Question

Consider the following term structure: Term Yield 1 1.5% 2 2.3% 3 3.5% 4 3.7% Compute the implied forward rate on a one-year security 1 year from now and 2 years from now. What is the economic interpretation of these rates according to the pure expectations theory? …according to the liquidity preference (modified expectations) theory? Suppose that you believe that the actual future one-year rates will be greater than the implied forward rates. How would you alter your desired borrowing pattern to take advantage of your forecast?

Answer 1

Let us assume that the given term structure of yields is given on a yearly basis. This essentially implies that yield for Year 1 is 1.5%, for Year 2 is 2.3% and so on.

Let the 1 year forward rate for a period of one year be denoted by f1

Then, (1+0.015) x (1+f1) = (1.023)^(2)

1+ f1 = {(1.023)^(2)}/(1.015) = 1.031063

f1= 0.031063 or 3.1063 %

Let the 2 year forward rate for a period of on year be denoted by f2

Then, (1.023)^(2) x (1+f2) = (1.035)^(3)

1+f2 = [(1.035)^(3)/(1.023)^(2)]

1+f2 = 1.059424

f2 = 0.059424 or 5.942% per annum

The pure expectations theory states that the term structure of yield or long-term interest rates holds a forecast for short-term interest rates. In simpler term, it essentially implies that if an investor invests in a one year bond, followed by rolling over the one-year investment proceeds into another annual bond, the overall investment returns would be equivalent to that made by investing directly in a two-year bond.

In the context of the example given above, if an investor makes an investment of $1 at the 1-year rate of 1.15 %, the investment proceeds should be 1 x (1.015) = $ 1.015 after one year. This, when rolled over into another year-long investment at the 1-year forward rate of f1= 3.1063 %, would give proceeds of 1.015 x 1.031063 = $ 1.04653.

The same investment proceeds can be made by investing directly in a two year zero coupon bond so as to yield 1 x (1.023)^(2) = $ 1.046529 or approximately $ 1.04653 at the end of Year 2.

The liquidity preference theory states that current long-term interest rates are in fact a geometric average of the expected short-term rates plus a liquidity premium demanded by an investor for holding long-term securities instead of more liquid short-term less risky securities.The liquidity premium is demanded as investors feel that long-term securities are less liquid and hence riskier. Any riskier investment would lead to investors demanding a premium and that is what happens in real life as well.

If the actual future rates are higher than the implied forward rate then the same provides an arbitrage opportunity in the following way:

- Borrow a sum of say $ 100 for a period of two years. This implies that one needs to payback 100 x 1.023 x 1.023 = $ 104.6529

- Take the $ 100 and invest in a one-year security at 1.5% to yield 100 x (1.015) = $101.5 after one year. Now, this should be invested for another year at the one-year interest-rate existing then (the one year implied forward rate one year from now). However, the actual future rate at that point in time is higher than the implied one-year forward rate of 3.1063%. Let this rate be 3.3% instead of the predicted forward rate of 3.1063 %. Then the one year proceeds of $ 101.5 become 101.5 x (1.033) = $ 104.8495

- The total investment proceeds are $104.8495 whereas the outstanding debt is $ 104.6529, thereby ensuring a riskless arbitrage profit of 104.8495 - 104.6529 = $ 0.1966