ISMA Bond Pricing

The following bond pricing method is based on the ISMA redemption yield formula. It is applicable to bonds with:

1.

Fixed coupon payments that do not vary as a function of the actual dates between payments (with the possible exception of the first and last payments).

2.

No allowance for the precise timing of the cash flows. That is, bonds are conventionally priced by assuming that each coupon payment falls on the nominal payment date with no adjustment for holidays or weekends.

3.

A single redemption date and a fixed redemption value (bullet bonds).

4.

No call or put provisions.

With these restrictions, the settlement (dirty) price of a bond with a face value of 100 can be determined as a function of redemption yield, i.e.,

P = dirty price (clean price plus accrued interest) of the bond

r₁ = number of days from the settlement date to the next nominal coupon payment date (based on the appropriate accrual convention).

r₂ = number of days from the date of the last 'normal' coupon payment to the maturity date
(based on the appropriate accrual convention). This is equal to zero if the bond does not have an odd final period.

s = number of days in the relevant coupon payment period (based on the appropriate accrual convention).

d₁ = the first/next coupon payment. For an odd first period, this amount may differ from the standard coupon payment. It may also equal zero if the bond is trading ex-dividend.

d₂ = the coupon payment due on the next nominal payment date. For bonds with an odd first period, this amount may differ from the standard coupon payment.

c = annual coupon payment per 100 units of face value.

c* = the final coupon amount for a bond with an odd final period. This is zero for all other bonds.

h = number of coupon periods in a year

n = number of full coupon periods remaining until redemption. The number of remaining coupon payments is therefore equal to .

v = the discount factor for one period, 

y = the required annual nominal redemption yield, expressed as a decimal.

As a basis for deriving some of the risk statistics, it is also useful to give an expression for bond price when there are only 1 (n = 0) or 2 (n = 1) outstanding coupon payments. In these cases the ISMA formula simplifies to:

Because many markets trade bonds on a price basis, it is often necessary to compute yield to maturity as a function of a given price. As the above equations cannot be rearranged to express yield as a algebraic function of price, the yield to maturity must therefore be determined iteratively using a trial and error approach