Flat Price, Full Price and Accrued Interest: What You Actually Pay for a Bond
A bond quoted at 97.40 does not settle at 97.40. The buyer wires a larger number, and the difference is interest that the seller earned but has not yet been paid.
That gap is the flat price against the full price, and it is one of the few places in fixed income where the arithmetic is trivial and the marks are still lost routinely. The reason is that three different quantities travel under six different names, the day count that converts between them changes with the instrument, and questions are written to test whether the candidate noticed which of the two prices was being asked for.
Key Takeaways
- The flat price is what is quoted; the full price is what is settled, and the difference is accrued interest.
- Accrued interest = coupon × (days since the last coupon / days in the coupon period), accrued in a straight line even though the bond itself is valued by compound discounting.
- Government bonds conventionally use actual/actual and corporate bonds 30/360, and under 30/360 a settlement date on the 31st is treated as the 30th.
- On a 6% semiannual bond settling on 31 August, the two conventions gave 0.766 and 0.750 of accrued interest, a gap worth about 1,600 on 10 million of face value.
- Discounting the remaining cash flows at the yield produces the full price, not the flat price; the flat price is obtained afterwards by subtraction.
- The full price falls by the coupon amount on the payment date, which is a cash transfer rather than a loss of value, exactly as an equity price falls ex-dividend.
What the flat price is, and what you actually pay
A bond quote of 97.40 is a flat price. It is the price of the bond stripped of the interest that has built up since the last coupon date. Nobody settles at that number unless the settlement date happens to fall exactly on a coupon date.
What changes hands is the full price, and it has two parts.
Full price = flat price + accrued interest
the flat price is what is quoted, the accrued interest is what has built up since the last coupon, and the full price is what the buyer wires
The vocabulary is unhelpfully rich, because the market and the curriculum use different words for the same three quantities. The flat price is also the clean price or the quoted price. The full price is also the dirty price, the invoice price, or the price plus accrued. None of these is a different calculation. They are different names for the same two numbers and their sum.
| Quantity | Also called | What it is |
|---|---|---|
| Flat price | Clean price, quoted price | The price excluding interest earned since the last coupon |
| Accrued interest | AI | The seller’s share of the coupon now approaching |
| Full price | Dirty price, invoice price, price plus accrued | The cash actually settled on the trade |
One point matters more than the naming. Every valuation formula in fixed income, the present value of the cash flows discounted at the yield, produces the full price. The flat price is the derived figure, obtained afterwards by subtracting accrued interest. Candidates usually meet the two in the opposite order, see the quoted price first and treat the full price as an adjustment to it, and the arithmetic then feels arbitrary. It is the other way round.
Accrued interest: the seller’s share of the coupon
A coupon is paid in full to whoever holds the bond on the payment date. If a bond changes hands two months into a six-month coupon period, the buyer will collect the whole coupon four months later, including the two months that belonged to the seller. Accrued interest settles that at the point of sale.
Accrued interest = coupon payment × (t / T)
t is the number of days from the last coupon date to the settlement date, and T is the number of days in the full coupon period
Two features of that fraction deserve attention. The first is that accrual is linear, a straight-line share of the coupon, even though the bond itself is valued by compound discounting. That inconsistency is deliberate and universal: a market convention chosen because it is simple and unambiguous, not because it is theoretically clean.
The second is that t is counted from the last coupon date, not from the trade date and not from the date the buyer decided to buy. Settlement is what fixes t, which is why a trade settling on a different day, even one business day later, produces a different invoice.
Day count conventions, and where the numbers diverge
Counting days sounds like the part that cannot go wrong. It is the part that most often does, because there is more than one way to count them and the market picks the method by instrument type rather than by logic.
| Convention | How days are counted | Typically used for |
|---|---|---|
| Actual/actual | Real calendar days, over the real length of the period | Government bonds |
| 30/360 | Every month is 30 days, every year 360 | Corporate bonds, many municipal and agency issues |
| Actual/360 | Real days, over an assumed 360-day year | Money market instruments in several markets |
| Actual/365 | Real days, over an assumed 365-day year | Money market instruments in others |
The 30/360 convention has a further wrinkle that catches people. A settlement date falling on the 31st of a month is treated as the 30th, because the convention does not admit a 31st day. That single rule is enough to make two correct calculations disagree.
A bond pays a 6% annual coupon in two instalments of 3.00 per 100 of face, on 15 January and 15 July. It settles on 31 August at a quoted flat price of 97.40. What is the full price under each convention?
- Under actual/actual, count real days from 15 July to 31 August: 16 in July plus 31 in August = 47 days.
- The coupon period from 15 July to 15 January is 184 actual days, so t / T = 47 / 184 = 0.2554.
- Accrued interest = 3.00 × 0.2554 = 0.766, and the full price is 97.40 + 0.766 = 98.166.
- Under 30/360, the 31st becomes the 30th, so the count is one full month of 30 days plus (30 − 15) = 45 days, over a period of 180.
- Accrued interest = 3.00 × 45 / 180 = 0.750, and the full price is 97.40 + 0.750 = 98.150.
Answer: 98.166 under actual/actual and 98.150 under 30/360. The gap of 0.016 per 100 of face looks trivial until it is scaled. On a position of 10 million of face value it is about 1,600 of cash, settled or not settled depending on which convention the confirmation says applies. This is the reason the convention is written on the term sheet rather than assumed.
Notice which way the difference ran. Actual/actual gave the larger figure here because August has 31 real days and the 30/360 method throws one of them away, twice over, once in the month count and once in the capped settlement date. There is no general rule that one convention always exceeds the other. The sign depends on the calendar, which is exactly why it cannot be guessed.
The sawtooth: why the market quotes flat
Watch the full price of a bond through a coupon period and it climbs steadily, day by day, as accrued interest builds. On the coupon date it falls by the amount of the coupon just paid, then starts climbing again. Plotted across a few periods it is a sawtooth.
If bonds were quoted at the full price, a trader comparing today’s screen with last week’s would have to work out how much of the change was a genuine repricing and how much was three days of accrual. Quoting flat removes the mechanical part and leaves the part that carries information. A flat price that moves has moved because yields moved, or because the issuer’s credit changed, not because the calendar advanced.
The drop in the full price on the coupon date is not a fall in the value of the bond. The holder received the coupon in cash, so total wealth is unchanged. The same logic explains why an equity price falls on the ex-dividend date, and it is worth carrying the parallel, because both are cases where a price change is an accounting event rather than an economic one.
The same bond is quoted at a flat price of 97.40 on 31 August and again at 97.40 on 30 November. Under actual/actual, has the buyer of the second trade paid more or less than the buyer of the first?
- From 15 July to 30 November is 16 + 31 + 30 + 31 + 30 = 138 days, against the same 184-day period.
- Accrued interest = 3.00 × 138 / 184 = 2.250.
- Full price on 30 November = 97.40 + 2.250 = 99.650, against 98.166 on 31 August.
Answer: the November buyer pays 1.484 more per 100 of face, and the bond has not repriced by a single basis point. The flat price is identical on both dates. The whole difference is accrued interest, and the November buyer recovers it six weeks later when the January coupon arrives in full.
Where candidates lose marks
Four mistakes account for most of the lost marks on this topic, and none of them is conceptual.
The first is discounting to the wrong date. Present-valuing the remaining cash flows at the yield gives the full price on the settlement date, not the flat price and not a price at the last coupon date. A question that asks for the flat price is asking for two steps, and the second one is a subtraction that is easy to forget once the first has been done correctly.
The second is counting days from the trade date. The convention is the last coupon date to settlement. On a question that supplies a trade date and a settlement date, the trade date is there to be ignored.
The third is applying the wrong convention because the instrument was not read carefully. Government bond, actual/actual. Corporate bond, 30/360. The question states the instrument type for a reason, and where it states the convention explicitly, that instruction overrides the default.
Adding accrued interest to a price that already contains it. If a question gives the present value of the remaining cash flows and asks for the invoice amount, the work is already done. Adding accrued a second time produces a number that is out by exactly the accrued interest, which is a plausible enough figure to survive a quick check and is almost always one of the four options offered.
The fourth is treating the accrued interest as a cost. It is a reimbursement, recovered in full at the next coupon date. What it does affect is the timing of cash, and for a leveraged position that timing has a financing cost attached, which is where the topic connects to repo and to bond futures invoicing rather than ending at the invoice.
Expect one of three question shapes. Compute accrued interest given the dates and the convention. Convert between flat and full price in either direction. Or identify which of four figures is the quoted price, where the distractors are the full price, the present value at the last coupon date, and the price computed under the other day count convention. Reading the question for the word quoted, flat, clean, full, dirty or invoice is what settles which of the two numbers is being asked for.
The short version
- Quoted price means flat price. Settled price means full price. Accrued interest is the bridge between them.
- Accrued interest is the coupon multiplied by the fraction of the coupon period that has elapsed at settlement, counted from the last coupon date rather than the trade date.
- Actual/actual for governments, 30/360 for corporates, and the 30/360 rule that turns a 31st into a 30th is the detail that makes two correct answers disagree.
- Valuation formulas produce the full price. If the question asks for the quoted price, one subtraction remains.
- Accrued interest is reimbursed at the next coupon, so it is a timing item rather than a cost, which is where the topic connects to repo financing.
- Read the question for the words quoted, flat, clean, full, dirty or invoice. That single word decides which of the two numbers is being asked for.
Frequently Asked Questions
What is the difference between the flat price and the full price of a bond?
The flat price is the quoted price, which excludes interest earned since the last coupon date. The full price is what the buyer actually pays at settlement, and it equals the flat price plus accrued interest. The flat price is also called the clean price and the full price is also called the dirty price or the invoice price, but there are only two calculations behind the six names.
How is accrued interest calculated?
Accrued interest is the coupon payment multiplied by the number of days from the last coupon date to settlement, divided by the number of days in the full coupon period. The count runs from the last coupon date, not from the trade date, and the day count convention decides how those days are measured.
Which day count convention applies to which bond?
Government bonds conventionally use actual/actual, which counts real calendar days over the real length of the period. Corporate bonds, and many municipal and agency issues, conventionally use 30/360, which treats every month as 30 days and every year as 360. Money market instruments use actual/360 or actual/365 depending on the market. Where a question states the convention, that instruction overrides the default.
How much difference does the day count convention actually make?
On a 6% semiannual bond with coupons on 15 January and 15 July, settling on 31 August, actual/actual gives accrued interest of 0.766 per 100 of face and 30/360 gives 0.750. The gap of 0.016 is trivial per bond and about 1,600 on a position of 10 million of face value, which is why the convention appears on the term sheet rather than being assumed.
Why does the price of a bond fall on the coupon date?
Because the coupon has been paid out in cash. The full price includes accrued interest, which builds up through the period and resets to zero once the coupon is paid, so the full price drops by exactly the coupon amount. The holder is no worse off, having received that amount in cash. It is the same mechanism that makes an equity price fall on the ex-dividend date.
Does discounting the cash flows give the flat price or the full price?
The full price. Present-valuing the remaining cash flows at the yield to the settlement date produces the amount actually payable on that date, which includes accrued interest. To reach the quoted flat price, accrued interest has to be subtracted afterwards, and forgetting that subtraction is one of the most common errors on this topic.
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