Settlement, accrued interest, and clean versus dirty price

The price paid is not always the price quoted

A bond price quote commonly shows the clean-bond-price, also called the flat or quoted price. The buyer pays the dirty-bond-price, also called the full, cash, or invoice price. The dirty price is the clean price plus the accrued-interest:

$$\explain{dirty-bond-price}{P^{\mathrm{dirty}}} =\explain{clean-bond-price}{P^{\mathrm{clean}}} +\explain{accrued-interest}{AI}$$

(2.5.1)

On a coupon date, the accrued interest is zero, so the two prices are equal. For this reason, the earlier bond lessons, which settle on a coupon date, use one bond-price symbol.

Actual/actual within one coupon period

Under the convention of this lesson, let the local coupon-period-elapsed-days be $\explain{coupon-period-elapsed-days}{n_{\mathrm{elapsed}}}$ and the local coupon-period-total-days be $\explain{coupon-period-total-days}{n_{\mathrm{period}}}$. For one coupon-payment $\explain{coupon-payment}{C}$, the accrued interest is the coupon multiplied by the fraction of the coupon period that has elapsed:

$$\explain{accrued-interest}{AI} =\explain{coupon-payment}{C} \frac{\explain{coupon-period-elapsed-days}{n_{\mathrm{elapsed}}}} {\explain{coupon-period-total-days}{n_{\mathrm{period}}}}$$

(2.5.2)

Consider the coupon period from February 15, 2021 to August 15, 2021, which has 181 days. Settlement on May 17, 2021 is 91 actual days after the start of the period. With a coupon of USD 31.25, (2.5.2) gives the accrued interest:

$$\explain{accrued-interest}{AI} =31.25\left(\frac{91}{181}\right) =15.71132597\ \text{USD per bond}$$

(2.5.3)

The domain code checks that the dates are valid ISO dates and counts UTC calendar days. It is not a production schedule engine: business-day rules, ex-coupon periods, stubs, and other day counts require separate conventions.

Why this matters for derivatives

The cash amount paid at the settlement of a bond forward or a bond option depends on whether its price is clean or dirty. If a strike is quoted as a clean price and the modeled bond price is a dirty price, add the accrued interest at the exercise date to the strike before the payoff is calculated. Also, each coupon paid before delivery must be assigned to the party that owns the bond on the coupon date.

Knowledge check 2.5.1 Bond settlement

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

Sources

The identity between the clean and the dirty price (2.5.1) and the actual/actual calculation (2.5.2) follow Tuckman and Serrat, Chapter 1 §1.6 [1]. Hull’s European bond option discussion emphasizes matching cash versus quoted strike and bond prices [2].

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). draft ↩