Answer to Question #238215 in Macroeconomics for Comfort

Question

Assuming that the expectations theory is the correct theory of the term structure. Calculate the interest rates in the term structure for maturites of 1 to 5 years and plot the resulting yield curves for the following series of 1 year interest rates over the next 5 years.

I) 5%, 7%, 7%, 7%, 7%

II) 5%, 4%, 4%, 4%, 4%

Answer 1

Calculate the interest rates in the term structure:

Expectation theory: Expectation theory states that bonds of different maturities are perfect substitutes. This theory explains the relationship among the interest rates on bonds of different maturities.

Note: Here, we are assuming that the expectations theory is the correct theory of the term structure.

Determine the interest rates in the term structure for maturities of one to five years, and plot the resulting yield curves for the following series of one-year interest rates over the next five years.

(a) 5%, 7%, 7%,7%,7%

Interest rates:

$i = \frac{i_t +i_{t+1} + i_{t+2}+...+i_{t+(n-1)}}{n}$

where,

$i_t$ = Year 1 interest rate

$i_{t+1}$ = Year 2 interest rate

$i_{t+(n-1)}$ = Year n interest rate

n = Number of years

For a 1 year bond:

The interest rate for a one-year bond would be 5%.

For a 2 year bond:

Substitute 5% for year 1 interest rate and 7% for year 2 interest rate.

$i_{2t} = \frac{5 \% +7 \%}{2 } \\ = 6 \%$

For a 3 year bond:

Substitute 5% for year 1 interest rate, 7% for year 2 interest rate, and 7% for year 3 interest rate.

$i_{2t} = \frac{5 \% + 7 \% + 7\%}{ 3} \\ = 6.3\%$

For a 4 year bond:

Substitute 5% for year 1 interest rate, 7% for year 2 interest rate, 7% for year 3 interest rate, and 7% for 4^th year interest rate.

$i_{2t} = \frac{5 \% + 7 \% + 7\% + 7\%}{4} \\ = 6.5\%$

For a 5 year bond:

Substitute 5% for year 1 interest rate, 7% for year 2 interest rate, 7% for year 3 interest rate, 7% for year 4 interest rate, and 7% for year 5 interest rates.

$i_{2t} = \frac{5 \% + 7 \% + 7\% + 7\% + 7 \%}{5} \\ = 6.6\%$

Yield curve

If people preferred shorter-term bonds over longer-term bonds, the upward-sloping yield curve would be steeper due to the positive risk premium of longer-term bonds.

(b) 5%, 4%, 4%, 4%, 4%

For a 1 year bond:

The interest rate for a one-year bond would be 5%.

For a 2 year bond:

Substitute 5% for year 1 interest rate and 4% for year 2 interest rate.

$i_{2t} = \frac{5\% +4\%}{2} \\ = 4.5\%$

For a 3 year bond:

Substitute 5% for year 1 interest rate, 4% for year 2 interest rate, and 4% for year 3 interest rate.

$i_{2t}= \frac{5\% +4\% + 4\%}{3} \\ = 4.3\%$

For a 4 year bond:

Substitute 5% for year 1 interest rate, 4% for year 2 interest rate, 4% for year 3 interest rate, and 4% for 4th year interest rate.

$i_{2t}= \frac{5\% +4\% + 4\% + 4\%}{4} \\ = 4.25\%$

For a 5 year bond:

Substitute 5% for year 1 interest rate, 4% for year 2 interest rate, 4% for year 3 interest rate, 4% for year 4 interest rate, and 4% for 5th year interest rate.

$i_{2t}= \frac{5\% +4\% + 4\% + 4\% +4 \%}{5} \\ = 4.2\%$

Yield curve:

If people prefer shorter-term bonds over longer-term bonds, the downward sloping yield curve would be less steep and slightly positive upward due to the positive risk premium of longer-term bonds.