Chapter 2, Problem 14P

Summary Introduction

To determine: The annual rates from year 1 to year 4.

Answer to Problem 14P

The current rate of one-year is 5.65%, two-year is 6.22%, three-year is 6.47% and four-year treasury security is 6.67%.

Explanation of Solution

Determine the current rate of one-year and two-year treasury security

The current rate of one-year treasury security is already given, which is 5.65%.

$\begin{array}{c}CurrentRat{e}_{2-Year}=\left[{\left(\left(1+Rat{e}_{Year1}\right)\times \left(\begin{array}{l}1+ExpectedRat{e}_{Year2}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year2}\end{array}\right)\right)}^{\left(\frac{1}{n}\right)}-1\right]\\ =\left[{\left(\left(1+5.65\%\right)\times \left(1+6.75\%+0.05\%\right)\right)}^{\left(\frac{1}{2}\right)}-1\right]\\ =\left[1.062234-1\right]\\ =0.062234or6.22\%\end{array}$

Therefore, the current rate of two-year treasury security is 6.22%.

Determine the current rate of three-year treasury security

$\begin{array}{c}CurrentRat{e}_{3-Year}=\left[{\left(\begin{array}{l}\left(1+Rat{e}_{Year1}\right)\times \left(\begin{array}{l}1+ExpectedRat{e}_{Year2}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year2}\end{array}\right)\times \\ \left(\begin{array}{l}1+ExpectedRat{e}_{Year3}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year3}\end{array}\right)\end{array}\right)}^{\left(\frac{1}{n}\right)}-1\right]\\ =\left[{\left(\left(1+5.65\%\right)\times \left(1+6.75\%+0.05\%\right)\times \left(1+6.85\%+0.10\%\right)\right)}^{\left(\frac{1}{3}\right)}-1\right]\\ =\left[1.064651-1\right]\\ =0.064651or6.47\%\end{array}$

Therefore, the current rate of three-year treasury security is 6.47%.

Determine the current rate of four-year treasury security

$\begin{array}{c}CurrentRat{e}_{3-Year}=\left[{\left(\begin{array}{l}\left(1+Rat{e}_{Year1}\right)\times \left(\begin{array}{l}1+ExpectedRat{e}_{Year2}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year2}\end{array}\right)\times \\ \left(\begin{array}{l}1+ExpectedRat{e}_{Year3}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year3}\end{array}\right)+\left(\begin{array}{l}1+ExpectedRat{e}_{Year4}+\\ Liquidity\mathrm{Pr}emiu{m}_{Year4}\end{array}\right)\end{array}\right)}^{\left(\frac{1}{n}\right)}-1\right]\\ =\left[{\left(\begin{array}{l}\left(1+5.65\%\right)\times \left(1+6.75\%+0.05\%\right)\times \left(1+6.85\%+0.10\%\right)\\ \times \left(1+7.15\%+0.12\%\right)\end{array}\right)}^{\left(\frac{1}{4}\right)}-1\right]\\ =\left[1.066657-1\right]\\ =0.066657or6.67\%\end{array}$

Therefore, the current rate of four-year treasury security is 6.67%.

Summary Introduction

To determine: The yield curve for the current rate.

Explanation of Solution

The yield curve is as follows: