Distinguish clean bond pricing from dirty bond pricing and explain what accrued interest implies for bond prices.

FRM® Part I · Valuation and Risk Models · Pricing Conventions, Discounting, and Arbitrage · FRM1.VR.9.f

The idea

A bond's quoted price is the clean (flat) price, which leaves out accrued interest. What the buyer actually pays at settlement is the dirty (full or invoice) price. The two are linked by Dirty = clean + accrued interest, so the buyer pays the quote plus the interest earned since the last coupon. In the simple form used here, Accrued = coupon per period × (days since the last coupon ÷ days in the coupon period). The exact count depends on the bond's day-count convention, for example 30/360 or Actual/Actual, and the accrued amount pays the seller for the coupon earned but not yet received. Between coupon dates the dirty price climbs as interest accrues and then falls by the coupon when it is paid, whereas the clean price, at an unchanged yield, shows no coupon-date jump. Quoting clean prices keeps quotes comparable across bonds and dates. Pricing from a yield gives the dirty price first, and subtracting accrued interest gives the clean price. The common errors are comparing a dirty price with a clean one and treating a whole coupon as accrued.

Formulas

Dirty (invoice) price of a bond

Dirty = Clean + AI

Clean: the quoted (flat) price
AI: the accrued interest since the last coupon date

This is what the buyer actually pays at settlement. Pricing from a yield gives the dirty price first; subtract AI to get the clean price.

Accrued interest between coupon dates

AI = Cp × (t ÷ T)

Cp: the coupon per period, not the annual coupon
t: days since the last coupon date, counted under the bond's convention
T: days in the coupon period under the same convention

The day count follows the bond's convention, such as Actual/Actual or 30/360. Treating a whole annual coupon as accrued is the usual error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly relates clean and dirty bond prices?

The clean price equals the dirty price plus the interest accrued since the last couponWhy not: Reverses the relationship; accrued interest is added to the clean price to give the dirty price.
The dirty price, paid by the buyer at settlement, is the clean price plus accrued interest - correct
The dirty price is the price quoted in the market and it excludes accrued interestWhy not: Swaps the labels; the quoted price is the clean price.
The two prices differ only on coupon datesWhy not: Accrued interest builds daily, so the prices differ on most days between coupons.

Quoted (clean) prices exclude accrued interest. The buyer pays the full (dirty) price, clean plus accrued, so the seller is paid for the coupon earned.

A harder one

A portfolio manager buys USD 5 million face value of a bond quoted at a clean price of 98.50 per 100 face value. Accrued interest is 1.20 per 100 face value. The cash due at settlement is closest to:

USD 4,925,000Why not: Pays only the clean price and leaves out the accrued interest.
USD 4,865,000Why not: Subtracts the accrued interest instead of adding it.
USD 4,985,000 - correct
USD 5,045,000Why not: Adds the accrued interest twice.

Settlement cash is the dirty price times face: (98.50 + 1.20) / 100 x 5,000,000 = 4,985,000.

The hardest tier

A bond with face value 100 pays a 6% coupon semiannually and has two coupons remaining. The next coupon is due in 120 days, 60 days have passed in the 180-day coupon period, and the yield is 5% compounded semiannually. The clean price per 100 face value is closest to:

101.80Why not: Reports the dirty price without deducting the accrued interest.
100.80 - correct
102.80Why not: Adds the accrued interest to the dirty price instead of deducting it.
99.96Why not: Discounts over whole periods, 1 and 2, rather than 2/3 and 5/3, then deducts accrued interest.

Dirty = 3 / 1.025^(2/3) + 103 / 1.025^(5/3) = 101.80. Accrued = 3 x 60 / 180 = 1.00, so clean = 101.80 - 1.00 = 100.80.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →