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When a bond is first issued, it is generally sold at par, which is the face value of the bond. For instance, the par value of a bond with a face value of $1,000 is $1,000. The par value is the principal, which is received at the end of the bond’s term. Sometimes when the demand is higher or lower than an issuer expected, the bonds might sell higher or lower than par. In the secondary market, bonds almost always trade for either more or less than par, because interest rates change continuously. When a bond trades for more than par, then it is selling at a premium, and when it is selling for less, it is selling at a discount. The main determinant of most bond prices in the secondary market is the prevailing interest rates. When interest rates rise, bond prices decline, and vice versa.
When bond prices are listed, the convention is to list them as a percentage of par value, regardless of what the face value of the bond is, with 100 being equal to par value. Thus, a bond with a face value of $1,000 which is selling for par, sells for $1,000, and a bond with a face value of $5,000 that is also selling for par will both have their price listed as 100, which means their prices are equal to 100% of par value, or $100 for each $100 of face value.
This pricing convention allows different bonds with different face values to be compared directly. For instance, if a $1,000 corporate bond was listed as 90 and a $5,000 municipal bond was listed as 95, then it can be easily seen that the $1,000 bond is selling at a bigger discount, and, therefore, has a higher yield. To find what the bond’s price actually is, the listed price must be multiplied as a percentage by the face value of the bond, so the price for the $1,000 bond is 90% x $1,000 = 0.9 x $1,000 = $900, and the price for the $5,000 bond is 95% x $5,000 = .95 x $5,000 = $4,750.
A point is equal to 1% of the bond’s face value. Thus, a point's actual value depends on the face value of the bond. Thus, 1 point = $10 for a $1,000 bond, but $50 for a $5,000 bond. So a $1,000 bond that is selling for 97 is selling at a 3 point discount, or $30 below par value, which equals $970.
No commission is charged when buying or selling bonds. A bond dealer makes money through the spread—the difference between the bid price, which is what the dealer is willing to pay for a bond, and the ask price, which is what the dealer is selling the bond for. To keep the spread further apart, bond prices are generally listed in 1/32 increments of a point, or a higher multiple, although some Treasuries have price differentials as low as 1/64. (Another reason for this convention is that a point is not equal to a dollar, but a decimal base would still be more convenient.) The pricing convention is to list the point after a dash. Thus, a price listed as 102-04 is equal to 102 + 4/32 = 102 + 1/8 = 102.125% of par value. If this listed price were for a $1,000 face-value bond, then this price would be equal to $1,021.25. The integer point value, in this case 102, is known as the handle. When traders negotiate, the handle is usually known and not expressed. So a trader might say that he’ll offer 2 for the bond, meaning the handle + 1/16 (= 2/32).
Because the trading volume in Treasuries is much greater than for other bonds, Treasuries sometimes trade in 1/64 increments. A 1/64 increment is denoted by a plus next to the listed price. So a U.S. Treasury bond with a $1,000 face value that is listed as 101-1+ = 101 + 1/32 + 1/64 = 101 + 3/64 = 101.04875, so the bond’s price = 101.04875% x 1,000 = $1010.49 (rounded). 1,000 of these bonds would cost $1,010,487.50.
Listed bond prices are flat prices, which do not include accrued interest. Most bonds pay interest semi-annually. For settlement dates when interest is paid, the bond price is equal to the flat price. Between payment dates, the price of the bond will be the flat price + the accrued interest. Accrued interest is the interest that has been earned, but not paid, and is calculated by the following formula:
| Formula for Calculating Accrued Interest | |
|---|---|
| Accrued Interest = Interest Payment x | Number of Days Since Last Payment ───────────── Number of days between payments |
| Graph of the purchase price of a bond over 2 years, which is equal to the flat price + accrued interest. (It is assumed that the flat price remains constant over the 2 years, but would actually fluctuate with interest rates, and because of other factors.) The flat price is what is listed in bond tables for prices. The accrued interest must be calculated according to the above formula. Note that the bond price steadily increases each day until reaching a peak the day before an interest payment, then drops to minimum on the day of the payment. |
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When you actually buy a bond on the secondary market, you would have to pay the former owner of the bond the accrued interest. If this were not so, you could make a fortune buying bonds right before they paid interest then selling them afterward. Because the interest accrues every day, the bond price increases accordingly until the interest payment date, when it drops to its flat price, then starts rising again.
You purchase a corporate bond with a settlement date on September 15 with a face value of $1,000 and a nominal yield of 8%, that has a listed price of 100-08, and that pays interest semi-annually on February 15 and August 15. How much must you pay?
The semi-annual interest payment is $40 and there were 31 days since the last interest payment on August 15. If the settlement date fell on a interest payment date, the bond price would equal the listed price: 100.25% x $1,000.00 = $1,002.50 (8/32 = 1/4 = .25, so 100-08 = 100.25% of par value). Since the settlement date was 31 days after the last payment date, accrued interest must be added. Using the above formula, with 184 days between coupon payments, we find that:
| Accrued Interest = $40 x | 31 ─── 184 | = $6.74 |
Therefore, the actual purchase price for the bond will be $1,002.50 + $6.74 = $1,009.24.
Tip: In most cases, it will be more convenient to use a spreadsheet, such as Excel. With Excel, you can use these functions for determining number of days or the actual price:
| Number of Days since Last Payment = | COUPDAYBS(settlement,maturity,frequency,basis) |
| Number of Days Between Payments = | COUPDAYS(settlement,maturity,frequency,basis) |
| Bond Price = | PRICE(settlement,maturity,rate,ytm,redemption,frequency,basis) |
Search Help for more information. Below is another example of obtaining a bond's price by using Excel's PRICE function:
| 15-Feb-08 | Settlement Date |
| 15-Nov-17 | Maturity Date |
| 5.75% | Coupon Rate |
| 6.50% | Yield to Maturity |
| 100 | Redemption value |
| 2 | Number of Interest Payments per Year |
| 1 | Day Count Basis (Month/Year = Actual/Actual) |
| = | |
| 94.63544921 | % of Par Value of Actual Price for Corporate Bond, $1,000 Face Value |
| $946.35 | Actual Price for Corporate Bond, $1,000 Face Value |
To calculate the accrued interest on a zero coupon bond, which pays no interest, but is issued at a deep discount, the amount of interest that accrues every day is calculated by using a straight-line amortization, which is found by subtracting the discounted issue price from its face value, and dividing by the number of days in the term of the bond. This is the interest earned in 1 day, which is then multiplied by the number of days from the issue date.
Microsoft Excel has several formulas for calculating bond prices and other securities paying interest, such as Treasuries or certificates of deposit (CDs). These prices take into account the accrued interest, if any.
| PRICE, PRICEDISC, PRICEMAT, and DISC Functions. |
|---|
Bond Price (per $100 of face value) = PRICE(settlement,maturity,rate,yield,redemption,frequency,basis) Discounted Bond Price = PRICEDISC(settlement,maturity,discount,redemption,basis) Discount Rate of Security = DISC(settlement,maturity,price,redemption,basis) Price of Security that pays interest only at maturity = PRICEMAT(settlement,maturity,issue,rate,yield,basis)
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The following basic facts—where they apply—will be used for each of the example calculations for a 10-year bond originally issued in 1/1/2008 with a par value of $1,000:
What is the price of a bond selling for a yield to maturity of 8%?
Bond Price =PRICE("3/31/2008","12/31/2017",0.06,0.08,100,2,1) = 86.62092 = $866.21
What is the discount price of a zero coupon bond with a par value of $1,000 yielding 8%?
Price Discount =PRICEDISC("3/31/2008","12/31/2017",0.06,0.08,100,1) = 21.99288 = $219.93
What is the interest rate of a discounted zero coupon bond selling for $219.90 that pays $1,000 at maturity?
Interest Rate of Bond Discount = DISC("3/31/2008","12/31/2017",21.99,100,1) = 0.080003 = 8%
The last function, PRICEMAT, calculates the price of a security that pays all of its interest at maturity, which includes negotiable money market certificates of deposit (CD).
What is the price of a negotiable, 90-day CD originally issued for $100,000 on 3/1/2008 paying a rate of 8% with a current yield of 6% and a settlement date of 4/1/2008? Here we use the Microsoft Excel Date function, which takes the format DATE(year,month,day) to do some calendar arithmetic. We also use the banker's year of 360 days, so we choose a basis of 0, which we could have omitted, since it is the default.
Market Price of CD = PRICEMAT(DATE(2008,4,1),DATE(2008,3,1)+90,DATE(2008,3,1),0.08,0.06,0)= 100.3181 per $100 of face value = $100,318.10
Note: The above calculations were made using Microsoft Office Excel 2007. These functions are also available in earlier versions of Excel.
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