Distinguish a US Treasury bond's clean price from its dirty price, and compute its accrued interest and dirty price.
The idea
A Treasury bond is quoted at its clean price, which excludes interest accrued since the last coupon. The buyer pays the dirty price, also called the cash or invoice price: Dirty = Clean + Accrued interest. Accrued interest is the semiannual coupon times the fraction of the coupon period that has passed, counted on an Actual/Actual basis: Accrued = semiannual coupon × days since last coupon ÷ days in the coupon period. Quotes are in 32nds, so 98-16 means 98 and 16/32, or 98.50. Quoting clean keeps the price from saw-toothing. The dirty price rises as interest accrues and drops by the coupon when it is paid, while the clean price moves mainly with yields. Typical errors are using the annual instead of the semiannual coupon, subtracting accrued interest instead of adding it, counting days on the wrong basis and reading 32nds as decimals. Exam questions usually ask for the cash paid at settlement, which is the dirty price.
Formulas
Accrued interest on a Treasury bond
accrued interest = semiannual coupon × days since last coupon ÷ days in coupon period
- semiannual coupon
- half the annual coupon, per 100 of face
- days since last coupon
- actual days from the last coupon to settlement
- days in coupon period
- actual days between the two coupon dates
Actual/Actual. Add it to the clean quote to get the dirty price the buyer pays; using the annual coupon is the usual trap.
A practice question
Which statement about Treasury bond prices is correct?
- The quoted price is the dirty price, which includes accrued interest.Why not: Quotes exclude accrued interest; the dirty price is what is actually paid.
- The quoted price is the clean price, which excludes accrued interest. - correct
- Accrued interest is paid by the seller to the buyer at settlement.Why not: The buyer compensates the seller for interest earned since the last coupon.
- The clean price falls by the coupon amount on each coupon date.Why not: It is the dirty price that drops by the coupon when it is paid; the clean price moves mainly with yields.
Bonds are quoted clean so that the price does not saw-tooth with accrual. The buyer pays the clean price plus accrued interest to the seller.
A harder one
A Treasury bond with a 6% coupon pays interest on 15 April and 15 October. On 20 July (96 days after the last coupon, in a 183-day period) it is quoted at 98-16 (in 32nds of a point). The dirty price per 100 of face value is closest to:
- 100.07 - correct
- 99.73Why not: Reads 98-16 as 98.16 rather than 98 and 16/32 = 98.50.
- 101.65Why not: Accrues the annual coupon of 6 instead of the semiannual coupon of 3.
- 96.93Why not: Subtracts accrued interest from the clean price instead of adding it.
Clean = 98 + 16/32 = 98.50. Accrued = 3 x 96/183 = 1.5738. Dirty = 100.07.
The hardest tier
An investor buys USD 20 million face of a 4.5% Treasury bond (coupons 15 May and 15 November) at a quote of 101-08, settling 10 September (118 days after the last coupon in a 184-day period). The investor holds until 15 November, receives the coupon, and sells immediately after at the same 101-08 quote. The total gain is closest to:
- USD 161,413 - correct
- USD 450,000Why not: Counts the full coupon as gain and ignores the accrued interest paid to the seller at purchase.
- USD 288,587Why not: Reports the accrued interest paid at purchase (118/184 of the coupon) instead of the interest actually earned over the holding period.
- USD 0Why not: Assumes no gain because the clean price is unchanged, ignoring that the investor earns the 66 days of accrual to the coupon date.
Bought at clean + accrued; coupon 2.25 received; sold ex-coupon at clean. Gain per 100 = 2.25 - 2.25 x 118/184 = 0.8071, i.e. interest earned over 66 days. On 20m face: 161,413.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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