Coupon bond price calculator

This coupon bond price calculator allows you to calculate the price of a coupon bond based on inputs like face value, coupon rate, yield to maturity, years to maturity, and coupon frequency. You can enter fractional values into the years to maturity box (such as 9.58 years). The calculations are based on a 30/360 day convention.

As you enter or change values, the calculator automatically updates and displays the bond price. Additionally, the calculator generates a table showing the present values of each cash flow (coupon payments and face value) for each period until maturity, which also updates in real-time.

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What is a bond?

A bond is a fixed-income instrument representing a loan made by an investor to a borrower, typically corporate or governmental. Bonds are used by companies, municipalities, states, and sovereign governments to finance projects and operations. When you buy a bond, you are lending money to the issuer in exchange for periodic interest payments (called coupon payments) and the return of the bond’s face value when it matures.

Types of bonds

1. Government bonds:
   • Treasury bonds (T-Bonds): Issued by a national government with a maturity of more than 10 years. They pay periodic interest and are considered low-risk.
   • Municipal bonds (Munis): Issued by states, cities, or counties to finance public projects. Interest income is often exempt from federal taxes and possibly state taxes.
2. Corporate bonds:
   • Investment-grade bonds: Issued by companies with strong credit ratings. These are considered low to moderate risk.
   • High-yield bonds (Junk bonds): Issued by companies with lower credit ratings, offering higher interest rates to compensate for higher risk.
3. Agency bonds:
   • Issued by government-affiliated organizations like Fannie Mae or Freddie Mac. These are generally considered safe but slightly riskier than U.S. Treasury bonds.
4. Convertible bonds:
   • These can be converted into a predetermined number of shares of the issuing company’s stock. They offer the stability of fixed income with the potential for capital appreciation.
5. Zero-coupon bonds:
   • Issued at a discount to their face value and do not pay periodic interest. Instead, they pay the face value at maturity.
6. Inflation-linked bonds:
   • Designed to protect investors from inflation. The principal and interest payments are adjusted based on inflation rates. U.S. Treasury Inflation-Protected Securities (TIPS) are an example.
7. Foreign bonds:
   • Issued by a foreign government or company, usually in the currency of the country in which it is issued. They carry exchange rate risks.

Bond price calculation formulas

Present value of a bond: The price of a bond is the sum of the present value of its future cash flows, which include periodic coupon payments and the face value at maturity.

P = \sum_{t=1}^{T} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^T}

• P = Price of the bond
• C = Coupon payment
• r = Discount rate or yield
• T = Number of periods
• F = Face value of the bond

Yield to maturity (YTM): Yield to Maturity is the internal rate of return (IRR) earned by an investor who buys the bond at the current market price and holds it until maturity. It can be estimated by solving the following equation for r:

P = \sum_{t=1}^{T} \frac{C}{(1 + \text{YTM})^t} + \frac{F}{(1 + \text{YTM})^T}

Since this equation cannot be solved algebraically, it usually requires numerical methods or financial calculators.

Current yield: This is a simpler measure of the bond’s yield, calculated as:

\text{Current Yield} = \frac{C}{P}

• C = Annual coupon payment
• P = Current market price of the bond

Bond duration: Duration measures the bond’s sensitivity to changes in interest rates. It is the weighted average time it takes for the bond’s cash flows to be paid back.

D = \frac{\sum_{t=1}^{T} t \times\frac{C}{(1 + r)^t} + T \times\frac{F}{(1 + r)^T}}{P}

• PV_t = Present value of the cash flow in period t
• T = Total number of periods
• P = Price of the bond

Dirty Price

The dirty price of a bond is the price including accrued interest. It is the actual amount the buyer pays for the bond.

\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}

Clean price

The clean price of a bond is the price excluding any interest that has accrued since the last coupon payment. It’s the quoted price typically seen in financial markets.

\text{Clean Price} = \text{Present Value of Bond's Future Cash Flows}

This is calculated by discounting the bond’s future cash flows (coupons and principal) at the bond’s yield.

Accrued interest (Accumulated coupon)

Accrued interest represents the interest that has accumulated on the bond since the last coupon payment but has not yet been paid to the bondholder. This amount is added to the clean price to determine the dirty price.

\text{Accrued Interest} = \frac{\text{Coupon Payment} \times \text{Days Since Last Coupon}}{\text{Days in Coupon Period}}

where:

• Coupon payment is the interest payment made to the bondholder.
• Days since last coupon is the number of days that have passed since the last coupon payment.
• Days in coupon period is the total number of days in the coupon period.

Example

If a bond has a clean price of $980, an annual coupon payment of $40 (paid semi-annually), and 60 days have passed since the last coupon payment in a 182-day period, the calculations would be:

Accrued interest:

\text{Accrued Interest} = \frac{\$40 \times 60}{182} = \frac{\$2,400}{182} \approx \$13.19

Dirty price:

\text{Dirty Price} = \$980 +\$13.19 = \$993.19

This means the buyer would pay $993.19 for the bond, of which $13.19 is accrued interest.

Additional information about bonds

• Credit risk: The risk that the bond issuer may default on its payments. Higher credit risk typically results in higher interest rates.
• Interest rate risk: The risk that the bond’s price will decrease due to rising interest rates. Bond prices and interest rates have an inverse relationship.
• Callable bonds: Some bonds can be “called” (redeemed) by the issuer before their maturity date, usually when interest rates drop. Callable bonds typically offer higher yields to compensate for the risk of being called early.
• Bond ratings: Bonds are rated by credit rating agencies like Moody’s, S&P, and Fitch. Ratings range from AAA (highest quality) to D (default).
• Tax considerations: Interest income from bonds may be subject to federal, state, and local taxes. Municipal bonds, however, often offer tax-free interest.
• Liquidity: Bonds can vary in liquidity. Government bonds tend to be more liquid, while corporate or municipal bonds may be less so.

Bonds are an essential part of many investment portfolios, offering a balance of risk and return, especially for those seeking income stability and capital preservation.