 With reference to the yield curve, there are certain questions that may arise in the mind of a finance student like where does a yield curve come from or why is the yield curve sometimes upward sloping and other times downward sloping or how the interest rates are affecting the shape of today's yield curve. So these and other are the few questions that are related to the yield curve and its characteristics. Let's talk on these issues with a world of no uncertainty. So how yield curve work under certainty? We know that all investors already know the path of a future interest rate. This means that a bond cannot be expected to provide a higher rate of return than those of the other bonds available in the market. This means that all bonds and securities must provide an identical return to the bond holders. And if it's not the case, then the investors will bid up the price of high return bonds until its price or return become equal to the other bonds that are available in the market. An upward sloping yield curve may indicate short term rates going to be risen in the days to come. We have an example in this regard. Suppose you want to invest in bonds for two years. Now there are two strategies to work in this particular scenario. Strategy one is that buy and hold two years is zero that is offering yield to maturity of year two at the rate of six percent. Now with the face value of thousand dollars at current price the bond's value is equal to 890 that we get by dividing the face value over 1.06 which is the year two's yield to maturity. Then it matures in two years to its face value that is thousand. This means that two years growth factor for the investment would be equal to 1.12 and that is the solution of face value which is divided by the present value or it is this 1.06 square. The value of two years definitely if we multiply P naught with 1.06 square it comes to 2278.40. Then we have strategy two that says that invest 890 dollar in one year zero coupon bond with YTM of five percent and on maturity then reinvest the proceeds of this bond in another one year bond. So we have ruling over of investment in strategy two. The next year's interest rate offered by one year bond is called as R2 or the rate of return of year two. Then both strategies must provide equal returns. Therefore we can say that the proceeds after year two to any strategy should be equal to like buy and hold two years zero should be equal to roll over one year strategy. This means that the proceeds from either of the strategy should be equal to each other. This one or if we quantify this we would say that the 800 890 into 1.06 square should be equal to 890 into 1.05 into 1.06 square. Now in place of 1.06 square for the second strategy we have an R2 and that R2 is unknown. If we solve this equation for R2 we come to the next year's interest rate equal to 7.01 percent. So that is a rate that would be equalize our equation. One year bond offer a lower YTM that is five percent than the two years bond which is offering six percent. We see that the second technique has a compensating advantage in the sense that it allows rolling over of funds into another shorter period bond next year when rates will be higher that is six percent we see next year's interest rate is higher than the two days interest rate by just enough to make the rolling over one year bond equally attractive as investing in the second year bond and we see in the graph in the first phase we have timeline from zero to one and one to two that is two period timeline in the second phase we have depiction of strategy one that we have a standalone two years zero bond and in the picture three at the bottom of the screen we see our year two strategy in which we have two consecutive one year's investment bonds. In this scenario we need to have a distinction between spot rates and the short rates. Spot rates basically are the rates that are prevailing in today for a given maturity like the two days spot rate two years spot rate is basically the geometric average of the two days and the next year's short rates. So if we to have a short rate today and if we have next year's short rate then we have a geometric average of these two rates the resulting figure will be termed as the spot rate whereas the short rate is the rate that is for a given maturity for example a one year time period at different points in time in our example we see that today's short rate is the five percent and the short rate at the end of year two is seven point zero one. So how short rates are related with the yield curve slopes we see that if r2 is greater than r1 then the average of these two rates is higher than the two days rate this means that y2 or the yield at the maturity of the year two should be greater than the rate of the second year then the yield curve slopes upward and if r2 is lesser than the r1 then the yield curve would be sloping downward means that these yield curve basically reflect the market assessment of the coming periods interest rate we see in this regard that if the r2 is greater than r1 this means that it may indicate the market expects rate to rise in future and if r2 is greater than r1 less than r1 then it may indicate that market expects rates to fall in future what is the difference between then short rates and spot rates if we see this diagram we see that the top line represents the short rates for each year that is five percent r1 r2 seven percent r3 nine point zero two five percent and r4 eleven point zero six percent so these are the short rates the lower lines that we see at y1 five percent y2 six percent y3 seven percent and y4 eight percent these lower lines represent basically these spot rates or equivalently these are the ytms on zero coupon bonds for different holding periods as we see these holding periods are ranging from one year to four years and the conclusion on this discussion is that the yield or spot rate on the long-term bonds reflect the path of short-term rates anticipated by the markets over the bonds life what are the forward rates it is the forecast of a future short rate implied by the market this means that being a break-even interest rate this forward rate equates the return on an n period zero coupon bond to an n minus one period zero coupon bond that is rolled over into a one year bond in year n and in this particular equation this means that if we solve it for the short run short rate in the last period then the 1 plus r in that equation will refer to the forward rate we see that as the future interest rates are uncertain so the inferred or observed rate is the forward rate rather than the future short rate that equation if we repeat again it replaces y with the f so 1 plus fn refers to the forward rate and equivalently this equation can also be written as like 1 plus yn raised to power n and again we have the next expression that shows the forward rate the example we have to compute the forward rate for year 4 if the rate for year 4's maturity is 8% and the rate for third year's maturity is 7% then putting the values into the computation formula of forward rate the resulting figure is 11.06% and that is the forward rate for the year 4 available at right now