Chapter D, Problem 81P

To determine

The yield of the bond if held to maturity is in what range.

Answer to Problem 81P

Therefore the correct option is $\left(d\right)\text{Greater}\text{than}10\%$.

Explanation of Solution

Given:

Face value is $\$10,000$.

Bond interest rate is $4\%$.

Maturity time is $7\text{years}$.

Bond is bold at $\$6,950$.

Calculation:

Calculate the interest rate in 6 months.

$\begin{array}{c}\text{Interest rate in 6 months}=\frac{4\%}{2}\\ =2\%\end{array}$

Calculate the amount received after 6 months.

$\begin{array}{c}\text{Amount received after 6 months}=2\%\text{of}10,000\\ =\frac{2}{100}\times 10,000\\ =\$200\end{array}$

Write the expression to calculate the present worth of the bond.

$PW=-P+A\left(P/A,i,n\right)+F\left(P/F,i,n\right)$ ...... (I)

Here, current price of the bond is $P$, price of the bond is $F$, interest payment of the bond is $A$, interest rate is $i$, and time period is $n$.

Substitute, $\$6,950$ for $P$, $\$10,000$ for $\$10,000$, $\$200$ for $A$, $14$ for time period in Equation (I).

$\begin{array}{l}PW=-\$6,950+\$200\left(P/A,i,14\right)+\$10,000\left(P/F,i,14\right)\\ 0=-\$6,950+\$200\left(P/A,i,14\right)+\$10,000\left(P/F,i,14\right)\end{array}$

Apply trial and error method and determine the interest rate.

Interest rate	$\left(P/A,i,14\right)$	$\left(P/F,i,n\right)$	Value
$5\%$	$9.899$	$0.5051$	$80.8$
$6\%$	$9.295$	$0.4423$	$-668$

Interpolate the interest rate between $5\%$ and $6\%$.

$IRR=i_{\text{low}}+\left(i_{\text{high}}-i_{\text{low}}\right)\left[\frac{F_{\text{low}}-P}{F_{\text{low}}-F_{\text{high}}}\right]$ ....... (II)

Substitute, $5\%$ for $i_{\text{low}}$, $6\%$ for $i_{\text{high}}$, $\$80.8$ for $F_{\text{low}}$, $-\$668$ for $F_{\text{high}}$ in Equation (II).

$\begin{array}{c}IRR=5\%+\left(6\%-5\%\right)\left[\frac{\$80.8-0}{\$80.8-\left(-\$668\right)}\right]\\ =5\%+1\%\left(\frac{\$80.8}{\$748.8}\right)\\ =0.05+0.001079\\ =0.051079\end{array}$

$IRR=5.1\%$

Write the expression to calculate the nominal annual rate of interest.

$\text{Annual}\text{interest}\text{rate}=\left(\text{semi}\text{annual}\text{rate}\right)\times 2$ ....... (III)

Substitute, $5.1\%$ for semi annual rate in Equation (III).

$\begin{array}{c}\text{Annual}\text{interest}\text{rate}=5.1\%\times 2\\ =10.2\%\end{array}$.

Conclusion:

Therefore, the correct option is $\left(d\right)\text{Greater}\text{than}10\%$.