Clean Price, Accrued Interest and Dirty Price of a US Treasury Bond
We will cover following topics
Introduction
Understanding the pricing of US Treasury bonds involves the concepts of clean and dirty prices, as well as accrued interest. These components play a crucial role in determining the actual transaction price of a bond in the secondary market. In this chapter, we will delve into the differences between clean and dirty prices, explore how accrued interest affects bond pricing, and provide step-by-step calculations for both accrued interest and dirty price on a US Treasury bond.
Clean Price and Dirty Price
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Clean Price: The clean price of a bond refers to the flat price of the bond without accounting for any accrued interest. It is the price at which the bond is actually traded between buyers and sellers.
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Dirty Price: The dirty price (also known as the full or invoice price) includes the accrued interest that has accumulated since the last coupon payment date. In essence, the dirty price reflects the total cost of purchasing the bond, considering both its principal value and the interest earned since the last coupon payment.
Accrued Interest Calculation
Accrued interest is the interest that has accrued on a bond between the last coupon payment date and the date of sale or settlement. It compensates the seller for the portion of the coupon payment they earned while holding the bond. To calculate accrued interest, the following formula is used:
$$ \text{Accrued Interest = Days Since Last Coupon} \times\left(\frac{\text { Coupon Rate }}{\text { Days in Coupon Period }}\right) \times \text{Face Value}$$
Where:
- Days Since Last Coupon: The number of days from the last coupon payment date to the settlement date.
- Coupon Rate: The annual coupon rate of the bond.
- Days in Coupon Period: The total number of days in the coupon period.
- Face Value: The par or face value of the bond.
Dirty Price Calculation
The dirty price is calculated by adding the accrued interest to the clean price of the bond. Mathematically, it can be expressed as:
$$\text{Dirty Price=Clean Price+Accrued Interest}$$
Let’s consider a US Treasury bond with a face value of 1,000, a coupon rate of 5%, days since last coupon as 45 and clean price of USD 950. To calculate the accrued interest and dirty price:
Calculate Accrued Interest:
$$\text{Accrued Interest} = 45 \times \left( \frac{0.05}{180} \right) \times 1000 = \text{USD 12.50}$$
Calculate Dirty Price:
$$\text{Dirty Price} = 950 + 12.50 = \text{USD 962.50}$$
Conclusion
Understanding the distinction between clean and dirty prices, as well as the calculation of accrued interest, is essential for accurately valuing and trading US Treasury bonds. Accrued interest compensates for the interest earned but not yet paid, and the dirty price reflects the total cost of purchasing the bond. These concepts play a critical role in the world of interest rate futures, where the accurate valuation of bonds is pivotal for effective risk management and trading strategies.