Clean price
Clean price is the quoted price of a bond that excludes accrued interest since the previous coupon date. The corresponding total that includes accrued interest is the dirty or full price. Let
Overview
Clean price is the quoted price of a bond that excludes accrued interest since the previous coupon date. The corresponding total that includes accrued interest is the dirty or full price. Let
P
clean
{\displaystyle P_{\text{clean}}}
and
P
dirty
{\displaystyle P_{\text{dirty}}}
denote prices per 100 of nominal amount, and let
A I
{\displaystyle \mathrm {AI} }
denote accrued interest for the current coupon period. Then
P
dirty
=
P
clean
+
A I
{\displaystyle P_{\text{dirty}}=P_{\text{clean}}+\mathrm {AI} }
Concept and calculation
Let prices be quoted per 100 of nominal amount. Let
A I
{\displaystyle \mathrm {AI} }
denote accrued interest for the current coupon period,
C
{\displaystyle C}
the coupon due for that period per 100,
α
{\displaystyle \alpha }
the day-count fraction,
d
{\displaystyle d}
the number of days since the previous coupon date up to but excluding the settlement date, and
D
{\displaystyle D}
the number of days in the coupon period under the relevant convention. For standard coupon periods the accrued interest equals the period coupon multiplied by the elapsed fraction of the period.
A I
= C × α
{\displaystyle \mathrm {AI} =C\times \alpha }
α =
Market practice and conventions
In most cash bond markets prices are quoted on a clean basis for comparability across issues, while the amount paid at settlement equals the dirty price that adds accrued interest to the quote. This treatment is codified in official rules. In the United States the settlement amount for Treasury notes and bonds includes any accrued interest. In the United Kingdom the gilt market’s published formulae set out how accrued interest is calculated for settlement on a per 100 basis. In the UK an official end of day set of reference prices is produced for gilts. Tradeweb and FTSE Russell jointly calculate benchmark closing bid, mid and offer prices, which feed the FTSE Actuaries UK Gilts Index Series. The index calculation guide lists the analytics calculated each day, including a gross or dirty price index and accrued interest, and assumes replication at closing middle market prices. For repo operations it is best practice to value fixed income collateral using the middle clean price in the appropriate market and to include accrued interest up to but excluding the margin delivery date when fixing the collateral market value.
Worked examples
Assume a bond with a 6% annual coupon paid semiannually, so the coupon per period is
C = 3
{\displaystyle C=3}
per 100 of nominal, and use the 30/360 US convention for accrual. The day-count fraction is
α = d
/
D
{\displaystyle \alpha =d/D}
, where
D = 180
{\displaystyle D=180}
for a half-year and
d
{\displaystyle d}
is the 30/360 US day count from the last coupon to the settlement date. Accrued interest is
A I
= C × α
{\displaystyle \mathrm {AI} =C\times \alpha }
, and the settlement price per 100 equals
P
Example A: mid-period accrual
Coupons pay on 15 January and 15 July 2025. Settlement is 16 April 2025. Under 30/360 US, the day count is
d = 30 × ( 4 − 1 ) + ( 16 − 15 ) = 91
{\displaystyle d=30\times (4-1)+(16-15)=91}
, so
α = 91
/
180 ≈ 0.5056
{\displaystyle \alpha =91/180\approx 0.5056}
. The accrued interest is
A I
=
3 ×
91 180
=
1.516
6 ¯
≈
1.5167
per 100.
{\displaystyle \mathrm {AI} \;=\;3\times {\frac {91}{180}}\;=\;1.516{\overline {6}}\;\approx \;1.5167{\text{ per 100.}}}
Example B: eve of coupon
Using the same schedule, take settlement as 14 July 2025. The day count is
d = 30 × ( 7 − 1 ) + ( 14 − 15 ) = 179
{\displaystyle d=30\times (7-1)+(14-15)=179}
, so
α = 179
/
180 ≈ 0.9944
{\displaystyle \alpha =179/180\approx 0.9944}
and
A I
=
3 ×
179 180
=
2.983
3 ¯
≈
2.9833
per 100.
{\displaystyle \mathrm {AI} \;=\;3\times {\frac {179}{180}}\;=\;2.983{\overline {3}}\;\approx \;2.9833{\text{ per 100.}}}
Zero-coupon special case
For a zero-coupon bond,
C = 0
{\displaystyle C=0}
in every period, so
A I
= 0
{\displaystyle \mathrm {AI} =0}
and the clean and dirty prices are identical by definition.
See also
Bond index Bond valuation Duration (finance) Yield curve Yield to maturity
Further reading
Calculation of the clean price of a bond in a cash management bond repurchase operation, Bank of Canada. Retrieved 2015-11-10.
External links
Clean price By Investopedia Calculate clean and dirty price of a bond by Financetrain
Wikipedia contributors · Clean price · revision 1379621562 · CC BY-SA 4.0 · Plain text extracted and formatting adapted; tags derived from linked topics.