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ISMA Formula Example - Yield


Consider a 20-year French Government Bond (OAT) trading at a price of 99.75 with a coupon rate of 7.25%, a dated date of 25 June 1996, a maturity date of 25 June 2016, and a face value of $100,000. What is the yield of this bond assuming a settlement date of 1 May 2003?



Function Specification

=oBondFR_Yield(99.75, "1/5/03", "25/6/96", , , "25/6/16", 100000, 0.0725, 0)

For convenience, we have assumed no odd first or last coupon periods.



Referring to the ISMA formula, we get the following inputs:

P = 99.7500
r1 = 55 days (25/6/03-1/5/03)
r2 = 0 days (no odd final period)
s = 365 ('Actual' day count convention)
d1 = 7.25 (coupon per $100 Face Value)
d2 = 7.25 (coupon per $100 Face Value)
c = 7.25 (coupon per $100 Face Value)
c* = 0.00 (no odd final period)
h = 1 (annual coupons)
n = 13 (full coupon periods remaining)



As the ISMA formula cannot be rearranged to give yield as an algebraic function of price, the Newton-Raphson iteration procedure is used to solve for y.


When calculating yields, the Newton-Raphson iteration procedure uses a seed value as suggested by Brown (1998) for the initial estimate of the yield, y0 (see Newton Raphson Starting Values). The procedure will iterate using more and more precise estimates of the yield until the difference between the PPH derived from using the yield estimate and the given market PPH of the bond is less than the desired accuracy level. The desired accuracy level for yield iterations is 9 decimal places.



Referring to Newton Raphson Starting Values, the initial estimate of the yield is determined as follows:

Equation Template


Using the above inputs with our estimated yield, the ISMA formula becomes:

Equation Template



With the accrued interest of 6.1580 (PPH), the clean price is 99.6585. As this price is below the given PPH (99.75), the ISMA formula is recalculated with a lower yield, i.e., y1 < y0. The Newton Raphson procedure determines y1 as:

Equation Template

where -826.8183 is the partial derivative, i.e.,. Equation Template of the above ISMA formula with respect to yield.


Using the above inputs with our new estimate of the yield, the ISMA formula returns a clean price of 99.7501, slightly above the given PPH of 99.7500. Therefore the ISMA formula is recalculated with a marginally higher yield, i.e., y2 > y1. This procedure continues until the convergence criteria is met, which for this example occurs on the 3rd iteration at a yield of 0.072761. The final ISMA calculation would appear as:

Equation Template



As the difference between the equated PPH and the given PPH is less than the desired accuracy level (9dp) the equivalent annual yield of this bond is set equal to 7.2761%.




As the bond has a face value of $100,000, and the output flag has been set to 0 (see French Bond Yield Function), the following outputs are produced.



Clean Price


Accrued Interest


Dirty Price


Macaulay Duration


Modified Duration




Price Value of a Basis Point




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