Chapter 15, Problem 9PS

Summary Introduction

To determine: The market’s expectation of the yield curve as per the expectations hypothesis from now to 1 year and further calculate the expected values of next year yields on bonds after one year, two years and three years.

Introduction:

Yield-curve analysis: It is a technique to calculate the difference in interest rate between the note value and the term of to maturity.

Answer to Problem 9PS

The forward rate after one year is 7%; after two years it is 7.5% and after three years it is 8.51%.

Explanation of Solution

We have been given the following information:

Bond	Years to Maturity	YTM (%)
A	1	5
B	2	6
C	3	6.5
D	4	7

We have to calculate the expected rate. Expected rate is called as expected hypothesis. The following formula can be used for calculation.

  $1+Forwardrat{e}_{Expectedreturn}=\left(\frac{{\left(1+Yieldtomaturit{y}_{Numberofyears}\right)}^{Numberofyears}}{{\left(1+Yieldtomaturit{y}_{Numberofyears-1}\right)}^{Numberofyears-1}}\right)$

Calculation of expected rate after one year:

  $\begin{array}{l}Forwardrat{e}_{1year}=\left(\frac{{\left(1+Yieldtomaturit{y}_{Numberofyears}\right)}^{Numberofyears}}{{\left(1+Yieldtomaturit{y}_{Numberofyears-1}\right)}^{Numberofyears-1}}\right)-1\\ =\left(\frac{{\left(1+Yieldtomaturit{y}_2\right)}^2}{{\left(1+Yieldtomaturit{y}_1\right)}^1}\right)-1\\ =\left(\frac{{\left(1+6\%\right)}^2}{{\left(1+5\%\right)}^1}\right)-1\\ =\left(\frac{{\left(1+0.06\right)}^2}{{\left(1+0.05\right)}^1}\right)-1\\ =\left(\frac{{\left(1.06\right)}^2}{{\left(1.05\right)}^1}\right)-1\\ =\frac{1.1236}{1.05}-1\\ =1.0700-1\\ =0.0700\end{array}$

When converted into percentages, it becomes 7%.

Calculation of expected rate after two years

  $\begin{array}{l}Forwardrat{e}_{2years}=\left(\frac{{\left(1+Yieldtomaturit{y}_{Numberofyears}\right)}^{Numberofyears}}{{\left(1+Yieldtomaturit{y}_{Numberofyears-1}\right)}^{Numberofyears-1}}\right)-1\\ =\left(\frac{{\left(1+Yieldtomaturit{y}_3\right)}^3}{{\left(1+Yieldtomaturit{y}_2\right)}^2}\right)-1\\ =\left(\frac{{\left(1+6.5\%\right)}^3}{{\left(1+6\right)}^2}\right)-1\\ =\left(\frac{{\left(1+0.065\right)}^3}{{\left(1+0.06\right)}^2}\right)-1\\ =\left(\frac{{\left(1.065\right)}^3}{{\left(1.06\right)}^2}\right)-1\\ =\frac{1.2079}{1.1236}-1\\ =1.0750-1\\ =0.0750\end{array}$

When converted into percentages, it becomes 7.5%.

Calculation of expected rate after 3 years:

The percentages have to be converted into decimals by dividing it by 100.

  $\begin{array}{l}=\left(\frac{{\left(1+0.07\right)}^4}{{\left(1+0.065\right)}^3}\right)-1\\ =\left(\frac{{\left(1.07\right)}^4}{{\left(1.065\right)}^3}\right)-1\\ =\frac{1.31079601}{1.20794963}-1\\ =1.0851-1\\ =0.0851\end{array}$

When converted into percentages, it becomes 8.51%.

Therefore, the forward rate after one year is 7%; after two years it is 7.5% and after three years it is 8.51%.