Well, let's say a few words about evaluating risks debt and the use of option evaluation in this vast area. Well, first of all as we discussed earlier when we talked about riskless debt, we can say that the overall price for the bond looks like this. This is C over 1 + r1 + the general term C over 1 + rk to the kth power + the final term C + F over 1 + rT to the Tth power. Here, C is the coupon of the bond, F is the face value, and r1, rk, and rT, all these are T corresponding correct rates at which we have to discount these cash flows. This is the right formula. And as we said before, the good news about that is that this is all right. So it correctly values each coupon, and coupon and principal cash flow. Now the bad news is that unfortunately this formula contains a lot of parameters some of which are not nicely known. So when we talk about all of that was for riskless debt, for government securities. But now I would like to talk about risky debt. And the first big question arises, where does the risk of investing in risky debt come from? Well, first of all we can say that here are the changing rs. Because, for example, if you invest in a bond, you bought it and then if all rates would go up, then the price and the value of the bond will go down. Just because, well, we can say, this is clearly seen from the formula. But there is a very simple and clear straightforward explanation to that, because the bond produces some kind of return. And if all the rates now are higher, then to reach these rates, you will have to quote start from a lower base and that is why the price of the bottle go down. Now here there are a couple of areas where we have to take this changing rs into account. I would say that there are two major kinds of risk. This is reinvestment risk. And capital gain, Or loss risk. Again, with capital gain and loss, that is what I just said. But what is reinvestment risk? Let's say if you hold a fixed income security, that sort of provides you with some promised cash flows in the future, so you have a locked-in return in that. But what if for any reason these cash flows can be delivered to you earlier, or maybe at one of the same moment? Then, you are full of cash, but that most often happens when the rates go down, and you will be able to reinvest this cash only at a lower rate, to reach a lower return, and that's a huge risk. Again, in general, people who manage fixed income portfolios, here we talk about huge amounts of money. And therefore, even minor changes in some of the parameters that influence valuation they can result in significant changes in overall values of these portfolios. Now, all that is just about changing rs. But then there's another huge area that now we can talk about. And we can say that fixed income investments have a lot of embedded options. And that makes the evaluation a very challenging task. What kind of options are we talking about? For example, I will put like bonds can be, Well, callable, that means that the issuer, I am a corporation, I issue bonds. But I have the right to buy them back at an earlier moment than the maturity at a pre-specified price. Now this option is available for me, because I can go ahead for that if it's beneficial for me, and clearly this option is bad for the holder of these bonds. Then they can be convertible, convertible bond is the one that can be exchanged in some amount of shares of stock at the specified proportions. They can be puttable, that's the other case. At some point in time the option, I'm sorry, the bond holder can put the bond back to the corporation. And when we talk about convertible, in general, they can be exchangeable. The term exchangeable is used when bonds are exchanged not for the stock of this corporation. But into some other assets of some other corporations, most often linked to the one issuing one. So basically, you can see that here we deal with a huge universe of various risks and as we will see in just very quickly, that unfortunately the proper valuation of this risk is quite a challenging thing. Well, let me first of all go over some important parameters of risk adapt relation. You, I think, you have not completely forgotten the idea of the yield to maturity. So we said that if this is the time, and this is the corresponding r. So you have some curve that is called the term structure, so basically, this is the curve that shows the expected correct rates of return that normally are applied to riskless debt. And all rates for valid risky debt, they are taken or they are produced from this term structure. And the ideal yield to maturity of the use of a uniform, just one number to discount all cash flows goes like this. So you discount all cash flows at this rate which is yield to maturity. Now what you can see here is that this is a time and there are some cash flows that basically all cash flows of such a bond are mispriced. Probably with the exception of the one, if that goes exactly here. So the yield to maturity is a great instrument that allows you to use just one number, but that results in a huge compression of information, and therefore you have mispriced cash flows. This is sometimes called a simple finance approach. Now, the thing is that the analysis of fixed income securities is actually a vast area, and that in detail goes fundamentally beyond the scope of this course and this specialization. And this is a long and complex piece, but now I'm just showing some of the important things that are studied there. So here you can see all cash flows are mispriced and there is one idea that is very important here is that if, these rs change, what happens to the value of the bond? Let's say if all rs, they change by 1%, by a small percent each points, you can see that all cash flows their discounted, their present values are changed. And therefore, so is the value of the bond. And what people do they basically take some point of this interest rate and then going around this we study the small changes of the value of the bond with respect to changes in this r. Basically as in math, we can always expand that into serious and there are some approximations to that. And one idea is the idea of duration, well, duration is sort of, double quoted, effective maturity. So basically, the idea is as follows what if we put all cash flows of the bond, at one point. So as the PV of newly created security where all coupons and the principal, they come at this point, we discount that at the rate that is observed at this point, and then the PV is the same. So that is called duration. So the idea is if we put a chart here, again, this is T, and this is a term structure. So we basically put all of them at one point. This is a point D so that the distribution is like delta function. And all cash flows they just come here, sort of artificially. Now, you can say, well, it's kind of a strange way to do that. Well, this is basically an explanation, because the formula for the duration is like this. D is = to the sum from i = 1 fto T- 1. PV(C1) divided by the P times i, so this is the term. And the final number here is PV(CT + F) times T and divided by P. So in this formula we see that basically all present value of cash flows they come into that with the weight that is the time that must pass until this cash flow occurs. Now, this formula can be rewritten in a more mathematical way. D is = -(1 + y) where y is yield to maturity, and here goes the derivative, logarithmic derivative, with respect to y. And that can be rewritten in the form of small changes that delta P over P, so the percentage change in P is -D over 1 + y delta y, or if we rewrite that, -D* delta y. This D* is called modified duration, and this is the Macaulay duration. Now the question goes, why is that that people use this special thing, and why is that that this is kind of useful? Well, in, soon, what follows, I will explain that this is just the first term of the series, but now the important thing is that people who deal with managing portfolios, they are exposed to significant risks. And because their assets are very long term, if interest rates change, then the longer the term, and therefore, the duration of this asset and liability too, then the more acute is the impact of that. So therefore, people who are in managing portfolios, or fixed-income securities, they care about the so-called immunization. So they have to make sure that the duration of their assets and the duration of their liabilities are close. Otherwise there's going to be a gap, and they will be exposed to significant risk. Can say that, I'll put that in general, You can say that delta P over P = -D* delta y, this is sort of a first approximation, then + 1 over 2 C times (delta y) squared. Where the C has a special name, it's called convexity. So strictly speaking, convexity is just the second derivative, so C is 1 over P, D like this. So we are expanding this into a series and this is the first term, this is the second term. And now you can say, why the hell do people care about these convexities and other things? Now, we have to, because basically the idea is as follows, if this y changes, then the change in price, is it's not linear. And so we have to somehow approximate that, because this series has an infinite number of members. But the contribution of every next one is progressively smaller. So sometimes we can easily limit ourselves as just a first approximation. But with the second one, we are much better off. So all that is just some parameters that people use in analyzing bonds. And now come these embedded options. I mentioned before that we have all these callable bonds, puttable bonds, exchangeable bonds, and others. So the problem is that we cannot ignore options in fixed income securities at all. And you can say, why? Well, because there are so many of these options that are observed in fixed income markets. And we talked about that, but there is another very special kind of an option that most often people who are not really familiar with that, they do not recognize. For example, if we talk about mortgages, most mortgage owners have the ability to prepay these mortgages. That means that you can, at some point in time, pay off the balance of the mortgage and then you just retire your whole liability. You can say, well, you can sort of try to disenchant people by offering them some potential penalties. But the thing is that this is not happening only because of evil will of these people. Sometimes they quite certain sort of irrational ideas, oftentimes they're known as QFDR. This is quit, fire, die, or retire. Because mortgages get prepaid when people change their jobs or when they move. And therefore, they have to sell the house at this moment if their mortgage is prepaid or if God forbid they die. And in this case again, when are talking about irrational things. We are talking about something that is quite natural, but at the same time it's not because of the change in interest rates. So here, we have to say that ignoring these things unfortunately is impossible, but the problem is that with all these options if we talk about stock options, remember block controls. We said that the block controls will say that rF is constant. Now you can see that for many cases with bond options, the use of block controls is absolutely impossible. You're talking about rs that are random, and if they are random, now you can see that unfortunately this whole apparatus that we developed before for stock options here completely fails. And your dealing with the cases in which the valuation of bond options becomes very, very challenging. So this is really a problem and oftentimes you'll have to take into account that sometimes even the distributions of these small changes, it's not normal. So we cannot sometimes use a low normal distribution that is widely used for option evaluation. So that creates a lot of problem. Now, another thing that is sort of finals, I'm pronouncing frightening things is this volatility sigma. And remember that in black control is volatility is constant too. Here not only is it not constant but it's also random. So you are dealing with options that are random rs, random sigmas, and sometimes they are even path dependent. So all that results in a gigantic mess. And this mess is not only because people don't understand the apparatus. The apparatus is quite advanced, but not so very much incomprehensible, but you cannot deal with stochastic volatility. You can hardly deal with stochastic liquidity again. It's not an exaggeration to talk about that. So bond options, this is clearly one of most messy and advanced areas in asset valuation. So with this negative message, I'm wrapping up this episode. And then in the next one, I will give you some other ideas that unfortunately have to be taken into account when dealing with bond and fixed income options and securities.