Pricing Between Coupon Dates
Bonds don't only trade on coupon dates. Buy one partway through a period and you owe the seller a slice of the coupon that's already accruing to them — so a purchase mid-period involves three prices, not one: the flat (dirty) price you actually pay, the accrued interest owed to the seller, and the clean (market) price that's actually quoted.
By the end you'll be able to compute the flat price, accrued interest, and clean price at any point between two coupon dates, and explain why the clean price — not the flat price — is the one that moves smoothly through a coupon date.
Predict: as you drag t across a coupon date, does the clean (market) price climb smoothly, or does the flat price — which one actually jumps? Drag the slider across t = 1 to check.
This bond has book value 1147.20 right after purchase and pays an annual coupon of 80 at yield 6%. Drag t to move through the first two coupon periods. The flat price accumulates the book value forward at \((1+i)^t\) (theoretical method) and jumps down by the coupon the instant it's paid; the accrued coupon rises linearly as \(t \cdot Fr\) and resets to 0 at the same instant; the clean price (flat minus accrued) is what's left over — watch what it does at the coupon date.
The basic bond formula prices a bond exactly on a coupon date. A buyer who settles partway through a period needs three related prices to settle up fairly with the seller.
Let B be the book value at the last coupon date and let \(t \in (0,1)\) be the fraction of the period elapsed. The flat (dirty) price accumulates the book value forward at the yield rate: \(P_{\text{flat}} = B(1+i)^t\) (theoretical method; the practical method uses \(B(1+i \cdot t)\) instead — pick one convention and don't mix them). The accrued coupon is always a straight-line fraction of the coupon, even under the theoretical method: \(AI = t \cdot Fr\). The clean (market) price strips that out: \(P_{\text{clean}} = P_{\text{flat}} - AI\). At a coupon date (\(t=0\)), accrued interest is zero and clean = dirty = book value, so this reduces to the amortization schedule's book-value recursion.
Newspapers and exchanges quote the clean price so quotes don't visibly sawtooth as coupons approach and get paid. But the buyer doesn't get a discount for that — they actually pay the flat (dirty) price, which is clean quote plus accrued interest. That's how the seller gets reimbursed for the days they held the bond since the last coupon, without the quoted price itself jumping around.
Take the F = C = 1000, Fr = 80 (annual), i = 6% bond with book value B = 1147.20 at the last coupon. Bought t = 0.25 of the way through the year (theoretical method): \(P_{\text{flat}} = 1147.20 \times 1.06^{0.25} = 1164.03\). \(AI = 0.25 \times 80 = 20.00\). \(P_{\text{clean}} = 1164.03 - 20.00 = \) 1144.03.
Same bond, one period later: per the amortization schedule, the book value right after the first coupon is 1136.03. It's bought t = 0.5 into the SECOND period, still at yield 6% with the same 80 coupon. \(P_{\text{flat}} = 1136.03 \times 1.06^{0.5} = 1169.62\). \(AI = 0.5 \times 80 = 40.00\). Now finish it: \(P_{\text{clean}} = 1169.62 - 40.00 = \) ____
Reveal the answer
\(P_{\text{clean}} = 1169.62 - 40.00 = \) 1129.62. Notice this uses the UPDATED book value 1136.03 from the amortization schedule, not the original 1147.20 — the starting point resets at every coupon date. Set t = 0.5 in the second period on the slider above and check the readouts against these numbers.
More info — why the clean price is the one that's quoted
The flat price jumps down by the full coupon the instant it's paid, then climbs back up as the next coupon accrues — a sawtooth that has nothing to do with changing market conditions. Subtracting accrued interest removes exactly that sawtooth, so a change in the clean price reflects a real change in yield, not just the calendar. See the Wikipedia link in Dive deeper for how clean and dirty prices are defined and used in practice.
Check your understanding
A bond's book value at the last coupon date is 1000, and the yield rate is 8% convertible annually. Using the theoretical method, what is the flat price one-half period later (t = 0.5)?
The instant a coupon is paid, what happens to the flat (dirty) price, and why?
A 1000 par, 8% annual coupon bond bought to yield 6% has price 1147.20, and per its amortization schedule the book value right after the first coupon is 1136.03 (interest 68.83, write-down 11.17). Using the theoretical method, what is the flat price a quarter-year (t = 0.25) into the SECOND coupon period?
Recap
- Flat (dirty) price \(P_{\text{flat}} = B(1+i)^t\) (theoretical) or \(B(1+i \cdot t)\) (practical) — the amount actually paid.
- Accrued coupon \(AI = t \cdot Fr\) — always a straight-line fraction, even under the theoretical method.
- Clean (market) price \(P_{\text{clean}} = P_{\text{flat}} - AI\) — the quoted price, smooth through coupon dates because the sawtooth has been subtracted out.
- The flat price is what jumps down at each coupon date, resetting to the new book value from the amortization schedule.
Dive deeper
- thismatter.com — Bond Pricing, Accrued Interest, Illustrated with Examples See how accrued interest is added between coupon dates with examples.
- Clean price — Wikipedia Read the definitions of clean price, dirty price, and accrued interest.
Sources
- Pricing Between Coupon Dates — Clean and Dirty Price