Lecture 7: Linear Rates, Products, and Models

Key Takeaways

So, it's my pleasure to uh to introduce Andrew Garson uh who I had honor and pleasure to work with for for a long while at Morgan Stanley. Um and Andrew is uh actually a MIT alum. uh he got his PhD uh from in geo physics uh from MIT. Uh and uh after that uh he worked in a variety of uh uh roles in financial senior quantitative roles in financial industry and now Andrew uh is head of quantitative strategies at Mizukho in New York. So Andrew, welcome and please take it away. >> Okay. Uh thanks everyone. So I'm going to have a quick talk today about sort of linear uh models and rates in particular which is area I work in. Um we'll just have a quick overview. We're going to talk a little bit about some basic markets. Uh interest rates, what are they? Um what are some of the more liquid rates products, yield curves, you know, a little bit about hedging, how do you hedge your positions, uh P&L attribution, and then end with a little bit on E-rading and bonds. Uh I'm going to caveat this is going to be a very non-mathy presentation probably for this class. So um that's probably my personal bias is I don't find math necessarily uh that illuminating in some of these problems versus what problems are you actually solving. As I told Peter, you know, if I was you were talking about, you know, how to build a house, we wouldn't spend, you know, 80% of the time talking about hammers. You know, we're talking about how big you want your rooms and how the connectivity is. Uh I would highly encourage you to ask questions. I'm going to ask Peter if nobody else does if he can pipe in a couple just because I'd like to keep the discussion going and I'm generally happy to yak on and on about any particular topic that anyone uh is interested in. So, so please uh talk away. Okay. So, start with the basics. What are linear products? You know, they're things that are not nonlinear. Uh nothing too surprises there. Um you know, some of the major categories are things like bonds, high interest rate swaps, interest rate future, CDs. These are sort of the more traditional relatively simple products. These are not you know codo squared or colable spread range acrals or any of the super exotic stuff that has uh been around for a while and still continues to be done. You know the modeling is in some sense relatively simpler here. These are not you know huge you know multiffactor scastic calculus models you have to do in term structure models and things. It's relatively straightforward you know. So the question then is you know why do we care you know as quants? Why is this interesting? You know anybody want to hazard a guess as to why the stuff is still relevant today? Somebody take a guess. Look, you look like you're thinking. >> I was going to say it's still related to important stuff like interest rates. >> It is it's it's an important thing. It's the core underlying. What is what are some other reasons? Maybe you might care about this. >> It's traded a lot. >> It's a very big market and a very liquid market. True. A couple other ones are it's also generally used for hedging of the more complicated stuff and it's also an input to all the more complicated models which generally you start from the simple stuff and you build your models up. Um you know these are very large these are the largest markets on the planet. Um just a couple of statistics in the last few years. Equities is is you know getting close now particularly with the sort of run up in global equities last couple years. Um bonds have something north of $120 trillion outstanding. uh interest rate swaps are more like you know 500 plus trillion dollars outstanding you know just the main futures contract more than 10 trillion you know these are very large numbers um you don't need to have a huge profit margin and when a market this size to you know do very well in a market of this size that's very actively it trades a lot um you know the daily volumes treasuries approaching a trillion dollars a day uh interest rate swaps you know comparable uh again futures something like four million contracts a day and the sofur futures which are the main uh short-term interest rate contracts. And then I think some some a lot of the more complicated models start with the simple stuff and then sort of build up on top of that. And again, I have a tendency to talk faster and faster and faster if nobody comes back. So I'm going to apologize in advance and if you find me doing that, somebody just uh tell me to stop. um you know and again I as said they mentioned they're sort of uh based to the compllets and uh inputs to the more complicated model you know previously I think they were thought quite simple uh I think the global financial crisis in 2008 sort of brought a number of factors to the four that people hadn't considered before uh in particular around things like funding uh and what was previously say a very simple product so you know I'll pay you a dollar in 10 years how to value that became quite difficult and and subject to a a lot of uh thinking and discussion and um and interesting there's not a unique answer to that um which is also interesting. Couple of the other things that came in there was a lot of regulation that came in after uh 2008. Um you know some sense somebody said the regulators wanted to make banking boring. Um and you know it's not boring but they've changed where the interest is in it. um and sort of the super complicated exotic products that were really uh very much in vogue in the early knots and midnots really those things uh ended poorly for a lot of people. uh you sort of think about the causes of the financial crisis at least on on the market side a lot of it ultimately uh boiled down to US residential real estate uh which had a huge runup and then an enormous crash but off of that they'd been issued you know a lot of very complicated securization products uh very highly levered and multi-layered uh and things were you know very opaque and hard to value you know this was not uh financial modeling's finest hour uh so I think there was a lot of regulation that came in on top of that in terms of model risk management and that that really um sort of ended that little that section of the business. So, uh people went back a little bit to basics and and started thinking a little more about some of the fundamentals. Um which I think was a good thing because I think things were kind of silly there. Uh and I remember talking to some of my friends at a soccer game or something our kids were playing and he was mentioning they just done a mortgage securization issue which I believe defaulted without making a single payment. So, this stuff was just so, you know, so poorly supported and thought out. Um, but because it was so opaque, you know, you know, smiling fast talkinging sales guy could push a lot of the stuff to areas that really probably should not have been involved in this area. Uh, and you find little like municipalities in Norway having declared bankruptcy because they bought a whole bunch of CDOS's from Iowa, you know, strip malls or something crazy like that and you sort of think, you know, how did that happen? Um so I think there was quite a lot of uh regulation came in really to you put a little bit of fences around some of this stuff which is is uh you know it's quite powerful but you know could be used for good or bad in some sense. So okay so let's talk a little bit about what is an interest rate. You know an interest rate at its most basic um is a rate at which you can you know either borrow or lend money uh on a principal amount. So in this case, I'm going to have, you know, my interest I payment will be N, which is a notional typically times the rate R acrewed over whatever time I'm borrowing it for. So I have, you know, I'm borrowing $1,000 at 5% for a year. In a year, I'll have to pay back $1,000 plus 5% uh times a year times $1,000, $50 of interest. Um, relatively straightforward. You know, how do I value that in terms of what I'm willing to pay for that today? you know, I'm not willing to pay $1,000 for it because uh well, let me reverse that. So, if I get $1,000 in a year, how much am I willing to pay for that today? And whatever I'm willing to pay for today, I can invest. And if you work out arithmetic, it works out to be one over one plus the rate times the time. So, at a 5% rate, I'll pay something like $950 odd dollars for that. Uh and then in a year get $1,000. Now, this depends a lot on the discounting rate. for example. So, what rate am I going to discount at? Um, you know, that's hard to observe. Rates I can get for investing are easy to observe. You know, you walk into a bank and you look at the signs on the front door, IRCDs, four and a half% rates for a year. Uh, whatever. How much a discount is is hard. Um, typically, you know, in our world, we're going to assume one of the, you know, institutional rates, which is something like a sofa rate or an OS rate, which we'll talk more about in a little bit in the future. Um but these are sort of a standard uh assumption. Now one of the interesting things practically you find every person has a slightly different rate that they should be discating out because it depends on where they can borrow money and that that depends on your creditworthiness. So you know you may have a good credit rating and you can borrow money at 4%. You know you no offense I'm not picking you. You may have a terrible credit rating so you have to pay 8% to borrow money. You may have a medium one so you're borrowing at 6%. So uh and you know it all depends on the particular issues of the issuer. So this this makes it uh interesting because at this point you know the uh a contract for you guys to pay me a thousand bucks in a year has different prices depending on who you are. So there's no single price here makes it interesting. Uh a lot of the banks will deal with this because you know it's kind of a pain in the ass for a trading desk to have to manage you know a thousand different counterparty relationships each with their own funding. So usually you'll centralize that on a central funding desk of some sort. They will then assume all of the uh individual traders can fund at a relatively generic rate and the funding desk will deal with all these horrible little uh funding bases uh and small collateral exposures and things like that. Okay, going back a little bit in history. Um you may or may not have heard of Liebore. Uh probably not because it now is gone. But this was one of the first really common use rates. Stands for London interbank offered rate. Uh it was developed I believe by a Greek guy in the 60s uh as a method of standardizing syndicated lending in London. Most bank lending to corporates is floating rate lending. So you won't you won't do a fixed rate loan for 10 years. You'll do a rate that resets every month, every three months at the then current rate. And the question then was what rate should we be using to set the interest payments on these loans at? So what he came up with is a notion we're going to pull the banks as to where they lend money shortterm and we'll take the average of that you know kick out the highs and lows and then that'll be a standard rate everyone can use as a reference index you know was a very great very good idea incredibly successful first interest rate swaps which are really synthetic versions of those came out in the early 80s uh first swap I believe was 1980 or 81 which is a crosscurrency swap with IBM uh and then the first pure single currency swap came out the next here. Uh these products had you know phenomenal growth in the 90s and knots. Uh currently there's something about 350 or 400 trillion of them outstanding as of uh labor cessation and still continue to grow a little more than that. Um so you know very pro very popular product very liquid uh very actively traded. You know, liberate though, you know, again, another one of the fallouts from the financial crisis showed that it had some issues. You know, liber was set by polling banks. So, you'd go ask the funding department at each of the big banks. I think there's 20 odd people in the LIBOR pool, were you going to lend and borrow? Were you going to borrow money from they would come up with a number, they'd quote it. Note that it's not based on actual transactions. So, they're just making a number up. They're trying to in theory if they're all good players hitting it where they would actually do the trades. But this is not actually observable trades and I think in the sort of late knots early teens um group of traders of uh less than honorable intent realized that this is and this was a manipulatable rate. Now you may have for example you know every day you're going to look at how much money you know these rates set every day and based on your portfolio you'll have some impact on that. Some of them you can have millions of dollars a basis point. A basis point being a hundth of a percent. So if you can get that rate up or down just a little bit uh in your favor it can end up making you a lot of money. So what they did was they essentially all you know chatted with each other and they were chatting with whoever the people at their banks were responsible for setting these rates and convincing them look I need you to make this a high rate today or I need you to make this a lower rate today. Um and they could you know you can move this thing a little bit because it's not observable because nothing trades. And they, you know, they were doing that for a while to their advantage. Um, this was discovered. Number of them were in jail for some time. There was very substantial multi-billion dollar fines to the banks doing this. But I think the realization came that this rate didn't have any support. You know, one of the things is Liebore is a um, you know, it's an unsecured rate, if you will. It's it doesn't have anything backing it like a repo or a collateral rate. Um, and there's just not a lot of transactions. Post 2008, nobody wanted to lend unsecured because you know you had a number of bankruptcies. People like Leman's gone out and you just you don't want to lend to money who's nothing backing it. So uh people weren't happy with that. And you know some of the things in some of the teners and markets I think had you know three transactions in a six-month period. That's not enough but you have to quote this rate every day. So there's just nothing fixing it. So that was a real issue. So the Fed set up uh or the Fed plus the other regulators uh I guess it came out of London originally was the big driver for this. the FSA there um because L being uh London and Liber really decided you know you needed a new rate that was going to be defensible backing up based on actual transactions and a number of other criteria to make it hard to manipulate so that you know this would be a genuine real observed rate and not something that was just being made up uh by banks to favor their trading desks. >> See Andrew just a comment li sound a bit esoteric at this point, but at the time uh mortgages mortgage interest rates were often quoted as LIBOR plus a spread. And my experience was during the the time of the knots and the teams, you know, looking at >> possibly refinancing my mortgage to a lower rate because rates were coming down and the lie board did not move much at all during that period which was rather frustrating to watch if anyone understood why it was not moving. Well, it was partly that and partly um the libore to treasury spread which is the difference between libore rates which are the the bank rates and the treasury rates which are government rates uh blew out in that period around 20112 uh the fine institution I was at at that point wrote off 8.4 four billion dollars in structured notes because of that for example um because they were hedging their structured notes with LIBOR products and assuming the two would stay together and they just went uh most of it came back in the next couple years but it was quite unpleasant for a while there and and uh made them want to change how they were doing that. um you know when we went through sort of the liebore retirement you know when you look at how much of what kind of products are out there there's all sorts of weird stuff you know swaps are the biggest piece but then as you have floating rate mortgages uh you have a fair amount of corporate bonds out there um you have some really bizarre slightly esoteric stuff that is like you know I don't know how you redo that when you're going to go knock on somebody's door and tell my mother that you know her library plus 100 mortgage is now a sofa plus 150 I don't know what she's going to do with that other than be confused and call me and ask for help. I think but um so you know so they really the labor had to go because it was it was not not a good rate anymore. Um and this is also true of the equivalent libs in at least Japan, Canada, US, Switzerland, Europe and uh the UK I think all have uh replaced their rates with something a little more uh defensible. In the US, it's called the sofa rate, which is the securitized overnight funding rate. Um, which is essentially the average of overnight repo trades. So, a repo uh rate is, you know, if I have a bond, I can just buy it outright if I want to, but it's not usually the, you know, an efficient way to do. I can borrow money against the bond. So, if I have I buy 100 million US treasuries, I can then lend that to somebody, you know, post that to somebody and borrow say $98 million of cash. He he'll hold my bond. uh and then if we want we can swap it back in again but the rate at which you're going to pay in that borrowing is known as the repo rate or reverse repo rate depending which direction you're going uh and that's at yeah a real rate real money changes hands um US has about 800 billion of that a day of the main uh repo facilities so that's something that there's a lot of transaction data behind you can you know look at it measure it you know add it all up average it uh and come up with the rate so that's what they Um, you know, uh, so far does have some features that are not so fabulous. You know, first of all, it's a daily rate. Uh, nobody wants to have a loan where you have to pay interest every day. It's just a pain in the ass operationally and and not good. You want to pay every three months or six months or annually, for example. That just makes it just easier mechanically to deal with. Um, you know, you can compound it into term rates. So, we'll talk about that in a slide or two. Uh another thing if you do compound it into term rates so you do like the average so for over threemonth period and then pay that rate you don't know the rate until the end of the period because you haven't finished doing all your daily uh features. That's a bit annoying for uh people because a lot of people want to know the rate up front so they can just arrange you know guess operationally set all their payment systems make sure the payments get there schedule that have enough time to sort out make sure they're doing the right rate. Uh and the other interesting thing is it's sofur is a secured rate. Banks actually prefer to lend unsecured. Uh so I think when a bank when a bank makes a loan so it'll do a floating rate lend to a corporate they'll give them $100 million. Corporate will pay them the floating rate say Libbor plus you know spread or whatever. Now if you think about what happens when um bank credit goes bad. So if there's a crisis, typically the bank spreads will blow out, which means LIBOR will blow out. And if the majority of the income is coming from LIBOR products because that's what you've loaned at, you're now making more money. So you loan money at Liber plus 50, LIBOR's now 100, you got an extra 50 basis points of income because your credit's deteriorating. So it's sort of a rightway risk kind of feature. um that is you know I don't know if it's worth you know a whole lot of the bank but it's a nice feature to have that if bank credit goes bad you end up making more income so people like that said Libbor has mostly been phased out in the last few years um US ended lieore last June so June 2023 they're one of the biggies sterling went earlier um Canada went two months ago Mexico is going into this year uh and then there's a couple features in yen that are also happening I think uh early next year. You know this was an enormous migration and and uh you know you should hope in your careers you never deal with this kind of stuff because it is just painful and tedious. Yes. >> Do you mind introduced lending the relationship between uh bank credit and um the funding rate? >> Okay. So I'm I'm a bank. My my majority of my corporate lending portfolio is floating rate loans which are going to pay me an interest of liber plus a spread fixed spread. So it it's going to float. If LIBOR's higher, I make more money. If LIBO is lower, I make less money. I'm going to fund myself as something that's typically not, you know, less less uh floating than the rat. So bank credit goes bad. So banks get into trouble or somebody perceives that there's a a systematic issue out there with banking allot 2008 or a couple other episodes, the liber spreads are going to increase because, you know, larger spreads, higher rates, lower quality generally. So what that means is the spreads are going to increase. So my interest I'm going to earn on my loans has now gone up which helps me because I get more income. Make sense? Or >> sorry. >> Okay. Yeah. No, please again please. I'm talking fast on a big range of topics. So please jump right in. You're looking confused or anything? No questions? >> No. Yeah. Yeah. Okay. Um you know again you hope in your career you don't have to deal with this because it's a pain in the ass. It's just an enormous mechanical process. Uh there's been a few of them around. Euro conversion was another one in the late 90s we did where you know all of the European currencies converted to euros. Um yeah I remember sitting in Canary Warf at that one at like three in the morning because we'd screwed something up and it was like oh dear. Um but it all happened in Y2K is another classic one. There's been a bunch of other ones like that and but it all happened and fine. So no more library. Okay. So as I mentioned sulfur is a daily rate. you know, you don't want to pay daily. So, you know, you can compound it up. Two standard ways. I'm going to do daily compounding. So, each day I'll, you know, work out my discount factor for that day, which is one plus the rate over 360. And you just compound those up. Uh, and then subtract one off the end, and that'll convert it to an equivalent term rate. Fine. And you can pay the term rate. Or I can just do a simple average. So, just average um the sofur or average the daily sofers. In this case, the little d's are the number of days it applies for. So a rate setting in Friday will apply for three days over the weekend. A rate setting in Monday only applies for one day. Excuse that. You add it all up, pay that rate. Um the majority of products in the dollar market go on the compounding uh sofa because it's just it's a simpler rate to calorie. Uh it also doesn't have the average and you need to make a slight correction due to volatility which is just annoying. You know, it's a few basis points running so most people won't care. But um if you're trading this a lot as a market maker, you want to get all this stuff right because the bid offers are couple ten of a basis point. So uh if you have something that pushes you outside of that, you're going to lose money. You know, there's lots of other rates out there. Uh term sofer is one that CME created a couple years ago. Uh use they actually fit something. They have a little model. They do a best fit to all the futures prices which are again are very liquid, well traded stuff. and they work out one month, three months, six month rates and then publish those. It's a set in advance rate. So I set it now and it applies for the next three months say so I'll fix a term so for now pay in three months at that rate plus whatever. Uh this one is is quite common in corporate loans. Uh and again corporate lending you know um the banks in generally have much better systems than the users. not always the case, but they've been at this for a while where if I'm a corporate, I don't necessarily want to deal with a lot of the complexities uh and sort of nuances of this stuff. I just want to have something fairly simple. Every 3 months, I'll look a rate up on a screen. I'll put that on my loan and I'll pay that three months later. Yep. I don't know if this is like if I'm not understanding this correctly, but like let's assume like we go back to like um when rates were kind of now, but like when rates were there was more of a risk of rates um going above 6%. >> That like would directly influence how like the duration of the the futures contracts. >> Is that my >> sure I'm understanding like the cheapest deliver for the the futures contract changes because um like the price at 6%. >> Sure. So at 6% your futures contract will hit at 94 because the 100 minus the price is generally the rate. Fine. Yep. So then for in terms of thinking about how in terms of think about how that affects the term sofur like theoretically um >> like that would change the duration of the like the cheapest deliver which means that your term sofer represents a different rate >> term so it it is a slightly different rate. Um, so okay. So let let me doodle here for a little bit if you don't mind. Okay. So that's time that's rate. So what it is, the futures will each set a block of say three-month futures. We'll set an average over a three-month rate. So I might have a rate that does that. Another future that does that and that might have a little stub period in here which you know can go back to that future. So each future will be set and it's essentially the average over the term of the future. So what CME then did was they did a fit of this model to the futures contracts and they then interpolate the daily rate say for six months which will run from here to about here. So they look at that and they'll work out what the equivalent six-month term rate is based on these rates to compute that. Does that make sense? Yeah, I think I think I'm not asking this question. Oh, sorry. I think for a second. >> Anyway, so you know, CMA came up with that. The loan guys like it. Again, you know, end user corporates don't want to deal with all the mechanics of daily sets and operational stuff. It's just a pain in the ass for them. Um, so they're happy to let somebody else do that. Um, you know, it's the usual set in advance, pay six months or three months from now, fine. that makes everyone happier. Um the Fed however doesn't want this to become a de facto new LIBORE rate. So they actually have quite severe restrictions on how much the dealers can trade in it. Essentially dealers are only allowed to trade it to hedge their loan portfolio. Uh which also is an interesting feature which means it's an entirely one-way market because you know banks issue loans. They have to hedge it. It's all in the same direction. So there's a very substantial basis between sort of regular vanilla sofer swaps and the term sofer swaps. you know, something around four or five basis points, which again, in a market context where that's 20 or 30 times a bid offer, it's way outside the bid offer range. Um, there's another sort of older version of sofur called Fed funds, which is really average of of slightly different Fed funds drawing similar to sofur at still in wide use. And there's a whole lot of other crazy rates, prime, disco, you know, municipal or military housing rates and all sorts of other stuff. Disco is my favorite rate. Not for any reason other than I like the name but which is a Fanny made discount rate. Okay. Okay. So now discounting rates. One of the questions is how do you work out what the discounting rate is going to be? For example, what rate should you use to discount you know a future payment? So let's assume we have you know a simple case of a fixed payment f for fixed at a time t. We're assume some discounting rate and we'll value the payment today. give a present value of P. Now we have to fund this rate. So we have you know we generally in in this world you assume that you are not just you know spending money out of your back pocket. You're going to go borrow money from somewhere and use it to purchase this thing. So we'll buy P and we're going to buy at whatever our funding rate is. So whatever we can borrow money at that's what we will do to fund it. And now just think a little bit about from one day to the next day. What are the cash flows involved here? You have two of them essentially. you have you're going to have to pay money to borrow in your you will have to pay interest on your borrowing. So whatever interest rate you've done for one day you're going to pay that and then your underlying asset which is the fixed cash flow in the future will increase in value a little bit because you're getting one day closer to its payment. Okay. So the interest paid you know the interest is just going to be whatever the present value of that is be it t not times the borrowing rate um over 360. So that's how much you're paying to fund this. And the increase in value, it's going to go up by yesterday's value times 1 plus r over 360. That's sort of the the compounded rate occurral factor or the change in that is you know again the uh value times r over 360. Notice that the first and the third equation have very similar forms here. The only thing different are the two rates. So this is actually going to imply those rates have to be the same or there's an arbitrage here. So that means that your discounting rate that you should be using to discount un you know future cash flows is whatever your funding rate is. So whatever you can borrow money at that's the appropriate rate to discount future flows at. Now if those rates are different you know there's a trade. You can either borrow cheaper than you can invest or you can invest cheaper than you can borrow and you do one or the other and that'll come back. This actually this is a very you know if you only pick one thing away from the hour and a half we're spending here remember this because this is you know funding and discounting are the same. Okay let's talk a little bit about some of the products just going to talk about the vanilla ones. So the question is how do I compute what I think um forward rates should be and what projections of those should be. So what I'm going to do essentially is build a curve uh to do that I'm going to start with by thinking what are the easily observable instruments in the market. What can I see out there that I can think I you know whatever my pricing uh model is going to be these are the things it needs to hit because I can trade these things. Some of the common ones we're talking about are CDs which are certificate of deposits. These are very short-term uh bank lending borrowing. You know, you can walk into a bank and buy a one-mon CD at four and a half% if you want. Maybe a three-month one, six-month one. Doesn't go much beyond that. Maybe a year, you know, again, fairly liquid, quite common. Interest rate futures, very liquid products, uh, can do that. And finally, interest rate swaps. Now, so for each of these, we're going to work out what is the price of this as a function of my discounting curve. So, and then given that the cash rates, I can invert that to find what my discounting implied curve is. that prices all these instruments back to market. So CDs, you know, nothing too hard here. You know, I buy a CD, I get the amount I invested plus the rate accured over the time. The value of that is that forward uh interest amount plus the forward return of my principal discounted back or inverted. I just get, you know, Z is one over one plus R delta T. That's again, that's a thing you're going to see all over the place. These rates are well observable. They're a little little bit tricky because a swaps test typically can't necessarily trade them because these CDs are funded products. I have to actually, you know, buy them. I want a 100 million CD. I need to give somebody $100 million. I have to get that. Swap desks don't like that. They, you know, they prefer derivative instruments which are not unfunded instruments because you just have to put less cash up front and everybody likes less cash up front because it means you can do more, leverage yourself up more, uh, and so on. Okay. Second one is interest rate futures. Uh again, very liquid contracts. Uh US ones trade on the CME exchange in Chicago. Uh every country generally has, you know, a relatively active short-term rate. Canada trades on Montreal. Um the UK contracts trade on, I believe, the LME. The Euro contracts trade on the Euroex. Japan contracts trade on um Tokyo, I think, stock exchange. So, um two main contractors, a one-month contracts and a three-month contracts. They're slightly different, which is a bit annoying. Three-month contract trades on a compounded rate. One-month contract trades uh on a average rate. Three-month contract goes from ImMate to Imm. IMM is international monetary market, which was an old market in Chicago that sort of set the standards here, which is the third weddednesday of every March, June, September, December. Uh it's just when the contracts settle and that's fine. The one month contracts trade on a calendar month. So today's contract started in September 1st, we'll settle on October 1st. I think maybe off by a day there. The futures settle at 100 minus uh the rate. So you know, if I have a interest rate of 5% when I'm settling a future, that'll be a 95 price. And so and again, people just did it that way because they like to know markets rallying, contracts go up, markets selling off, contracts go down. For bonds, price and yield move in opposite directions. So yields go up, prices go down and so on. So that's why they set the uh 100 minus that line. There's a small convexity adjustment here as well dealing with futures margining and I'm going to just wave my hands and gloss over, but it's something you have to be conscious of. Uh futures do daily margining. It's a standard part of a futures contract. So if I have an interest rate futures contract and I you know I enter in into it and generally you'll enter into a futures contract at no upfront cost aside from some initial margin perhaps market moves up you know futures contract goes from 90 to 91 they'll give me one back in margin if it goes from 90 to 89 I have to pay one to remar it so at the end of every day you settle up the net move on that day and they do this to sort of minimize credit exposure between that but if I think if I have an interest rate contract you When the price of the contract goes up, I'm getting money back. But the the fact that the price went up implies rates have gone down. So that means I can reinvest my money and now at a lower rate or if the contract goes down in value, uh I have to fund that more because I need to anti up more margin to the exchange, but I'm having to do that at a higher interest rate. So this sort of correlation between the uh flows of money and what it's going to cost me to borrow or lend that introduces a small uh valuation impact. Okay, so let's talk about interest rate swaps, which are probably the biggest single uh product on the planet. Uh this is basically an agreement, a custom agreement between two counterparties to exchange two sets of cash flows. In the most uh simple case, one of these is a fixed rate. So I'm going to pay you 5% a year for 10 years. And the other one is a floating rate. I'm going to pay you in um compounded so far. I'll pay you annually every 10 years. We'll do that for 10 years. Uh and then we'll call it a day. Uh there many many variations of these structures. There's all sorts of crazy stuff and you you know imagination is the limit here. As long as you convin somebody else to do the other side of it. They're a tremendously useful instrument. Um they're great if you know if I need to hedge something or change my risk profile. It's a very efficient way to do it. These are all derivative products means they generally don't use balance sheet. They don't take a lot of upfront capital. They're cheap to fund. So they're quite popular uh for those reasons and I said they are the most liquid uh interest rate product out there. They're one of these things that you know there's a lot of mechanics to get it right. Um in particular you know if you if you enter you know you work in this field you will get very good at data arithmetic which is sort of a stupid thing but um I mean I remember once I was interviewing a guy and he just done a PhD in numerical methods I think and one of the questions he asked me which was a great question so you know keep this in mind when you're interviewing he said if I come and work for you a year from now what will I know now what will I know then that I don't know now and I said you will know how to do data arithmetic he looked at me kind of funny but went away and they actually gave a job and he took it and he's fabulous guy and he came back a year from now and said you know you were right you know I know how to tell you know between my birthday and today how many days are there it's not a unique answer to that one by the way just to make it come depends on your conventions um and I'm also going to guess in interest rate swaps at least 90% of all problems are date related that's the first thing you want to look at do I have the dates correct because you know am I using the right holiday calendars uh and holiday calendars again it's another one of these things that's occasionally annoying you They're they change. We had Junth added a couple years ago, for example. We had the Queen's funeral added. Every time a president or ex-president dies, there's a holiday and just these, you know, Canada had truth and reconciliation day added recently. So, you know, you have to make sure you build your models to account for, you know, the fact that holidays may be dynamic. So, I just put on there a simple sort of standard dollar swap roll out and you there's a half dozen different steps to actually get what are the acral periods and the cash flows and the payment dates and the reset dates. You know, you start with today, you go to the trade date. It's a forward starting date. From that, you typically add two days. You go to the effective date. You then add the tenor of the swap to get the maturity date, unjusted maturity date. You then roll that back in increments of the payment frequency. Get the unadjusted roll dates. And then you adjust everything based on a holiday calendar, which typically be New York banking or something. Um, and again, you got to be really precise on the calendars. Another one of the questions I like to ask people is how many distinct dollar calendars do you think exist in terms of swaps monitor which is a standard source for all calendar data. How many dollar calendars do you think they have? Any want to take a guess? So I have New York banking for example. I might have New York Fed. I might have somebody guess no penalties for getting it wrong. Okay, Peter, what's your guess? You've been around for longer. >> Let's see. Um, I would guess maybe 150. >> You're in the right ballpark. Last time I checked, there was 294. And they're all, you know, every exchange has got its own calendar and they're all different. Every major business center has its own calendar. And it's just, you know, if you get the wrong calendar, you're going to price stuff on the wrong date and you're going to mispric things. Uh, and then winner's curse rule says if you price something wrong, you only get the trade when it hurts you. So, um, and again, dates are just an enormous pain. Okay, value of the swap, we're going to do it in two pieces. We'll start with, um, the fixed side. So, fixed side is relative forward. We roll out all the cash flows. We work out how much we're paying on what dates we're paying. We then, you know, from our curve, get the uh, uh, discount factors there. Sum it all up. So in this case I've just got C which is the coupon rate. Delta is the acrual fraction. So if it's an annual swap that'll be our round one notional and then times the discount factor. Sum it up. That's a present value of the fixed side. Floating side's a little harder but same idea but instead of the um coupon rate I've got the forward rates uh which are little F. So I have to work out what the forward rates are but you know we'll get to that in the next slide. uh you know some mechanics are about what is the u what are the particular specifications is it set at the beginning at the end is it forward maybe a spread in there but you end up with there sort of two different uh curves you here I need one curve to generate the forward rates from and another curve to generate the discounting factors from in the vanilla swap those are both the sofur curve so they're actually only one curve okay so what do you want to do here there's another bit of magic here that I can actually generate an employee floating rate from fixed flows only. So if I assume I can borrow fixed rate bonds, I can borrow one at a given at a first date, invest it and pay it back at a later date, invest it at the unknown floating rate and then pay it back at the end. Uh and I'm assume I'm going to invest at a rate that's a fair trade. So PV of that is zero. So I end up here with, you know, the initial term there is me borrowing the money. I have time t0 me getting the interest rate at time one and me getting the principal back at time one. So given all that sums to zero I can work out what the implied forward rates from this discounting curve are. So I use that plug it through the swaps uh and now I've got a uh swap market. So for an implied swap markets you know generally when you do a swap it's usually done at a zero PV or something pretty close to a zero PV. So when I look at the two legs and sum them up, I'm making zero. I've got, you know, the fixed side plus the float side. You know, both just the sum of the fixed rates times the discounting factors. Uh and then, you know, subtracting the float side giving factors, do some arithmetic, and I get the uh what the implied swap rate coupon is. Now, this is useful because swap rates are observed. I go to my favorite Bloomberg screen, I can see all sorts of swap rates. So that's what I have to work out my zero curve to fit those. Any questions? I feel like I'm just motoring through this here right now. So, yep. >> Can we get these these slides? This is a question. This is a question. >> Sorry. What was the question? >> I was going to ask if we'd get the slides. I don't know if they posted. >> I can certainly send them. I don't think I have them, but >> we don't have them. >> There is there is nothing too top secret in any of this. Um that's how you do. >> See Andrew, one comment. Um I'm not a fixed income expert, but um the zero sort of present value at trade initiation is quite an important uh feature I think where it um encourages people interested in either side of the swap to be comfortable with uh making the trade. Yeah, that's sort of the so-called par rate, which is the rate that prices it at par, aka for a swap, that's zero. Um, you could price it at something other than that. And, you know, again, bid offers in this market are usually a tenth of a basis point, two ten of the basis point. So, it's around zero. Um, you know, some of the structured transactions will have swaps that are way off market and then, you know, that may get accounted for elsewhere or that just may be where you're taking your P&L out of. Just another comment that the a long time ago, this is like the late 80s, um I was at the Sloan School and uh swaps were becoming very popular. And one of the reasons they were becoming very popular was that certain corporations could access sort of fixed rates at very low rates and less uh creditw worthy companies could not have access to such good rates for borrowing but did have access to floating rates and so these corporations would sort of pair up uh with a swap. I don't know to what extent the swap market now is >> it's I think it's well past that but >> but at the time if I'm a lower rated guy and I can maybe borrow short term but nobody's going to l lend me money long term because they don't trust you know I'll be around in 20 years but maybe they're they're comfortable that I'll be here in three months and have another guy who can then fund for 20 years because he's a high credit rating. We can both bore in respective countries then do a swap. So I now have fixed rate swaps and he has floating rate swaps and depending how they want to manage their treasuries that may be a great trade for all sides. So yeah, again swaps are very flexible. They let you move the risk around in quite a flexible way and quite a customizable way. Okay. So let's talk now about the core sort of uh valuation model really in uh interest rate swaps which is the yield curve. So we're going to parameterize the yield curve. And a yield curve is used to it's just to get a function of the discounting as a function of time which then feeds all of the other valuation models. So, we're going to parameterize it by a set of times and zero uh zero coupon bond prices. You can parameterize it in a number of different ways, but this is one of the easy ones. I also need to choose a method to figure out what is my uh zero um disease if I'm in between not points. So, maybe I have annual knot points. So, I'll do a 1, two, three, four, five up to 10 years for my times and I'll have disease at each of that. But suppose I need a four and a half year discount factor. How do I do that? lots of p lots of possibilities here. Um first of all I'll talk about in the not points you know how do you choose which times to use to your interpolate that simplest one which is the most common one is you just use the maturity dates of all your instruments so I'm going to build it with say one two three you know five seven 10 years swaps I have not points 1 2 3 5 7 10 years use that that's easy one uh there's other possibilities like FOMC dates which we'll talk about in a minute um you also have to be a little careful on picking your knot points to work well with your interpolation method because some of them there's some you can make bad choices here which will give you unstable hedging. We then are going to fit a set of market observables here to get the Z's. So given that we've chosen an interpolation on knots, you know, we have hopefully n knots and n points we can solve. Let's talk a little bit about CDF spline. This is a constant daily forwards. This is one of the easy ones which you assume that between your knot points your daily forward rates are constant. So you end up with a curve that looks like sort of a skyline or a stair step or something. Just an example there simple one where you know in this case you can see what my uh not points are. I think I have six months a year uh two years, three years, 5 years, 7, 10 and 20 I believe and 30 and the forward rates are just flat in between them. This perfectly good this is really fast. Um this is a local interpolation method which is one of the key features. And what I mean by that is if I'm looking at you know I want to find a discount factor say there it only depends on the value there and the value there. It doesn't depend on anything here or anything there. So I'm just going to interpolate between those two. >> Why do you not choose the not purity based on minimizing error? >> No on what? >> On minimizing error. >> How are you going to do that? >> Like chubby chef. you could um what you want to do the thing in on the simple case is you want to maintain a connection between your not point and your calibration instruments. So in this case I'm choosing be maturities. You can do lots of other stuff but you could for example let's suppose you had this set of uh not instruments to the not points here you know if I choose uh say not points at 5 10 15 20 25 30 I'm oversp specified here and I'm underspecified there. So you need to choose it with regards to how you're going to calibrate u particularly uh and the other thing that comes out is when I work out the simplest cases what my hedges are. It's in terms of my yield curve input instruments. So I want to choose a good set of liquid observable stuff that I can actually transact in to do the hedging. There's a zillion ways to do this and that's still you know 30 years in that's still a very active area of research and discussion. So um lots of stuff and and you know some of the areas like some of the um uh computer graphics for example has a lot of advanced interpolation stuff you know beastplines and beyond and all of those are likely entirely uh relevant here just haven't looked at it anyway so this one gets you your very characteristic shape so this is you know probably the least smooth simplest fast method so it's got popularities for that on the other side the most smooth little slower cubic splines. So in this case I'm just going to fit a peiewise cubic polomial here. So I'll work out all what my little parameters are and then I'm just going to you know sum it up. I have four terms you know cubic in each interval. I impose constraints at the edge of the intervals on continuity and smoothness and do that. Again this is still a reasonably fast method is a very smooth curve. Now this curve is not local. This is a global curve and you know if any of you go into