Accrued Interest

Accrued interest ( $Fr \cdot t$ ) is the portion of the next coupon that has been earned by the seller of a bond between the last coupon date and the settlement date, where $Fr$ is the coupon amount and $t$ is the fraction of the coupon period elapsed ( $0 \le t < 1$ ).

$AI = Fr \cdot t$

$\text{Full Price} = \text{Clean Price} + Fr \cdot t$

Quoted clean prices (flat prices) exclude accrued interest; the full price (dirty price) actually paid adds it back and equals the present value of the remaining cash flows.
The convention splits the price into a clean component (the bond's intrinsic value) and an accrued component that compensates the seller for interest earned since the last coupon.
For Exam FM, bonds are usually valued on coupon dates, so $t = 0$ and $AI = 0$ ; the topic is mainly conceptual.

Accrued Interest Between Coupon Dates A $ $1{,}000$ face value bond pays semi-annual coupons at an annual rate of $6\%$ (so each coupon is $ $30$ ). A buyer purchases the bond 2 months after the last coupon date. The clean price is $ $985$ . Find the accrued interest and the full price paid.

Answer Fraction of coupon period elapsed: $t = 2/6 = 1/3$ (2 months out of a 6-month period). $AI = Fr \cdot t = 30 \times \tfrac{1}{3} = \$10.00$ $\text{Full Price} = 985 + 10 = \$995.00$ The buyer pays $ $995$ to the seller. At the next coupon date (4 months later) the buyer receives the full $ $30$ coupon, of which $ $10$ reimburses the accrued interest paid at purchase.