7.2 Calculating the Purchase Price of a Bond on an Interest Payment Date

A $50,000 bond with a 10.15% coupon is purchased when there are 10.5 years remaining to maturity.  If the yield to maturity at the time of purchase is 4.31%, what is the purchase price?  What is the amount of premium or discount?

Solution:

Step 1:  Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual.  The given information is

[latex]\begin{eqnarray*} FV & = & \$50,000 \\ P/Y & = & 2 \\  I/Y & = & 4.31\% \\ C/Y & = & 2 \\ t & = & 10.5 \mbox{ years} \\ \mbox{Coupon Rate} & = & 10.15\% \end{eqnarray*}[/latex]

Step 2:  Calculate the bond payment.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 50,000\times \frac{0.1015}{2}\\ & = & \$2,537.50\end{eqnarray*}[/latex]

Step 3:  Calculate the purchase price.

[latex]PV=\$74,452.86[/latex]

Step 4:  Calculate the premium or discount. Because the purchase price is greater than the face value, this is a premium bond.

[latex]\begin{eqnarray*} \mbox{Premium} & = & \mbox{Purchase Price}-\mbox{Face Value} \\ & = & 74,452.86-50,000 \\  & = & \$24,452.86 \end{eqnarray*}[/latex]

The purchase price of the bond is $74,452.86 and the premium is $24,452.86.

A $25,000 bond with a 8.92% coupon is purchased when there are 22.5 years remaining to maturity.  If the yield to maturity at the time of purchase is 9.46%, what is the purchase price?  What is the amount of premium or discount?

Solution:

Step 1:  Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual.  The given information is

[latex]\begin{eqnarray*} FV & = & \$25,000 \\ P/Y & = & 2 \\  I/Y & = & 9.46\% \\ C/Y & = & 2 \\ t & = & 22.5 \mbox{ years} \\ \mbox{Coupon Rate} & = & 8.92\% \end{eqnarray*}[/latex]

Step 2:  Calculate the bond payment.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 25,000\times \frac{0.0892}{2}\\ & = & \$1,115\end{eqnarray*}[/latex]

Step 3:  Calculate the purchase price.

[latex]PV=\$23,751.28[/latex]

Step 4:  Calculate the premium or discount. Because the purchase price is less than the face value, this is a discount bond.

[latex]\begin{eqnarray*} \mbox{Discount} & = & \mbox{Face Value}-\mbox{Purchase Price} \\ & = & 25,000-23,751.28 \\  & = & \$1,248.72 \end{eqnarray*}[/latex]

The purchase price of the bond is $23,751.28 and the discount is $1,248.72.

An investor purchased a $10,000 bond with a coupon rate of 6.5% when there was 15 years to maturity and the yield rate was 4%. Four years later, the investor decided to sell the bond when the yield to maturity was 5.2%.

1. At what price did the investor purchase the bond?
2. At what price did the investor sell the bond?
3. What was the investor’s gain or loss on the sale of the bond?

Solution:

Step 1:  Calculate the investor’s purchase price.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 10,000 \times \frac{0.065}{2}\\ & = & \$325\end{eqnarray*}[/latex]

[latex]PV=\$12,799.56[/latex]

The investor paid $12,799.56 for the bond.

Step 2:  Calculate the selling price of the bond.  The selling price of the bond for the investor equals the purchase price of the bond for the person the investor is selling the bond to.  The purchase price is based on the time to maturity and the yield rate at the time the bond is sold.  The investor purchased the bond when there was 15 years to maturity and then sold the bond four years later.  So at the time of the sale there was 11 (15-4) years to maturity.

[latex]PV=\$11,078.66[/latex]

The investor sold the bond for $11,078.66.

Step 3:  Calculate the gain or loss on the sale.  Because the investor sold the bond for less than they paid for it, the investor realized a loss on the sale.

[latex]\begin{eqnarray*} \mbox{Loss}  & = & 12,799.56-11,078.66 \\  & = & \$1,720.90 \end{eqnarray*}[/latex]

The investor’s lost $1,720.90 on the sale of the bond.