Dirty Price and Accrued Interest
Dirty Price and Accrued Interest
The dirty price is the clean price plus accrued interest. It is the true economic price you pay or receive in a bond transaction, and it is the basis for all yield and return calculations.
Key takeaways
- The dirty price is the economic price: clean price plus accrued interest.
- Accrued interest reflects the coupon earnings from the last coupon date to the settlement date, prorated daily.
- The buyer pays the dirty price to the seller; the accrued interest portion compensates the seller for time held.
- Dirty price is used in all yield to maturity and return calculations; clean price is used only for market quotation.
- Understanding dirty pricing is essential for calculating actual returns and managing transaction costs.
The relationship: dirty = clean + accrued
At any point in time, a bond's price relationship is:
\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}
The dirty price is the all-in cost to the buyer (or all-in proceeds to the seller). It reflects the true economic value exchanged. The clean price is a market convention that strips out the accrued-interest component for ease of comparison and public quotation.
Why accrued interest belongs to the seller
Consider a bond paying $40 semi-annually (semi-annual coupon periods of 180 days). The seller holds the bond for 120 days into a coupon period and then sells it. Over those 120 days, they have earned:
\text{Accrued Interest} = 40 \times \frac{120}{180} = \$26.67
This is the interest the seller has earned by holding the bond. When they sell, they deserve to receive this interest. The buyer, who will hold the bond for the remaining 60 days until the coupon payment, will earn the final $13.33 of the $40 coupon.
The buyer pays accrued interest upfront so the seller receives their earned interest. In exchange, when the coupon payment arrives 60 days later, the buyer receives the full $40 coupon, of which $26.67 effectively compensates them for the accrued interest they paid.
From the buyer's perspective, they have paid an extra $26.67 upfront and will receive an extra $26.67 in the coupon. The net effect is zero over time, but the timing differs. This is why accrued interest is necessary: it ensures that interest earned is matched to the holder who earned it.
Worked example: buying a bond between coupon dates
Scenario: A bond with a 5% annual coupon ($50 semi-annually, $25 per period) is bought 60 days into a 180-day coupon period. The clean price is $1,010.
Accrued interest calculation: Accrued = $25 × (60 / 180) = $8.33
Dirty price: Dirty = $1,010 + $8.33 = $1,018.33
Settlement: The buyer pays $1,018.33 to the seller. Of this:
- $1,010 is the clean price (the price change due to market yields).
- $8.33 is the accrued interest (compensation to the seller for time held).
Coupon payment (120 days later): The bond pays its full $25 semi-annual coupon to the buyer. The buyer has now held the bond for 120 days and earned:
- The accrued interest they paid upfront: $8.33
- The remaining coupon earned during their hold: $25 - $8.33 = $16.67
- Total earned: $25
But they also received the full $25 coupon, so they have a net of $25 - $8.33 = $16.67 in actual cash earnings. This matches the 120 days they held the bond. Meanwhile, the seller received $8.33 in accrued interest compensation for the 60 days they held the bond.
Dirty price in yield calculations
When a bond's yield to maturity (YTM) is calculated, it uses the dirty price, not the clean price. The YTM is the annualized discount rate that makes the present value of all future cash flows equal to the price the investor actually paid (the dirty price).
Example: A bond quotes a clean price of $1,020 (104 points) with a 4% coupon and 4 years to maturity. The accrued interest is $8. The dirty price is $1,028.
To calculate YTM, a financial calculator or spreadsheet solves:
1,028 = \frac{40}{1+y} + \frac{40}{(1+y)^2} + \frac{40}{(1+y)^3} + \frac{1,040}{(1+y)^4}
The yield y that satisfies this equation is the YTM. It is less than if calculated on the clean price of $1,020, because the investor has paid more ($1,028) than the clean price suggests.
This is why a bond quote includes both the clean price and the YTM: the YTM is calculated using the dirty price, but the quoted price is clean for market consistency.
Impact on fund pricing and performance
Bond mutual funds and ETFs (like BND, AGG, TLT, IEF) calculate their net asset value (NAV) based on dirty prices. Each bond holding is marked to market at its dirty price (clean price plus accrued interest). The sum of all dirty prices, divided by the number of shares outstanding, gives the fund's NAV per share.
When you buy or sell fund shares, you transact at the NAV, which reflects the dirty prices of all holdings. The fund does not separately charge you accrued interest because the NAV already accounts for it. The fund's return (income and price appreciation) is calculated from dirty prices.
For example, if a bond fund holds a bond at a dirty price of $1,018.33 and it moves to $1,019.50 (due to a small yield change and accrual), the fund's NAV increases accordingly, and your share price reflects this gain. You do not pay accrued interest separately; it is baked into the share price.
Accrued interest and taxes
For individual bond buyers (not fund shareholders), accrued interest has tax implications. If you buy a bond between coupon dates, the accrued interest you pay is not a capital loss or deduction. It is treated as interest income recovered when you receive the coupon. In your first year owning the bond, you will report the accrued interest as interest income (even though you did not receive it in cash).
This can be surprising: you paid $8.33 in accrued interest, but you owe taxes on $25 in coupon income (including the accrued portion), even though you only held the bond 120 days. In a taxable account, this accelerates your tax liability. This is one reason to hold individual bonds in tax-deferred accounts (IRAs, 401(k)s) when possible, or to use tax-loss harvesting strategies.
In a tax-deferred account (like a 401(k) or IRA), accrued interest is irrelevant to your after-tax returns. In a taxable account, it is a consideration for buy-and-hold investors.
Dirty price and holding period returns
If you buy a bond at a dirty price and sell it later, your return is calculated as:
\text{Total Return} = \frac{\text{Coupons Received} + \text{Dirty Price at Sale} - \text{Dirty Price at Purchase}}{\text{Dirty Price at Purchase}}
For example, if you buy a bond at a dirty price of $1,018.33, hold it for one year, collect $40 in coupons, and sell it at a dirty price of $1,015, your total return is:
\text{Return} = \frac{40 + 1,015 - 1,018.33}{1,018.33} = \frac{36.67}{1,018.33} = 3.6\%
The dirty price at sale is lower than at purchase (prices fell due to rising yields), so even though you collected coupons, your overall return is modest. If you had not accounted for accrued interest, the calculation would be distorted.
Flowchart
Next
Accrued interest is calculated using the number of days between coupon dates and the settlement date. But how are those days counted? Different markets use different conventions (Actual/Actual, 30/360, Actual/360), which seem like minor details but affect pricing and yields in meaningful ways. In the next article, we explore these day-count conventions and why they matter.