Chapter 23, Problem 1PS

Expected yield You own a 5% bond maturing in two years and priced at 87%. Suppose that there is a 10% chance that at maturity the bond will default and you will receive only 40% of the promised payment. What is the bond’s promised yield to maturity? What is its expected yield (i.e., the possible yields weighted by their probabilities)?

Summary Introduction

To determine: The bonds promised yield to maturity and expected yield.

Yield to maturity (YTM) is the overall return anticipated on a bond throughout its maturity period and it is considered as a long-term bond yield and represented as an annual rate.

Explanation of Solution

Computation of bonds promised yield to maturity and expected yield is as follows:

$\begin{array}{c}\text{Price}=\frac{{\text{CF}}_{\text{1}}}{\left(\text{1}+\text{IRR}\right)}+\frac{{\text{CF}}_{\text{2}}}{{\left(\text{1}+\text{IRR}\right)}^2}\\ \$870=\frac{\$50}{\left(\text{1}+\text{IRR}\right)}+\frac{\$1,050}{{\left(\text{1}+\text{IRR}\right)}^2}\\ \text{By solving,}\\ \text{IRR}=\text{0}\text{.1277}\text{or}\text{12}\text{.77\%}\end{array}$

Therefore, to ascertain the expected yield, the expected pay off at maturity is needed and given a 10% probability that only 40% of the promised payment will be received.

$\begin{array}{c}\text{Expected pay off}=\left(0.90\times \$1,050\right)+0.10\left(0.40\times \$1,050\right)\\ =\$945+\$42\\ =\$987\end{array}$

Therefore, the expected yield is as follows:

$\begin{array}{c}\text{Price}=\frac{{\text{CF}}_{\text{1}}}{\left(\text{1}+\text{IRR}\right)}+\frac{{\text{CF}}_{\text{2}}}{{\left(\text{1}+\text{IRR}\right)}^2}\\ \$870=\frac{\$50}{\left(\text{1}+\text{IRR}\right)}+\frac{\$987}{{\left(\text{1}+\text{IRR}\right)}^2}\\ \text{By solving,}\\ \text{IRR}=\text{0}\text{.0942}\text{or}9.\text{42\%}\end{array}$

Therefore, the bonds promised yield to maturity and expected yield is 12.77% and 9.42% respectively.