Chapter 10, Problem 40PS

Summary Introduction

(a)

To calculate:

The implied one-year forward rates when yield to maturity for$1^{st},2^{nd}{\text{and 3}}^{rd}$year is$10\%,11\%\text{ and }12\%$respectively.

Introduction:

The implied forward rate is the rate which helps in determining the movements of the interest rate.

The yield to maturity is the rate which provides the amount of expected return on a bond held till its maturity.

Answer to Problem 40PS

The implied one-year forward rate for$2^{nd}$year is$12.01\%$and for$3^{rd}$year is$14.03\%$

Given:

The yield to maturity for$1^{st}$year is$10\%$, for$2^{nd}$year is$11\%$and for${\text{3}}^{rd}$year is$12\%$.

Explanation:

The formula for computing forward rate as follows:

  $\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1$

For computing forward rate of zero-coupon bond with$11\%$yield to maturity and maturity of$2$years:

  $\begin{array}{c}\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1\\ =\frac{{\left(1+0.11\right)}^2}{\left(1+0.10\right)}-1\\ =12.01\%\end{array}$

For computing forward rate of zero-coupon bond with$12\%$yield to maturity and maturity of$3$years:

  $\begin{array}{c}\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1\\ =\frac{{\left(1+0.12\right)}^3}{{\left(1+0.11\right)}^2}-1\\ =14.03\%\end{array}$

Thus, the implied one-year forward rate for$2^{nd}$year is$12.01\%$and for$3^{rd}$year is$14.03\%$

Explanation of Solution

Given:

The yield to maturity for$1^{st}$year is$10\%$, for$2^{nd}$year is$11\%$and for${\text{3}}^{rd}$year is$12\%$.

The formula for computing forward rate as follows:

  $\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1$

For computing forward rate of zero-coupon bond with$11\%$yield to maturity and maturity of$2$years:

  $\begin{array}{c}\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1\\ =\frac{{\left(1+0.11\right)}^2}{\left(1+0.10\right)}-1\\ =12.01\%\end{array}$

For computing forward rate of zero-coupon bond with$12\%$yield to maturity and maturity of$3$years:

  $\begin{array}{c}\text{Forward Rate}=\frac{{\left(1+\text{YTM for current year}\right)}^{\text{Corresponding year}}}{{\left(1+\text{YTM for previous year}\right)}^{\text{Corresponding year}}}-1\\ =\frac{{\left(1+0.12\right)}^3}{{\left(1+0.11\right)}^2}-1\\ =14.03\%\end{array}$

Thus, the implied one-year forward rate for$2^{nd}$year is$12.01\%$and for$3^{rd}$year is$14.03\%$

Summary Introduction

(b)

To calculate:

The yield to maturity of a zero-coupon bond having maturity of one year for the next year i.e.$2^{nd}$year if the expectations hypothesis is assumed to be correct and expectation of market is accurate.

Introduction:

The implied forward rate is the rate which helps in determining the movements of the interest rate.

The yield to maturity is the rate which provides the amount of expected return on a bond held till its maturity.

Answer to Problem 40PS

The yield to maturity of zero-coupon bond for$2^{nd}$year is$12.01\%$

Explanation:

The formula for computing yield to maturity as follows:

  $\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}$

Here,

PV is Current price of the bond

FV is Face value of the bond

  $r$is yield to maturity

  $n$is no. of periods

For computing Price of the bond

  $\begin{array}{c}\mathrm{Price}\text{ of the Bond}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)}\\ =\$892.78\end{array}$

For computing yield to maturity:

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}\\ \$892.78={\displaystyle \sum _{t=1}^1}\frac{\$0}{{(1+r)}^1}+\frac{\$1,000}{{(1+r)}^1}\end{array}$

Using function in excel, effective yield is calculated.

Enter the data as shown below:

Thus, the yield to maturity of zero-coupon bond for$2^{nd}$year is$12.01\%$

Explanation of Solution

The formula for computing yield to maturity as follows:

  $\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}$

Here,

PV is Current price of the bond

FV is Face value of the bond

  $r$is yield to maturity

  $n$is no. of periods

For computing Price of the bond

  $\begin{array}{c}\mathrm{Price}\text{ of the Bond}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)}\\ =\$892.78\end{array}$

For computing yield to maturity:

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}\\ \$892.78={\displaystyle \sum _{t=1}^1}\frac{\$0}{{(1+r)}^1}+\frac{\$1,000}{{(1+r)}^1}\end{array}$

Using function in excel, effective yield is calculated.

Enter the data as shown below:

Thus, the yield to maturity of zero-coupon bond for$2^{nd}$year is$12.01\%$

Summary Introduction

(c)

To calculate:

The yield to maturity of a zero-coupon bond having maturity of two year for the next year i.e.$3^{rd}$year if the expectations hypothesis is assumed to be correct and expectation of market is accurate.

Introduction:

The implied forward rate is the rate which helps in determining the movements of the interest rate.

The yield to maturity is the rate which provides the amount of expected return on a bond held till its maturity.

Answer to Problem 40PS

The yield to maturity of zero-coupon bond for$3^{rd}$year is$13.02\%$

Explanation:

The formula for computing yield to maturity as follows:

  $\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}$

Here,

PV is Current price of the bond

FV is Face value of the bond

  $r$is yield to maturity

  $n$is no. of periods

For computing Price of the bond

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)\left(1+0.1403\right)}\\ =\frac{\$1,000}{1.28}\\ =\$782.93\end{array}$

For computing yield to maturity:

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}\\ \$782.93={\displaystyle \sum _{t=1}^2}\frac{\$0}{{(1+r)}^1}+\frac{\$1,000}{{(1+r)}^2}\end{array}$

Using function in excel, effective yield is calculated.

Enter the data as shown below:

Thus, the yield to maturity of zero-coupon bond for$3^{rd}$year is$13.02\%$

Explanation of Solution

The formula for computing yield to maturity as follows:

  $\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}$

Here,

PV is Current price of the bond

FV is Face value of the bond

  $r$is yield to maturity

  $n$is no. of periods

For computing Price of the bond

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)\left(1+0.1403\right)}\\ =\frac{\$1,000}{1.28}\\ =\$782.93\end{array}$

For computing yield to maturity:

  $\begin{array}{c}\text{PV}={\displaystyle \sum _{t=1}^n}\frac{\text{PMT}}{{(1+r)}^1}+\frac{\text{FV}}{{(1+r)}^n}\\ \$782.93={\displaystyle \sum _{t=1}^2}\frac{\$0}{{(1+r)}^1}+\frac{\$1,000}{{(1+r)}^2}\end{array}$

Using function in excel, effective yield is calculated.

Enter the data as shown below:

Thus, the yield to maturity of zero-coupon bond for$3^{rd}$year is$13.02\%$

Summary Introduction

(d)

To calculate:

The expected total rate of return over the next year when purchase of a two-year zero-coupon bond on current day.

Introduction:

The expected rate of return is used to determine the total expected return on the bond for the holding period.

The yield to maturity is the rate which provides the amount of expected return on a bond held till its maturity.

Answer to Problem 40PS

The expected rate of return for two year bond is$10\%$.

Explanation:

The formula for computing price of the bond as follows:

  $\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}$

For computing price of the bond with$10\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.10\right)}^1}\\ =\$909.09\end{array}$

For computing price of the bond with$11\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.11\right)}^2}\\ =\$811.62\end{array}$

Price at which the bond will sell in the next year:

  $\begin{array}{c}\text{Price}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1}\\ =\frac{\$1,000}{\left(1+0.1201\right)}\\ =\$892.78\end{array}$

The expected rate of return for two-year bond is as follows:

  $\begin{array}{c}\text{Rate of Return}=\frac{\$892.78}{\$811.62}-1\\ =10\%\end{array}$

Thus, the expected rate of return for two-year bond is$10\%$.

Explanation of Solution

The formula for computing price of the bond as follows:

  $\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}$

For computing price of the bond with$10\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.10\right)}^1}\\ =\$909.09\end{array}$

For computing price of the bond with$11\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.11\right)}^2}\\ =\$811.62\end{array}$

Price at which the bond will sell in the next year:

  $\begin{array}{c}\text{Price}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1}\\ =\frac{\$1,000}{\left(1+0.1201\right)}\\ =\$892.78\end{array}$

The expected rate of return for two-year bond is as follows:

  $\begin{array}{c}\text{Rate of Return}=\frac{\$892.78}{\$811.62}-1\\ =10\%\end{array}$

Thus, the expected rate of return for two-year bond is$10\%$.

Summary Introduction

(e)

To calculate:

The expected total rate of return over the next year when purchase of a three-year zero-coupon bond on current day.

Introduction:

The expected rate of return is used to determine the total expected return on the bond for the holding period.

The yield to maturity is the rate which provides the amount of expected return on a bond held till its maturity.

Answer to Problem 40PS

The expected rate of return for three-year bond is$10\%$.

Explanation:

The formula for computing price of the bond as follows:

  $\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}$

For computing price of the bond with$12\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.12\right)}^3}\\ =\$711.78\end{array}$

Price at which the bond will sell in the next year:

  $\begin{array}{c}\text{Price}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)\left(1+0.1403\right)}\\ =\frac{\$1,000}{1.28}\\ =\$782.93\end{array}$

The expected rate of return for two-year bond is as follows:

  $\begin{array}{c}\text{Rate of Return}=\frac{\$782.93}{\$711.78}-1\\ =10\%\end{array}$

Thus, the expected rate of return for three-year bond is$10\%$.

Explanation of Solution

The formula for computing price of the bond as follows:

  $\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}$

For computing price of the bond with$12\%$yield to maturity:

  $\begin{array}{c}\text{Price of the Bond}=\frac{\text{FV}}{{\left(1+\text{r}\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.12\right)}^3}\\ =\$711.78\end{array}$

Price at which the bond will sell in the next year:

  $\begin{array}{c}\text{Price}=\frac{\text{FV}}{{\left(1+\text{Forward rate}\right)}_1{\left(1+\text{Forward rate}\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)\left(1+0.1403\right)}\\ =\frac{\$1,000}{1.28}\\ =\$782.93\end{array}$

The expected rate of return for two-year bond is as follows:

  $\begin{array}{c}\text{Rate of Return}=\frac{\$782.93}{\$711.78}-1\\ =10\%\end{array}$

Thus, the expected rate of return for three-year bond is$10\%$.