A bond price that includes accrued interest from the latest coupon payment
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Written byCFI Team
What is Dirty Price?
Dirty price is when a bond price includes interest that has accrued since the latest coupon payment.
When investors buy fixed-income securities, such as bonds, they expect to receive coupon payments based on a fixed schedule. However, the price of a bond is dependent on the present value of future coupon payments. Unless a bond is purchased on the coupon payment date, the bond price likely includes the interest that has accrued since then.
Therefore, the buyer will miss out on one coupon payment, and the seller will pocket the accrued interest – this would be a dirty price. This contrasts with a clean price, which excludes any accrued interest.
Summary
- Dirty price is when a bond price includes interest that has accrued since the latest coupon payment.
- It is seen as “dirty” because the accrued interest included in the bond price goes to the seller.
- To calculate the dirty price, sum the clean price and the accrued interest.
Understanding Dirty Price
To understand dirty price, it’s important first to understand how bonds work. Like other fixed-income assets, bonds provide a coupon payment to the bondholder on a fixed schedule. Coupon payments can occur monthly, quarterly, or annually. However, most bonds make coupon payments on a semi-annual basis (every six months).
A bond is priced based on the present value of its future cash flows. Once a coupon payment has been made, there will be no further payments until the next payment date. The interest that accrues between each payment date is known as the accrued interest. On the payment date, the accrued interest resets back to zero again.
If an investor decides to purchase a bond after the payment date, they won’t receive any coupon payments until the next payment date. However, the bond seller may price a bond to include any accrued interest up to the sell date – it is known as the dirty price.
The price is referred to as “dirty” because the buyer pays for the price of the bond and the accrued interest but won’t receive any coupon payments until the next payment date. Therefore, the accrued interest that was included in the bond price goes to the seller.
How to Calculate the Prices
In the U.S., it is typical to provide clean bond prices by excluding any accrued interest. After the purchase has been completed (settled), the accrued interest is then added back to the clean price to reflect the bond’s true market value.
Dirty Price
Calculating the dirty price is quite simple; we just need to add the accrued interest to the clean price.
The formula is as follows:
Dirty Price = Clean Price + Accrued Interest
Clean Price
If we wish to find the clean price, we simply separate the effect of the accrued interest from the dirty price.
The formula would be as follows:
Clean Price = Dirty Price – Accrued Interest
Both Prices
To calculate both prices, we would also need the formula for the accrued interest:
Where:
- F = Face value
- C = Total annual coupon rate
- M = Number of coupon payments per year
- D = Days since last payment date
- T = Accrual period (number of days between payments)
Real-World Example
Let’s suppose a government bond pays a coupon rate of 5% and reaches maturity in 2022. The coupon is paid semiannually on December 1 and June 1. Let’s suppose an investor buys the bond on January 1, 2020, for a price of $1,500.
To calculate the dirty price, we first need the interest that has accrued since the last payment date. If the bond was settled on January 1, then 31 days have passed. Using the formula from above:
Solving the above equation provides an accrued Interest of $6.37.
To find the dirty price, we would use the formula given above:
Dirty Price = Clean Price + Accrued Interest
Dirty Price = $1,500 + $6.37 = $1,506.37
Therefore, the dirty price of a bond sold on January 1 would be $1,506.37.
Related Readings
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