7.5 Quotation of Bonds

Calculating the Purchase Price of a Bond on a Non-Interest Payment Date

In a previous section, we learned how to calculate the purchase price of a bond when the bond is purchased on an interest payment date.  But, bonds can be bought or sold on any day of the year.  How do we calculate the purchase price when a bond is purchased on a non-interest payment date?

For each day that elapses after an interest payment date, interest for the next payment starts to accrue.  When the bond is sold in between interest payment dates, the interest, called the accrued interest, that accrues on the bond since the interest payment date before the purchase date up to but not including the purchase date is paid to the seller.  This accrued interest is included in the purchase price paid to the seller.  That is, the buyer has to pay the seller the purchase price of the bond on the previous interest payment date, called the market price, plus the accrued interest.  The market price plus the accrued interest is called the flat price, and is the actual amount the buyer pays the seller on the purchase date.

Why does the buyer have to pay the seller the accrued interest?  Suppose a bond makes semi-annual interest payments of $50 and the buyer acquires the bond two months after the last interest payment date.  Because the seller held the bond for two months of the six-month payment interval, it is fair and reasonable for the seller to receive the interest earned during those two months. However, the next interest payment occurs four months after the sale of the bond, at which time the buyer, who now owns the bond, will receive the full $50 interest payment for the full six month payment interval.  Thus, at the time of buying the bond, the buyer has to pay the seller the bond’s market price plus the portion of the next interest payment that legally belongs to the seller.  In this example, an interest amount representing two of the six months needs to be paid.

Arriving at the bond’s purchase price in between interest payment dates is a little complex because the flat price is increasing according to a compound interest formula, but in practice the accrued interest on the bond is increasing according to a simple interest formula.  If this seems peculiar to you, you would be right, but that is just how bonds happen to work!

Follow these steps to find the flat price, the actual purchase price of a bond between interest payment dates.

1. Identify four important dates:
   • Purchase date.
   • Maturity date.
   • Interest payment date immediately before the purchase date.
   • Interest payment date immediately after the purchase date.
2. Calculate the market price on the interest payment date immediately before the purchase date.  The market price is the purchase price if the bond is actually purchased on the interest payment date immediately before the purchase date.  This calculation is the same purchase price calculation we learned in a previous section.
3. Calculate the flat price on the purchase date by adding the accrued interest to the market price.  This is a compound interest only calculation that moves the market price from the payment date immediately before the purchase date to the purchase date. In this calculation, you calculate the future value with the present value set to the market price from the previous step, the payment is 0, the interest rate is the yield rate, and N is

   [latex]\begin{eqnarray*} & & \\ N & = & \frac{\mbox{Number of days from previous interest payment date to purchase date}}{\mbox{Number of days in between interest payment dates}} \end{eqnarray*}[/latex]

EXAMPLE

A $1,000 bond with a 7% coupon has a maturity date of February 1, 2035.  The bond was purchased on May 20, 2023 when the yield to maturity was 5%.  Calculate the purchase price.

Solution:

Step 1:  Identify the four dates.  Because February 1 is the maturity date, February 1 is one of the two payment dates.  The other payment date is six months later, in this case August 1.

• Purchase date:  May 20, 2023
• Maturity date:  February 1, 2035
• Payment date before purchase date:  February 1, 2023
• Payment date after purchase date:  August 1, 2023

Step 2:  Calculate the market price on February 1, 2023. The time for this calculation is the number of years between February 1, 2023 (the payment date before the purchase date) and February 1, 2035 (the maturity date), which is 12 years.

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 & = & \$1,000 \\ P/Y & = & 2 \\  I/Y & = & 5\% \\ C/Y & = & 2 \\ t & = & 12 \mbox{ years} \\ \mbox{Coupon Rate} & = & 7\% \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 1,000\times \frac{0.07}{2}\\ & = & \$35\end{eqnarray*}[/latex]

PMT Setting	END
N	[latex]2 \times 12=24[/latex]
PV	?
FV	[latex]1,000[/latex]
PMT	[latex]35[/latex]
I/Y	[latex]5[/latex]
P/Y	[latex]2[/latex]
C/Y	[latex]2[/latex]

[latex]PV=\$1,178.85[/latex]

The market price on February 1, 2023 is $1,178.85.

Step 3:  Calculate the flat price on May 20, 2023.  This is a compound interest calculation that moves the market price from the previous step forward in time to the purchase date.  The value of N for this compound interest calculation is

[latex]\begin{eqnarray*}N & = & \frac{\mbox{Number of days from February 1, 2023 to May 20, 2023}}{\mbox{Number of days from February 1, 2023 to August 1, 2023}} \\ & = & \frac{108}{181}\end{eqnarray*}[/latex]

N	[latex]\frac{108}{181}[/latex]
PV	[latex]1,178.85[/latex]
FV	?
PMT	[latex]0[/latex]
I/Y	[latex]5[/latex]
P/Y	[latex]2[/latex]
C/Y	[latex]2[/latex]

[latex]FV=\$1,196.35[/latex]

The flat price on May 20, 2023 is $1,196.35.  This is the purchase price of the bond on May 20, 2023.

_{Calculating Dates and Days (Month-Day-Year) by Joshua Emmanuel [3:32] (transcript available).}

EXAMPLE

A 25-year, $2,000 bond with a 3% coupon was issued on April 1, 2018. The bond was purchased on January 27, 2023 when the yield to maturity was 8%. Calculate the purchase price.

Solution:

Step 1:  Identify the four dates.  The bond was issued on April 1, 2018 and reaches maturity 25 years later.  So, the maturity date is April 1, 2043. Because April 1 is the maturity date, April 1 is one of the two payment dates.  The other payment date is six months later, in this case October 1.

• Purchase date:  January 27, 2023
• Maturity date:  April 1, 2043
• Payment date before purchase date:  October 1, 2022
• Payment date after purchase date:  April 1, 2023

Step 2:  Calculate the market price on October 1, 2022. The time for this calculation is the number of years between October 1, 2022 (the payment date before the purchase date) and April 1, 2043 (the maturity date), which is 20.5 years.

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 & = & \$2,000 \\ P/Y & = & 2 \\  I/Y & = & 8\% \\ C/Y & = & 2 \\ t & = & 20.5 \mbox{ years} \\ \mbox{Coupon Rate} & = & 3\% \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 2,000\times \frac{0.03}{2}\\ & = & \$30\end{eqnarray*}[/latex]

PMT Setting	END
N	[latex]2 \times 20.5=41[/latex]
PV	?
FV	[latex]2,000[/latex]
PMT	[latex]30[/latex]
I/Y	[latex]8[/latex]
P/Y	[latex]2[/latex]
C/Y	[latex]2[/latex]

[latex]PV=\$1,000.35[/latex]

The market price on October 1, 2022 is $1,000.35.

Step 3:  Calculate the flat price on January 27, 2023.  This is a compound interest calculation that moves the market price from the previous step forward in time to the purchase date.  The value of N for this compound interest calculation is

[latex]\begin{eqnarray*}N & = & \frac{\mbox{Number of days from October 1, 2022 to January 27, 2023}}{\mbox{Number of days from October 1, 2022 to April 1, 2023}} \\ & = & \frac{118}{182}\end{eqnarray*}[/latex]

N	[latex]\frac{118}{182}[/latex]
PV	[latex]1,000.35[/latex]
FV	?
PMT	[latex]0[/latex]
I/Y	[latex]8[/latex]
P/Y	[latex]2[/latex]
C/Y	[latex]2[/latex]

[latex]FV=\$1,026.11[/latex]

The flat price on January 27, 2023 is $1,026.11.  This is the purchase price of the bond on January 27, 2023.

Quotation of Bonds

The flat price we calculated in the previous examples is the actual price a buyer would pay to purchase those bonds on the given purchase date.  The flat price is the market price (on the payment date before the purchase date) plus any interest that has accrued since the previous payment. However, two different bonds purchased on the same day can have very different amounts of accrued interest, depending on where the purchase date is in relation to each bond’s payment date.  This can make it very difficult for an investor to directly compare the price of the bonds.

To make a meaningful comparison of bond prices, the flat prices are adjusted for their differing amount of accrued interest.  The quoted price of bond is the flat price excluding the accrued coupon interest.

[latex]\begin{eqnarray*}\mbox{Quoted Price} & = & \mbox{Flat Price}-\mbox{Accrued Interest}\\ \\ \mbox{Accrued Interest} & = & PMT \times \frac{\mbox{Number of days from previous interest payment date to purchase date}}{\mbox{Number of days in between interest payment dates}} \end{eqnarray*}[/latex]