today we're going to talk about uh a very important topic Topic in economics which is expectations we have barely mention expectations when we talk about the Philips curve we talked about expectations when we when we discussed the uip and so on but expectation is a much bigger issue in economics in fact most Decisions by firms by consumers governments involved considerations of the future and it plays an even bigger role in finance which essentially everything is about the future the price of an asset today is meaningless in itself you have to compare it with what you expect to get out of that asset in the future so it's all about expectations and so on so that's what we we're going to do today want to talk about expectations the how to Value things that that you expect to receive in the future H and how to compare those things with things that you have in the present um but before doing that actually let's talk a little bit about the news who knows who First Republic Bank is remember that a few weeks ago I told you that the Silicon Valley Bank I mean you read it I I I I just mention it that uh ER or discuss it that that you know we had the second largest bank by asset in in US history it was Silicon Valley Bank was the second largest asset Bank in terms of assets to collapse in the US the first one was many years ago and and then we had this bank that had more than 200 billion dollars in asset that essentially collapsed in a few days was run on deposits they had proba before but what really did as it's always the case with banks is they had a run on deposit funding H well it's no longer the second largest collapse in US Bank history now we have the weekend the the new second largest bank to collapse which is First Republic Bank that was essentially liquidated and sold to JP Morgan over today morning very very early in the morning okay so yeah an account in First Republic Bank you soon likely to have an account in JP Morgan but again what made it collapse was something very similar to what made Silicon Valley Bank collapse which is that they had invested on on a series of things that were very vulnerable to to the fast pace of hikes ER in in interest rates in the US and when they had those losses depositos become became worried about it and eventually they decided not to wait just run and see what happened they First Republic Bank lost about hundred billion dollar in deposit just last week okay the last few days of last week so so so so it was obvious that that it was not going to survive and that's a reason something was arranged over the weekend to avoid the panics Associated to collaps of bank and so okay but anyways by the way this is all about expectations you know is is if people had expected the deposit to remain in the bank then probably this bank would not have collapsed it's all about people anticipating what other people will do and so on so forth okay but now let me get into the specific of H of this lecture so there you have this is the most important index of of equity Equity index in the US S&P 500 it's a very inclusive index that captures all the large most of the large companies in the US all of the large I think companies in the US and and that's an index an average weighted average by by this cap capitalization value of each of these shares is a weighted average of the maor the main shares in the US Equity shares in the US and one thing you see is that it moves a lot around you know here for example when when we became aware that Co was going to be a serious issue the US Equity Market collapsed by 35% or so that's a very large collapse in a very short period of time and then as a result of lots of policy support actually we had a massive rally so up to the end of 2021 the equity Market had rally by 114% so big rally then we got inflation and the FED began to worry about inflation so they began to hike interest rates and when they hike interest rates that eventually lead to a very large decline in in in asset prices of the order of 30% or so 25% or so actually from the peck to the bot botom and then since the bottom which was more or less October of last year we have seen a recovery of about 16% or so of the equity Market okay and if you look at the NASDAQ which is another one index that is very loaded towards technology companies and you can see swings are even larger than that now why do these prices move so much well a lot of it has to do with expectations you know are things going to get worse in the future future will the FED cause a recession how much higher will be the interest rate and things like that matter a great deal another thing that matters a great deal is how much people want to take risk in any morning time I if you're very scared about environment you're unlikely to want to have something that to invest on something that can move so much you know and so risk is well known so it's called risk off when when people don't want to take risk these asset prices tend to collapse okay of the risky assets Equity is a very risky asset but that's not the only thing that moves these assets around it's not just the risk that the companies underlying companies may go bankrupt or anything like that here you have for example the movement of a for an ETF but it doesn't matter it's a portfolio of bonds of US Treasury bonds of very long duration maturity is beyond 20 years and so so this is incredibly safe bonds no because it's us treasuries so there's no risk of default like that still the price RS can be pretty large I mean in over this period you know there you see an increasing value of 45% then a declining value of of of about 20% another increase in 15% here there was a huge decline 40% since since essentially what do you think happen here why is this big decline in in in in bonds you're going to be able to answer that very precisely later on but but I can tell you in advance that that was essentially the result of monetary policy tightening you know increasing interest rate caus the bonds to the client so even these instruments are very safe in the sense that you if you hold it to maturity you'll get your money back and all the promised coupons along the path well still their price can move a lot and it's obvious that that movement in price is something you need to explain in terms of expectations what people expect things to to happen in this case is not whether people expect to get paid or not because you will get paid but it's expect but in this particular case is about expectations about future interest rate you think the interest rate will be very high then the price of bonds will tend to be very low and so on but it's all about the future okay so the a key concept H that we're going to discuss today and then we're going to use it to price specific assets is a concept of expected present discounted value this this is a load that concept there's lots of terms in there and we need to understand what each of these terms means so the key issue that we're going to discuss how how do we decide for example if you see the price of an asset out there that is 100 how do you decide whether that price is fair not looks cheap or or not okay and and and and and that question means you have to decide whether that price that you're paying today is consistent with the future cash flows that you're going to get from these assets I that's the reason you buy an asset is because you'll get something in return in the future okay but how do we compare that how do we compare the price today with those things that will happen in the future so answering that question which is we're going to do in this lecture involves the following concepts first expectations big thing that's a you know this is expected PR discounted value the E part is for expectations that comes there you you expectations really crucial because these are things that happen in the future you have to expect even if if it's a bond that promis you to pay you know 50 cents per dollar every six month you still may have an expectation that you know if it is a bond issue by First Republic ban it may not pay so so so you need to have an expectations about that ER so crucial term is expect ation then you need some method to compare payments receive in the future with payments made today I mean you buy an asset you pay today but you're going to receive things Returns on for that asset in the future so how do I compare that that suppose I pay one today and I receive one one year from now does that seem like a good asset probably not I mean you know probably not ER and that's what the word Des counted really means you know when you say expected present discounted value it says somehow that things they receive in the future I value less than things I have today okay so if you're going to tell me that you're going to pay me a dollar in the future and I have to pay you a dollar today most likely I won't take that deal so I need in other words I'm discounting the future how do we discount the future well something that we not have to figure out so let's let me first shut down this part the expectations and then we'll introduce it so assume for now that you know the future okay and I'm going to derve all the equations with assuming that you know the future so there's no issue of trying to figure out what the future is you know it but still you have to decide whether ER what is the right value for for an asset um okay so let's start with a case where you know the future sorry and uh and let's do a comparison uh let's try to understand how do we move flows how do we value Flows at different points in times the easiest thing is think first about comparing an asset that gives you a dollar in the future how much do you think is worth today well the easiest way to get to that value is to think on the alternative say suppose I have a dollar today what can I do with it well in terms of investment well suppose that you have available onee bonds treasury bonds and that the interest rate is i t that's the interest rate on an oneyear Bond so if you want if you if you have a dollar you have the option to invest it in that asset in that bond which give will give you one plus I dollar H next year well that means that I can get $1 next year by investing one over one plus I doll today no because if if I invest one plus one rather than $1 I invest one over 1 plus I today then I multiply this by 1 plus I and I get my Dollar in the future so that tells me that say the interest rate is 10% then with $1 today I can get $1.1 in the future that means that investing 90 90 cents today more or less I can get $1 in the future that tells me that a dollar in the future is equivalent to 90 cents today assumption okay so that's the reason when I told you the deal of look I have an asset that cost cost you a dollar but gives you a dollar in the future well that's not a good deal if the interest rate is positive if the interest rate is 10% then then Fair comparisons is 90 cents with $1 not $1 with $1 okay so that's the discounting of the future you can dis the most obvious way of discounting the future future is to discounted by the interest rate which interest rate to pick that's more subtle that depends on risk it depends on many other things which are going to discuss to some extent here but for now let's make it very simple and in a world which you really know the future really the right interest to use is the safe interest the interest rate of of Treasury bonds and things like that okay so that's that's that what about the dollar that you receive what about if you're thinking about what is the value of a dollar two years from now well you know if I get a dollar I can do the same logic if I if I I can use the same logic if I get a dollar today I can convert that into 1 plus it Time 1 plus I t+1 dollar okay so say 10% and 10% I get 1.1 next year and then I get 1.1 * 1.1 1.21 or something like that okay that's my final result so well then how much is it worth to have a dollar an asset that gives you a dollar to years from now well it's going to be that dollar divided by the product of these interest rates okay why is that well because with this amount of dollars today it's 80 cent or something like that I can generate a dollar two years from now that means a dollar two years from now is worth about 80 cents today okay we're going to use a lot this type of logic so and and I know that that it may not be that intuitive the first time you see it but ask questions you want me to repeat it okay remember the final goal is the following we're going to in the what comes next we're going to see which happens again with many decisions in life but it particularly for financial assets we're going to try to Value something that whose payoff happens at different times in the future and the question is how do I value an asset that pays me you know $5 to one year from now $25 three years from now ER minus $10 10 years from now plus $50 $100 from now what is the value of that of having an asset like that and so I need some method to bring it to today's value because today I have a minimum of what the dollar is you know and and therefore I can compare it with whatever price i i i i people are asking me for that asset so what this is doing is is is that is doing that it's telling you how to convert a dollar at different parts in the future into a dollar today and by that logic the recipe is well use the rate because you could always go the other way around you could always with a you can ask the question with a dollar today how many dollars can I get two years from now say that well say x well then I need one over X then $1 there is worth one overx today you know that's that's the logic because 1 /x * X is one so that's too fast probably so you know with $11 today oops I can generally say [Music] $1.1 at equal two okay then I'm the question I want to know is how much is a dollar worth how much is a dollar receive at time tal 2 worth today that's the question I'm trying to answer you know because an asset will be something that will pay you in the future so I want to know how much is $1 receiving the future worth today and then the answer is well then is I know the answer from this logic because I know that with one if I have one over $1.1 today I can convert it into one how do I know that because 1 over 1.1 time 1.1 is equal to 1 okay this if I invest these dollars today I'm going to get this return on that and the product of this thing gives me my dollar okay so if I tell you do you prefer to have a dollar two days from two years from now or or today you say I prefer it obviously I prefer it today because I can get $1.1 two years from now but then then the more relevant question is no no but then you do you prefer to have 90 cents today versus a dollar in the future H then I need to do my multiplication because I have to multiply the 90 cents by the 1.1 and see whether I get something comparable to a dollar or not okay but that's that's the logic behind that and and that's a so they interest rate is what we discount the future by and it's natural because if the interest rate is very high if the interest rate is zero say then a dollar receive two years from now a dollar receive today is the same because I can't if I invest a dollar today and the interest rate is zero I'm going to get my dollar two years from now if the dollar if the interest rate is 50% it makes a big difference receiving the dollar today versus receiving it two years from now if you're in Argentina interest I don't know what it is it's 700 % it makes a huge difference whether you receive it you know one year from now than today and and and so that's that's the role of the interest rate the higher is the interest rate the less is a dollar receive in the future worth relative to a dollar receipt today because you can get a much higher return from the dollar you have today if the interest rate is high the interest is low you don't get that much okay much difference okay good so this this is a big principle and and I I mean everything I'll say next Builds on this logic so let me give you a general formula so let's ask what is the value of an asset that gives payouts of ZT this year ZT plus one one year from now ZT plus 2 two years from now and so on so forth for end period more okay well I just need to do several of these operations I know that the dollar receed this year is is worth a dollar okay that's ZT a dollar received one year from now is not is not the same as a dollar received today it's the same as one over 1 plus it dollars received today so that cash flow I'm going to receive from this asset is worth this amount for a two something that I received two years from now then it's not it's not certainly it's much less than receiving a dollar today it's going to be one over 1 plus it 1 + i t + one and that I have to multiply by the number of dollars I will receive two years from now okay and I keep going so that's the that's the the present value present discounted value present because I'm bringing all these future cash flows to the present that's what each of these terms is doing the one over that is bringing it to the present discounted because the interested is discounting things making them smaller and value because I'm trying to reduce them to the current value okay that's a general formula so it's a formula you need to understand it's just so that that was an asset that gives you Z dollar today zt+ one one year from now so you use this formula ZT plus 2 two years from now so you use this formula and then you keep going okay what if we don't know the future you know I had remove the expected part well if we don't know the future then the best we can do in fact we do fancier things but that's what we want to all that we do in this course H all that you can do is just replace the known quantities we have here for the expectations okay so that's the closest so you know I know ZT that's flow I get now but I don't know ZT plus1 so I can replace it by expectation I do know the interest rate on a one-year bond from today to one years that's a reason I don't need an expectation here but I don't know what the one year rate will be one year from now that's a reason I need an expectation there and I don't know what the cash flow will be two years from now I have an expectation about what the cash flow will be but I don't know it so I have an expectation there okay so so all that I've done here is say Okay ack knowledge that this guy knew a little bit too much no he knew exactly what the cash flows were going to be in the future and he knew what the oneyear rates were going to be in the future this guy here knows less he knows the cash flow today he knows the interest rate today but it doesn't know the cash flows really has a hunch but it doesn't know the cash flows one year two years three years and so on for the future and it doesn't know the one year interest rate in the future so all these expectations here is important the concept of time these are expectation as of time T at time T you have some information and you make forecast about the future okay use whatever you want machine learning whatever but you have information at time T and then you have a forecast for the future at T plus1 you have you'll have more information so you make another forecast and so on so forth but in this we're valuing an asset at time T then all this expectations are taken as of time T that means given the information you have available at time T that's the reason these guys don't have expectations in front of them because you know this at time T had we taken the value at T minus one we would have not known that and we have had to expectation because it would have been expectation as of T minus one okay so that's your big formula there so there are some examples that are sort of well known and and and the and the and so let me let me show you the they have nicer Expressions so that's that's an example of valuation of of this the same asset but when they interestate is constant then then obviously I don't need all these products in the denominator I I I have a constant interest rate then I just get powers of that interest rate that's one in which you have constant payments so the interest rate may be different but the payments are the same over time okay that's that those are two easy formulas that's one in which you have both constant the interest rate and the payment then you get a nice expression that's just the that okay you recognize that if if you have a constant H constant interest rate here you see that the value is is declining is a it's a geometric series no the value of two years from now is a square of 1/ 1 plus some is a square of a number less than one no 1 over 1 plus I is some number less than one this is a square of that then the cube and so on so it's a geometric series that is declining in the rate 1 plus I one over 1 plus I okay or declining at the rate 1 plus I so that's your geometric series okay that's the value of that constant rate and payment forever suppose you have an asset that that lives forever there are some bondes like that called perpetuities a ER the US has an issue the UK has and so on so that's an asset for example that pays you fixed amount forever and if the interest rate is constant that's a trickier thing then the value of that asset you can see that this this is going to zero so the value of that asset is that and actually a formula that you may see that is very oftenly used as a as a first approximation is this one this is is it's the same asset but it's called X dividend or X coupon is is after the coupon of this year has been paid okay so it's an asset that starts paying at T plus1 ZT plus1 ZT plus2 and so on well that is the same as this minus the first coupon so is equal to that okay that's an interesting thing look what happened to this asset as the interest rate goes to zero so this is an asset that lasts for a very long time and and and look we got to evaluation formula what have what is happening as the interet goes to zero to the value the value BEC very LGE very large it goes to infinity and a lot of what has happening in Global Financial markets in the last few years has to do with that interest were very very very low and so most assets that had long duration had very high values okay and it has a lot to do that monetary policy had a lot to do whether it was the right monetary policy or not that's something to be discussed I think on average it was a right monetary policy but one of the things it did it increased the value of many Assets in fact part of that's one of the mechanism through which monetary policy Works in practice it's is not something we have discussed but you can begin to see here because if the value of all assets go up a lot people feel wealthier and that they will tend to consume more and so on this is one of the channels monetary policy does by the way this effect happens also to the asset that have finite M it's just a this go it's maximized when this asset lasts forever no this this asset literally goes to Infinity if the interet goes to zero while if an acid lasts for n periods it doesn't go to Infinity it goes to n * Z no it's a sum if the inter is zero just Su things you see that if a if if an asset last for n periods and it gives me a payment of Z in every single period Then when the interest is zero that asset is worth n * Z because I will receive Z coupons and I don't discount the future because the interest rate is zero what happens is when the asset lasts forever then n * Z is a very large number you know and that's that's what this pressure captures here okay so let's talk about bones now to start pricing bones so bones differ along many dimensions but one of them is is very important for bones is maturity the N that I had there in the previous expression okay H so so maturity means essentially how long the bone lasts okay when when does it pay pay you back the principal the bonds typically pay coupons and then there is a final payment which we call face value of the bond something like that and and when that final payment takes place that's a maturity of a bond okay so a bond that promises to make a $1,000 final payment in six month has a maturity of six month a bond that promises to pay $100 for 20 years and then $1,000 a final payment in 20 years has a maturity of 20 years maturity is different from duration I don't think I want to talk about duration here but but that's maturity just when when is the final payment of of a of a loan of a of a bond okay bonds of different maturities each have a price and an Associated interest rate we're going to look at those things and the associated interest rate is called the yield to maturity or simply the yield of a bond this is terminology but we're going to calculate this later on the the relationship between maturity and yield is called the yield curve very important concept big fuss about the yield curve these days I'll talk a little bit more about that or sometimes it's called the ter structure of interest rate term in the language of bonds is really maturity so term structure of interest rate really tells you what is the yield in a oneyear bone twoyear bone threeyear bone four five six so you PL them and that gives you a curve okay so H for example look at the those these are two different yield curves this is November 2000 and this is June uh 2001 so this tells you what the yield is in on a three-month Bond so a bond that matures in three in three month on a six Monon bonds and so forth up to 30e bonds okay what is the big difference between this what is happen here in between notice that these two curves are more or less the same longterm interest rate but they have very different these curves this is a very steep curve this is a very flat or even inverted curve what do you think may have happened there between November 2000 and June 2001 people change their expectations yeah that's true that's for sure true about that but look also that but that that this 3 month there is very little uncertainty about 3 month it was a lot lower than that so yes people change expectation but why do you think they change their expectation now Rising inflation from here to here these are these are nominal interest rates up to now I've been talking about nominal interest rate what happens here is there was a mini recession so the FED cut interest rate when you are in recessions the curve tend to look like this because the central bank is cutting interest rates in the in the short run to deal with the current recession what happened 30 years from now has nothing to do with the business cycle today so that interest rate doesn't need to move a lot but the FED is bringing interest rate down a lot in the front end okay so that's a typical shape of a curve in a recession that's a typical shape of a of a curve in the opposite situation where the inflation is too high and so on because what happens the FED is trying to the FED really controls the very front end of the curve that's what the FED really control the Central Bank in general but the FED they control the very front end of the curve because they're setting the very short-term interest rate so this is a situation where they're tighten the monetary policy is very tight because they have a situation of overheating in the economy and in fact they got two car it away that's the reason that we ended up in a recession here okay how do you think it looks today that do you think it looks more like this or more like that is inflation low high today High I mean that's a problem no the FED is trying to hike interest rate now recently because of the the mess in the banking sector then expectations of Interest began to decline a little but but but the situation was was very inverted here you are that's the green line is today okay so it's very inverted okay a year ago it looked like that so you see the long hasn't changed much but a year ago there was no sense that the inflation was getting so much out of line it happened a little later than that there was some concern that interest would would rise but but but now it's very clear that the economy is overheating and this I should have plot you something for for a month ago it would have been even steeper okay anyways but that's because the FED is trying to slow down the economies it's hik in interest rates that's the reason the curve is very very inverted today so let me let me calculate these rates how do we go about it so the first thing we're going to do is we're going to use this the spected present discounted value formula to calculate the price of a bond and then we want to start doing it for different bonds and we're going to construct the yield curve so suppose you have a bond that pays $100 nothing in between $100 one year from now so this is a bond with maturity one year maturity I'm going to call that bond with one year maturity P1 the price of a bone with a onee maturity at time T p1t well that's easy to calculate it's expected present discounted value for if you have the interest rate whatever you say one year interest rate then I know that the price of the bond is 100 divided by one plus the interest rate the one year interest rate today okay that's the price that's expected discounted value so I tell you what I'm showing you is the relationship between interest rates and prices okay and price of a bone the price of that bold is just 100 divided by one plus the one plus the onee interest rate today okay so important observation is that the price of a one-year Bond varies in verely with the current oneyear nominal interest rate this all nominal why is that invers relationship why is it the price of a one-year bond is inversely related to the oneyear interest rate in other words I'm asking what do you think happens to the price as a nominal as a nominal interest rate Rises and why do you think that's what happens to the price well the first question doesn't have a I mean it's very easy the answer to the first question what happens if I goes up well it's obvious that this price comes down but why and and and use the concept we have developed here remember we spend like 20 minutes in one slide well use that slide for that answer hint these $100 you're not receiving today you're receiving a year from now what happens with a dollar received a year from now what is the value of a year dollar received one year from now when the interest is high it's low because you know you much have the dollar today invest it and get this big return on the on on the dollar that means naturally the bond that is paying you $100 tomorrow is going to be worth less when the interest rate is very high it's going to be worth less today when the interest is very high you'd rather have the money today invest it in in the interest rate and get the interest rate and and uh no I need to invest one over one plus i1t doll to get $100 that's another way of saying what about a bond that pays $100 in two years well I need to Discount that by this which is a you know it's a product of the two interest rate and since I don't know what the one year rate will be one year from now I have to use expectation here rather than the actual rate but look at the notation I'm calling p2t dollar p2t the price of a 2-year Bond a bond with vity of two years as of time T okay and this is a bond that has no coupons they just pay you $100 at the end of the two years now not note that this price is inversely related to both the onee rate today and the spect of the one year rate one year from now if either one of these goes up the bond is worthless today you discount more a dollar receipt um two years from now I don't care which one you know either of them that goes up is is bad news for the for the price of a bone okay clear so there's an alternative so this is the way you price a bond bonds using just expected discounted value ER approach now it turns out that in practice a lot of the asset pricing is done by Arbitrage meaning you you compare different assets and that that have similar risk they should give you more or less the same return that's what you so let me let me do this arbitrary thing suppose you're consider investing $1 for one year so that's your decision I'm going to invest one I need I have a dollar which I want to invest for one year but I but I I have two options to do that I can invest a dollar in a one-year Bond I know exactly what I'm going to get you know in that Bond or I can invest in a two-year Bond and sell it at the end of the first year that's those are two ways of you know investing for one year Arbitrage has to be compared over the same period of time and everything it's not the return of a b that you hold for 10 years versus one that you hold for one year it has to be something a similar investment suppose I need to invest for one year or you know then then okay then if I have these two bonds the option is not buy one or the other and then hold to maturity because that would be comparing an investment of one year with an investment of two years I need to compare the strategies of getting my return in one year in the Oney year bond that's trivial because I get my return at the end at the maturity of the bone in the two-year bone it means I need to sell it in between after one year okay so those are the two strategies I want to compare and since I'm not take considering risk here as a central element those two strategies are going to have to give me the same expected return okay that's Arbitrage that's what we call Arbitrage okay to the two strategist have to give him the same Spector return so what do we get from this strategies well if I go through the oneyear bond I know I'm going to get my dollartimes 1 plus i1t that's what I get of a one year out of investing a dollar in a one-year Bond if I go through the two-year Bond strategy buy it and sell it at the end of the year then I'm going to get I I I I invest a dollar today no I'm going to pay p2t that's what I pay today for a 2-year Bond that's what I pay here for a 2-year Bond and I expect to get the price of a one-year Bond one year from now I mean the two-year Bond will be a one-ear bond after a year has passed no it's a two-year bone today but after one year it's going to have only one year to mature so that's the reason the price I need to forecast is this is the price of a one-ear bond one year from now that's what this is here okay and that's my return on this strategy because I'm going to pay this today this dollar and I expect to get that one year from now okay so Arbitrage means I need to set this two equal okay so that means I have to get the same r with the two strategies that means I'm investing the same so I only need to compare the the the the the returns this needs to be equal to that that's what I have here which tells you that you're solving from here that the price of a 2-year Bond at time T is equal to the expected price of a one-ear bond at t plus one discounted by 1 plus the one year interest rate this was like my cash flow my cash flow now is not the cash flow is is just the price I'm going to get a price for that asset that's like the Z's I had in my formulas okay and for a one-ear strategy I only needed to worry about the ZT plus one there was no dividend at Day Zero okay and that's exactly that formula but notice that at t + one that will hold no so at t plus when I'm at T plus1 I don't need expectations I know that P1 t+1 will be equal to 100 divid 1+ i1 the onee rate at t+1 therefore the expected is something like this approximately the expected price is something like that okay I expect I mean this will be without the e will be the price of this one year bond and t plus one I don't know exactly what the interest rate will be next year so I have the best I can do is have an expectation that's my expectation approximately okay but now I can stick this expression in here no I have this I'm out and I can stick that in there here and I get this expression so that's a price for the two-year bone do you recognize this you saw it before you know that's the same expression that we got when we use the expected present discounted value formula we said we going to go 0000 100 years 100 two years from now I know that this count factor for that is 1 over 1 + i1 T * 1 + i1 t + 1 expected well that's what I got just from Arbitrage okay from an Arbitrage logic this is used a lot in finance I I I'm going to say something complicated but but um just ignore it if it's not for the for the for the your quiz or anything but you know there's a big debate in the US today about not big debate a big concern about the the the US treasury debt because there is a debt ceiling meaning there's a maximum amount that the government can of that they can issue and and uh and that ceiling has been moved over time but every time we get close to a deadline when this needs to be agreed again again there's a concern and there's negotiations and so on and uh and uh and well I mean everyone at this moment at least thinks that as everyone as in every instance in the past they're going to reach some sort of agreement that day before of the deadline or not but if they don't and there is a mess this is huge for finance is huge for finance because US Treasury bonds especially short-term bonds are used for pricing everything through Arbitrage and so on so you get a mess there that's a mess in every single Financial Market you wouldn't know how to price many many financial asset actually so it would be a disaster but uh but the reason I I mentioned this here is because again lots of prices are pricing reference in in finance are pricing reference especially derivatives options and stuff like that you price them relative to something using this type of logic so if the thing you use as a base as a reference becomes highly unstable and uncertain and risky then obviously everything becomes very complicated very risky and and financial markets do not like Risk that's for sure anyway ignore that that's irrelevant for your quiz but that's the reason this the whole discussion can over the summer can get to be very very tricky for finance so the yield to maturity remember I mentioned this concept before of an end-year bond is also what we when whenever you see whenever you hear the threeyear rate is that is the yield to maturity which is different from okay let me tell you show you a formula EAS to expain then and it's Define it's important as the constant annual interest rate that makes the bond price today equal to the present discounted value or expected discounted value so notice notice the Highlight that is defined as the constant annual interest rate that makes the bond price today equal to the present discounted value of future payments of the bond okay so for example in our two-year Bond that's the price right this is the price of the asset we that we know the price we already got the price from the pre previous slides of the bond which was based on the short-term interest rate one year interest rate and our forecast of the shortterm the onee interest rate one year from now I know that price take that as a number so then I the yield the yield to maturity is calculated as that constant interest rate constant how do I see is constant because well I'm using the same interet for the first period and the second period i and now I'm calling it i2t it's a two-year interest rate but it's constant constant doesn't mean that it doesn't move over time it means I'm discounting all the cash flows as a Conant interest r that means I'm using I'm using this equation okay so the yield to maturity is find an interest rate that allows me to use this constant thing constant assume use this formula and get back the same price as I got by using the the the expected discounted value or the Arbitrage or something like that okay so that's that's the definition okay you have this price now you you look for that interest rate that allows you to match that price okay and that's called the yield that's the thing I remember I plot this the some curves well those those interest rate in those curves were computed this way now notice that we know what this price is this price is by the expected discounted value or the Arbitrage approach is equal to 100 divided by this so I know that these two things are this is equal to that which means that this denominator is equal to that and that implies for a small interest rate that this 2-year interest rate is approximately equal to the average of the expected interest rate one year ratees okay so this called actually the expectation of hypothesis by the way is that the two-year rate is approximately equal to the average of the oneyear rate this year plus the expected one year rate one year from now okay so that's an important concept and I'm going to start from here again the next picture