Chapter 10, Problem 40PS

Summary Introduction

(a)

Calculate the forward rate of zero-coupon bond with 11% yield to maturity and 2 years maturity.

Answer to Problem 40PS

The forward rates obtained for 2^nd year is 12.01% and for 3^rd year is 14.035, the yield to maturity for 3^rd year is 13.02% and the expected return for three year bond is 10%.

Explanation of Solution

Given Information:

Maturity(years)	Y TM
1	10%
2	11%
3	12%

Implied forward rates refer to the expected future movement of the interest rates by the market.

the rate of return expected on a bond that is held until its maturity is known as yield to maturity. It is fundamentally IRR (Internal Rate of Return) on the bond.

Yield to maturity of the bond can be calculated using the following formula:

The following information goes with all the parts of this question:

Table showing the current yield curve for zero-coupon bond:

Maturity(years)	Y TM
1	10%
2	11%
3	12%

  $\begin{array}{l}\text{Forward}\text{rate=}\frac{{\left(\text{1+YTM}\text{for}\text{current}\text{year}\right)}^{\text{Correspondingy}\text{year}}}{{\left(\text{1+YTM}\text{for}\text{current}\text{year}\right)}^{\text{Correspondingy}\text{year}}}\text{-1}\\ \text{=}\frac{{\left(\text{1+0}\text{.11}\right)}^{\text{2}}}{\left(\text{1+0}\text{.10}\right)}\text{-1}\\ \text{=}\frac{\text{1}\text{.23}}{\text{1}\text{.1}}\\ \text{=12}\text{.01\%}\end{array}$

Calculate the forward rate of zero-coupon bond with 12% yield to maturity and 3 years maturity

  $\begin{array}{l}\text{Forward}\text{rate=}\frac{{\left(\text{1+YTM}\text{for}\text{current}\text{year}\right)}^{\text{Correspondingy}\text{year}}}{{\left(\text{1+YTM}\text{for}\text{current}\text{year}\right)}^{\text{Correspondingy}\text{year}}}\text{-1}\\ \text{=}\frac{{\left(\text{1+0}\text{.12}\right)}^{\text{3}}}{{\left(\text{1+0}\text{.12}\right)}^{{}^{\text{2}}}}\text{-1}\\ \text{=}\frac{\text{1}\text{.40}}{\text{1}\text{.23}}\text{-1}\\ \text{=14}\text{.03\%}\end{array}$

Thus, the forward rates obtained for 2^nd year is 12.01% and for 3^rd year is 14.035

Summary Introduction

(b)

Calculate the YTM of the zero coupon bond 2^nd years.

Answer to Problem 40PS

Theyield to maturity for 3^rd year is 13.02%

Explanation of Solution

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Using RATE function in excel, Calculate the effective yield to maturity.

Enter the corresponding value in the field as below:

Calculate the YTM of the zero coupon bond 3r^d years.

$\begin{array}{l}\mathrm{Pr}ice=\frac{FV}{{\left(1+Forwardrate\right)}_1\times {\left(1+Forwardrate\right)}_2}\\ =\frac{\$1,000}{\left(1+0.1201\right)\times \left(1+0.1403\right)}\\ =\frac{\$1,000}{1.28}\\ =\$782.93\\ PV={\displaystyle \sum _{t=1}^n\frac{PMT}{{\left(1+r\right)}^1}}+\frac{FV}{{\left(1+r\right)}^n}\\ \$782.93={\displaystyle \sum _{t=1}^2\frac{\$0}{{\left(1+r\right)}^1}}+\frac{\$1,000}{{\left(1+r\right)}^2}\end{array}$

Using RATE function in excel, Calculate the effective yield to maturity.

Enter the corresponding value in the field as below:

Hence, the yield to maturity for 3^rd year is 13.02%.

Thus, from the calculate it can be seen that the yieldcurve is increasing and upward sloping, and it implies that according to hypothesis expectationthere is a shift upward in the curve of the next year.

Summary Introduction

(c)

Calculate the price of the bond with 10% yield to maturity.

Answer to Problem 40PS

The expected return for three year bond is 10%.

Explanation of Solution

$\begin{array}{l}\mathrm{Pr}ice=\frac{FV}{{\left(1+r\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.10\right)}^1}\\ =\$909.09\end{array}$

Calculate the price of the bond with 12% yield to maturity.

$\begin{array}{l}\mathrm{Pr}ice=\frac{FV}{{\left(1+r\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.11\right)}^2}\\ =\frac{\$1,000}{1.23}\\ =\$811.62\end{array}$

Calculate the price of the bond with 13% yield to maturity.

$\begin{array}{l}\mathrm{Pr}ice=\frac{FV}{{\left(1+r\right)}^n}\\ =\frac{\$1,000}{{\left(1+0.12\right)}^3}\\ =\frac{\$1,000}{1.40}\\ =\$711.78\end{array}$

Calculate the total expected rate of return.

In the next year, the two year zero bond will be the one year zero bond, and thus it will sell for

$\begin{array}{l}\mathrm{Pr}ice=\frac{FutureValue}{\left(1+r\right)}\\ =\frac{\$1,000}{\left(1+0.1201\right)}\\ \\ =\$892.78\end{array}$

In a similar manner the three year zero bond will become a two year bond and thus it will sell for $782.93 as calculate in part (b).

Expected return for two year bond is as below:

$\begin{array}{l}Rateofreturn=\frac{\$892.78}{\$811.62}-1\\ =1.1000-1\\ =10.00\%\end{array}$

Thus, expected return for two year bond is 10%

Expected return for three year bond is as below:

$\begin{array}{l}Rateofreturn=\frac{\$782.93}{\$711.78}-1\\ =1.1000-1\\ =10.00\%\end{array}$

Thus, expected return for three year bond is 10%.