Clean Price and Dirty Price of a Bond

Clean price refers to the price of a bond excluding any accrued interest, while dirty price includes this accrued interest. This distinction is important for investors as it affects the total amount paid when purchasing a bond, with the dirty price reflecting the actual cost at the time of the transaction.

A bond's clean price is its quoted price multiplied by its face value, excluding any accrued interest. In this case, a quoted price of 98 means the bond is selling for 98% of its face value. Therefore, 98% of $1,000 equals $980, which is the clean price of the bond.

Accrued interest represents the interest earned on a bond since the last coupon payment. It is calculated using the bond's coupon rate, which indicates the interest payment, and the time elapsed since the last payment, reflecting how much interest has accumulated during that period.

The dirty price of a bond includes both its clean price and accrued interest. In this case, the clean price is $950, and the accrued interest is $25. Adding these two amounts together gives a dirty price of $975, which represents the total amount a buyer would pay for the bond.

Clean price refers to the price of a bond excluding any accrued interest. It is the standard pricing method reported in financial press and bond markets, providing a clearer representation of a bond's value. This method allows investors to compare bond prices without the variability introduced by interest accumulation.

Since the bond pays semi-annual coupons, each coupon period lasts 6 months. With 3 months having passed since the last payment, half of the coupon period has accrued. Therefore, the fraction of the coupon that has accrued is 1/2, reflecting the elapsed time relative to the total coupon period.

The invoice price of a bond, also known as the dirty price, includes the bond's clean price plus any accrued interest since the last coupon payment. This total amount reflects what a buyer must pay to acquire the bond, ensuring that the seller receives compensation for the interest earned up to the sale date.

Accrued interest increases linearly between coupon payments as the bondholder earns interest over time. By the time the coupon payment date arrives, the accrued interest reaches its maximum and is then paid out, making it zero after the payment is made. Thus, both statements about linear increase and being zero at the payment date are accurate.

In bond trading, the quoted price typically refers to the "clean" price, which excludes any accrued interest. This allows investors to assess the bond's value without the influence of interest that has accumulated since the last coupon payment, providing a clearer picture of the bond's market price.

Clean price is preferred in bond markets because it excludes accrued interest, enabling investors to make direct price comparisons regardless of the settlement date. This standardization simplifies the evaluation of bonds, ensuring that price variations reflect only the bond's value rather than the timing of interest payments.

Accrued interest is calculated based on the time elapsed since the last coupon payment. Since the bond pays $40 every six months, the monthly interest is approximately $6.67 ($40/6). After 4 months, the accrued interest amounts to about $26.67 (4 months x $6.67), reflecting the interest earned during that period.

Just after a coupon payment is made, the bond's clean price (which excludes accrued interest) aligns with the dirty price (which includes accrued interest). Since the coupon payment has been made, there is no accrued interest left to account for, making both prices equal at that moment.