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Par coupon from dated discount factors

For equal annual coupons and redemption at year n, the par coupon rate is (1−Dn) divided by the sum of dated factors.

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Specify the coupon schedule

Let face amount be F and the annual coupon amount be Fc. Coupons occur at years 1 through n, with F redemption at n. D1 through Dn are factors for exactly those dates. The example excludes accrued interest and irregular first periods.

Separate coupons from redemption

At par, F = FcΣDt + FDn. Divide by positive F and isolate c. The final coupon is already included in the sum; redemption is the separate FDn term.

c=1−Dn∑t=1nDtc=\frac{1-D_n}{\sum_{t=1}^{n}D_t}

c is an annual coupon-to-face ratio for this annual schedule.

Insert the coupon back into the price

Using the three illustrative factors 0.961538, 0.907029, 0.839619, c = 5.922069%. With F = 100, the annual coupon is 5.922069. Its coupon present value plus redemption present value sums to exactly 100 before rounding.

Change the schedule before changing the label

For equal semiannual coupons, a periodic coupon-to-face amount is solved from the actual half-year factor sum. A nominal annual coupon rate would be twice that periodic ratio. Do not use three annual factors as if they were six half-year factors.

Further references