the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu let me begin by first asking whether there are any questions from last class which was a a week ago hope you had a good break any any questions okay before we begin a today's topic question now before we begin today's topic on arbitrage and the pricing of multiple fixed income securities I want to just take a few moments to talk a bit about what's going on in financial markets also to welcome the perspective students that we have sitting here in class today so for once over the weekend unprecedented things didn't occur and so I'm glad to report that we're still here financial markets are still around and as you know the government has proposed some measures to deal with this current financial crisis and at this point it's still unclear as to what they're proposing but we can actually see from the data what the markets reaction is last time remember we looked at the yield curve and literally it was just a week ago that the yield curve looked like this now remember that we focused on what happened at the very short end of the yield curve which is three-month Treasury bills and last week when we looked at this graph it was the yield was about three basis points for a three-month Treasury bill and we pointed out that that was telling us something about market in particular it was telling us that the market is panicking yeah because uh well I wouldn't say that it was looking at the market closely miss price yeah well there are a number of things that are priced into a security it's not just the risk but it's also a reflection of supply and demand right so in other words what's going on here the question that we want to answer from looking at the price is what do we know about what's going on in the marketplace based on that price what is it telling us the cost of borrowing over a three-month period when it goes down to three basis points that's telling you that the price of that security the price of Treasury bills is extraordinarily high relative to historical standards now let's take a look at what happens more recently in particular today if we go to the any of these websites so in particular let's go back to the Bloomberg site where we originally look at this first of all this is now the yield curve and it's hard to compare because I've got a different slide for the last week's this is just yesterday's the orange not not last week's but the one thing you'll note is that at the very short end now instead of three basis points the three-month Treasury is yielding 41 basis points what does that tell you about the price now so Treasury securities short-term Treasury securities have declined in price over the last week and that's one sign that perhaps markets are not as panicked as they were last week there isn't this mad rush to get into Treasury securities in the short-term all right so short-term means have gone up yeah yeah whatever one thing is when the track you know it's eclectic as I said there's a flight to liquidity yes now why was there such a huge movement or whatever the price dropping so much rent I guess it could be reasonably expected that the price would come back up and so people would just wait it out which price are you talking about dropping the yield dropping and and the price going up well so there are a number of factors at play but the the current perspective that most of us have in financial markets about last week and this is just perspective remember last week is not that far away what happened last week by most accounts is that there was a very significant rush to the exits by investors by rushing to the exits I mean getting out of risky securities and into safer securities and at this point it doesn't seem like there's much of a safe haven other than US Treasury securities and a particular short-term security because you know you can get out get the money out over a relatively short period of time so that's what the yield curve told us last week that three basis points means that basically people didn't care about the yield all they wanted to do was to get into US Treasuries at at almost any price okay this week it's different in particular not only has the short-term yield gone up so now instead of three basis points we're up to 41 but look at the long end of the yield curve before the long end of the yield curve let me just go back and remind you what that looked like at the 30-year a week ago the 30-year yield was 4% okay let's take a look at it what it is today it's now the 30-year yield 4.37 that's another big movement why is that why would the yield for the long-term bond go up what what is the market thinking today okay so inflation has now been incorporated just over the last seven days so question is this price correct or was last week's price correct getting to your point I mean what do we do is is is the short end of the yield curve appropriate today at 41 basis points or was it really appropriate three basis points there's no answer to that question because there's no right answer these prices are a reflection of the current expectations of all the market participants right or wrong it really reflects the combined either wisdom or fear or greed of the marketplace and so our approach is to try to understand what that is we want to explicate the information that happens to be in prices but you have to understand that these are the same imperfect kind of prices that we came up with on day one when you bid for that little package and you know it turned out that you got lucky and you know got an iPod for whatever $45 but it could have gone the other way and in fact in the second class it did go the other way so we won't have talked about that the prices reflect all aspects of the economy the rationale as well as the irrational and so last week was it irrational for people to pull their money out from all sorts of investments and put the bit of Treasury bills well this goes to the heart of why the Treasury acted so quickly and why Chairman Bernanke has said that he wants to get a quick resolution something very significant happened last week and I don't know how many of you really got wind of it certainly the Treasury knew what was going on and the Fed did but it wasn't really highlighted in the newspapers in the way that that I would have thought it should have been given the importance anybody know what I'm talking about yeah well that was one piece of news the SEC mandated that for a period of time to be determined we are not allowed to short sell financial stocks because they wanted to stop the the kind of run that there's been on these securities I'm going to come back and talk about it in the end of this lecture because we're going to talk about short sales but that's not what I was referring to that's certainly a concern but that's not the major concern that I think the market was responding to yeah that's right where was that coming from what what was going on with that why did that happen and was there a reason for that loss of confidence I mean you know money funds what does that have to do with mortgages and Lehman and Goldman what's that there's what's the connection it was a psychological reaction but did something trigger that is that psychological reaction completely unreasonable if your grandmother asked you what she should do now with her money market fund should you tell her don't worry about it you know just you know stay the course and see what happens something happened last week that is related exactly to that issue so you're onto something what is that make it broke the buck exactly which money fund you remember it was the reserve fund the reserve fund is one of the first if not the first money market funds and what is the money market fund do we do we know what that is you all know what that money market fund is you all probably have money in a money market fund whether you know it or not right money market fund is a fund that contains relatively short term and supposedly riskless securities like CDs Treasuries and other kinds of very very safe assets and what does it mean to break the buck whatever they okay right right breaking the buck means that when you put in a buck money funds are supposed to be so safe that at the very least when you withdraw the money you're going to get a buck back breaking the buck means that if you withdraw there is a possibility that what you withdraw is less than a dollar now that's scary because think about a bank when you put your money into a banking a checking account for a bank you expect to get that money out maybe not with a lot of interest maybe even with no interest if things don't go well but you expect to get what you put in you expect to get the principal back right well money market funds are very much the same way people use them as if they were checking accounts in fact there are money market funds where you can write checks on them right and so breaking the buck has been a major concern not just among the money funds but among regulators because if it turns out that retail investors ordinary consumers are scared about what's going on with their money market accounts they will do in masse what happened last week which is pull out huge sums of money from these money market funds and as I mentioned earlier no business can sustain a massive withdrawal of all capital all at once it's just not possible for a business to be able to be conducted in that manner if that happens we will see mass failures of financial institutions that will make the last four weeks look like the good old days and that's what the Fed is concerned about that's what the Treasury is concerned about and so the hope is that the measures that they put in place will calm the fears of the public that's the first order of business it's calming the psychological kinds of effects that these headlines have produced and so the hope is that once they put these measures in place that will take care of the concerns what they've done is to propose the guarantee money market funds the same way that the FDIC guarantees your banking accounts and the way that the SI PC guarantees your brokerage accounts and there are other measures that have been proposed we won't have time to talk about them here but over the next couple of weeks the the Finance Group at the Sloan School will be putting together some kind of a panel discussion that will focus exactly on these issues so we'll let you know when that happens and we'll take on these issues head-on in that session okay but it looks like for the moment at least from the U curves that we that we saw that things are actually quieting down we'll see on a day-by-day basis so you know this is obviously last week this week we have yields going up a little bit so that suggests that there isn't the same kind of pressure but every day is you know another day and with another set of revelations so by looking at these pieces of information we can actually glean what the market is thinking is it right of course not all forecasts are by construction incorrect to some degree but it's a window on exactly what's going on in the marketplace and what people are thinking yeah right yeah sure so the money market concern is that what people thought were safe apparently is not as safe as people thought and the reserve fund breaking the buck by the way breaking the buck in that case meant that if you put in a dollar when you withdrew as of last week you would have drawn 97 cents she lost three cents to the dollar which may not seem like much but if you went to your Bank of America atm and you did a withdrawal and for every dollar you put in you get 97 cents back you'd be pretty ticked off right so you know it's something that is of great concern to retail investors anyway so what happened last week was that actually the estimate if it's 90 billion dollars 90 billion dollars of money came out of these funds in a week and were put into either cash in the mattress or were put into Treasury securities like the three-month t-bills that's what pushed the prices of those t-bills up and therefore depress the yields and now we're back to a somewhat more reasonable level I say reasonable having this kind of a short-term yield of 40 basis points by historical standards is still pretty low so there are still many nervous investors out there that are trying to figure out what's going on and are waiting for the Treasury to come up with something this is another reason why Chairman Bernanke said we have to act quickly because markets are not going to stand and wait for the Treasury or the Fed to do something markets will react and if we wait too long the fear of this breaking the buck could actually return and then once you have a mass panic it's very very hard to stop that anybody who's ever seen a you know one of these old animal kingdom type of movies about a stampede if you've got you know water buffalo stampeding it's pretty hard to try to just say Oh calm down stop it you know don't you know slow down you know you can't easily do that once it begins so you've got to stop it before it actually gets to that critical point and that's exactly what the government is trying to do yes like the prices went down so now there's less demand so it means people are more we go yeah what if you say like if the interest rate went up it means they are comparing good pressure reviews to be more risky well remember that Treasury bills don't have any default risk at least as far as we know we have to be careful about staying stating less all these unprecedented things have happened the reason that Treasury bills don't have any default risk is because what the Treasury Security is is an IOU from the government that says I owe you a certain number of US Dollars and because the Treasury owns the printing press they can always print out more dollars to give it to you as long as you're willing to take it okay and at least from this graph it seems like a lot of people are willing to take it they really want Treasury bills right now and they're happy with that maybe they're not happy with it but that's the sort of the less the smallest of all the evils that they can think of in terms of putting their money yeah we hope not I mean you know I guess it could be possible that people are betting that the United States is going to default in 30 years but my sense is that what's more likely given that these are default free in the sense that their nominal bonds so these bonds are going to be paid off in the little certificates called u.s.

Dollars that the printing presses can always come up with there's no risk that they can't print up more dollars the risk is that a dollar 30 years from now isn't going to be worth as much as we thought it was going to be because of inflationary expectations so as of today that 30-year yield is not four percent it's four point three seven and the point three seven one could attribute to a inflationary expectations by the marketplace change and yield has anything to do with the recent devaluation of the dollar against four currencies you know it could be that because of that devaluation dollars are cheaper and people are putting more money you know into us security that's also a possibility but another way of putting that is that foreign investors are now finding Treasuries more attractive for whatever reason so yes that's also part of that supply and demand story okay last question yeah well I'm glad you asked that question because we have a graph these are the historical yields of the three-month six-month one-year two-year five-year ten-year 30-year from 1962 to 19 2006 2004 and it depends on what flavor you're looking at and it's kind of hard to read this graph because of the colors but if you look at the the dark blue line they actually all move together pretty much but at one point that short-term yield was 4% 4% for a three-month Treasury bill now these are all annualized remember so 4% doesn't mean 4% over three months it means 4% on an annualized basis which is why 41 basis points that's an annualized yield for a three-month loan just seems ridiculously small by historical standards but when people are scared about not getting paid that kind of fear that psychological pressure can be overwhelming and do you believe in these prices does it make sense well that I'm hoping to get you not to ask the question in that way but rather to ask the question given market prices and what I know about it what can I interpret from what's going on and what how does that affect me in terms of the financial decisions that I want to make so if you are thinking about pricing other securities based upon these kinds of numbers you need to ask yourself whether you believe the numbers make sense or are they just completely out of whack and the only way to do that is to understand what the basis is for these numbers so that's where we're going next in trying to understand how to measure the various different characteristics of these numbers to get a sense of what's reasonable what's not question well you know obviously it's very difficult to tell because we don't see who's making the purchases and sales but we can tell from certain mutual funds and other money market flows that it seems like most of the flows last week came not from hedge funds but rather from retail investors that were taking their money out of these money market accounts and then putting them into certain mutual funds that buy only Treasury securities as well as Treasury securities directly you know all of you can actually purchase Treasury securities directly there is a website called Treasury Directgov and you can give me a credit card and register as a user and actually participate in Treasury auctions and buy Treasury securities so it was a combination of those but it really seemed like it was the retail sector not institutions not sophisticated hedge funds that were trying to do some kind of complex arbitrage it was just investors saying I'm really scared I want to put my money into something that's that's real and that will will be there and so short-term Treasuries seemed like an answer and as we saw from last week gold was the other answer it's not an answer that I would propose for the typical investor because gold prices are quite volatile and so you have to be very careful when you make an investment in that security or in that in that particular asset but it is something that reflects the state of panic of the marketplace and and by the way I'm telling you something that you probably already know in a sense that my guess is it deep down inside all of you are feeling stressed out right I mean you probably stressed out about things like what does this mean for the job market for you know career prospects and so on I would urge you all to take a deep breath and not not get panicked about that because as I said this is the kind of dislocation that while very traumatic for market participants today and for those on the losing end there are as many opportunities created as there are taken away and so my guess is that in a year's time the job market is going to look extraordinarily attract particularly for those individuals that are trained in the the science and art of financial analysis so you're all going to be very well equipped for that even though you may not feel that way right now because of what's going on in the marketplace so I wouldn't panic certainly and by the way you can see the opportunities that are already being created you know Warren Buffett just spent five billion dollars purchasing a stake in Goldman Sachs and we're going to talk about that in a couple of lectures when we do common stock because I want to use as an example of getting a good deal uh you know in markets I mean first of all warren buffett given as the the success he's enjoyed as an investor you know that when he plunks down five million five billion dollars in cash he's probably doing it for a good reason not out of charity and by the way goldman raised another five billion dollars from additional rights issues that's ten billion dollars of capital that they raised relatively quickly also Nomura is in the negotiations to purchase certain assets of Lehman Brothers Lehman has a terrific franchise and has some very significant operations in Asia as well as in the US and it's a very smart move on the Morris part to take advantage of that so these are the kind of opportunities I'm talking about and when Nomura buys Lehman they're going to have to hire people in order to run the operations because you can bet that the whole dislocation ended up shaking loose a number of very talented professionals from those organizations so they got a higher it's going to mean the next two or three months uh there may be some difficulties in getting the attention of these organizations because they're in the midst of trying to figure out exactly what kind of organization they are going to have when all of this when all the dust settles but there are plenty of opportunities that are being created today including by the way the opportunity for the US government to take advantage of all of these distressed assets so one of the things that you should be careful about when you read that there's a seven hundred billion dollar bailout that is somewhat misleading in the sense that first of all we don't really know at this point exactly what the seven hundred billion dollars we'll be for how it will be used or how its dispersed or is it really 700 billion a lot of it depends upon how the money is spent and also what happens to housing markets there is a scenario that I can imagine where the actual amount expended is either zero or negative in other words the government actually makes money from the current state of the markets because they can buy assets very cheaply hold on to them forever until they pay off and then gain the kind of profit that the original financial engineers were expecting but could not take advantage of because of this liquidity crunch so we'll talk about that over the next few lectures because we're going to develop some techniques to be able to to illustrate how these kind of arbitrage strategies work yeah is it more certain that it's going to happen than me well certainly there are a number of banks I wouldn't say many simply because there aren't that many banks that are well capitalized enough to be able to take on you know a large unit of a business as big as Lehman Brothers so Nomura is one of a handful of banks that are engaged but they seem to be the front-runner at this point kind of it on the vision and that is a man who can do it guys quiet lady that my phone oh absolutely I mean there are a number of issues that will come up in any kind of a deal and so you don't know whether or not something is going to go through until it actually goes through the same thing could be said for what happened with Merrill Lynch with AIG all of these deals are sort of put together at the last minute and you know either they get consummated or there's some hitch at the end that makes it difficult so yeah I mean with a grain of salt you should take all of these news reports and literally until the deal is signed you will not know whether or not something's wrong but the point I'm illustrating is that these these assets are not completely worthless what's happened is a very significant liquidity crunch and panic when that happens the pricing of all these assets becomes questionable okay we're gonna actually see an example of that so if you would my let me put that off for a few minutes and then if you have further questions about this we can come back to it let me start uh so this is lecture six and what I want to do is to start where we ended last time which was a discussion of coupon bonds and how you price coupon bonds simply as a package or a portfolio of pure discount bonds underlying this approach to pricing coupon bonds is a very important principle that's a principle that was given to you in the very first day of class where we talked about the six fundamental principles of financial markets and it's the principle of the law of one price so what I want to do today is to focus on that and talk about the law of one price and what it means for things like arbitrage leverage short selling and relative pricing those are the key concepts we're going to cover today so let me talk about the law of one price and remind you all what it is it's a very simple idea it's so simple that you might think it's obvious but it's got some very very dramatic implications the law of one price says that two identical cash flows must have the same market price okay let me repeat that two identical cash flows must have the same price now remember that when we think of an asset we think of an asset as just the sequence of cash flows that's what an asset is so all I'm saying is that when you have two identical assets they have to have the same price that's not a very controversial statement and this principle is one of the most important ideas in all of modern finance because it leads to the pricing of all sorts of securities including all the derivatives that have ever been priced on Wall Street they use this idea of the law of one price okay yeah no no I don't have to qualify that at all first because this is a free country and I don't do anything I don't want to do but but more importantly it's because I don't want to restrict it to an equilibrium by equilibrium you mean when supply equals demand right I don't care about supply and demand supply may very well not equal demand that's okay with me this principle of law one price that two identical cash flows have to have the same market price the only assumption that I need in order for that law to be true is that people prefer more money to less money and it's not even people I just need one person in the economy that prefers more money to less money and I'm happy to volunteer for that position okay why is that it's because if that law is violated if you can show me two identical cash flows that's sell for different market prices first of all tell nobody but me about it what I'm going to do is I'm going to buy the cheaper asset I'm going to sell the more expensive asset so as of today I make money right because I bought the cheap I sold the more expensive that difference is positive for me and then I argue that from that point on I have no further risk and in fact no further obligations I can just forget about the deal and take my money and spend it why because I've bought and sold identical cash flows so for that point on in the future all the cash flows cancel out so I'm done that's called an arbitrage or more technically a free lunch I've been able to create money out of nothing it doesn't assume supply equals demand it doesn't assume any kind of mathematical formula for any kind of instrument all it assumes is that more people prefer more money to less options the iPhone yes then the only difference between an open package and the concealed package was that in one case you basically know give enough information right and it wasn't reaching a fair value and so wait wait wait wait wait well when you say fair value that's a loaded term what do you mean by fair value it was a fair value given all the information that the market had okay when the person by logic does package what are they opening that the cashflow that object is the same as if it were not packaged and they're going to reach it that's right but don't you think there's a difference between the package wrapped and the package unwrapped yeah one of them is I mean it's just like a factory we could produce the same stuff with a value that the market at the lower price that would be right well what if one factory didn't tell you how it produced it and another factory did you think that they would sell for the same price factories of them but their future cash flow in the revenue could be the same only if it turns out that as a matter of fact it is identical but you don't know that ahead of time you can only price something with the information you have okay so when I say to cash flows are identical I'm saying that we acknowledge that the cash flows are in fact identical and we know that they're identical if I put two packages up front in the room one is an iPod and the other one is wrapped so it looks like an iPod it's got the same shape the same dimensions but it's wrapped would you say that they're identical you can't know that if you knew that then you would price it accordingly by the way that gives you a very important piece of information suppose that I did that experiment I had the iPod that was clearly unwrapped and it was an iPod and then I had another package that was the identical dimension but it was wrapped okay and I action them off and it became clear initially that the two were priced at about the same well if you saw that you would then know that it was quite likely that what was in the wrapped package was the same what was in the unwrapped package right but the only reason you would know that is because somebody in the audience apparently did the same price now why would they do that either they're knuckleheads or they know something that you don't know and now that you look at the price you actually learn something about what's in there so what matters for pricing is what the entire market knows not just what you know but what the entire market knows now going back to this arbitrage let's not worry about information on the symmetries so let's assume that we all know exactly what there is to know which is that these two securities have the same cash flow if they have the same cash flow they've got to have the same price and that's exactly what we saw last time with this example same cash flows and therefore they have to have the same price if they don't have the same price then instead of feeling upset and despondent that somehow finance theory is in question the most exciting thing for a finance professor is to see that this theory breaks down because then we actually can go transact in the market place and make money so if the law of one price fails instead of calling me up and complaining about it you should call me up and tell me what it is so I can take advantage of it all right it's a great thing and here's the example this price of a pure of a coupon bond has got to be equal to the prices of the discount bonds multiplied by the number of bonds you need in order to yield an identical cash flow okay now if you have an identical cash flow the left hand side has to equal to the right hand side if it doesn't if the price of the left hand side is greater than the prices of the right hand side then you should feel very happy because what you're going to do is to make money now how much money you're going to make well let's see what you're going to do is you're going to you're going to buy the right hand side you're going to buy those bonds and you're going to sell the left hand side bond so you what you make is that difference right because you're buying or you're spending a certain amount of money that's the right hand amount of money and the left hand side is what you're going to get and so you're actually going to you're actually going to be able to make that difference but here's the key when you make that difference it's not a risky investment there is no risk by the way how much money did you have to spend how much of your own personal wealth did you have to commit to this trade just the risk what risk did you have to spend that that's money that you get not that you have to spend is there any money that comes out of your pocket to do this trade no because what you're buying is actually financed by what you sold and then on top of that you have a little extra leftover okay so this is like one of these infomercials at two o'clock in the morning you know the real estate how to make a million dollars in real estate with no money down right you've put no money down you haven't spent any money because what you've done is you've sold the bond and you've gotten this amount of cash with that cash you can buy these bonds and how do you know you have money left over by assumption I'm assuming that this price is greater than that price so now you have money left over you've put no money down you have cash in the pocket you have no risk you have no obligations because all the future cash flows that you owe are financed completely by the bonds that you bought one for one dollar for dollar it matches so literally as of today you walk away from this transaction a richer individual yeah well that let me get back to that that's an interesting point the question is can these things really exist because once they do it's almost too late because they're gone arbitrage desks absolutely exist in financial firms and what they do is look for this stuff all day long that's what they do and so is it true that by the time you identify it's gone well it's true if you're a finance academic as well as a retail investor but you guys you're going to go out in the job market and you're going to be hired by some of these arbitrage desks so you're going to be doing this in fact you're going to be doing something quite a bit more complicated I'll get to that in one minute before I do that want to make sure that everybody's with me about this arbitrage yes very good point yes in this context there's an assumption that you can buy and sell freely with no transactions cost obviously in the real world there are transactions costs you've got to stick those in and figure out whether you can still make money with transactions cost now one form of transactions cost is a is not a fan numerical cost but it's a friction what do you have to be able to do in order to do this transaction what financial operation do you need in order to be able to get this deal done yeah short sell right remember we talked about short selling last time that's selling something you don't know so you have to borrow the bond from somebody and then sell it and get those proceeds what if it were the case that somebody imposed a constraint that you can't short sell who would ever do that well we've seen what happens in the marketplace that can generate that kind of a constraint what happens if you can't short sell yeah what's that mean to put up the cost of e well in fact yes you need to put up the cost but it doesn't it really defeats the purpose because if you're going to buy the bond and then sell the bond you haven't really done anything right this arbitrage argument relies on the fact that you can sell something you don't own by borrowing and shorting it and getting the proceeds and then giving it back to the person you borrowed whenever they want okay if I don't allow you to short sell this argument doesn't work anymore and what that means is that this pricing relationship this left hand side has to equal the right hand side that relationship goes out the window for the next several weeks and possibly several months finance theory is going to be on vacation because the government has suspended short sales for certain securities and so the kind of force of markets that drive prices towards not an equilibrium but a pricing relationship that doesn't not depend on equilibrium but depends on the law of one price that goes out the window if you can't short sell yeah I'll show you first so hi and portfolios you know that's a fantastic alternative I think that wouldn't be far better you're absolutely right increase the cost of short selling by raising borrowing costs by the way when you short sell as we said last time it's not free people aren't going to lend you the security for free they're going to charge you for it so what if instead of forbidding it all together why not just triple the cost or quadruple the cost there therefore only people who really really need to do it will do it the problem with that is more of a political one the political problem is that we want these evildoers these short sellers they're driving the prices of financial securities to stop their bad activities and so what we're going to do is to mandate by law that they can't do that now that may be a reasonable thing from a political perspective but it's not a reasonable thing from an economic perspective because what it does is it disrupts relationships like this pricing relationships like this yeah by seven Finnish audience doctor well if you're the government and you're in a time of crisis it seems like you could do anything you want so I mean practically yes you can say for all financial stocks that are on this list we will now charge an extra high rate of interest for borrowing those stocks and in fact there exists a mechanism even before this rule was put in place that for certain stocks that are quote hard to borrow that's a technical term that Wall Street firms brokerage firms use securities that are hard to borrow means that they're very actively traded and it's very hard to find a counterparty that's willing to let you borrow it from them the hard to borrow stocks typically are lent out at a premium so they are charged higher prices for those that are very very popular and you know by simply eliminating short sells you basically make those costs infinite that's exactly what's going on so it would have been better to make them finite but bigger that would be better than then what they did which is making an infinite but from the purely financial markets perspective it would be best if there were no restrictions and no transactions cost at all obviously transactions costs are inevitable so when you price these relationships the Equality is going to be up to transactions cost there might be a little bit of a wedge between the left-hand side of the right-hand side but the difference will have to be small enough that apart from transactions cost they are not significantly different yeah well if it were illegal transaction the way they should have done is to prosecute the illegal parties as opposed to stopping short sales for everybody right that would be a far more effective way of dealing with illegal activities that's not what they were concerned about primarily the SEC has an enforcement division whose sole job it is to check on all the kind of transactions that may have occurred to see whether or not there's any kind of illegal activity going on it wasn't the illegal activity that prompted the short sales restriction it was really the concern that financial firms were being pounded by short sellers that were betting on them failing and the more pressure that they imposed on the stock price going down down down the more likely it is that people would lose confidence and then all of a sudden stop doing business with it you know in many ways that's what happened with Bear Stearns at least from the historical record it seems like what happened was that there was a rumor that Bear Stearns was not going to be solvent even though they didn't have any particular pressure to to pay certain debts at that point in time and somehow that rumor grew into general fear and you know the short sellers got in and started shorting the stock the stock price went down people looking said oh my god the stocks going down I better take my business elsewhere and everybody started doing that once everybody started doing that the firm began having great difficulties in maintaining a business yeah yes the investor more expensive that's right that's right that's right so again do top you do the yo that's that's a very good point let me repeat it the point is that even if you're not allowed to short sell then the folks who own the left-hand-side who own the coupon bond they can say hey my coupon bond is worth 110 but I can get the exact same payoff by buying a bunch of pure discount bonds and it only and it cost me a hundred to buy them I'm going to sell my coupon bond at 110 and I'm going to buy a hundred dollars worth of these discount bonds and I've just made 10 bucks now that can happen as long as two things occur one the person who owns the bond knows about this and two they actually want to take the trouble to do the trade or are they're able to do the trade the problem is that most of the folks that bought the bond on the left hand side are pension funds that are not in the process and not in the business of doing these kind of arbitrage transactions so they're not trying to do that all they want to do is they've got you know a bunch of pension plan participants they need to pay out their benefits they've got contributions they just want to match assets the liabilities they're not in the business of doing this kind of high-speed transaction so what that means is that you're still right that if they realize this relationship is there and they are in a position to do the trades they will sometimes and then that will force prices closer but it's not going to be the same kind of numerical identity that has to hold when you've got greedy people like myself trying to do the trades and being able to do the trades at a moment's notice exactly right there are all sorts of additional complexities of being able to do the trade that as a pension plan you're not even allowed to do never mind whether you want to do them or not so these are the kind of restrictions that would present the the this gap okay but let's for now assume that there's no gap we're going to assume that there's no frictions we're going to assume that there's no short sales prohibitions and when that happens this pricing will actually pass the whole now I promised you I want to show you something more complicated that all of you may do when you get out of here and that's this instead of looking at just one bond what if we looked at a whole bunch of coupon bonds now let's take a look using this pricing relationship that's based on the law one price and basic greed we've got n bonds here one through n and each has its own coupon whatever that is I'm just going to use notation to write down that each bond has its own particular coupon right so you've got a three percent 10-year bond you've got a four percent 30-year bond you may have a five and a half percent five-year bond so some of these coupons may be zero because capital T I'm going to assume is the most extreme thirty year period right so a five year bond would have coupons for the first five years and then for all of the coupon payments after that I'm going to have zero zero zero zero zero right so this should harken back to your high school algebra days where you have multiple equations with multiple unknowns now what's unknown here in these equations what are the unknowns well you observe the prices right from the market place so the left-hand side is not on home what about the right-hand side what are the unknowns what do you what do you observe what are the knowns how about that let's start with that yeah the coupons right if I tell you that I've got a thirty-year four and a half percent bond you know what the coupon is going to be right what's that no the yield is what we don't know all right that we either get from the marketplace or we can try to solve it from these prices but how many different yields do we need in order to price each bond these are two year bonds how many yields yeah we need tea party yields one for each year we have tea yields in these equations and they're the same across the different bonds right because we're using the same pure discount bonds to replicate each of these coupon bonds I've got tea unknowns how many equations do I have n bonds right now on any given day you might have 200 to 300 bonds that are trading but you've only got 30 unknowns so let's think back to your high school algebra or college linear algebra days if you've got 200 equations and 30 unknowns how many solutions do you have well let's let's get simpler suppose you had two equations and two unknowns you have one solution assuming certain kind of conditions that hold like invertibility which we're going to talk about in a minute two equations two unknowns you have one solution how about one equation and two unknowns how many solutions infinite okay that's right because you've got that extra degree of freedom right there's lots of different solutions what about three equations and two unknowns now how many solutions do you have yeah it's linear are you guaranteed to have one you might have zero what are the conditions under which you would have one right there is a possibility you have one solution if the solution of the two equations actually applies to the third do you remember under what condition that what exhausted that's not quite the term that that mathematicians use there's something called linear dependence that's a very complicated word that describes the fact that if you have these two equations and two unknowns and you have one solution you can just take an average of those two a combination of those two and you'll get the third one you can actually replicate the third equation from those two what happens if you don't what happens if you cannot take combinations of the first two equations to get the third what does that mean one of the numbers wrong what do you mean by that it can't be a solution okay now this is getting really interesting because on the one hand you're probably getting confused about what any of this has to do with money in a minute I'm going to tell you has everything to do with money as you might expect if we've got two equations and two unknowns we have one solution that basically says that there are two yields a one-year yield and a two-year yield that is able to price both bonds and that better be the case because we're going to use these two yields to replicate the bonds and so this kind of a relationship has to hold now if we add a third bond if we add a third bond and those same two yields that work for the first two bonds they don't work for the third bond something's wrong right it means that this third bond it's price does not satisfy the relationship between those two yields that's evidence of a mispricing something is wrong but in this case what's wrong is that there's something miss priced between those two bonds and the third so when you run into a situation like that and we're to give you an example in the problem set when you come across that what that means is you should be extremely excited as opposed to depressed in math class you know Jesus there's no solution in finance what no solution means is that there is a transaction there exists a linear combination of the first two bonds and the third that a costs you no money down B will generate cash flow today C will require no future payments of any sort so it's riskless it is a free lunch it is an arbitrage now that's with three equations and two unknowns anybody can do that right that's easy what if it worth 200 equations and 30 unknowns now not so easy now you actually have to know some think about linear algebra now the whole notion of what invertible matrix is what the eigenvalues are all the kind of infrastructure that you can build for understanding this becomes relevant as a quant trading on a proprietary trading desk in the 1970s a number of MIT graduates were hired by Salomon Brothers they knew very little about fixed income securities they knew the basics which is what was taught here but they didn't know much about market realities in practice and so when they went to solve their three equations and two unknowns they observed inconsistencies and they didn't do three equations in two unknowns they did 200 equations in 30 unknowns and in the 1970s that was not easy to do because we didn't have pcs we didn't have Excel we didn't have a lot of the tools that we have today and what they did was they took these simultaneous linear equations this is high school algebra even back then it was high school algebra okay and they just cranked through and looked for Miss pricings looked for no solutions and they found a lot of cases with no solutions in one of the years during the 1970s or 80s one of these MIT grads was paid an annual bonus of 222 million dollars for doing this for solving simultaneous linear equations so this is an extruded for Salomon Brothers it was a lot more than 22 million dollars this activity is known as fixed income arbitrage there are many other versions of it but this is the plain vanilla version and by the way the plain vanilla version it still works in the sense that occasionally if you're quick enough and you have the right tools you can identify MS pricings and take advantage of them quickly it doesn't last long so you're absolutely right it doesn't last long but the person who gets paid is the person who can do this the fastest and able to understand the interrelationships among these securities and you can understand now why retail investors have no chance of doing this kind of thing on their own this does something you definitely don't want to try at home okay I know you guys have MATLAB and you can do matrix inversions but there are lots of other frictions transactions costs and imperfections that you have to build into this analysis but once you do all sorts of interesting things start popping out right question yeah this I am the arms race between everyone well it's actually much narrower now than it used to be so in fact the market has gotten much much more competitive but it's not down to zero precisely because there are frictions and other aspects for example some of these bonds they have weird features like their callable or in some cases they may have certain types of other requirements and market institutional imperfections that require you to build in those constraints when you do the analysis and so it's really about who has the better model and frankly there's an arms race that's been going on for the last three decades as to who has the the fastest computers the first supercomputer that was ever installed on Wall Street a Cray two was installed at Salomon Brothers doing simultaneous linear equations among other things and so this is where technology has played a really big role in the developments of market prices another question okay so this very very simple idea and you know it again it is simple has all sorts of important ramifications for the pricing of bonds and other securities what that tells us is that market prices have all sorts of information that are incorporated into it and one of the things that we want to understand is how to interpret that information in particular the first thing I want to do is to understand the risks alright because we talked about the fact that some of these market prices may have missed the risks that were implicit in some of the trades and so the question is why and how how do we measure the risk of a bond portfolio so what I want to do is to now look at the market price of a bond in terms of a function that has inputs and the price is the output and I want to ask the question what kind of fluctuation in the input will yield fluctuation in the output you know we now know how to price these things we now know how the relationships must work from a present value approach and now what I want to do is to ask whether we can measure the sensitivity so one way to do it is to just graph the price of a bond as a function of its yield and we know that there's an inverse relationship right just like we saw with Treasuries this week versus last week right last week there was a big run on Treasuries lots of people wanted to get into them price goes up yield goes down and this week less pressure so we have yielded going up and the prices coming down that provides us with one kind of measure of risk so the kind of measure I'm talking about is the slope of this line the instantaneous slope that tells us for a bit of a tweak and interest rates or yield what does that do to the market price because obviously when you're an investor you're focusing on the price right yield is a convenient way of summarizing the properties of a bond but ultimately what you care about in your portfolio is price and so the question is for a move in the interest rate what does that do to the price well it turns out that there is one way to get at that that is somewhat more intuitive than just looking at this kind of fluctuation it's called duration and it's named after a person Macaulay who first proposed it as a way of measuring how risky a bond is what Macaulay noted was that the longer the maturity of a bond the more sensitive is the bond price to the yield so for example a 30-year bond when you move the interest rate by one basis point will have a much much larger price fluctuation than a three-month bond right why is that anybody give me some intuition for why that should be why that makes sense why should a longer maturity bond be more sensitive to changes in yield yeah because the cash is kind of yeah because the risk the opportunity costs hide out for that long can't spend it on that's right you're investing for a longer period of time so in fact the opportunity cost as measured by the forgone interest is going to be much larger in other words the discount rate that you use this one plus are you've raised that to the 30th power not to the 1/4 power and so something that's raised to the 30th power in the denominator it has a much larger impact when you perturb that denominator by a little bit okay because you're actually increasing that that power you're increasing that quantity by that 30th power well so if that's the case if it's the case that longer the maturity the more at risk you are per basis point of interest rate fluctuation then why don't we just measure the average duration of the bond by average duration I mean how long does the bond last when you wait it by the coupon payments that it pays you so for a pure discount bond the duration is simple it's just the maturity date right if you've got a 30-year strip that pays you $1 in 30 years the duration of that bond is just 30 years but what if you had a coupon bond that paid you coupons all along the way well then you should take a weighted average of the 30 years and give some weight to the early years to because the early years are years where you're going to receive cash and therefore interest rate fluctuations are going to affect the value of those cash flows so the weighted average term to maturity is simply equal to the sum of the date 1 2 3 4 5 6 7 multiplied by a weight factor that sums to 1 and let's just use as a weight factor the proportion of present value of cash flow on that date okay so if you take the weighted average where you take the weights as the present value of the coupon divided by the present value of the bond those weights certainly sum to one and then you're waiting that by the date the year that you get paid that number will give you what's called Macaulay duration okay so it turns out that the reason that Macaulay duration is interesting is that when you take a look at bond prices and you ask the question for a certain percentage change a certain basis point changing the interest rate what does that do to that to the bond price as a percentage of its current price so this is the sensitivity it turns out that you can show that that's actually the negative of the Macaulay duration divided by one plus the bond yield so this is a long way of stating that the duration gives you a measure of how sensitive the bond price is to changes in yield okay the longer the duration the more sensitive the bond is to changes in yield and duration now just means a weighted average of all of the payout dates that a bond will have if a bond pays a lot of its cash upfront and very little in later periods is a duration high or low low and so if a duration is low what that says is that there's not a lot of sensitivity to changes in yield on the bond price itself if on the other hand all of your payment is out in the future way out in the future that's going to make it very interest rate sensitive okay and the negative number here indicates that there's an inverse relationship between price and yield so when bond investors look at a particular portfolio of bonds and this is key they look at a portfolio of bonds because that's what they're going to be investing in when you put your money in a money market fun or in a bond fund medium-term long-term you're not putting it in one bond you're putting it in a whole set of bonds your natural question is how sensitive is that portfolio to changes in interest rates and the answer is it's related to the duration and so if I told you that this portfolio has a duration of five and a half years that will give you some intuition for how sensitive or how risky that portfolio is so duration is a measure that I've defined for a bond but you can define it for a portfolio of bonds right simply by taking the cash flows at every period and then computing a weighted average where the weights are the present value of those cash flows as a function of the entire portfolio okay so here's an example uh where I've got a four year Treasury note with a face value of $100 seven percent coupon selling at one hundred and three dollars and fifty cents which yields 6% that's the yield that's the why that makes this 103.5 equal to the present value of all of those coupon payments plus the return of principal okay and so here what I've done is to calculate for you the cash flows of the bond the present value of those cash flows and then the respective product of T times the present value of cash flows so you can actually compute for yourself that duration number and you can get a sense of exactly what that is so the duration is about seven point one three years so duration is a measurement that is in units of years or half-year units sorry seven point one three half-year units and the modified duration is going to be given by six point nine two price risk at a yield of three percent therefore is going to be given by just this expression right here what that says is that if the yield moves up by one a tenth of a percent or ten basis points the bond price is going to decrease by 68 basis points that will give you a sense of how exposed you are if you have very long duration portfolios that means that you're going to be in for a wild ride as interest rates swing around a lot and if you're willing to take on that risk that's great but you're going to be at least compensated for that okay Macaulay duration you know you can compute it in this way for entry year coupons that's a very straightforward calculation so you want to make sure you know how to do that you simply just use the usual discounting and dividing the yield by the appropriate payment periods and now the last concept that I want to cover today is convexity convexity is another measure of risk it's the second derivative what it measures is how the sensitivity itself changes and it turns out that convexity as a measure gives you a sort of a higher order approximation both to the price of the bond as well as how the bond is going to move with respect to interest rates so let me just show you what the derivation is it's pretty straight forward but if have any questions happy to discuss it you take the price of a bond and you take the second derivative with respect to the yield what you're going to get out of it is an expression that looks like this now that expression in and of itself has relatively little intuition but let me just write down the percentage change in the first derivative as V sub M okay and then I'm going to show you an interesting relationship this for those of you who remember your high school calculus or college calculus class you remember there's something called a Taylor approximation right Taylor series this is a method of approximating nonlinear relationships using polynomials right powers of the variables in question if you took the bottom price as a function of the yield right it's a nonlinear function of course because you've got that discounting going on then you can ask the question how can i approximate the bond price as a function of the yield if I'm willing to take a couple of terms of the approximation and when you do that you get what they look like an awful expression but actually it's quite beautiful in its own right what this says is that the price of the bond at a new interest rate at a new yield if the yield changes then the price of the whoops price of the bond at the new interest rate is going to be equal to the price of the bond at the old interest rate multiplied by a factor of something-or-other and that factor is going to be given by one minus a term that's a linear function of the yield plus another term that is a quadratic function of the change in the yield so this is what I mean when I say this is a second order approximation to the pricing relationship and this basically gives us a way to figure out what yields move what does that do to my portfolio now you might be thinking gee this is an awful long way to go to try to figure that out can't we just you know use an Excel spreadsheet and then you know move the interest rate and then you know recalculate and see what that does today you can but in the 1970s you couldn't you didn't have that ability so much of this framework was developed in the 1970s because people wanted to have easy ways of not only pricing bonds but figuring out what the risk of their positions were as bond traders and to do that quickly is very difficult you remember the story I told you about when I got a mortgage you know 20 years ago and the bank loan officer you know just could not figure out what my mortgage payments or she had to go look for a book and try to thumb through what those calculations are well imagine if you're a bond trader trading literally every minute of the day and there was bond trading going on in the 1970s believe it or not you have to figure out these pricing relationships and it was not so easy so a number of mathematicians came up with these kind of relationships for bond prices in fact it's funny bond pricing is actually quite a bit more of a mathematical art than equity pricing simply because with a bond there aren't that many moving parts and so they are subject to mathematical analysis like this whereas with stock prices since there are so many factors impinging on it the actual tools that we use are quite a bit less complex at least from a historical perspective so the purpose of this is really just to approximate the risk of a bond portfolio you want to know when you change yields what that does to the portfolio and you also want to know if you if the yield changes in volatility what does that do to the portfolio so this first term here this term tells you about shifts in the interest rate what this term does is tell you about fluctuations or volatility of interest rates and it turns out that volatility has an impact on bond portfolios as does changes in the level of interest rates just looking at this can anybody tell me off the top of their head what direction this goes in other words if you are holding a bond and interest rates rise we know that bond prices will fall but what if the volatility of interest rates rise as they have over the last few weeks yields have bounced around a lot more lately than they have in the past what does that do to a bond does it make it more valuable or less valuable to put it that way how do you know less or we have some volatility here okay take a look at that expression V sub M is going to be the second derivative and then you've got a term there that's going to be the change in the yield right squared if the change in the yield squared goes up other things equal and other things are never equal but economists like to say other things equal it actually makes bonds more valuable in this respect owning a bond actually is sort of like owning an option now you don't know about options yet we're going to come to that in a few lectures but it turns out that having an option when volatility goes up can be very valuable and similarly for bonds bonds have option like characteristics and we see that here with this approximation that second term will actually yield some actual value when volatility goes up now I do a numerical example here that I'd like you to look through on your own where I compute for you both the first and second terms of that approximation and you can actually see how good the approximation is okay so I'd like you to take a look at that make sure you understand it and next time be happy to answer questions about this what we're in a cover on Monday is risky debt which of course is directly related to what's going on in markets today and so I'd like you to read up on that in the textbook and when we come back on Monday we're going to talk about debt ratings and possibly what went wrong with all of these subprime securities