Chapter 9, Problem

1.

To determine

Bonds

Bonds are a kind of interest bearing notes payable, usually issued by companies, universities and governmental organizations. It is a debt instrument used for the purpose of raising fund of the corporations or governmental agencies. If selling price of the bond is equal to its face value, it is called as par on bond. If selling price of the bond is lesser than the face value, it is known as discount on bond. If selling price of the bond is greater than the face value, it is known as premium on bond.

To Identify: If the market rate is 8%, will the bonds issue at face amount, a discount or a premium and calculate price of the bonds.

Answer

The bond issued at a premium and price of the bonds is $45,057,518.65.

Explanation of Solution

Stated interest rate (9%) is greater than the market interest rate (8%) means, the bonds issue at a premium.

$\begin{array}{c}\text{Price of bonds}\}=\left\{\begin{array}{l}\text{Present value of principal}+\\ \text{Present value of interest payments}\end{array}\right\}\\ =\$8,539,850.83+\$36,517,667.81\\ =\$45,057,518.65\end{array}$

Working notes:

Calculate the present value of face value of principal.

Particulars	Amount ($)
Face value of bonds (a)	$41,000,000
PV factor at an annual market rate of 4% for 40 periods (b)	$\times$ 0.20829
Present value of face value of principal $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$8,539,850.83

Note: The present value of $1 for 40 periods at 4% is 0.20829 (refer Table 2 in Appendix).

Calculate present value of interest payments.

Particulars	Amount ($)
Interest payments amount (a)	$1,845,000
PV factor at an annual market rate of 4% for 40 periods (b)	$\times$19.79277
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$36,517,667.81

Note: The Present value of an ordinary annuity of $1 for 40 periods at 4% is 19.79277 (refer Table 4 in Appendix).

Calculate the amount of interest payment.

$\begin{array}{c}\text{Interest payment}=\text{Face value of bonds}\times \text{Stated interest rate}\times \text{Time period}\\ =\$41,000,000\times \frac{9}{100}\times \frac{6}{12}\\ =\$1,845,000\end{array}$

Conclusion

Therefore, price of the bonds is $45,057,518.65.

2.

To determine

To Identify: If the market rate is 9%, will the bonds issue at face amount, a discount or a premium and calculate price of the bonds.

Answer

The bond issue at a par and price of the bonds is $41,000,000.

Explanation of Solution

Stated interest rate (9%) is equal to the market interest rate (9%) means, the bonds issue at a par.

$\begin{array}{c}\text{Price of bonds}\}=\left\{\begin{array}{l}\text{Present value of principal}+\\ \text{Present value of interest payments}\end{array}\right\}\\ =\$7,049,076.74+\$33,950,923.26\\ =\$41,000,000\end{array}$

Working notes:

Calculate the present value of face value of principal.

Particulars	Amount ($)
Face value of bonds (a)	$41,000,000
PV factor at an annual market rate of 4.5% for 40 periods (b)	$\times$ 0.17193
Present value of face value of principal $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$7,049,076.74

Note: The present value of $1 for 40 periods at 4.5% is 0.17193 (refer Table 2 in Appendix).

Calculate present value of interest payments.

Particulars	Amount ($)
Interest payments amount (a)	$1,845,000
PV factor at an annual market rate of 4.5% for 40 periods (b)	$\times$18.40158
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$33,950,923.26

Note: The Present value of an ordinary annuity of $1 for 40 periods at 4.5% is 18.40158 (refer Table 4 in Appendix).

Calculate the amount of interest payment.

$\begin{array}{c}\text{Interest payment}=\text{Face value of bonds}\times \text{Stated interest rate}\times \text{Time period}\\ =\$41,000,000\times \frac{9}{100}\times \frac{6}{12}\\ =\$1,845,000\end{array}$

Conclusion

Therefore, price of the bonds is $41,000,000.

3.

To determine

To Identify: If the market rate is 10%, will the bonds issue at face amount, a discount or a premium and calculate price of the bonds.

Answer

The bond issue at a discount and price of the bonds is $37,482,387.30.

Explanation of Solution

Stated interest rate (9%) is equal to the market interest rate (10%) means, the bonds issue at a discount.

$\begin{array}{c}\text{Price of bonds}\}=\left\{\begin{array}{l}\text{Present value of principal}+\\ \text{Present value of interest payments}\end{array}\right\}\\ =\$5,823,872.97+\$31,482,387.30\\ =\$37,482,387.30\end{array}$

Working notes:

Calculate the present value of face value of principal.

Particulars	Amount ($)
Face value of bonds (a)	$41,000,000
PV factor at an annual market rate of 5% for 40 periods (b)	$\times$ 0.14205
Present value of face value of principal $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$5,823,872.97

Note: The present value of $1 for 40 periods at 5% is 0.14205 (refer Table 2 in Appendix).

Calculate present value of interest payments.

Particulars	Amount ($)
Interest payments amount (a)	$1,845,000
PV factor at an annual market rate of 5% for 40 periods (b)	$\times$17.15909
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	$31,658,514.32

Note: The Present value of an ordinary annuity of $1 for 40 periods at 5% is 17.15909 (refer Table 4 in Appendix).

Calculate the amount of interest payment.

$\begin{array}{c}\text{Interest payment}=\text{Face value of bonds}\times \text{Stated interest rate}\times \text{Time period}\\ =\$41,000,000\times \frac{9}{100}\times \frac{6}{12}\\ =\$1,845,000\end{array}$

Conclusion

Therefore, price of the bonds is $37,482,387.30.