Bonds seem like a sleepy fixed-income investment to many people but there are calculations, rules, and pricing conventions concerned with how the market price of a bond is determined that makes the bond market exciting and sometimes esoteric.
Basic Bond Price
A bond’s face value that is on the bond certificate when it is purchased is known as its par value. The amount of the bonds market price that is over its par value is known as a bond premium. When the market price is below the par value it is said to be selling at a bond
discount. The interest rate that it pays is known as the coupon rate.
When calculating the price of a potential bond investment an investor is determining the maximum amount of money that he would be willing to pay compared to the average rate currently being received by investors in the market. Usually a bond premium or bond discount are used to adjust a bonds rate of return to the current prevailing interest rates to make it more a more attractive investment choice for investors.
Bond Price Formula
There is a basic formula for calculating a bonds price that is just the sum of the present value of all of the coupon payments and the par value when the bond is redeemed at maturity. Instead of calculating all of these values to arrive at their sum the shorter bond price formula is used. This is:
Bond Price = C × [1 - [ 1÷ (1 + i)-n ]÷ i] + M ÷ (1 + i)n
Bond Par Value received at maturity (M) = 1,000
Term: paid in 10 years
Coupon Rate = 10%
Required Yield = 12%
Coupon Payments = semi-annual (paid every six months)
Value of each coupon payment (C) = $50 ($1,000 × 0.05) or ($1,000 × .10) ⁄ 2
Semi-annual yield (required yield) (i) = 6% (12% / 2)
These amounts can be plugged into a formula:
Bond Price = 50 × [1 - [1 ÷ (1 + 0.06)20 ]] ÷ 0.06 + 1000 ÷ 1 + (0.06)20 = 50 × 11.47 + 311.82
$885.32 = 50 × 11.47 + 311.82
This particular bond is trading at a discount that gives a rate of 12% instead of the 10% at par value. This is necessary to make it competitive with any other bond investment that is selling at the markets prevailing rate which is 12%.
The bond in the previous example pays out coupon payments twice a year. Some bonds can pay more frequently. Some might pay monthly, quarterly, or annually. The following formula accounts for this:
Bond Price = C ÷ F × [1 - [1 ÷ (1 + (i ÷ F)n × f)] ÷ (i ÷ F)] + M ÷ (1 + i ÷ F)n × F
The variable F is added to the formula and represents the frequency per year of the coupon payments. Semi-annual payments for 10 years would make F equal to 20.
Zero coupon bonds are bonds that have no coupon payments until the maturity date. This means the formula is shortened to:
Zero Coupon bond Price = M ÷ (1 + i)n
When a bond is bought or sold between coupon payment periods an investor must know how to calculate accrued bond interest for the seller and the bond interest earned after the transaction date to the date of the next coupon payment. These calculations use a day count convention of either an actual/actual day count or a 30/360 day count. The actual/actual day count uses the actual number of days in a 365 day year (366 days in a leap year). This is commonly used for Treasury bond transactions. A Treasury bond purchased on March 1, 2005 with the next coupon payment due 122 days later on July 1, 2005 would have 60 days of interest accrued on it that would be paid to the seller.
Other bonds, such as corporate and municipal bonds, use a 30/360 day count convention for calculating accrued interest. If the bond in the previous paragraph was a corporate bond purchased on March 1, 2005 there would have been 120 days left until the coupon payment date. The number of days of accrued interest remains at 60 days. This convention assumes that each month has 30 days and is simpler to use than the actual/actual day count.
The accrued interest calculation is:
Accrued Interest (AI) = Coupon × [Days between settlement and last coupon payment ÷Total days in period]
Therefore, if the bond in the previous paragraph was a $1,000 bond that carried an annual interest of 7% paid semiannually (7% ÷ 2), the accrued interest would be:
Accrued Interest = $35 × [60/180] = $11.67
Bonds are usually quoted at “clean prices”. This means that the price being quoted is based on the cash flows starting from the transaction day. The buyers purchase bonds at the “dirty price” meaning that the price that they pay includes accrued interest paid to the seller.
Yield and Price
Another factor that affects bond prices is its required yield. This is not the yield-to-maturity. It is the yield that a seller must offer to a buyer that makes the price competitive with other similar bonds of a similar maturity and credit quality.
There are debt-rating agencies that provide ratings for the credit quality of corporate bonds. Among them are Standard & Poor’s, Moody’s Investor’s Service, Fitch Ratings, and A.M. Best.
When a required yield has been determined by the investor a current yield calculation must be made for the potential bond investment. Current yield is the percentage return that an annual coupon payment will provide between the purchase date and the maturity date for the investor. An adjustment for capital gains or losses from premiums and discounts must also be included. Using a
$100 bond purchased at a discount for $95.92 at 5% interest maturing in 30 months as an example, the formula is:
Current Yield = [Annual Coupon ÷ Market Price] × 100% + [(100 - Market Price) ÷ Years to Maturity]
6.84% = [$5 ÷ $95.92] × 100% + [(100 − 95.92) ÷ 2.5]
The most commonly used term that investors and analysts coin when referring to yield is the yield to maturity. This is a yield calculation that takes into account the time value of money which is the present value of all the future cash flows received from a bond that equal the bonds market price. Determining the interest for this calculation is a rather hit or miss proposition as different interest rates must be plugged into the formula to try eventually find the best fit. Usually analysts use a calculator or programs that can arrive at this number with speed and little effort. If these are not available then there are tables that can be used to find the closest interest rates with corresponding prices. Usually more than one table must be used for carrying out the estimate to adequate decimal places.
The basic formula is:
Bond Price = Cashflow × [1 - [1 ÷ (1 + interest rate)n] ÷ interest rate] + [Maturity Value × 1 ÷ (1 + interest rate)n ]
Yield curves, the term structure of interest rates, are used to compare a group of similar bonds that have different maturities. Each point on the graph is calculated with a zero-coupon rate known as the spot rate and conveys how a bond’s yield changes as it approaches maturity. The uses of yield curves are various and can be used to help price new bond issues, how the market price of a bond is determined when comparing bonds, and provide a better understanding of bond risks.
Duration is another factor that enters into a bond price. Basically, this indicates how long it will take for a bond’s internal cashflows to repay the cost of the investment. It is a complex calculation and there are various ways to calculate it. A higher number usually indicates a greater interest-rate risk, or volatility.
Bond Pricing Conventions
The bond market itself consists of many kinds of bonds that can be broken down into classifications that have their own bond pricing conventions. Corporate bonds use the on-the-run U.S. Treasuries as a pricing benchmark for bonds with corresponding maturities. A curve on a graph is determined using the maturities of the underlying securities, in this case the Treasuries. The points of the curve are plotted by using the yields of the benchmark securities with their maturities. In the case of Treasuries they would range from 3 months to 30 years.
When comparing a bond yield to it’s benchmark’s yield there is usually a difference between the two known as a spread. This is useful
when comparing the value of bonds in relation to each other. A 10 year corporate bond might be trading at a 70 basis point spread above it’s 10 year Treasury bond benchmark which would be a return 7% greater than the return on the Treasury bond. Any corporate bond with the same maturity and all other factors being the same (credit rating, duration, etc.) would be a better value if it’s spread was greater than 7% or a lesser value if it’s spread was below 7%.
Different types of bonds have different benchmarks and spread calculations. These bond types are:
- U.S. Treasuries
- Corporate Bonds
- High Yield Bonds
- Mortgage-Backed Securities (MBS)
- Asset-Backed Securities (ABS)
- Agency Bonds: Debt issued by government-sponsored enterprises (GSEs)
- Municipal Bonds (Munis)
- Collateralized Debt Obligations (CDOs)
Popular benchmarks are:
- Spot Rate Treasury Curve
- Swap Curve
- Eurodollars Curve
- Agency Curve
The four primary yield spreads are:
- Nominal Yield Spread
- Zero-Volatility Spread (Z-spread)
- Option-Adjusted Spread (OAS)
- Discount Margin (DM)
How the market price of a bond is determined can be a very involved process. Investors have various mathematical formulas for calculating yields, bond prices and other relevant factors. Also, over the years the bond market has developed its own bond pricing conventions for determining the prices of bond issues and comparing bonds in homogenous groups. Understanding it might seem like a daunting task at first but it’s not insurmountable if you keep in mind some of the the basics.
Barry Nielson, CFA, How Bond Market Pricing Works
Joanna Place, Basic Bond Analysis Centre for Central Banking Studies, 2000. Bank of England, London.
Bankrate.com Corporate Bond Risks and Rewards
Investopedia Advanced Bond ConceptsUniversity, 2011
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