Price & Yield

Clean vs Dirty Price

Quoted clean price plus accrued interest gives the dirty invoice price.

highlighted = computed this step

Quoted price vs invoice price

A bond can be quoted clean but settled dirty. The $1,000.00 face at 10% annual pays $50.00 semiannually. In this example the clean quoted price is $980.00, and settlement is between coupon dates.

Pdirty=Pclean+AIP_{\text{dirty}}=P_{\text{clean}}+\text{AI}

Day-count fraction

Using the 30/360 convention, the last coupon date is January 1, 2024, settlement is May 1, 2024, and the next coupon date is July 1, 2024. Accrued days are 120 out of 180, so the exact fraction is 2/3.

accrued daysperiod days=120180=2/3\frac{\text{accrued days}}{\text{period days}}=\frac{120}{180}=2/3

Accrued interest

Accrued interest is the period coupon times that day-count fraction. The exact value is 100/3 dollars, which rounds half-up to $33.33.

AI=$50.00⋅2/3≈$33.33\text{AI}=\$50.00\cdot 2/3\approx \$33.33

Dirty price

The invoice, or dirty, price is clean price plus accrued interest: $980.00 plus $33.33 equals $1,013.33.

Pdirty=$980.00+$33.33=$1,013.33P_{\text{dirty}}=\$980.00+\$33.33=\$1,013.33

Convention note

The 30/360 day count is one convention and an idealization. Real markets also use actual over actual and actual over 360, among other conventions.

dirty price depends on the chosen day-count convention\text{dirty price depends on the chosen day-count convention}