Clean price

A bond pays its coupon once or twice a year, but interest builds up every day in between. If you sell halfway through a coupon period, you have earned half a coupon that the buyer will collect.

The market handles this by quoting two prices. The clean price is the price of the bond itself, without that built-up interest. The dirty price is what the buyer actually pays: clean price plus accrued interest, which compensates the seller.

Quoting clean keeps the price steady. If accrued interest were included in the quote, the price would creep up every day and then drop sharply on each coupon date, even when nothing about the bond had changed.

From Chapter 18:

• For secondary-market trades of coupon bonds there are two prices: clean price, "the price at which the bond is negotiated", and dirty price, "the price at which the bond is settled".
• Dirty price = Clean price + Accrued interest. The dirty price is always higher by the accrued interest.
• Accrued interest is interest at the coupon rate from the previous coupon date to the settlement date. The buyer pays it to the seller at settlement and gets it back in the full coupon on the next coupon date.
• Trading and quotation are done at the clean price; settlement happens at the dirty price.
• The day count convention decides the accrued interest: Actual/Actual, 30/360, Actual/365 (Indian money market) or Actual/360 (widely used in swap valuation).
• Current yield is coupon income as a proportion of the clean price.
• In Interest Rate Futures (Chapter 20), quotation is price based, with trading and settlement at clean price for a face value of Rs 100.

Pricing between coupon dates: value the bond as of the previous coupon date, carry that value forward to the settlement date (the dirty price, which is the present value), then deduct accrued interest to reach the clean price.

A terminology conflict in the workbook: page 449 says the dirty price is "also known as invoice price". Page 460, describing the pricing steps, says deducting accrued interest gives "the clean price or trade price or invoice price". The first usage, dirty price as invoice price, is consistent with invoicing being what the buyer pays at settlement.

Dirty price      = Clean price + Accrued interest
Accrued interest = Coupon rate (per period) × Days since last coupon ÷ Days in period
                   (days counted per the day count convention)
Current yield    = Annual coupon ÷ Clean price × 100

Workbook example 1 (Chapter 18): a 6.90% bond paying semi-annually; previous coupon 13 January, next 13 July, trade settling 5 March.

Actual/Actual:  AI = (6.90 ÷ 2) × 51/181 = 0.9721
30/360 (E):     AI = (6.90 ÷ 2) × 52/180 = 0.9967

Clean price = Rs 101.50
Dirty price = 101.50 + 0.9967 = Rs 102.4967   (30/360 E)

The day count convention alone changes accrued interest by about 2.5 paise per Rs 100.

Workbook example 2: 5.79% bond maturing 11 May 2030, yield 5.90%, settlement 12 August 2020, 30/360 European. Last coupon 11 May 2020, so 91 days have passed.

Accrued interest = 5.79 × 91/360 = Rs 1.4636
Dirty price (PV of all cash flows at 5.90%) = Rs 100.6464
Clean price = 100.6464 − 1.4636 = Rs 99.1828

The bond is quoted at Rs 99.1828 and settled at Rs 100.6464.

Scaling (illustrative): for Rs 5 crore face value, the buyer pays 100.6464% × 5 crore = Rs 5,03,23,200, of which 1.4636% × 5 crore = Rs 7,31,800 is accrued interest owed to the seller.

Chapter 18 (Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Markets) covers accrued interest, day count conventions, clean and dirty price and the between-coupon pricing method with worked examples; Chapter 20 (Exchange Traded Interest Rate Futures) specifies that IRF trade and settle at clean price and computes a theoretical futures clean price. Expect: dirty = clean + accrued interest, which price is quoted and which settles, and an accrued interest calculation under a stated day count.

• Quoted at clean, settled at dirty for cash bond trades.
• Dirty price is always higher than clean price by the accrued interest (between coupon dates).
• Accrued interest runs from the previous coupon date to the settlement date, not the trade date or next coupon date.
• The buyer pays accrued interest to the seller, and recovers it from the full coupon.
• Invoice price: the workbook uses it once for dirty price and once alongside clean price. Dirty price as invoice price is the consistent reading.
• IRF are quoted and settled at clean price.
• Day count matters: Actual/Actual and 30/360 give different accrued interest for the same dates.