 The shape of the yield curve has two major theories to interpret its characteristics These theories are the expectation theory and the liquidity preference theory First is the expectation hypothesis theory We see that the forward rate is the market consensus Expectations of future short interest rate. This means that the forward rate is the Subject of the expected future rate and it also assumes zero liquidity premium The observed long-term rate is a function of today's short term rate and the expected future short term rates if we Express this in equation form we will see that 1 plus y2 raised to part 2 is equal to 1 plus y1 into 1 plus f2 Here 1 here f2 is the forward rate and y2 is the Rate at the end of year 2 or it is the yield at maturity The expectation hypothesis may also conclude then that 1 plus y2 Scare is equal to 1 plus y1 into 1 plus expected return at the end of year 2 so it is proving that the Return forward rate is the function of the expected rate of return for the year to come or the second year The ytm can be determined solely by the current yield which is y1 and the expected rate of return Which is e r2 or the expected return on second year Which it is future single interest rate an upward slope a sloping yield curve may indicate that the investor anticipates and Increases in the interest rate or rising interest rates This theory doesn't is not limited to the Nominal bonds only It can also be applied to the term structure of real interest rates as well It helps in learning the market expectation of coming in inflation rates Second is the liquidity preference theory We see that short-term investors hold long-term investment or long-term bonds if F2 is greater than the Expected return of the year to long-term investors will hold short-term on if the scenario is reversed Means if expected return is greater than the F2 This means that in investors in fact demand high interest rate on bonds with long-term Maturities having greater risk. So they demand risk premium for the Maturity risk whereas short-term investors dominate the market so that the forward rate or F2 will Generally exceed the expected or e r e r 2 Short rates this means that this liquidity premium is predicted to be positive And that is the liquidity premium which this short-term investor is demanding to understand this we have an example where We can see the implications of term structure interest rate theories We see that the short interest rate is expected to be constant indefinitely like if we have R1 equal to 5% then we assume that the R2 or R3 will also be equal to 5% and so on Now under the expectation hypothesis the year to yield to maturity can be derived using the equation that is We multiply the 1.05 Into 1.05 to determine yield at the maturity of two years and we have a value of 1.1025 This means that y2 is equal to 5% so the bonds yield on all the maturities would also be equal to 5% Now under the liquidity preference theory the y2 would exceed e r2 This means that there would be the liquidity premium now assume that liquidity premium is 1% So F2 is 5 plus 1 is equal to 6% then the value of two years bond will be determined using the similar formula that 1 plus r1 into 1 plus F2 and we have the value of r1 equal to 5% and the value of 2 is 6% that we have just computed So the yield at the year 2 would be equal to now 1.13 This means that 1 plus y2 is equal to 1.055 Similarly we can determine the value of F3 which is equal to 6% So the yield on yield three years bond can also be determined in the similar way So now we have again the repeated formula that 1 plus r1 into 1 plus F2 into 1 plus F3 So the yield on year 3 would be equal to 1.05 into 1.06 into 1.06 That is 1.1798 So that is the yield in year 3 So how these yield two maturities are related with the expected rate of return and the forward rates In the left panel we see that we have a constant expected short rate and that is this thick black line This shows that there is a liquidity premium of 1% and that is constant between the thick blue and the thick black line The result is that we have a rising yield curve in this particular case In the second case where we have a declining expected short rates in the form of this thick black line We see that we have an increasing trend in our liquidity premium and that is increasing from this onwards The result is that we have a rising yield curve which shows that the falling expected interest rate and that is the rising yield curve The expected rates are falling in this way In the third case we have a scenario where we have the expected short rates at the declining state and that are declining here We see a constant liquidity premium which is available between the blue and black thick lines The result is that we have a hump shaped yield curve in this particular case And in another case where we see the increasing expected short rates in the form of this rising black line We see an increasing liquidity premium which is available between this thick blue and the thick black line The result is that we see a sharp increase in the yield curve in this particular case The conclusion of this discussion is that if interest rates are expected to change over time The liquidity premium may be used with the expected spot rates to determine the forward rate of interest And the yield to maturity for each rate will be an average of the single period forward rate of interest