Welcome back. So 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. And 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. So let me return it, so that is 1.3439. Now let me do this. Take a1 and I take the tenth root of this. Remember, we are doing compounding so if it's over ten years without compounding, I do the tenth root of it, its point raise to power 0.1. 1.03 so remember the 3% you subtract one out of this calculation and you get the yield to maturity, because it's future value over present value. And you have to subtract the one. Because the one buck you put in is your investment. The rate of return is [INAUDIBLE]. Now, what's the difference between this, and if I make compounding happen every six months? I'm taking now 1 to the 20th. So take a1 and now 1 to the 20th is what? 0.05. Okay, there I can see what I did. Minus and put those equals, you see I make very silly mistakes. So 1.01489. So 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. So let's move on. And I got 1.489. I would encourage you to be very familiar with this stuff. So 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. And it's any of this every morning if you pick up any newspaper to do with money or reporting about the economy you'll see this and I want to do to get a flavor of what that means. It's the relationship between the majority of a bond and the yield and it's for government bonds. And clearly 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 bond. So if you throw in coupons, it's not really picking up the relationship clearly. Typical relationship and why, let me just show you the typical relationship and why. So drawing a graph, if this is 0, and this is yield and suppose this is 1 year, this is 2 years, this is 30 years. And the reason I'm doing 30 is, believe it or not, you can buy a bond that promises to pay you 100 bucks 30 years from now. And it is traded, and it has a price. So that's why I kind of find it really cool. So the typical relationship is something like this. It tends to go up and, I want to make sure that you get it before we do coupon bonds. So the typical relationship is going up. Why is that? The reason is very straightforward. If you buy a 1 year bond versus a 2 year bond, or compare a 1 year bond with a 10 year bond, whose price is likely to be varying 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're talking about real world investments. Loan or a stock. Risk has to be at the back of your mind. We'll stay away from it in explicit manner, in explicit treatment, but bring it forth as we go along. So let me ask you this, very simple. Let me draw a timeline. 1 bond, 1 year from now gives 1,000. And it's government. The other government bond gives you 10 years from now, 1,000. Which of these is perceived to be more risky? So suppose you just bought this bond and you are at some point beyond 0. 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 don't answer to almost 99% of the question. The answer is compounding. So, this will fluctuate less and the reason is its price is simply 1 plus r. 1000 divided by 1 plus r. This price is 1000 divided by 1 plus r raised to power 10. So imagine how r changes. And for a common bond which doesn't have much risk, hopefully. The main reason r is changing is because of inflation. Remember I told you r's job is to keep up with inflation. So the main reason is inflation. And there's a little bit of what we call real return built into it. So, if the interest rate goes up per period, what happens to 1 plus r versus 1 plus r raised to power of 10? 1 plus r raised to power of 10 is going to be much larger than 1 plus r. So the price of a 10 year bond fluctuates much more than the price of a 1 year bond. And maybe they have to sell these bonds at some point. So 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 adverse. I don't like risk. I being the average person. In fact everybody almost in the world so 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. And the reason I'm going to coupon paying bonds is this is the nature of most loans, that most loans don't just borrow, you don't just get money today and then paid back one shot right at the end. Most loans, even corporate loans, have coupons built into it. So let's start with government bonds. Most government bonds do have coupons. So, and it's the most common type of bond out there. These bonds pay periodical bonds and a larger face value at maturity. All payments are explicitly stated in the IOU contract. So we talked about the fact that this is an IOU. So, the difference between the zero coupon and a coupon paying bond is simply the coupon part. And 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. And the reason is we are not doing something profoundly different than what we have just done. Having said that, the mechanics and the inclusion of this is very important. And I'll take a break when we think you've gotten over its first few steps of understanding this. So does, everybody please pay attention to this for a second. Suppose a government bond has a 6% coupon of face value of $1,000 and 10 years to maturity. What is the price of this bond given that similar bonds yield and annual return of 6%? What if the similar bonds yield 4%? And what if they yield 8%. So let’s, 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. So what I’m going to do is I’m going to develop the timeline and the formula and then we can take a break and come back and do the number crunching, so 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. And the nature of the preemptive payment process determines the compounding intervals. So zero through, how much? 20. So that's the first thing. What will happen at the year point 20, which is year 10. What will happen here? We'll get 1000 bucks, and this is called face value. Very clear. Until 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 6% coupon. And many time, 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, doesn't belong to any entity. So please, I'm going to be painful and call it coupon. And the coupon rate of 6% is this, C over F, and it's a percentage. So we know F is 1,000 so what does the coupon? Very simple. 6% of 1,000 is 60 bucks. However, although this is all written in on the IOU, you know that the compounding interval is what? Every six months. So what really is happening is you're getting 30 bucks and 30 bucks. And the reason is over 1 year then you're getting 60 bucks, so 3, 30. And the nature of this bond is such, that you also get 30 at the end. So how many 30s are you getting? You're getting 20, 30s. And how many 1000s? One. Doesn't this remind you of the loan? So 30 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, PMT, PMT, PMT. So this is the nature of the timeline. Do I know n, yes. Do I know coupon, 30 bucks per 6 months. Remember, I have to match n with the coupon. I can't say 60 here. And what is r? r was 6% per year. Which is what? 3% per 6 months. Every got the details? So it's a very straightforward problem to do and the two components of this. The price today will have a PMT component. Right of 30 bucks. How many times? 20 times. And the interest rate is how much? How much is the interest rate? Interest rate is 3%. Remember half of six. And this is the PMT flow. And you will do the PV of this. So this is the nature of the PMT and you'll do the preview of this plus you'll do the PV of 1,002, how many years from now? How many periods, sorry. 10 years, probably it's 20 and then to straight it 3%. So the way to think about a coupon paying bond is, it has two chunks. The first chunk is a PMT chunk, a present value for PMT chunk. The second is, present value over one shot. So, 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 thousand 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 $1,000. So 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 today come back and crank through some numbers. Take care.