Chapter 10, Problem 16BE

To determine

Ascertain the issue price if the bonds are priced to yield (a) 6% (b) 8% and (c) 4%.

Explanation of Solution

Bonds:

A bond (long term debt) is a written document and a signed promise to pay interest periodically as well as the principal amount borrowed at the time of maturity to the bond investor.

(a)

Calculate the issue price of the 6%, 5-year, $100,000 bonds if the market interest rate 6% is payable semiannually.

$\begin{array}{c}\text{Issue price of bonds}=\left\{\begin{array}{l}\text{Present value of principal (2)}\\ +\text{Present value of interest payments (6)}\end{array}\right\}\\ =\$74,400+\$25,600\\ =\$100,000\end{array}$

Therefore, the issue price of the bonds is $100,000.

Working notes:

Calculate the present value for 5 periods and 6% interest compounded semiannually:

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{1}{{\left(1+0.03\right)}^{10}}\\ =0.744\end{array}$

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.

(1)

Calculate the Present value of face value of bonds:

Particulars	Amount ($)
Face value of bonds (a)	100,000
PV factor at a semiannual market rate of 3% for 10 periods (b) (1)	0.744
Present value of face value of the bonds $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	74,400

(Table 1)

(2)

Calculate stated semiannual interest rate:

$\begin{array}{c}\text{Semiannual stated interest rate}=\frac{\text{Stated interest rate}}{\text{2}}\\ =\frac{6\%}{2}\\ =\text{ 3}\%\end{array}$ (3)

Calculate the interest payable:

$\begin{array}{c}\text{Interest payable}=\text{Face value of bonds}\times \text{Semiannual interest rate}\\ =\$100,000\times \frac{3}{100}\\ =\$3,000\end{array}$ (4)

Calculate the present value factor for semiannual market of 3% for 10 periods:

$\begin{array}{c}\text{Present Value of Annuity factor= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\\ =\frac{\text{1}-{\left(1+0.03\right)}^{-10}}{3\%}\\ =8.530\end{array}$ (5)

Calculate present value of interest payments:

Particulars	Amount ($)
Interest payable amount (a)	$3,000
PV factor at a semiannual market rate of 3% for 10 periods (b) (5)	8.530
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	25,600

(Table 2)

(6)

(b)

Calculate the issue price of the 6%, 5-year, $100,000 bonds if the market interest rate 8% is payable semiannually.

$\begin{array}{c}\text{Issue price of bonds}=\left\{\begin{array}{l}\text{Present value of principal (8)}\\ +\text{Present value of interest payments (10)}\end{array}\right\}\\ =\$67,600+\$24,333\\ =\$91,333\end{array}$

Therefore, the issue price of the bonds is $91,333.

Working notes:

Calculate the present value for 5 periods and 8% interest compounded semiannually:

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{1}{{\left(1+0.04\right)}^{10}}\\ =0.676\end{array}$

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.

(7)

Calculate the Present value of face value of bonds:

Particulars	Amount ($)
Face value of bonds (a)	100,000
PV factor at a semiannual market rate of 4% for 10 periods (b) (7)	0.676
Present value of face value of the bonds $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	67,600

(Table 3)

(8)

Calculate the present value factor for semiannual market of 4% for 10 periods:

$\begin{array}{c}\text{Present Value of Annuity factor= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\\ =\frac{\text{1}-{\left(1+0.04\right)}^{-10}}{4\%}\\ =8.111\end{array}$ (9)

Calculate present value of interest payments:

Particulars	Amount ($)
Interest payable amount (a)	$3,000
PV factor at a semiannual market rate of 4% for 10 periods (b) (9)	8.111
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	24,333

(Table 4)

(10)

(c)

Calculate the issue price of the 10%, 5-year, $100,000 bonds if the market interest rate 4% is payable semiannually.

$\begin{array}{c}\text{Issue price of bonds}=\left\{\begin{array}{l}\text{Present value of principal (12)}\\ +\text{Present value of interest payments (14)}\end{array}\right\}\\ =\$82,000+\$26,949\\ =\$108,949\end{array}$

Therefore, the issue price of the bonds is $108,949.

Working notes:

Calculate the present value for 5 periods and 4% interest compounded semiannually:

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{1}{{\left(1+0.02\right)}^{10}}\\ =0.820\end{array}$

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.

(11)

Calculate the Present value of face value of bonds:

Particulars	Amount ($)
Face value of bonds (a)	100,000
PV factor at a semiannual market rate of 2% for 10 periods (b) (11)	0.820
Present value of face value of the bonds $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	82,000

(Table 5)

(12)

Calculate the present value factor for semiannual market of 2% for 10 periods:

$\begin{array}{c}\text{Present Value of Annuity factor= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\\ =\frac{\text{1}-{\left(1+0.02\right)}^{-10}}{2\%}\\ =8.983\end{array}$ (13)

Calculate present value of interest payments:

Particulars	Amount ($)
Interest payable amount (a)	$3,000
PV factor at a semiannual market rate of 2% for 10 periods (b) (13)	8.983
Present value of interest payments $\left(\text{a}\right)\text{×}\left(\text{b}\right)$	26,949

(Table 6)

(14)