Dirty price

A bond pays its coupon on fixed dates. Somebody who buys it three months into a six-month coupon period is buying a security on which the seller has already earned three months of interest but has not yet been paid.

That interest has to be handed over. So the buyer pays two things: the clean price, which is the market's valuation of the bond, and the accrued interest, which is the seller's share of the coupon that is about to arrive. Added together they are the dirty price.

On the next coupon date the buyer collects the whole coupon, including the part that economically belonged to the seller — and is square, because he paid for it at settlement.

The market quotes clean. The workbook is explicit: the market typically trades a bond on the basis of clean price, and all yield and price formulas are on the basis of clean price. If quotations included accrued interest, a bond's quoted price would sawtooth upwards through every coupon period and drop on every coupon date, telling you nothing about whether it had cheapened.

Accrued interest is not discounted. The workbook gives the reason: there is no intervening cash flow before the first coupon to be received after the investor buys the bond.

Pricing between coupon dates follows three steps in the workbook:

1. Take the settlement date back to the previous coupon date and value the bond using the coupon, the yield and the residual maturity from that last coupon date.
2. Bring that value forward to the settlement date using a future value factor.
3. Deduct the accrued interest from the total value.

A caution on the naming, and it is the workbook's own. In one paragraph it says the amount the buyer pays the seller — clean price plus accrued interest — is often called the full price or invoice price. A few lines later, step three describes deducting accrued interest to arrive at the clean price or trade price or invoice price. Those two sentences put "invoice price" on opposite sides of the accrued interest. Section 9.4.3 contradicts itself here; treat full price and dirty price as the settlement amount, clean price as the quoted price, and do not rely on "invoice price" to tell you which is meant.

Dirty price = Clean price + Accrued interest
Clean price = Dirty price - Accrued interest

Accrued interest = Coupon per period x (days since last coupon / days in coupon period)

A government security with a 7.26% coupon on a face value of Rs 100, paying half-yearly, so Rs 3.63 per period. The last coupon fell 73 days ago in a 182-day coupon period. The bond is quoted at a clean price of Rs 98.50.

Accrued interest = 3.63 x (73 / 182) = Rs 1.456
Dirty price      = 98.50 + 1.456     = Rs 99.956

On a real ticket of Rs 50,00,000 face value:

Now follow it to the coupon date, 109 days later. The buyer receives the whole half-yearly coupon:

Coupon received = 50,00,000 x 3.63%  = Rs 1,81,500
less paid away at settlement          = Rs   72,800
Net interest genuinely earned          = Rs 1,08,700

And Rs 1,08,700 is exactly 109/182 of Rs 1,81,500 — the portion of the period he actually held the bond. The accrued interest mechanism has divided one coupon between two owners, to the day.

Why the quotation stays clean. Had the market quoted dirty prices, this bond would have appeared to rise from Rs 98.50 to Rs 102.13 over the coupon period and then fall Rs 3.63 overnight — all of it accounting, none of it a change in value.

Chapter 9 (Investing in Fixed Income Securities), section 9.4.3 on valuation and pricing of bonds, under "Valuing Bonds at Non-Coupon Dates", immediately after the coupon rate / current yield / YTM relationship table. Questions ask for the dirty price given a clean price and accrued interest, ask which price the market quotes, and ask which price the yield formulas use.

• Dirty price is the larger number. Clean plus accrued. If your dirty price is below your clean price, the addition went the wrong way.
• Yields are computed on the clean price. Using the settlement amount inflates the price and understates the yield.
• Accrued interest is not discounted — there is no intervening cash flow before the next coupon, so it is added at face.
• The workbook's own naming is inconsistent. Section 9.4.3 calls the settlement amount the "full-price or invoice price" and then, three lines later, calls the clean price the "trade price or invoice price". Answer using clean and dirty, or clean and full; do not build an answer on "invoice price".
• Count days on the actual coupon period, not on a 30-day month, unless the question supplies a convention. The workbook warns that money market and bond market conventions differ within the same country.
• The buyer receives the entire next coupon, not a pro-rated one. He has already compensated the seller in cash at settlement.
• Clean prices do not sawtooth; dirty prices do. A chart of a bond that steps down sharply on coupon dates is plotting dirty prices.