 The expectations, this section is basically based upon the expectations theory. So expectation theory says that the interest rate on long-term bonds is always equal to the average of the short-term bonds that cover the same time period with different maturities. In other words, we can say that the investor is indifferent between the bonds that mature over different time periods. So if the investor has to decide that he has to invest in a short-term bond or a long-term bond, he stays indifferent. What he would do is he will simply take into account the average of the short-term bonds, the return which is going to be earned, that can be earned on the short-term bonds and that average is then matched with the yield or the interest rate on the long-term bonds. So if the two things are equal to one another, then the investor will take the two things equally. So we need to make it sure that the average of the short-term bonds turns out to be the same as somebody can earn from the long-term bond. It is necessary because according to the investment analyst or the investment experts or the proponents of the expectation theory, if the long-term bond yield is smaller as compared to the average of the short-term bonds, then nobody will invest in the long-term bonds. So in order to make the long-term bonds equally desirable, what is to be done is that the yield on the long-term bond must be equivalent to the average of the short-term bonds. So that's the basic concept of the expectation theory, which says that you need to make it at par. Otherwise people will prefer investing only in those types of securities or bonds whose yield is higher as compared to others. So if you want the other options the same level of desirable for the investor, then you need to consider this aspect that the yield should be at least the same for the long-term bonds. Why? Because we have observed that most of the people or the investors are risk averse and they prefer to invest in the short-term investments. So if they are planning to or intending to invest in the long-term bonds, then it is important that whatever they can earn from the long-term bonds should be equivalent to the short-term bonds. So people prefer to they will not hold the long-term bonds unless and until the return or the yield on the long-term bonds becomes lower than what they can earn by investing in the short-term bonds. So basically what we are trying to say is if this is so the yield is at par for the long-term and the short-term bonds, then the two types of bonds or the bonds with multiple different types of maturities are considered as perfect substitutes of each other and the investors become indifferent to choose the short-term or the long-term bonds, right? So that's the basic concept behind that. So I'm going to explain the expectations theory with the help of an example. For example, if there is a certain short-term bond that will mature in a year and the expected yield for that short-term investment is for example, 6%. And if we invest in another short-term bond in the next year or the second year, then we are expecting that the yield that will be that can that can be earned investing in the second year for a short-term bond for another short-term bond or the same short-term bond will be 8% in year two. So if I have to invest in a long-term bond, the return on the long-term bond which is expected to mature in two years must be equal to the average of these two short-term bonds. So how we find out the average we know we will consider the yield from the first year short-term bond that is 6% and then in the next year, second year, we are expecting to earn 8% from the by investing in one year term bond. So 6% plus 8 will give you divided by 2 will give you a 7% average. So this average is then compared to the long-term bonds. Here we are assuming that the long-term bond will mature in two years. So if the long-term bonds yield is 7%, only then the investor will consider the long-term bond as a viable option and this is how the planners that plan or they decide the yields on the short-term bond and the long-term bond, they do consider this particular aspect explained by this theory that they need to be at par and this is the investors consider the two types of instruments that are maturing in just one year or over two years or like for example in five years, whatever expected yields are there, they should be equivalent to each other. So when we explain this expectation theory in terms of the three facts which we identified earlier and those facts were defined by looking at the historical data, explaining the relationship between the yields or the rate of return over different maturity time periods. So we have observed that there is fact number one which says that whether the bonds are short-term or long-term, they are going to mature in a year or they are going to mature in 20 years, the overall yields from the short-term bonds and the long-term bonds more or less move in the same direction. So if the short-term yields are moving up, the long-term yields will also move up and if the short-term yields are going down, the long-term yields also follow the same pattern. So there are ups and downs over bonds of different maturities that move in almost the same direction. So this has been observed in the historical data and that has been considered as fact number one. So if we take into account the expectation theory, we are saying that the investor considers the average of the short-term bonds and it matches or compares the yield or the return which he is going to expect from the long-term bonds. So if the short-term bond yields are going up, the long-term bonds will also go up because they are to be compared and they have to be matched. So we say that the expectation theory is in line with fact number one. Then the fact number two that was observed from the historical data was that when the short-term yields are low, people expect that in future the short-term yields will rise. So there is an upward movement whereas when the short-term yields are considerably low, they are expected to rise in future, they will move in upward direction. But if it has been observed in the market, according to the historical data, that the short-term yield or the rate of interest that is earned on the short-term investment instruments or any type of bonds that are going to mature in just one year. For example, if they are high, then we expect that the yields are going to fall in future. So there is a downward movement. So we say that the yield curve becomes inverted. So this is fact number two which has been observed in the historical data that shows the relationship between the yield and of different types of bonds that are going to mature in different time periods. So when we look at the expectation theory, it also confirms fact number two. But when we consider the fact, the third fact, fact number three, which says that most of the times the yields move in the upward direction. So the rate of return, the expected rate of return is observed mostly going up. So if we look at this particular aspect, we are saying that in the case of the short-term and the long-term bonds, the two things are considered to be equal. Otherwise people will not invest in the long-term investment opportunities. Hence this particular theory fails to explain fact number three.