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Bond Price

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Bond Price Calculator

FinPricing provides probably the most comprehensive valuation models for financial products, including computation of:

• Bond Accrued Interest Calculator
• Bond Coupon Rate Calculator
• Bond Price Calculator
• Bond Yield Calculator
• Bond Duration Convexity Calculator
• Bond Spread Calculator
• Check FinPricing valuation models

An entity can raise capital in financial markets either by issuing equities or bonds. A bond is a debt instrument in which an investor loans money to the issuer for a defined period of time. An institution's long-term debt is usually involves interest-only loan. The security that guarantiees interest and principal payments is called a bond. A bond may involve more than one interest payment during a year.

There are more than 1,500 different types of bonds from valuation perspective. The differences are specified by calculation type. Each calc type defines a method used to determine the accrued interest, price, yield of the bond based on specified market conventions and security structures.

For example, South African bonds are yield-quoted and may be entered to five decimal places. They have two fixed ex-dividend dates per year. Exdividend dates have a fixed offset from the coupon date and, therefore, will not roll forward to account for holidays. The settlement date is automatically calculated as the second Thursday following the trade date.

Over the life of the bond, the investor will receive coupons paid by the issuer at fixed/floating interest rate. The bond principal will be returned at maturity date. Bonds are usually issued by companies, municipalities, states/provinces and countries to finance a variety of projects and activities. Based on issuing types, bonds can be categorized into

• Government bond or sovereign bonds
  A government bond is issued by government or government agency to support central government spending. Government has obligation to guarantee to pay back the interest and principal. Typical government bonds are treasury bills, treasury notes, treasury bonds, etc.
• Municipal bonds
  Municipal bonds are issued by states/provinces or municipalities to fund special spending or projects.
• Corporate bonds
  Corporate bonds are issued by corporation, institutions, and commercial entities. They usually have higher yield.
• Foreign bonds
  Foreign bonds are issued by foreign entities, such as Eurobonds.

From valuation perspective, we can also divide bonds into several major types based on instrument types.

• Fixed rate bond
  Fixed rate bonds generally pay higher coupons than interest rates. An investor who wants to earn a guaranteed interest rate for a specified term can choose fixed rate bonds. The benefit of a fixed rate bond is that investors know for certain how much interest rate they will earn and for how long. Due to the fixed coupon, the market value of a fixed rate bond is susceptible to fluctuation in interest rate and therefore has a significant interest rate risk. Most issued fixed rate bonds are treasury bonds and treasury notes.
• Floating rate note
  A floating rate note is a bond that has variable coupons, depending on a money market reference rate, such as treasury floating rate notes.
• Zero coupon bond
  A zero coupon bond is a bond that doesn’t pay interest/coupon but instead pays one lump sum face value at maturity. Typical zero coupon bond is treasury bill.
• Callable bond
  A callable bond is a bond in which the issuer has the right to call the bond at specified times from the investor for a specified price.
• Inflation linked bond or real return bond
  An inflation indexed bond is a bond whose principal is indexed to inflation or deflation on a daily basis in terms of a reference index, such as real return bond issued by Canadian government. Check inflation curve

The bond pricing formula relies on discounting future cash flows to find the present value of each cash flow. With limited modifications, the present value formula can be applied to most other financial instruments.

Therefore, a bond may be priced knowing the discount factors, principal, and coupon interest rate. The value of the bond is directly proportional to the principal value P. For most bonds, the price is normalized by setting P=100, and quoting its price as a percentage of par.

The value of a bond between coupons will thus have two components: its “clean” price and the accrued interest. The value must equal the future cash flows discounted by the appropriate discount factors. However, using a constant rate as the discount factor (a constant yield to maturity), we can determine a relationship between price and yield to maturity.

The present value can be calculated in two steps: first, all future cash flows are discounted using the next coupon payment date as the valuation date, and, second, the resulting present value for all future cash flows is multiplied by a second discount factor for the remaining fraction of the coupon.

In general, the cash flows can be expressed as an interest coupon rate times the principal.There are two types of bond valuation models in the market: yield-to-maturity model and credit spread model.

2.1 Yield-To-Maturity Model

The present value of a bond under the yield-to-maturity model is given by

where c = C / nFreq and C is the annual coupon rate and nFreq is the payment frequency per year.

When purchasing a bond, the investor will pay the asked price plus the accrued interest since the last coupon payment. The accrued interest is computed by taking the size of the last coupon payment and multiplying by the ratio of the accrual days since the last coupon payment and the actual number of days in the coupon period.

The total price paid for a bond is referred to as the dirty price, whereas the quoted price is often called the clean price. The dirty price is equal to the clean price plus the accrued interest. The accrued amount, is equal to the accrued interest times the bond notional. The accrued interest is a fraction of the next coupon to be paid, pro rata temporis.

Practical Notes

• The present value of a bond computed by any pricing models is the bond dirty price. To purchase a bond, the buyer pays the dirty price.
• Although investors pay dirty prices, bonds are typically quoted in terms of clean prices 
  Dirty Price = Clean Price + Accrued Interest
• The Yield-To-Maturity Model is a good tool to compute the present value or the fair value of a bond. But it is impossible to calculate risk, such as sensitivities, that is more important than the fair value in trading, hedging and risk management. Therefore, we introduce the Credit Spread Model for computing both risk and fair value.

2.2 Credit Spread Model

The present value of a fixed rate bond under the credit spread model can be expressed as

where

• t is the valuation date
• i is the i-th cash flow
• r_i is the continuoups compounded interest rate for period i
• T_i is the coupon payment date of the i-th period
• s is the bond spread
• P is the principal or face value
• c is the coupon rate

The credit spread model can be used to compute both risk and fair value for even very exotic bonds. From now on, we will focus on the credit spread model for the rest of products.

Practical Notes

• Intuitively,  can be regarded as a credit risk adjusted discount factor.
• First, one should generate cash flows based on the start time, end time and payment frequency of the bond, plus calendar (holidays), business convention (e.g., modified following, following, etc.) and whether sticky month end.
• Second, one needs to construct yield curve by bootstrapping the most liquid interest rate instruments in the market. FinPricing provides useful tools to build various curves, such as swap curve, basis curve, OIS curve, bond curve, treasury curve, etc.  View market data analytic tools
• Then calibrate the model price to the market quoted price by solving the credit spread. Comparing to yield curve construction or calibration for exotic products, the solving here is very simple.
• After making the model price equal to the market price, one can calculate sensitivities by shocking yield curve and credit spread.
• We use LIBOR curve plus credit spread rather than bond specific curves for discounting because bond specific curves rarely exist in the market, especially issued by small entities. Using LIBOR curve plus credit spread not only accounts for credit risk but also solve the missing data issue.

FinPricing provides a simple interface to price a bond and calculate its sensitivities.