Question

Assuming the expectations theory is the correct theory of the term structure, calculate the interest rates in the term structure for maturities of one to four years, and plot the resulting yield curves for the following paths of one-year interest rates over the next four years:

a. 4%, 6%, 11%, 15%

b. 3%, 5%, 13%, 15%

How would your yield curves change if people preferred shorter-term bonds to longer-term bonds?

Short answer

If short-term bonds are preferred over long-term bonds due to the addition of a positive liquidity premium in the interest, the steepness and slope of the yield curve will change.

Step-by-step solution

Given information

The expectation theory of the term structure states that the interest rate on a long-term bond will equal an average of the short-term interest rates expected to occur over the life of the long-term bond. The interest rates in the term structure for maturities recalculated with the following equation:

$i_{nt}=\frac{i_t+{i^e}_{t+1}+...+{i^e}_{t+(n-1)}}{n}$

i_t is today's interest rate on a one-period one, ${i^e}_{t+1}$is the interest rate on a one-period bond expected for the next period, $i_{2t}$is today's interest rate on the two-period bond expected for the next period, and so forth.

Explanation (part a)

Using the above equation, we are able to calculate the interest rates in the term structure for maturities of one to four years as follows:

$$\begin{gathered} One-yaer=\frac{4\%}{1} \\ =4\% \\ Two-year=\frac{4\%+6\%}{2} \\ =5\% \\ Three-year=\frac{4\%+6\%+11\%}{3} \\ =7\% \\ Four-year=\frac{4\%+6\%+11\%+15\%}{4} \\ =9\% \end{gathered}$$

Representation of yield curve:

Explanation (part b) 

Interest rate for four-year maturity:

$$\begin{gathered} \text{One-year=7\%/1} \\ =7\% \\ Two-year=7\%+5\%/2 \\ =6\% \\ Three-year=7\%+5\%+3\%/3 \\ =5\% \\ Four-year=7\%+5\%+3\%+5\%/4 \\ =5\% \end{gathered}$$

Representation of yield curve:

If short-term bonds are preferred over long-term bonds due to the inclusion of a positive liquidity premium in the interest rates of years $2,3and4$the steepness and slope of the yield curve will be altered.