# Dirty Price

## Introduction

When it comes to investing in bonds, understanding the concept of “dirty price” is crucial. The dirty price of a bond refers to its market price, which includes both the clean price and the accrued interest. This article will delve into the details of dirty price, explaining how it is calculated, its significance for bond investors, and how it differs from the clean price. By the end, you'll have a clear understanding of why dirty price matters in the world of finance.

## What is Dirty Price?

Dirty price, also known as the full price or the market price, is the actual price at which a bond is traded in the market. It includes both the clean price and the accrued interest. The clean price, on the other hand, refers to the price of the bond without any accrued interest.

Accrued interest is the interest that has accumulated on a bond since its last coupon payment. Bonds typically pay interest semi-annually or annually, and the accrued interest is calculated based on the number of days since the last coupon payment. When a bond is traded between two parties, the buyer pays the seller the dirty price, which compensates the seller for the accrued interest.

## Calculating Dirty Price

Calculating the dirty price of a bond involves adding the clean price and the accrued interest. The clean price is the present value of the bond's future cash flows, discounted at the bond's yield to maturity. The accrued interest is calculated based on the number of days since the last coupon payment and the bond's coupon rate.

Let's consider an example to illustrate the calculation of dirty price. Suppose we have a bond with a face value of \$1,000, a coupon rate of 5%, and a remaining time to maturity of 3 years. The bond pays interest semi-annually, and the current yield to maturity is 4%. If the last coupon payment was made 30 days ago, we can calculate the dirty price as follows:

• Step 1: Calculate the accrued interest
• Accrued interest = (Coupon rate / Number of coupon payments per year) * (Number of days since last coupon payment / Number of days in a year)

Accrued interest = (5% / 2) * (30 / 365) = \$0.2055

• Step 2: Calculate the clean price
• Clean price = Present value of future cash flows

Clean price = (Coupon payment / (1 + Yield to maturity / Number of coupon payments per year)) + (Coupon payment / (1 + Yield to maturity / Number of coupon payments per year)^2) + … + (Coupon payment + Face value / (1 + Yield to maturity / Number of coupon payments per year)^n)

Clean price = (\$25 / (1 + 0.04 / 2)) + (\$25 / (1 + 0.04 / 2)^2) + (\$25 / (1 + 0.04 / 2)^3) + (\$1,025 / (1 + 0.04 / 2)^6) = \$1,034.62

• Step 3: Calculate the dirty price
• Dirty price = Clean price + Accrued interest

Dirty price = \$1,034.62 + \$0.2055 = \$1,034.83

## Significance of Dirty Price for Bond Investors

Understanding the dirty price of a bond is essential for bond investors as it provides a more accurate representation of the bond's value in the market. The dirty price takes into account the accrued interest, which compensates the seller for the interest earned since the last coupon payment. By considering the dirty price, investors can make informed decisions about buying or selling bonds based on their desired yield and market conditions.

For example, if an investor is interested in purchasing a bond and wants to know the total cost, including the accrued interest, they would need to consider the dirty price. On the other hand, if an investor is selling a bond, they would receive the dirty price, which compensates them for the accrued interest they have earned.

## Dirty Price vs. Clean Price

While dirty price includes both the clean price and the accrued interest, the clean price only represents the price of the bond without any accrued interest. The clean price is often used to calculate the yield to maturity, which is the total return an investor can expect to earn if they hold the bond until maturity.

It's important to note that the clean price and the dirty price will converge as the bond approaches its next coupon payment date. At that point, the accrued interest will be reset to zero, and the dirty price will equal the clean price.

## Conclusion

Dirty price plays a significant role in the world of bond investing. It represents the actual market price of a bond, including both the clean price and the accrued interest. By understanding how to calculate the dirty price and its significance for bond investors, individuals can make informed decisions when buying or selling bonds.

Remember, the dirty price takes into account the accrued interest, compensating the seller for the interest earned since the last coupon payment. It differs from the clean price, which only represents the price of the bond without any accrued interest. Both the dirty price and the clean price are essential concepts for bond investors to grasp in order to navigate the bond market effectively.

So, the next time you consider investing in bonds, don't forget to consider the dirty price. It's a crucial factor that can impact your investment decisions and ultimately determine your returns in the bond market.