Bond Yield


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Bond Yield Calculator


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

A bond is a debt instrument in which an investor loans money to the issuer for a defined period of time. Bonds are traded and quoted based on yield to maturity (YTM). The actual settlement clean price depends on the number of coupons available. For bonds with a single remaining coupon, the bond trades at a pure discount (i.e., like a money market instrument). For bonds with multiple remaining coupons, these are priced with a special formula.

Yield is the rate of discount at which the present value of the promised future cash flow equal the price of the bond. When the Coupon Daycount is ACT/ACT, the standard yield-to-maturity is employed using equal-length discounting periods; for any other Coupon Daycount, actual time to each payment date is used in discounting (sometimes called "Actual Yield").

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. The large variety is the main challenge for pricing bonds.

For instance, Australian and New Zealander bonds are traded on a yield basis. Bond prices in these markets are quoted as 100 - yield. The price per $100 is rounded to nine decimal places. However, entered yields will be rounded to one basis point.

The yield to maturity of a bond is the internal rate of return on a bond held until maturity. In other words, it is the discount rate that will provide the investor with a present value V equal to the price of the bond. The yield to maturity does not account for the actual term structure of interest rates.

The present value formula, as well as the yield to maturity expression, can be modified to generate alternate measures of yield that might be common in certain markets. For example, in the US Treasury market, yields of bills are quoted in terms of equivalent bond yields. This corresponds to the coupon of a US Treasury bond which, when traded at par (i.e., when traded at price P=V) would give the same return as the bill.

The bond price from yield is the price obtained by discounting all cash flows of the bond with the given fixed yield. The yield is assumed to be quoted in the compounding frequency corresponding to the coupon frequency, and thus the coupon frequency must be chosen even for zero-coupon bonds.

For bonds with more than one coupon date remaining, there is no closed-form expression for the yield. However, there is an approximation that can be used to seed a Newton-Raphson search very accurately from the bond price formula below, so that usually only a single iteration is required to find the yield to four decimal digits.

bond yield calculation

where

  • τ0 is the time from the settlement date to the next coupon date.
  • τ is the coupon interval as a year fraction
  • ν=1/(1+yτ)
  • n is the number of coupon periods after the settlement date
  • c is the bond coupon rate
  • y is the bond yield

The equation allow the transformation of yields expressed with different compounding methods to a common basis for evaluation.

The formula above is applied to a regular bond. But for a large number of irregular bonds, we need to calculate yield-to- maturity differently, such as, a Canadian government bond or a US treasury security.

Canadian Government Bonds are traded and quoted based on yield to maturity (YTM). The actual settlement clean price depends on the number of coupons available. For bonds with a single remaining coupon, the bond trades at a pure discount (i.e., like a money market instrument). For bonds with multiple remaining coupons, these are priced with a special formula.

Canadian government with a single coupon remaining are quoted using simple interest conventions. In this case, yield is quoted as a money market yield-to-maturity.

For a Canadian govenment bond with more than one coupon remaining, the unadjusted dirty price of the bond (at settlement) is

Canadian bond price

where

  • N denotes the number of remaining coupons,
  • E denotes the number of days in the first coupon period that includes the settlement date,
  • DSC denotes the number of days between settlement date and coupon payment, and
  • Y denotes the semiannual yield-to-maturity of the bond

US Treasury securities play a central role in US financial system. A Treasury bill is a zero-coupon bond with a maturity less than one year. A Treasury note is a coupon bond with a maturity less than 10 years, while a Treasury bond is a coupon bond with a maturity greater than 10 years.

Unlike corporate bonds, the interest earned on Treasury securities is exempt from state taxes. Since Treasury securities are default-free, they usually have lower yields than corporate bonds.

A Treasury bill is quoted by discount yield that reflects the return of the face value rather than the purchase price. Since the purchase price is normally cheaper than face value, the discount yield is understating the yield. Alternatively investor can calculate the investment yield to correspond to the return to the purchase price.

T-bill discount yield

where

  • F denotes the face value,
  • B denotes the purchase price,
  • m denotes the number of days to maturity

To price floating rate bonds, such as a bond paying LIBOR plus a fixed spread (LIBOR is an interest rate widely used as an index for floating rate instruments), the Infinity applications rely on replicating the cash flow profile of the security. Based on the no-arbitrage assumption, any security may be evaluated using an equivalent group of securities.

To further simplify the example, consider a bond that pays LIBOR a year from now, with no spread. The cash flow profile can be replicated with a one-year deposit of the principal in the money market. The resulting cash flow would be one-year LIBOR in interest plus the principal back. The appropriate discount factor in this case is the inverse of one plus LIBOR (otherwise arbitrage opportunities would arise). Thus the present value of the one-year bond is identical to its principal.

A spread, or fixed amount, is often added to the rate that a floating interest instrument pays. One example is the addition of a spread to reflect default risk. Consider a counterparty promising to deliver a certain cash flow in the future: if there is a possibility that the counterparty may not deliver the cash flow, we should expect that the present value of that cash flow should be reduced, or that the interest rate used in the present value calculation should increase, for example, from LIBOR to LIBOR plus a certain spread or premium.

The coupon rate for a floating rate note changes each period depending on some reference rate, such as treasury curve or LIBOR curve or SOFR curve. Since the coupon rate in future is unknown, it is impossible to determine the cash flows. That means that a yield to maturity cannot be computed. Instead, a discount margin is used for potential return for floating rate bonds. The discount margin reflects the average spread over the reference rate.

There are situations where forward yield or volatility adjustments are required. For example, to obtain the expected bond yield in a world that is forward risk neutral with respect to a zero coupon bond maturing at time , due to the non-linearity between bond price and bond yield, a convexity adjustment is required.


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