nominal & real interest rates + term structure of interest rate

The rates of US Treasury securities at the beginning of August was 0.94% annual return on bonds with a 3 year maturity and 0.47% with a two year maturity. What is the implied forward rate for i21?

(1+ i03)3 / (1+i02)2 = (1+ i21)(1.0094)3 / (1.0047)2 = 1.02847/1.009422 = (1+ i21) = 1.01886 (or 1.8%).

1. Using a supply and demand function, explain the logic of the Fisher equation. Assume that nominal and real rate of interest equal 5% when the expected inflation rate is zero. The average lender and borrower now expect inflation to be 3%.

1The Fisher equation is (1+i) = (1+r)(1+ π) which is approximately given by i = r +πe where πe is the expected inflation rate. The implication of this equation is that an increase in the expected inflation rate will lead to an increase in the nominal interest rate ( we are treating r as exogenous, that is, as a constant, in this equation). Lenders point of view: The original supply curve tells us that some lender was willing to supply the qth unit of funds if he or she received at least a real return of 5%. If this "average lender" now expects a 3% inflation rate, then to still be willing to provide this qth unit of funds would require a NOMINAL return of 8%, since 8% adjusted for inflation would be 8-3 =5%. This argument could be made at any unit of funds, so the upshot is that it causes the supply curve to move VERTICALLY up by the change in the expected inflation rate. Borrowers point of view. The original demand curve tells us that some borrower was willing to pay up to a real rate of interest of 5% to borrow the qth unit of funds. If this "average borrower" now expects a 3% inflation rate, then to say this borrower is willing to pay up to 5% in real terms is the equivalent of saying this person is willing to pay up 8% in NOMINAL terms since 8-3 is 5% in real terms. This argument could be made at any unit of funds, so the conclusion is that it causes the demand curve to move VERTICALLY up by the change in the expected inflation rate. In combination, these two shifts result in exactly a three percentage point increase in the nominal rate of interest, just as the Fisher equation predicts.

Assume that the Expectations Theory is correct. How might it be useful for helping us predict annual inflation rates for future years?

By the Fisher equation we know that i = r + π . So if the real rate is roughly constant and with (a) an estimate of r based on recent data (realized r = i - actual inflation), (b) then we can deduce i11 from the Expectations Theory equation, (c) plug that estimate along with our estimate of r into i11 = r + πe and solve for the expected inflation rate.

Compare a one-year zero-coupon bond of 10,000 with a perpetuity of 1,000 per year. The current interest rate (YTM) is 10%. Find price. Now suppose the interest rate rises to 20%. Compare prices. What does this outcome suggest about the Expectations theory?

P = $9090.9 and P = 1000/0.10 = $10,000; now P = 10,000/1.2 = 8333.33 (8.3 % decline in price) P = 1000/.20 = 5000 (50% decrease in price)

What is meant by the "term structure of interest rates"? What does the Expectations Theory of the term structure assume and based on these assumptions explain the logic of this equilibrium condition:(1+ i02)2 = (1+i01)(1+ i11)

This refers to the relationship between short-term and long-term interest rates on bonds of similar default risk. According to the Expectations Theory, a two-year bond purchased today a perfect substitute for purchasing one one-year bond today and another one-year bond one year from today (assuming similar default risk, same tax treatment etc). This can of course be generalized to 3years, 4 years etc. Suppose (1+ i02)2 > (1+i01)(1+ i11). Since the average annual yield is greater holding one two-year bond rather than two consecutive one-year bonds, the demand for 2-year bonds will rise (supply of LF in the 2-year loan market will rise) and the demand for 1-year bonds will decline (supply of LF in the 1-year loan market will fall). The increase in the supply of LF in the two-year loan market will drive down i02 and the decrease in the supply of LF in the 1-year loan market will cause i01 to be bid up. This process occurs until i02 again equals (i01 + i11 )/2 .

2. Using the Fisher equation, evaluate the following claim: "Present value calculations ignore inflation in that future dollar amounts are likely to have a lower purchasing power due to an increase in the price level."

This statement is not correct. The interest rate would use to take the PV of the future nominal dollar amounts is the nominal rate of interest, i. But the nominal rate, in turn, is equal to the real rate plus the expected inflation rate. So, for example, the PV of some amount one year from today (V1) is V1/(1+i) which by substitution can be written as V1/[(1+r)(1+ π)].

Suppose the yield curve has an upward slope. Under the Liquidity Premium Theory, what can we conclude about the public's interest rate expectations? Provide numerical examples (the use approx. version and assume 2 periods).

Without knowing the risk premium we cannot conclude anything with certainty. For example, supposei02 =10%, i01 =9%,and i11 =7%.IFtheriskpremiumis2%thenthiswouldbean equilibrium under LPT. 10 = (9+7)/2 + 2. Yet, even though the yield curve slopes upwards, it is thecasethepublicexpectsSTratestodecline(noticethati01>i11).Anotherexample,supposei02 = 10% , i01 = 9% , and i11 = 10%. IF the risk premium is 0.5% then this would be an equilibrium under LPT. 10 = (9+10)/2 + 0.5. In is the case the public expects ST rates to rise (notice that i01 <i11). Athirdexample,supposei02 =10%, i01 =9%,and i11 =9%.IFtheriskpremiumis1% then this would be an equilibrium under LPT. 10 = (9+9)/2 + 1. In is the case the public expects ST rates to remain unchanged (notice that i01 = i11). As the above examples indicate, when the yield curve is upwards slopping, we are unable to determine the relationship between i01 and i11 without knowing the risk premium.