Welcome back. We did some simple examples of the simplest security or bond there is out there, which is a government bond which pays no coupon and we tried to calculate the yield to maturity. Let me tell you a little simple way of doing this. Let's take $1,000 and divide by $744.09. What is this telling me? This is telling me future value 1,000 and present value, the only two things I know. However, the key here is the number of periods that pass. Let me return it. So that is 1.3439. Now, let me do this. Take a1 and I take the 1/10 root of this. Remember we're doing compounding. So if it's over 10 years without compounding, I do the 1/10 root of it. It's raised to power 0.1, 1.03. You remember the three percent? You subtract one out of this calculation and then you get the yield to maturity because it's future value over present value. You have to subtract the one because that's the one buck you put in as your investment. The rate of return is not [inaudible]. Now, what's the difference between this and if I make compounding happen every six months. I'm taking now one to the 20th. Take a1, and now 1/20 is what? 0.05. Okay, there, I can see what I did. Minus and put two equals. You see, I make very silly mistakes. So 1.01489. You see how I did it. I'm just showing you that the fact of compounding is the only reason that calculations become a little bit tricky and difficult. Okay, let's move on. I got 1.489. I would encourage you to be very familiar with this stuff. Something that you'll see all the time in the press, and towards the end we'll go to a website and see all of this, is something called the yield curve. It's in your face every morning. If you pick up any newspaper to do with money or reporting about the economy, you'll see this, and I wanted you to get a flavor of what that means. It is the relationship between the maturity of a bond and the yield. It's for government bonds and purely should be zero-coupon bonds because it's trying to show you the connection between the length of time and the interest rate on government bonds. If you throw in coupons, it's not really picking up the relationship keenly. Typical relationship and why? Let me just show you the typical relationship and why. Drawing a graph, if this is zero and this is yield and suppose this is one year, this is two years, this is 30 years. The reason I'm doing 30 is, believe it or not, you can buy a bond that promises to pay you a 100 bucks 30 years from now, and it is traded and it has a price. That's why I find it really cool. Typical relationship is something like this. It tends to go up. I want to make sure that you get it before we do coupon bonds. The typical relationship is going up. Why is that? The reason is very straightforward. If you buy a one-year bond versus a two-year bond, or compare a one-year bond with a 10-year bond, whose price is likely to be wearing a lot? Always keep risk at the back of your mind. But now I'm increasingly going to pull that concept out and bring it to you because we are talking about real-world investments, a loan, or a stock. Risk has to be at the back of your mind. You just stay away from it in explicit manner, in explicit treatment, but bring it forth as we go along. Let me ask you this, very simple. Let me draw a timeline. One bond, one year from now gives $1,000. Its government. The other government bond gives you 10 years from now, 1,000. Which of these is perceived to be more risky? Suppose you've just bought this bond and you are at some point beyond zero and you bought both of them. Whose price will fluctuate more? Think about it. Very simple. Whose price will fluctuate more and because of what? You know the answer to almost 99 percent of the questions, right? The answer is compounding. This will fluctuate less. The reason is, its price is simply 1 plus r, 1,000 divided by one plus r. This price is 1,000 divided by 1 plus r raised to power 10. Imagine how r changes. If we're a government bond which doesn't have much risk, hopefully. The main reason r is changing is because of inflation. Remember I told you, our job is to keep up with inflation. The main reason is inflation, and there's a little bit of what we call real return built into it. If the interest rate goes up per period, what happens to one plus r versus 1 plus r raised to power 10? One plus r raised to power 10 is going to be much larger than 1 plus r. The price of a 10 year bond fluctuates much more than the price of a one year bond. Maybe may have to sell these bonds at some point. Because of that, what happens is the interest rate built into a 10 year bond has to compensate me for risk because I am risk-averse. I don't like risk, I being the average person. In fact, everybody almost [inaudible]. What happens? The interest rates are higher for 10 year bonds and that's why the yield curve is going up. That doesn't mean always going up. There's a second component is, how much do we expect the interest rate in the future to be and stuff like that. But I just wanted to give you a flavor of this and we'll talk about it and see some data later. Now, let's move away from zero-coupon bonds to coupon paying bonds. The reason I'm going to coupon paying bonds is, this is the nature of most loans. That most loans, you don't just give money today and then pay it back one shot right at the end. Most loans, even corporate loans, have coupons built into it. Let's start with government bonds. Most government bonds do have coupons. It's the most common type of bond out there. These bonds pay periodic coupons and a larger face value at maturity. All payments are explicitly stated in the IOU contract. We talked about the fact that this is an IOU. The difference between a zero-coupon and a coupon paying bond is simply the coupon part. We'll just do some examples. I'm going to spend a lot of time on this example and I think you should stay with me. The reason is we are not doing something profoundly different than what we've just done. Having said that, the mechanics and the intuition of this is very important. I'll take a break when we think we've gotten over the first few steps of understanding this. Please pay attention to this for a second. Suppose a government bond has a 6 percent coupon, a face value of $1,000, and 10 years to maturity. What is the price of this bond given that similar bonds yield an annual return of 6 percent? What if the similar bonds yield 4 percent and what if they yield 8 percent? Before we take a break and you get away for coffee or just go for a swim. Just let's go through the mechanics of this a little bit and try to understand what it's talking about. What I'm going to do is I'm going to develop the timeline and the formula. Then we can take a break and then come back and do the number crunching. Let's draw the timeline. The timeline is, if I remember right, how many years of this bond. Ten years. However, what do you remember about bonds? The bonds of government bonds of the US and I'm going to stick with those because that's what the data I'm showing you but you should be able to see this very clearly, is that they pay coupons every six months. The nature of the PMT payment process determines the compounding intervals, so zero through how much? 20. That's the first thing. What will happen at year 0.20, which is year 10? What will happen here? You'll get a 1,000 bucks and this is called face value. Very clear. Till now, what are we talking about? A zero coupon bond, we just priced it. Here's the twist. It says what? You'll get a six percent coupon. Many times in the real world, the word interest is used for coupon, I don't like that at all. To me, interest always belongs to the market, it doesn't belong to any entity. Please, I'm going to be painful and call it coupon. The coupon rate of six percent is this, C/F, and it's a percentage. We know F is a 1,000, so what is coupon? Very simple, six percent of a 1,000 is 60 bucks. However, although this is all written on the IOU, you know that the compounding interval is what? Every six months. What really is happening is, you're getting 30 bucks and 30 bucks. The reason is, over one year, then you're getting 60 bucks. So three, 30, and the nature of this bond is such that you also get 30 at the end. How many 30s are you getting? You're getting 20, 30s, and how many thousands? One. Doesn't this remind you of the loan? Thirty reminds you of what? The payment you pay on the loan. The only difference between this and a standard loan is this, that the face value in a standard loan is not there, you're just paying PMT. This is the nature of the timeline. Do I know m? Yes. Do I know coupon? Thirty bucks per six months. Remember, I have to match m with the coupon. I can't say 60 here. What is r? R was six percent per year, which is what? Three percent per six months. Everybody got the details? It's a very straightforward problem to do. The two components are these; the price today will have a PMT component of 30 bucks, how many times? Twenty times and the interest rate is how much? Interest rate is three percent. Remember, half of six. This is the PMT flow and you'll do the PV of this. This is the nature of your PMT and you'll do the PV of this plus you'll do the PV of a 1,000 too. How many periods? Ten years per periods, 20, and the interest rate is three percent. The way to think about the coupon paying bond is, it has two chunks. The first chunk is a present value of a PMT chunk. The second is present value of a one shot. You remember in the first day of class, I broke up the introduction of PV and FV into two parts. First day, we talked about single payments, the 1,000 chunk. The next day we talked about the loans and so on, the PMT chunk. This is a combination of the two just because the nature of the beast is such that you have a final payment of a $1000. If you understand the timeline, the formula and as I told you, all of this is explicitly stated in an IOU. Let's take a break and come back and crank through some numbers. Take care.