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# Clean and Dirty Price

When trading coupon bonds, you’ll see two types of bond prices: the clean price and the dirty price. These are Bond 101 and essential for bond valuation and understanding how interest payments affect bond prices. In this post we’ll break down the definitions, formulas and examples.

## What is a Clean Price?

The clean price of a coupon bond is the price before any accrued interest has been added. In other words, it’s the price of the bond without any interest that’s been added between coupon payments. Financial news sites and market quotes will show the clean price as it’s a clean and consistent way to compare bonds.

The clean price is important because it allows investors to focus on the bond’s value without the accrued interest which can vary depending on the time since the last coupon payment. This price is the present value of future cash flows, the face value of the bond and the future coupon payments, discounted back to the present using the bond’s yield.

## What is a Dirty Price?

The dirty price is the clean price plus the accrued interest. This is the price the buyer pays the seller of the bond. The dirty price is the total amount the buyer needs to pay the seller for the interest that’s been added since the last coupon payment.

The dirty price is important for investors because it’s the actual cost of buying the bond including the interest added up to the purchase date. When bonds are traded between coupon payment dates the dirty price ensures the seller gets paid for the interest earned but not yet paid.

## Clean Price vs. Dirty Price: The Formula

To differentiate between the clean and dirty prices, you can use the following relationship:

Clean Price=Dirty Price−Accrued Interest\text{Clean Price} = \text{Dirty Price} – \text{Accrued Interest}Clean Price=Dirty Price−Accrued Interest

Where:

• Clean Price is the price of the bond without accrued interest.
• Dirty Price is the total price paid by the buyer, including accrued interest.
• Accrued Interest is the interest that has accumulated since the last coupon payment.

## Example of Clean and Dirty Price:

The dirty price is the price the bond seller must pay to give up ownership. It includes the present value of the bond plus the accrued interest. The clean price is the dirty price less accrued interest:

clean price = dirty price – accrued interest

To calculate both prices, we would also need the formula for the accrued interest:

$Accrued Interest = F*\frac{C}{M}*\frac{D}{T}$

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)

Consider a \$100 par value bond that pays a 3% coupon semiannually. This means a coupon of \$1.50 is paid every six months. If the bond is sold (and settles) 41 days after the last coupon, the buyer will need to pay the seller \$1.50 × 41 182 = \$0.3379 for every \\$100 purchased. When calculating the discount factor, the amount of the accrued interest needs to be added to both the bid and asked for a quote before calculating the midpoint.

## Why Clean and Dirty Prices Matter

Knowing clean and dirty prices is important for both buyers and sellers in the bond market. For buyers it helps in evaluating the total cost of buying a bond including accrued interest. For sellers it ensures they get fair compensation for the interest added during their holding period.

## Practical Applications of Clean and Dirty Prices in Bond Trading

Knowing the difference between clean and dirty prices is not only bond 101 but also has practical implications for trading and investment strategies. Here’s how these concepts impact investors and traders: