 in the bank at 7.19, you will get $1,320. That's why it's more accurate. So it's showing you the compounding effect, the compounding effect using only percentages. And why am I emphasizing percentages? Because you're going to see in a moment, we're not taken into account the dollar amount. So geometric mean always use percentages. Now, how do we compute the geometric mean using Excel sheet? So basically what you do is you put your return 0.1, 0.25, negative 0.2, and 0.2. Those are the four rate of return. And what you do is you use this formula, just as simple as that. Let me just show you the formula. You would go to GeoMean and you highlight the rate and you add plus 1. Then what you do is because you added plus 1, then you subtract minus 1. So let's take out the one here. It doesn't. So geometric mean you have to put plus 1. The formula would ask you to put plus 1. Then what you do is you take the formula and you subtract 1 from it. And you'll get 7.19. So it's a very, very simple way to compute it using Excel sheet. So once again, you have the function GeoMean. And let's pull the function. Why don't I do that? So this way, in case you are. See, how did we get here? Just go to the function and you go to the Geo function. Right here, Geo function. You highlight the rate and you put plus 1. Now, y plus 1, I'm not going to worry about this for now. Let me just because F13, not F14. F13. Okay. Then you will get 1.071. Then you'll take the answer minus 1. There's a way to do it in a formula to take the one out. But I just, I don't want to complicate this. Equal to 7.19. So this is your geometric mean. So geometric mean doesn't care about the dollar amount. It tells you if you took your money and placed it in another investment, you would have earned 7.19 equivalent to that fund. Now let's take a look at the third measurement, which is the dollar weighted return. So from the word dollar, from the word dollar, you should understand that this takes into account, this takes into account how much was invested in every period, how much was invested, because it's taken into account the dollar amount. In contrast, in contrast, geometric. Geometric, all what geometric cares about is the percentage. Here it's going to find the percentage, but it's going to take into account the weight, how much money you had in the account. To account for the varying amount, which is, we did have varying amount under management, and that's why, before I proceed, that's why they use the geometric mean for the mutual fund, because the manager don't have control. If the investors want to take their money out, they have no control under how much money, how much money do they have under management. All what they can control is how much they can earn. This is why we use geometric return, not dollar weighted average, because the dollar weighted average of the investors take their money out, and the stock market goes down, then it would look bad for the manager. So that's why we use the geometric. To account for the varying amount under management, we treat the fund cash flow as would be a capital budgeting problem in corporate finance and compute the portfolio's internal rate of return. Here what we're going to look at this problem is an internal rate of return. Hopefully you know what this is. If not, I'm going to show it to you briefly. If not, go to my corporate finance, but basically we're looking this as a capital budgeting. To find out what would be the rate of return to make this investment equal to zero. In other words, when you put your money, you kind of, what's your rate of return based on the money that you invested? Okay, so the initial one million and the net cash inflows are treated as cash flow associated with an investment project. And the year for liquidation value, at the end of the year, you're going to take out 1.56 is the final cash flow of the project. So this is how you set up the problem to find out the dollar weighted return. Year one, you invested one million. Also, this is at the beginning of the period. This is 0.0. At the end of the year, you invested, so this is the one million, negative one million, then you invested 100,000. And this is this 100,000 here. Year two, you invested another half a million. Year three, you invested, you took out 800,000. Notice it's a plus. Year three, it's a plus, 800,000. And year four, you invested 600,000. Then you took out 1.56 million. So the net was a positive 9.6. Now what we do is we compute IRR. And obviously you don't want to do this manually, but this is the formula if you're computing IRR. Basically you set the formula equal to zero because NPV equal to zero for IRR. And you say, you invested a million, then you invested 100,000, one plus the rate, the internal rate of return that your net present value equal to zero. So this is the actual formula though. You don't want to do this manually. I'm going to show you how to do this on an Excel sheet. Excel sheet. So let's go back, let's go to the Excel sheet and show you how you can quickly do this on an Excel sheet. Then tell you exactly what does that mean? What does the N-P-N-I-R-R mean when we find the IRR? So back to the Excel sheet. Now we're going to look at period zero and we have four periods, four periods. Period zero, we invested a million dollar. Then let me show you what we did in period one. We invested 100,000, we invested half a million. We took out 800,000 and in year four remember the net was 960,000. We put some money and we took out the 1.56. Now we need to learn how to compute the IRR. It's pretty simple, very simple computation. We take, we say IRR, basically we go to the function and we go IRR, the IRR function. Let me go down and you highlight the cells where the rate of returns are and you click on OK. And you find out it's 3.38. It means based on the dollar amount invested. Now based on the dollar amount 3.38, you might be saying this does not make any sense. I only earned, the arithmetic was, if you remember the arithmetic was 8.75. The geometric was 7.19. 3.38 is pretty low. Why? Here's the question here. Why is the IRR is low? Here's why. Because when you had $2 million under investment, you suffered a loss of 20%. What does that mean? It means when you had a lot of amount invested, the manager suffered a huge loss. As a result, it lowered your internal rate of return. You did earn 25% at some point, but you did not have that much money under management. So that's why it did not, so your rate of return really suffered with that 20% decrease when you had $2 million under management. Now you earned another $2 million, another 20%, but at that point the investors took their money out. So that's why this is called a dollar, a dollar weighted return, a dollar weighted average. So it's taken into account how much money you had. So notice we had three different returns for this mutual fund. Again, which return do they quote for mutual fund? They quote the geometric. Why? Because it's a time series return. It ignores all other factors. I will take a look at this exercise. And as a practice, I will work three different, I will compute three different returns. Just to kind of make sure you are comfortable with this. Also, if you want to try this on your own, you can try it. In the next session, I will have the example. If you have any questions, any comments, by all means, let me know. As always, I'm going to invite you to visit my website, farhatlectures.com, like this record and share it, put it in playlist, study hard and stay safe.