 Hello and welcome to the session. This is Professor Farhad in which we would look at the yield curve and specifically we're going to look at the expectations theory that explained, that's one explanation of the yield curve. This topic is covered on the CPABEC section, the CFA exam, as well as essentials or principles of investments. As always, I would like to remind you to connect with me on LinkedIn and subscribe to my YouTube channel where I have over 1,800 plus accounting, auditing, tax as well as Excel tutorials. If you like my lectures, please like them and share them with others. If they benefit you, it means they might benefit other people as well. Connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources to supplement this course as well as your other accounting and finance courses. I strongly suggest you check out my website. Let's talk about the yield curve. What is the yield curve? It's a graphical relationship between the yield to maturity, which is the yield to maturity rate and the term, the time. And specifically, we're going to be dealing with treasury bonds because we're dealing with government bond. It's a risk-free rate. Bonds, generally speaking, bonds with short-term maturity offer lower yields to maturity than long-term bond. And hopefully this makes sense. If the government is borrowing money for short-term, they usually pay less in interest. Why? Because if you're borrowing money for three months, the lender is willing to give you that money because there is less uncertainty versus giving you that money for 10 years. If the lender wants to give you the money for 10 years, they want to be compensated because a lot of things could happen between now and 10 years. So for them to be compensated, they would require a higher rate of return. The relationship is this relationship is also called the structure of interest rate because it relates yield to maturity to the term, to the maturity. And this is what the yield curve would look like. Treasurely, it means the government yield curve. And this one here is the rising yield curve. This is the kind of, in a sense, the normal yield curve. What does that mean? Does that mean by normal or normal in quote or rising yield curve? It means shorter borrowing would require lower interest rate. And as time goes by, one year, three years, five years, 10, 30, the government will have to pay more. So up to a year, they pay close to 0%. It's not zero. It's close to 0%. That's why the graph is like this. For example, this is a sample of the yield curve from October 2014. This is September 11, year 2000. We have an inverted yield curve. What happened in the inverted yield curve? Short-term borrowing the three months is, if the government wants to borrow it, they have to pay approximately 6.2% versus a 30 year. They have to pay less. This is an inverted. It means short-term borrowing is higher than long-term borrowing. This is usually, this is a signal that we are going into a recession and the inverted yield. It happened last year. We really did not go through a recession, but we don't know because COVID came and we don't know what happened because COVID threw everything off-chart, so we really don't know. But this is the inverted yield curve. This is a humped yield curve. This is October 4, 1989. It's upward sloping. Then it goes down. It looks like a hump. In January 2006, the yield curve looked like this almost flat. Now, to understand the yield curve, you have to put it into a economic historical perspective. For our purposes, you just want to know how to read the yield curve. One theory to explain the yield curve is called the expectation hypothesis or the expectation theory. It starts with the assertion that the bonds are priced, so the treasury bonds are priced, so that buy and hold investments in the long-term bond provide the same thing as rolling over a series of short-term bonds. So if you buy a bond for five years or if you buy every year, bonds for year one, year two, year three, year four, and year five, you keep rolling over your bond, you should be earning the same rate because the five-year bond is the expectation of that one year roll over. Simply put, let's assume to start this illustration, let's suppose that the current one-year investment rate is 8%. So for one year, it's 8%. And year two, if you keep your money for year two, you would earn 10%. So year one is eight, year two is 10%. What can we imply from this? If we know this much, what can we imply from this? Well, we can imply that if you have a two-year investment, let's assume this is a two-year investment, the two-year investment should be approximately 9%. Why it should be 9%? Because if in year one, you are earning eight, in year two, you are earning 10%. On average, between the two years, you are earning 9%. So simply put, if you already know that the two-year bond is offering 9%, and the one-year bond is offering eight, you will be able to find out what's the year two bond, what should be the year two bond when you roll over your money. Because when you roll over your money, it should compensate you in total of 9%. So what it should be? It should be approximately 10% to be compensated. So it looks something like this. So year one, the bond is offering 8%. Year two, if you invest your money in year one, and then you roll it over year two, you already know that much, 1.08 times 1.1%, it's going to give you in total 1.18. If we find the square root of this, it's going to give us, you know, basically 8.95. Because this is 1.18, you square root it, then you subtract one, the rate is approximately 8.95, or let's just say for simplicity, 9%. So the two-year investment should be 9%. So simply put, this two-year investment, let's call this two-year investment, investment B, and this is investment A. Investment A is putting your money in an investment for 8% in the bond, then roll over it, roll over the money for year two. So for year one, those two, technically they should equal to each other in the long run for two years. So year one, this is paying 8%. This is paying a little bit more than 8%. This is paying 8.99. But in year two, investment A will compensate, will earn you 10%. So in other words, for year one, if you really think about it for year one, they're equal. They're equal. All what happens in year two, the reason year two, this is 10% is to compensate you because you want to earn the two-year investment. And this is basically, in a sense, the expectation hypothesis, which is asserts that the slope of the yield curve is attributable to the expectations of changes in the short-term rate. So by knowing the 9%, you would know year two is 10%. So sometime, relatively high yield on the long-term reflect expectation of future increases, and obviously low yield on the long-term, it indicates a downward slope reflecting expectation of falling interest rate. So even if we don't know year two rate, we can find year two rate. So the final proceeds of the rollover strategy depend on the interest rate that actually transpires in year two. We know over two years we should earn 8.99%. So how can we find year two mathematically? We can solve for the second-year interest rate that makes the expected payoff of the two strategies, whether you have two-year 8.99% or one-year at 8% plus some unknown for year two. Okay, but we're going to find year two. So the break-even value is called the forward rate for the second year. And how can we find the forward rate? Remember, for two years it's 1.899 raised to the second power. This is your total return for two years. It should equal your year one, which is 1.08 times one plus some forward rate. Now if you solve for the forward rate, you will find the forward rate equal to 10%. This is basically the expectation theory. Based on the two-year, the two-year, based on the two-year bond, we were able to find the rollover rate or the forward rate. Let's take a look at another example. Suppose a two-year maturity bond offers yield to maturity of 6%, and the three-year bond has yields of 7%. So what does that mean? It means we have two bonds. One is offering 7, 7, 7, and we have one is offering 6%. And we don't know the third year. Now, can we find what's the rate in the third year? Yes, we can. What is the rate in the third year? The rate in the third year should equal, so if this is investment A, this is investment B. So investment B, the rate of year three, this rate of year three, should equal, should compensate us to be equivalent to the bond in year A. So that's all what we're saying. Why? Because the expectation theory, because those bonds, if they have the same risk, although you're earning 6 here, you're earning 7 here, this is a short term. For three years, they should equal to each other. So let's see how we can solve this. Let's see how we can solve this. So buy, so what should you do? If you buy a three-year bond, the total proceeds will be 1.07 raised to the third power, which is 1.1225. Now, that's one way to do it. Then you buy a two-year bond. You don't have to buy. Just think about it. First, you earn 6%. You reinvest the proceeds at 6%. Now, the question is how much you should earn in year three? This is the expectation. Well, the expectation is you should earn in year three enough that your total will be 1.225. So you invest all the proceeds in one-year bond in the third year, which will provide a return for that year, or three, the return. Total proceeds per dollar invested will be the result of two-year growth of invested at 6% plus the final year, which should be whatever that final year is. So it's whatever you invested a dollar times 1.06 for the first two years times some other rate for the third year, which is simply put, we can summarize the formula for the two-year bond is 1.1236 times some third year rate. Now, we know what should be the total return. The total return should be this one here. This is the total return 1.2250. So all this formula should equal to 1.2250. All we have to do now is solve for F3, the interest rate for period three, which is if we do that. So in year three, you should earn 9.2%. So year three, now we find out year three should be 9.02. So how did we find this out? It's based on the expectations. If investment A earns 7.7 and 7, investment B for the third year should earn you enough. That's the expectation should earn you enough that over a three-year period, those two investments should earn the same amount. So what's the implication of the expectation theory? One of the expectation is that expected holding period on bonds on all maturities ought to be about equal. Even if the yield curve is upward sloping, so the future rate are higher than the current rate. This does not mean that investors expect higher rate of return on the two years. Over the three-year period, you're going to earn the same rate. Or earn the same rate. So the higher yield to maturity on the two-year bond is necessarily to compensate investors for the fact that the interest rate will rise next year. So yes, if you invest for two years, yes, you might earn a little bit more. But if I invest for one year and when my one year is up, I would reinvest again. Well, I would know what should I reinvest at based on your two-year rate because the expectation theory I should earn two years the same that you are earning. That's basically where it is. It doesn't have to be over two years. Over the two-year period and indeed over any holding period, this theory predicts the holding period return will be equalized across bonds of all maturity. So whatever you invest, so simply put, if you have an investment for two years and if you have an investment for five years, knowing if you know the investment like this is seven, seven, seven, seven and seven. Let's make it three years. Let's not make it too long. Well, I'm using the same numbers. This is eight, eight and eight. And here you have six and six. You will be able to know what should be the third year. Why? Because for three years, you should earn in total 1.08 raised to the third tower. So this is R3. It should be 1.06. So this should be 1.06 rate to the second power times, times one plus R3s. And you just set the formulas. So 1.06 rate to the second power plus 1.R3 should be 1.08 raised to the third power. All we have to do now, and you'll be able to find the rate in the third year. So this is the, simply put, the expectations theory. In the next session, we would look at another explanation of the yield curve, and that's the liquidity preference theory. As always, if you like this recording, please like it and share it. And don't forget to visit my website, farhatlectures.com for additional resources that's going to complement and supplement your accounting, finance, as well as your CPA preparation. Good luck and study hard.