Now let's proceed with bond cash flows and present value. Again, I will briefly recall. So this is a pattern, so 0, 1, t, c, c, c, and then C + F here. So the PV formula looks like this, PV of the bond is equal to C over 1 + r1. If here we have the rate of r1 + C2 over 1 + r2 squared + the general firm is Ck over 1 + rk for the kth hour, and then the final payment is C. I'm sorry, here, these Cs are equal, like this, C + F over 1 + r t to the tth power. That would be too much of a good thing, if we. So the formula looks great, if we knew all these r1, r2, r3, and so on to rt. And if we did know all of them, we could say, great. This should be equal to the market price of the bond and 0.1. The problem is that on the market we observe ps, we do not observe rs as anything seen. So our problem is to somehow extract rs from market prices of bonds. Now, you can immediately see that the use of this formula is, well, you cannot get all these Rs from not only one equation. But you can have many equations and these equations are not linear, so it's kind of difficult to find these Rs. And on top of that the key story is that these rs are expected and the long of the horizon the wider the range of these expectations. So we can see that, unfortunately, even for riskless bonds, we see that we cannot use our standard shortcuts, because these are all different. And we have to somehow find these rs, that are by the way called spot rates, and that really creates a problem. Like I said, we do not expect anything we do not expect anything else. But the question is, let's say I'm an investor and I look at this formula. And I say, well, that's really cool but I wish I understood what these rs are and normally investors have more specific and more straightforward question to answer. And the question goes like, if I held this bond for a certain period of time, what would be my realized rate of return over this period of time? And there's this specific case when I hold the bond until maturity, because investors normally do not have to hold bonds until maturity. We talked about that in greater detail in our capital market scores. What the idea is that bonds are public liquid instrument. So basically people buy and sell them freely, and therefore even if I bought a bond at the moment of the issue. Then I might change my mind, and then sell it in market at any point in time. Then my return would be calculated from the time that I bought it to the time when I sold it. But oftentimes, people buy bonds to hold them until maturity and that specifically is the case with bonds with no options. Because for example, if I bought a callable bond, and I did have an idea to hold it until maturity, but it might be called by the issuer. So this question of how much will I get if I held these bonds to maturity, actually can be treated in a way by introducing special parameter. I will specifically start from the next flip chart, and that would be called yield to maturity. Let's see, what if, in the general formula, I forcefully replace all these rk with one and the same parameter. You can say, well wait a minute, you just told us that you can not, assume that rs are equal. Well, they are not, this is a special parameter. This parameter is called Y, or yield to maturity. And this is not any kind of an average or, this is a very complex severance if you will. So I feed the formula and say PV is equal to C over 1 + Y plus C over 1 + the same Y squared + the general term will be C over 1 + Y to the K power. And the final payment will be C + F 1 + Y to the tth power. So this is sort of an artificially designed formula. But the good news about this formula is that, first of all, from this equation, if we set this PD to the price of the bond, we can say that we sold it. Because these Cs are, for bonds known. P we observe on the market. We can solve this equation, well, this is a bad equation, it's nonlinear. It has a. But it can be solved using numerical methods, and we can receive this Y as a single thing. Well, people who studied Calculus and advanced algebra can say, well, this equation has two solutions, but it can be shown that for us only one would be a good one. Without going deep with what this good means. So the important thing is the meaning of this yield to maturity. And again, the meaning is, like I said, this is what will be my expected annual rate of return if I held the bond from the moment I bought it, until maturity. This is very nice, and again this is a great advancement for us in the ability to use the approach. However, the yield to maturity is widely used, but unfortunately as some problems. Well, and all these problems, they stem from the fact that when I replace the more general formula with a known RK, with this formula. I sort of compress the information. It can be said in a metaphorical way that Y is a very specific average of all these expected rs. And therefore the information has been lost. And therefore if I see yields to maturity, that is not enough to oftentimes find these market prices. So the story goes like this, we observe expected rs from them we must calculate the right PV based on the formula that was on the previous list on the flip chart. Only then from PV, we can go to yield to maturity, not the other way around, because on the first way, we have a lot of information. And then we get it from crest. We cannot go backwards, because from only one parameter, you can by no means find all these rs. So often times people say, well, that yield to maturity is sort of equivalent to the price of the bond, because you can find one from the other. Well, this is not perfectly right, it's sort of approximately right, in most cases. But you have to really keep in mind that on this way you might find some aggravations. And therefore like I said, strictly speaking P over 1 is not the function of yield maturity. But as I said before, the good news is that most often we can use the yield to maturity as a very nice proxy. Or better to say approximation to the realized return that investors would actually get. I'm wrapping up here, and in the next episode, we will go a little bit further. Remember, I told you that we observe prices, and we need these RK. How can we extract these expected rates of return, RK from observed prices? This is called calculating spot rates, where we'll study the example of how this is done in the very next episode.
