In the previous episode, we saw that the prices of bonds observed on the market allow us to extract spot rates. But, what if we would like to make an investment? Not from zero to a certain point in the future but from one point in the future to the other point in the future. Why would we need so? Well, let's suppose that we expect to receive some amount of money but not now, but let's say in a year and we would like to invest it for a year starting from point one. But for any reason we would like to lock in the terms of these investments and investments right now. Well, that is called forward investments. So I will put the chart here. Forward rates; zero, one, two and again, simple investments are like this. Let's see. This is the investment from zero to one and this occurs at the rate r1 at the spot rate. Then alternatively, we can make a two year investment and that would occur at the spot rate r2. Now you can see that actually this is the forward investment it is the f1, 2 but we just make it this way. Now, how can define f1, 2 if we know r1 and r2? Well here we use a classic approach that is called the absence of the riskless arbitrage. So, we basically state that, if an investor goes over the upper path then the amount of money that he receives here that he gets on this path. Otherwise, there is the ability to make money from nothing, I will show some example from that in just a moment. But if this is the case, we can write a very simple equation. We can say that 1 plus r1 this is applied rate, multiplied by 1 plus f1, 2 is equal to one plus r2 squared. Again, all these rates are one, f1, 2 and r2 they refer to one period. Well, clearly this formula can be easily generalized and it goes like this, to be consistent I will put 1 plus rk to the k power times 1 plus f km to m minus k power and that should be equal to 1 plus rm to the m power here, here m is greater than k. Now, this is a very simple formula and oftentimes it allows us to calculate all these future rates like 2, 3, 2, 4,2 5 whatever. And now, I promised to you to show that the absence of the risk arbitrages it presupposes some relationship between Rs. Let's for example try to analyze the situation. I put it like, If r1 is five percent and r2 is two percent. Well, if we use the formula, we would see that f1, 2 will be equal to minus 09 percent. What does that mean? That means that here the rate is negative. Well, it's very easy to show what goes on here. In this box I'll show some cash flows. Let's say this is zero, this is one, this is two and see what happens. If this is the case, then I borrow for two years at r2. So here from borrowing, I get, I'll be consistent with powers, so I get $100 that they borrowed for 2 years. So, here I will have to pay out minus 104.04. This is 100 times one plus r2 squared. Now, so I take this amount and then I immediately invested at rate r1 and here I get plus 105. See what my cash flows are 0, here I have 105 and here I have to pay out minus 104.04. If I did nothing to this 105 at all, then here I will get plus $0.96. So, I started with zero and ended up with a positive amount of money. That is called Whistler's arbitrary. So I made money from out of nothing. So, clearly this this can exist for some time but not for a long period of time. Well, now with the development of technologies we know that oftentimes trading in capital markets occurs at a very, very low spreads. For example, high frequency traders, this is a very special kind of traders. They use nearly, on this order flow, the minor differences in quotes and then they just move very close to make sure that the market is clear in the best way. And that to a first approximation seems to contradict the market efficiency, but it doesn't. Well, this is a story that goes well beyond the scope of this course. Maybe someday we'll do a course about [inaudible] too. But for now, the important thing is that we have found the way based on riskless bonds to extract both spot and forward rates. Now, I will wrap up this short discussion of bonds in this first part of a second week of our corporate finance course by saying a few things about the term structure of interest rates. So this is basically the function. If this is t and this is r, this is what people expect to see what will happen to these Rs. There are a couple of theories by which people tried to explain this structure. Again the term structure of interest rates very often is upward sloping, but sometimes there are also periods of negatives slopes. And some of these theories deal with expectations, some of them deal with the preferences, some of them are much more advanced. Now, it can be stated that no one theory is completely prevailing and this is actually one of the areas of modern finance, that is still under thorough investigation and people really try to find some new pieces of biblical evidence to go in favor of this or that approach. We will come back to the theory of the term structure in our six week when we talk about advanced bond valuation again. But for now, it just must be kept in mind, that so far we have seen and that was a good sign for us, that our approach of calculating PB is consistent with what is observed in the market and with the use of that we were able to bring some numerical discipline in the universe of bonds. From the next episode, we will switch to stocks. And for stocks, we will with a sign of relief. We will no longer really worry about these different Rs. But unfortunately, we will indeed worry about CDs.