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Consider the attached discussion from Berk and Demarzo's Corporate Finance.

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I am confused about the calculation of a bond's "clean price". It seems that the procedure described above seems to tacitly "linearly discount" the accrued interest, rather than discounting cash flows with an appropriate factor. Put differently, why does the given graph suggest that Dirty Price grows linearly in between coupon payments, rather than according to some $(1+r)^t$ factor as would be expected based on (rigorous) discounting?

EE18
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  • The clean price is a convention which provides the buyer and seller a "common language" when they bargain with each other. We could equally agree that the price must be quoted in hexadecimal notation or whatever. Just a way to encode information. From an economic point of view what matters (i.e. what determines profit and loss) is the dirty or all-in price. At least that's what I was taught. (Though the original purpose of the convention does seem to have been a rough and ready way to make the chart look less jumpy). – nbbo2 Dec 04 '23 at 13:59
  • Thank you very much @nbbo2 I think your last (bracketed) sentence is exactly what I was wondering about :) – EE18 Dec 04 '23 at 15:15

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In most, but not all, markets, the convention is to quote a bond's clean price (ex accrued). When a bond is actually traded in a secondary market, the proceeds are the dirty price: cum accured until the settlement date, and also taking into account the factor for amortizing and PIK bonds.

Traditionally, the accrued interest is calculated as your book describes: a period fraction times the outstanding principal. The calculation of the period fraction may get quirky in practice - some bonds use actual days, but a few accrue only on business days, etc. Some floaters linked to an index observed daily do use daily compounding. I don't recall any examples of fixed-coupon bonds paying "interest on interest". This is just the traditional market practice. Here is a 2005 paper https://www.economics-finance.org/jefe/fin/Secrestpaper.pdf arguing that daily compounding affects a bond's fair price.

Dimitri Vulis
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  • Thank you very much for your linked paper. After reading it, it seems (along with your answer) to confirm the suspicion that true bond price (or transaction price, as the paper says) is found my correctly discounting, but that this subtraction of accrued interest from true/transaction price is done by convention in financial markets? – EE18 Dec 04 '23 at 15:14
  • It may also help to consider what happens if you buy a stock, rather than a bond, when a divided has already been declared, but before the ex date, Everyone expects the fair price to drop by the dividend amount on the ex date. But we don't really separate the price into "dividend certain" and "other uncertain" :) With bonds, the information conveyed by the clean price and the dirty price is exactly the same. You pay the dirty price, but, unlike stocks, there is a traditional methdology, approved by tax authorities and accountants, to split up the dirty price into a part that (cont) – Dimitri Vulis Dec 05 '23 at 12:19
  • you pay for the principal and the part that you pay for the accrued coupon. My understanding of the Secrest et al article is that they suggest that when looking for a bond price to analize, rather than use the traditional clean price methodology, we start from the dirty price, but do something more complicated. Anyway, most of the time, unless the bond is very distressed and trades on price, people mostly analyze based on its yield, which still conveys the same information as either the clean price or the dirty price. – Dimitri Vulis Dec 05 '23 at 12:24