Now, let's take a look at this expression and it's important that we make a distinction between this expression and the valuation expression that we looked at before which I will bring in a minute once again. What do we know about this expression? Well, we know the left-hand side. The left-hand side is the price that I have to pay today. Bonds like stocks have market prices. Bonds like stocks trade and I can actually observe what that market price is. Every bond at every point in time will have a price and that price is objective and observable as all prices are. What else do we know about these bond? Well, remember, we know the coupon that is going to pay, we know when those coupons are going to be paid and we know when and how much we're going to get back in terms of principal. We know the left-hand side, we know the numerators of the right-hand side. The one thing that we don't know is the discount rate and that is exactly the yield to maturity that we're going to calculate. Now, that yield to maturity that is, if you look at the expression, it's a little complex to calculate that. You cannot certainly do it by hand, you need to do it in Excel, you need to do it in a financial calculator, you need something to calculate those yields to maturity. I want to make sure that we define it and we understand exactly what it is. First, it is the annualized return that we get from buying the bond at the market price and holding the bond until maturity. There's a couple of things that are important here. Number 1 is we take the market price as given. In order to know what my return is going to be, I need to know how much I'm going to pay for the bond. In other words, the bond tells me when it's going to pay me and how much it's going to pay me. Once I put that together with how much I have to pay to get access to those cash flows, then I can calculate the return. It's very important that the market price is known. The market price is taken as given in the expression that we have here. Point number 2 which is also extremely important, this is the return, the annual or annualized return that we're going to get if we buy the bond at that market price and we hold it until maturity. That second part is important too because if I buy a 10 year bond and I calculate the yield to maturity and that yield to maturity is 3 percent per year and I sell the bond two years down the road, that doesn't mean that I'm going to get 3 percent and 3 percent. I can get a return that is much higher than that or much lower than that. I can even lose money when I do that simply because market prices will be fluctuating over time and what is going to determine the return that I put in my pocket is the price that I paid for the bond, the coupons that I put in my pocket and the price at which I sell the bond if I sell it before maturity. A critical component of the yield to maturity and that's why it's called yield to maturity is that we buy the bond at the market price and we hold it until the bond expires. That's what gives me the annualized return of buying the bond today and holding it until maturity. If you sell the bond anytime before maturity, your return can be higher or lower than the number that comes out of this expression. It's very important that you keep it in mind most of the time for what I mentioned before, we tend to refer to a bond's yield but the actual name is yield to maturity which indicates that you pay today the market price, you hold the bond until maturity and you pocket all those cash flows until the bond expires. Again, we're going to calculate that number in a minute for at least one of the bonds in the case and I'm going to give you the numbers for the rest of the bonds in the case. Second very important thing about the yield to maturity and this goes back also to a concept that we talked about in our previous course which is the internal rate of return. Think about the way we evaluate projects. We typically say well, in order to start this project I need to put down this amount of money today and then what I expect to get is this much and this much and this much and this much until the project is basically, it's depreciated or the project ends or however you put an end to this particular project which it doesn't have to have an end but typically we do put an end when we do investment evaluation. Now, we can calculate a net present value to determine whether we want to invest in this opportunity or not, but we can also calculate an internal rate of return. Remember the way we think about internal rates of return in project evaluation. Once again, I put some money down today and I expect to put some money in my pocket in the future. When I put all these thing together, I can back out the return that I'm going to put in my pocket. If you think about that, it's nearly exactly what we said about the bond. I have to take money out of my pocket to buy this bond and what this bond is going to do for me is going to be paying me cash flows for a specified period of time at a specified points in time and then the bond is going to expire. Another way of saying that is that the same internal rate of return we calculate for an investment project, we can calculate for a bond. That internal rate of return we can calculate for a bond is identical to, it's exactly the same as the yield to maturity that we're talking about. In other words, calculating a bond's internal rate of return and calculating a bond's yield to maturity are exactly the same thing. In other words, the bond's yield to maturity is the bond's internal rate of return. When we calculate for a given bond the yield to maturity in a minute, we're going to use the idea of internal rate of return. But it's very important that you put in the back of your mind that idea that a bond's yield to maturity and a project internal rate of return are the same thing applied to two different types of assets. The expression that you have there is the same one we've seen before for valuation of a bond, but there's a fundamental distinction with the expression that we have on top. Fundamental distinction is that in the expression that we have on top, remember we know the market price, we know all the numerators, we know all the coupons and the face value and we need to back out the discount rate that is the yield to maturity. In the lower expression, we know all the right-hand side and we need to calculate the left hand side, meaning that we input all the coupons and face value that we're going to put into our pockets, we also input the discount rate, what we think is the proper discount rate for this bond and then we get as a result, what we think we should pay for a bond. Remember that's why it has a v, it doesn't have a price. If markets are efficient you would expect p to be very similar to v. But again, when we're doing valuation we're calculating an intrinsic value. We're calculating how much I think that this bond is worth. I'm using the word causality here in quotations, I don't want to push that expression too much but it's important that you keep in mind that in the expression for the calculation of the yield to maturity that we have on top, we have the market price is given and the coupons are given and the principal or face value are given, we back out the yield to maturity. But in the expression below, we know the coupons, we know the principle, we know the discount rate and then we get as a result, the value of the bond. It's important that although the two expressions look very similar to each other, what we calculate from one and what we input from one is different from what we input in one and we calculate from the same expression. What we're going to calculate in a minute is not how much we should pay for each of the bonds in the case, we're going to take the market price as given and we're going to ask the question, if I pay this market price and I hold the bond until maturity, what is the return that I'm going to put in my pocket? That's the question we'll be answering in just a minute.