Bond Pricing

Bond prices are determined by 5 factors:

  1. par value
  2. coupon rate
  3. prevailing interest rates
  4. accrued interest
  5. credit rating of the issuer.

Generally, the issuer sets the price and the yield of the bond so that it will sell enough bonds to supply the amount that it desires. The higher the credit rating of the issuer, the lower the yield that it must offer to sell its bonds. A change in the credit rating of the issuer will affect the price of its bonds in the secondary market: a higher credit rating will increase the price, while a lower rating will decrease the price. The other factors that determine the price of a bond have a more complex interaction.

When a bond is first issued, it is generally sold at par, which is the face value of the bond. Most corporate bonds, for instance, have a face and par value of $1,000. The par value is the principal, which is received at the end of the bond’s term, i.e., at maturity. 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, bond prices are almost always different from par, because interest rates change continuously. When a bond trades for more than par, then it is selling at a premium, which will pay a lower yield than its stated coupon rate, and when it is selling for less, it is selling at a discount, paying a higher yield than its coupon rate. When interest rates rise, bond prices decline, and vice versa. Bond prices will also include accrued interest, which is the interest earned between coupon payment dates. Clean bond prices are prices without accrued interest; dirty bond prices include accrued interest.

Although prevailing interest rates are usually the main determinants of bond prices in the secondary market, as a bond approaches maturity, the present value of its future payments converges to the par value; therefore, the par value becomes more important than the prevailing interest rates, since the bond price, whether at a premium or discount, converges to the par value, as can be seen in the diagram below.
Graph of price changes for discounted and premium bonds.

Bond Value Equals the Sum of the Present Value of Future Payments

A bond pays interest either periodically or, in the case of zero coupon bonds, at maturity. Therefore, the value of the bond is equal to the sum of the present value of all future payments — hence, it is the present value of an annuity, which is a series of periodic payments. The present value is calculated using the prevailing market interest rate for the term and risk profile of the bond, which may be more or less than the coupon rate. For a coupon bond that pays interest periodically, its value can be calculated thus:

Bond Value = Present Value (PV) of Interest Payments + Present Value of Principal Payment

Bond Value = PV(1st Payment) + PV(2nd Payment) + ... + PV(Last Payment) + PV(Principal Payment)

Bond Price Formula
Clean Bond Price=C1
(1+r/k)1
+C2
(1+r/k)2
+ ... +Cn
(1+r/k)kn
+P
(1+r/k)kn

C = coupon, or interest, payment per period

k = number of coupon periods in 1 year

n = number of years until maturity

r = annualized market interest rate

P = par value of bond

Example — Calculating Bond Value as the Present Value of its Payments

Suppose a company issues a 3-year bond with a par value of $1,000 that pays 4% interest annually, which is also the prevailing market interest rate. What is the present value of the payments?

The following table shows the amount received each year and the present value of that amount. As you can see, the sum of the present value of each payment equals the par value of the bond.

YearPaymentAmount ReceivedPresent Value
1Interest$40$38.46
2Interest$40$36.98
3Interest + Principal$1040$924.56
Totals$1120$1,000.00

The above formula can be simplified by using the formula for the present value of an annuity, and letting k=2 for bonds that pay a semiannual coupon:

Simplified Bond Price Formula for Semiannual Coupon Bonds
Clean Bond Price=C
r
[11
(1+r/2)2n
]+P
(1+r/2)2n

C = Annual payment from coupons

n = number of years until maturity

r = market annual interest rate

P = par value of bond

Note that the above formula is sometimes written with both C and r divided by 2; the results are the same, since it is a ratio.

Example: Using the Simplified Bond Pricing Formula

Given:

Case 1:

Then:

Bond Price=5
.04
[11
(1.02)10
]+100
(1.02)10
=104.49
Bond Price=5
.04
[11
(1.04)5
]+100
(1.04)5
=104.45

In the primary bond market, where the buyer buys the bond from the issuer, the bond usually sells for par value, which is equal to the bond's value using the coupon rate of the bond. However, in the secondary bond market, bond price still depends on the bond's value, but the interest rate to calculate that value is determined by the market interest rate, which is reflected in the actual bids and offers for bonds. Additionally, the buyer of the bond will have to pay any accrued interest on top of the bond's price unless the bond is purchased on the day it pays interest.

Bond Price Listings

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 the actual price of the bond, 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% × $1,000 = 0.9 × $1,000 = $900, and the price for the $5,000 bond is 95% × $5,000 = .95 × $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 (= $1,000 × 102.125% = $1,000 × 1.02125). 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 = 101.046875, so the bond’s price = 101.046875% × 10 = $1010.47 (rounded). Thus, 1,000 of these bonds would cost $1,010,468.75.

Accrued Interest

Listed bond prices are clean prices (aka 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, accrued interest must be added to the flat price, which is often called the dirty price (aka all-in price, gross price):

Dirty Bond Price = Clean Price + 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 (Actual/Actual Day-Count Basis)
Accrued Interest = Interest Payment ×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, such as changes in the credit rating of the issuer.) 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 back to the flat price on the day of the payment.
Graph of a bond price that shows how the price goes up with accrued interest, then drops to its flat price on the interest payment day.

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 accruing interest again.

Day-Count Conventions

In calculating the accrued interest, the actual number of days was counted from the last interest payment to the value date. Most bonds use this day-count basis, which is referred to as actual/actual basis, because the actual number of days are used in the calculations. However, some bonds use a different day-count basis, which will cause the accrued interest to be slightly different from that calculated using the actual/actual convention. Closely related to actual/actual are the following conventions, which are only used for bonds that have 1 annual coupon payment:

Actual/360: Accrued Interest = Coupon Rate × Days/360

Actual/365: Accrued Interest = Coupon Rate × Days/365

Note that the accrued interest calculated under the actual/360 convention is slightly more than the interest calculated under the actual/actual or the actual/365 method.

There are 2 other methods where each month counts as 30 days, regardless of the number of days in the month and each year is considered to have 360 days. Although these methods are rarely used nowadays to calculate accrued interest, they did simplify calculating the number of days between a coupon date and the value date, which was valuable before the advent of calculators and computers, especially since the calculated interest differed little from that calculated with the actual/actual method. So, under these methods, there is always 3 days between February 28 and March 1, because each month counts as 30 days, including February, even though February has either 28 or 29 days. By the same reasoning, there are 25 days between January 15 and February 10, even though there are actually 26 days between those dates. When figuring accrued interest using any day-count convention, the 1st day is counted, but not the last day. So in the previous example, January 15 is counted, but not February 10.

30/360 and 30E/360 Day-Count Conventions
Start Date:M1/D1/Y1
End Date:M2/D2/Y2
Day Count Fraction= Day Count/360
Day Count= (Y2 – Y1) × 360 + (M2 – M1) × 30 + (D2 – D1)
30/360 Day-Count Convention (aka US 30/360)
If (D1 = 31)Set D1 = 30
If (D2 = 31) and (D1 = 30 or 31)Set D2 = 30
30E/360 Day-Count Convention (aka European 30/360)
If D2 = 31Set D2 = 30

So the number of days between December 29, 2014 and January 31, 2015 is 32 under the 30/360 convention, but 31 days under the 30E/360 convention. This is determined thus:

The number of days are then divided by 360, then multiplied by the coupon rate to determine the accrued interest:

30/360 and 30E/360: Accrued Interest = Coupon Rate × Days/360

Day Count Conventions Used in US Bond Markets

Bond Market

Day-Count Basis

Treasuries

Actual/Actual

Corporate and Municipal Bonds

30/360

Money Market Instruments

Actual/360

  • So a 1% bond would earn 365/360 × 1% of interest in 365 days.

As already stated, most bond markets outside of the U.S. use the actual/actual convention except:

Bond Markets Not Using Actual/Actual

Bond Market

Day-Count Basis

Eurobonds

30/360

Denmark, Sweden, Switzerland

30E/360

Norway

Actual/365

Example — Calculating the Purchase Price for a Bond with Accrued Interest

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. Accrued interest is determined using the actual/actual convention. 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% × $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 ×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, that provides several functions for determining the number of days or the dirty bond price, with the settlement and maturity dates expressed as either a quote (e.g., "12/11/2012") or as a cell reference (e.g., B12):

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-08Settlement Date
15-Nov-17Maturity Date
5.75%Coupon Rate
6.50%Yield to Maturity
100Redemption value
2Number of Interest Payments per Year
1Day Count Basis (Month/Year = Actual/Actual)
=
94.63544921% of Par Value of Actual Price for Corporate Bond, $1,000 Face Value
$946.35Actual 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.

Steps to Calculate the Price of a Zero Coupon Bond

  1. Total Interest Paid by Zero Coupon Bond = Face Value - Discounted Issue Price
  2. 1 Day Interest = Total Interest / Number of Days in Bond's Term
  3. Accrued Interest = (Settlement Date - Issue Date) in Days × 1 Day Interest
  4. Zero Coupon Bond Price = Discounted Issue Price + Accrued Interest

Bonds with Ex-Dividend Periods may have Negative Accrued Interest

Interest accrues on bonds from one coupon date to the day before the next coupon date. However, some bonds have a so-called ex-dividend date (aka ex-coupon date), where the owner of record is determined before the end of the coupon period, in which case, the owner will receive the entire amount of the coupon payment, even if the bond is sold before the end of the period. The ex-dividend period (aka ex-coupon period) is the time during which the bond will continue to accrue interest for the owner of record on the ex-dividend date. (The ex-dividend date and the ex-dividend period are misnomers, since bonds pay interest and not dividends, but the terminology was borrowed from stocks, since the concept is similar. Although ex-coupon is more descriptive, ex-dividend is more widely used.) If a bond is purchased during the ex-dividend period, then any accrued interest from the purchase date until the end of the coupon period is subtracted from the clean price of the bond. In other words, the accrued interest is negative. Only a few bonds have ex-dividend periods, which are usually 7 days or less. The UK gilt, for instance, has an ex-dividend period of 7 days, so if the bond is purchased at the beginning of that 7-day period, then the amount of interest subtracted from the clean price will be equal to the coupon rate × 7/365.

Most bond markets do not have ex-dividend periods except:

PRICE, PRICEDISC, PRICEMAT, and DISC Functions in Microsoft Office Excel for Calculating Bond Prices and Other Securities Paying Interest

Microsoft Excel has several formulas for calculating bond prices and other securities paying interest, such as Treasuries or certificates of deposit (CDs), that include accrued interest, if any.

Microsoft Excel Functions: PRICE, PRICEDISC, PRICEMAT, and DISC
  • Calculates the price, given the yield.
    • Bond Price (per $100 of face value) = PRICE(settlement,maturity,rate,yield,redemption,frequency,basis)
    • Discounted Bond Price = PRICEDISC(settlement,maturity,discount,redemption,basis)
  • Calculates the yield, given the price.
    • Discount Rate of Security = DISC(settlement,maturity,price,redemption,basis)
  • Calculates the price of a security that pays interest only at maturity, such as a negotiable Certificate of Deposit:
    • Price of Security = PRICEMAT(settlement,maturity,issue,rate,yield,basis)
  • The following dates are expressed in quotes (e.g., "1/1/2012") or are listed as cell references (e.g., A1):
    • Settlement = Settlement date.
    • Maturity = Maturity date.
    • Issue = Issue date.
  • Rates listed in decimal form (5%=.05):
  • Price = Price of security as a percent of par value (but without the % or $ sign, so if $1,000 par-value bond is selling for $857.30, then the corresponding percentage value is 85.73).
  • Redemption = Value of security at redemption per $100 of face value, usually = 100.
  • Frequency = Number of coupon payments / year.
    • 1 = Annual
    • 2 = Semiannual (the most common value)
    • 4 = Quarterly
  • Basis = The number of days counted per year.
    • 0 = 30/360 (This U.S. basis is the default, if omitted)
    • 1 = actual days in month/actual days in year
    • 2 = actual days in month/360
    • 3 = actual days in month/365 (even for a leap year)
    • 4 = European 30/360

Examples—Using Microsoft Office Excel for Calculating Bond Prices and Discounts

The following listed variables — where they apply — will be used for each of the example calculations that follow, for a 10-year bond originally issued in 1/1/2008 with a par value of $1,000:

Price of a bond with 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

The discount price of a zero coupon bond with a $1,000 par value yielding 8%:

Price Discount =PRICEDISC("3/31/2008","12/31/2017",0.06,0.08,100,1) = 21.99288 = $219.93

The interest rate of a discounted zero coupon bond paying $1,000 at maturity, but that is now selling for $219.90:

Interest Rate of Bond Discount = DISC("3/31/2008","12/31/2017",21.99,100,1) = 0.080003 = 8%

PRICEMAT calculates the prices of securities that only pay interest at maturity:

What is the price of a negotiable, 90-day CD originally issued for $100,000 on 3/1/2008 with a nominal yield of 8%, a current yield of 6% and a settlement date of 4/1/2008? Using the Microsoft Excel Date function, with format DATE(year,month,day), to calculate the maturity date by adding 90 days to the issue date, and choosing the banker's year of 360 days by omitting its value from the formula, yields the following results:

Market Price of CD = PRICEMAT("4/1/2008",DATE(2008,3,1)+90,"3/1/2008",0.08,0.06) = 100.3181 (per $100 of face value) × 1,000 = $100,318.10

Related Links

Tax Treatment of Accrued Interest