Dirty Price

The full price is derived by discounting every remaining coupon and the redemption amount back to the settlement date, not back to a coupon date. The standard mechanic does this in two steps. First, value the bond as of the previous coupon date using the usual present value formula, which produces a price for a bond with a whole number of periods remaining. Second, carry that value forward to settlement by multiplying it by one plus the periodic yield, raised to the fraction of the coupon period that has elapsed.

That fraction is the number of days from the last coupon date to settlement divided by the total number of days in the current coupon period. Compounding forward reflects the simple fact that the buyer receives the entire next coupon despite holding the bond for only part of the period, so the price must be grossed up for the time value already consumed. Subtracting accrued interest from this full price yields the clean price that appears on dealer screens.

Accrued interest is calculated on a straight-line basis: the coupon payment for the period multiplied by the elapsed portion of that period. The convention used to measure elapsed time is not cosmetic. Government bonds in most markets, including US Treasuries, use actual over actual, counting real calendar days in both the numerator and the denominator. Corporate and agency bonds typically use thirty over three-sixty, which treats every month as thirty days and every year as three hundred sixty days, producing slightly different figures for the same trade date.

Because accrual is linear while the true present value of a bond compounds, the clean price backed out of the full price is not perfectly smooth; it contains a small periodic distortion. At Level I depth the accepted treatment is to accept the linear accrual convention, apply the day count specified for the security, and recognise that market quotes and invoice amounts are reconciled through this convention rather than through exact discounting.

Trade confirmations and settlement instructions always show three lines: principal at the clean quote, accrued interest, and the total consideration wired to the seller. Custodians, clearing systems and repo desks all settle on the full amount, and repurchase agreements are typically struck on full-price collateral values so that the lender is secured against the economic value actually transferred.

Quoting conventions differ by market. Most bond markets quote clean, but some instruments trade dirty by convention, and certain markets impose an ex-coupon period in which the seller retains the upcoming coupon, producing negative accrued interest for a short window before payment. Under IFRS and US GAAP, a purchaser records the bond investment at the clean amount and treats the accrued portion as a receivable that is extinguished when the coupon arrives, which is why interest income is not overstated in the first period of ownership.

Expect to move fluently among the three quantities: full price, accrued interest and clean price. Given a yield, coupon, settlement date and day count, compute the present value at the previous coupon date, compound it forward for the elapsed fraction, calculate accrued interest, and report the clean quote. Given a clean quote instead, reverse the sequence to obtain the invoice amount, and be able to solve for the yield to maturity implied by a full price.

Equally important is knowing which price feeds which calculation. Yield, duration and convexity are properties of the full price, because that is the amount actually invested and the true present value of the remaining cash flows. Money duration and price value of a basis point should therefore be applied to the full price when estimating the currency gain or loss on a position, even though risk is often quoted against clean prices for convenience.