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. $5\%;7\%;12\%;12\%$

b.$7\%;5\%;3\%;5\%$

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

Short answer

The steepness of the yield curve along with the slope of the curve will be changed if short-term bonds are preferred over long-term bonds because of the addition of a positive liquidity premium in the interest rate.

Step-by-step solution

Formula

The interest rates in the term structure for maturities recalculated with the following equation:

$i_{nt}=\frac{i_t+{i^e}_{t+1}+{i^e}_{t+2}+...+{i^e}_{\text{t+(n-1})}}{n}$

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:

$$\begin{gathered} One-year=\frac{5\%}{1} \\ =5\% \\ Two-year=\frac{5\%+7\%}{2} \\ =6\% \\ Three-year=\frac{5\%+7\%+12\%}{3} \\ =8\% \\ Four-year=\frac{5\%+7\%+12\%+12\%}{4} \\ =9\% \end{gathered}$$

Explanation (part b) 

Interest rate for four-year maturity:

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

The interest rate is the proportion of the amount borrowed or lent that is due over a specified period of time. The steepness and slope of the yield curve will change if short-term bonds are preferred over long-term bonds due to the addition of a positive liquidity premium in the interest rates of years$2,3and4.$