The first three lessons in this topic priced bonds where settlement fell exactly on coupon dates. In practice, that is never the case: a bond bought on any given day settles one business day later, and coupons are paid only twice a year. This note generalises the pricing framework to handle the full generality of real-world fixed-income markets.
Three generalisations are needed. First, accrued interest: when a bond is sold between coupon dates, the buyer compensates the seller for the fraction of the current coupon that has already accrued — giving rise to the clean/dirty price distinction that all Bloomberg screens display. Second, compounding conventions: different money-market segments and different countries quote rates under different compounding assumptions; continuous compounding, in particular, is the natural language for the dynamic models developed in No-Arbitrage Pricing and the Risk-Neutral Framework and beyond. Third, curve fitting: once bonds mature on arbitrary dates (not neat six-month intervals), the bootstrapping algorithm of Lesson 1 cannot be applied directly and a smooth discount function must be fitted to the full set of market prices.
Learning objectives. By the end of this note you should be able to:
- compute accrued interest and distinguish flat (clean) price from full (dirty) price;
- relate semiannual, continuous, and simple interest rates via the discount factor;
- write the generalised price-yield formula for a bond settling between coupon dates;
- explain why curve fitting is necessary and compare spline-based with parametric approaches at an introductory level.
In this lesson
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- Accrued interest: the clean/dirty price split
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- Compounding conventions
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- Pricing between coupon dates
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- Curve fitting: from sparse prices to a smooth discount function
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- Practical conventions in the U.S. Treasury market
Before you start
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