Zero Coupon Bond Price Calculator
This tool calculates the present value (price) of a zero coupon bond based on its face value, years to maturity, and yield to maturity (required rate of return).
A zero coupon bond does not pay periodic interest (coupons). Instead, it is sold at a discount to its face value, and the investor receives the full face value when the bond matures. The difference between the purchase price and the face value represents the investor's return.
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Understanding Zero Coupon Bonds & Pricing
What is a Zero Coupon Bond?
A zero coupon bond is a debt instrument that does not pay interest (coupon payments). Instead, it is sold at a significant discount from its face value. The investor's return comes from the difference between the discounted purchase price and the full face value received at maturity.
Zero Coupon Bond Price Formula (Present Value)
The price of a zero coupon bond is calculated as the present value of its face value, discounted at the yield to maturity for the number of years until maturity.
Price = Face Value / (1 + Yield)Years
Where:
- Face Value: The amount paid to the bondholder at maturity.
- Yield (Yield to Maturity, YTM): The annual rate of return an investor expects to receive if the bond is held to maturity. This is used as the discount rate in the formula. It must be used as a decimal (e.g., 3% = 0.03).
- Years: The number of years remaining until the bond matures.
This formula is a basic present value calculation for a single future lump sum payment.
Zero Coupon Bond Price Examples
Click on an example to see the step-by-step calculation:
Example 1: Simple Calculation
Scenario: Calculate the price of a bond with a Face Value of $1,000, maturing in 5 years, with a Yield of 3%.
1. Known Values: Face Value = $1,000, Years = 5, Yield = 3% (0.03).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0.03)5 = 1000 / (1.03)5 ≈ 1000 / 1.15927
4. Result: Price ≈ $862.61
Conclusion: The theoretical price is about $862.61.
Example 2: Longer Maturity
Scenario: What is the price of a bond with a Face Value of $5,000, maturing in 10 years, with a Yield of 4.5%?
1. Known Values: Face Value = $5,000, Years = 10, Yield = 4.5% (0.045).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 5000 / (1 + 0.045)10 = 5000 / (1.045)10 ≈ 5000 / 1.55297
4. Result: Price ≈ $3,219.55
Conclusion: The price is significantly discounted due to the longer term and higher yield.
Example 3: Higher Yield
Scenario: Find the price of a bond with a Face Value of $10,000, maturing in 7 years, with a Yield of 6%.
1. Known Values: Face Value = $10,000, Years = 7, Yield = 6% (0.06).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 10000 / (1 + 0.06)7 = 10000 / (1.06)7 ≈ 10000 / 1.50363
4. Result: Price ≈ $6,650.57
Conclusion: A higher required yield results in a lower current price.
Example 4: Shorter Maturity
Scenario: Calculate the price of a bond with a Face Value of $2,000, maturing in 2 years, with a Yield of 2.5%.
1. Known Values: Face Value = $2,000, Years = 2, Yield = 2.5% (0.025).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 2000 / (1 + 0.025)2 = 2000 / (1.025)2 ≈ 2000 / 1.05063
4. Result: Price ≈ $1,903.50
Conclusion: Shorter maturities have prices closer to the face value, assuming a positive yield.
Example 5: Very Low Yield
Scenario: Find the price of a bond with a Face Value of $1,000, maturing in 20 years, with a Yield of 0.5%.
1. Known Values: Face Value = $1,000, Years = 20, Yield = 0.5% (0.005).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0.005)20 = 1000 / (1.005)20 ≈ 1000 / 1.10495
4. Result: Price ≈ $904.91
Conclusion: Even with a long maturity, a very low yield keeps the price relatively high.
Example 6: Yield Equals Zero
Scenario: What is the price of a bond with a Face Value of $1,000, maturing in 10 years, with a Yield of 0%?
1. Known Values: Face Value = $1,000, Years = 10, Yield = 0% (0).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0)10 = 1000 / (1)10 = 1000 / 1
4. Result: Price = $1,000.00
Conclusion: If the required yield is 0%, the price equals the face value. This is theoretical as bonds typically yield something positive.
Example 7: Bond Maturing Today (Years = 0)
Scenario: Calculate the price of a bond with a Face Value of $1,000, maturing in 0 years, with a Yield of 3%.
1. Known Values: Face Value = $1,000, Years = 0, Yield = 3% (0.03).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0.03)0 = 1000 / (1.03)0 = 1000 / 1 (Any non-zero number to the power of 0 is 1)
4. Result: Price = $1,000.00
Conclusion: If the bond matures today, its price is its face value, regardless of the yield.
Example 8: Large Face Value
Scenario: Find the price of a bond with a Face Value of $100,000, maturing in 15 years, with a Yield of 5.2%.
1. Known Values: Face Value = $100,000, Years = 15, Yield = 5.2% (0.052).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 100000 / (1 + 0.052)15 = 100000 / (1.052)15 ≈ 100000 / 2.14774
4. Result: Price ≈ $46,560.73
Conclusion: The price is significantly below the large face value.
Example 9: Fractional Years
Scenario: Calculate the price of a bond with a Face Value of $1,000, maturing in 3.5 years, with a Yield of 4%.
1. Known Values: Face Value = $1,000, Years = 3.5, Yield = 4% (0.04).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0.04)3.5 = 1000 / (1.04)3.5 ≈ 1000 / 1.14866
4. Result: Price ≈ $870.60
Conclusion: The formula works for fractional years as well.
Example 10: Impact of Doubling Yield
Scenario: Compare the price impact if the yield in Example 1 doubled from 3% to 6%.
1. Known Values: Face Value = $1,000, Years = 5, New Yield = 6% (0.06).
2. Formula: Price = Face Value / (1 + Yield)Years
3. Calculation: Price = 1000 / (1 + 0.06)5 = 1000 / (1.06)5 ≈ 1000 / 1.33823
4. Result: Price ≈ $747.26
Conclusion: Doubling the yield significantly lowers the price (from $862.61 in Example 1 to $747.26).
Frequently Asked Questions about Zero Coupon Bonds
1. What is the main characteristic of a zero coupon bond?
It does not pay regular interest (coupons). Its return comes solely from the difference between its purchase price and its face value received at maturity.
2. Why are zero coupon bonds sold at a discount?
The discount is how the investor earns a return. The difference between the discounted price paid today and the higher face value received in the future represents the accumulated interest over the bond's life.
3. How is the price of a zero coupon bond determined?
Its price is the present value of its face value, discounted back to the present using the required rate of return (Yield to Maturity) and the time until maturity.
4. What is Yield to Maturity (YTM)?
YTM is the total return anticipated on a bond if it is held until it matures. For a zero coupon bond, it's the discount rate that equates the present value of the future face value to the current price.
5. How does YTM affect the bond's price?
There is an inverse relationship: If YTM increases (market interest rates rise), the price of an existing zero coupon bond decreases. If YTM decreases (market rates fall), the price increases.
6. How does the time to maturity affect the bond's price?
The longer the time to maturity, the lower the present value (price) of the face value, assuming a positive yield. Longer-term bonds are also more sensitive to changes in interest rates (YTM).
7. Are zero coupon bonds riskier than coupon bonds?
Zero coupon bonds have higher "interest rate risk" or "duration risk" compared to coupon-paying bonds of the same maturity, because the entire return is received at maturity, making the price more sensitive to YTM changes.
8. What are zero coupon bonds typically used for?
They are often used by investors planning for a specific future expense, like college tuition or retirement, because they provide a known lump sum payment on a specific date.
9. Are there different types of zero coupon bonds?
Yes, many types of bonds (like Treasury bonds, corporate bonds, municipal bonds) can be issued as zero coupon bonds. There are also "stripped" bonds, where the principal and interest payments of a traditional bond are separated and sold as zero coupon securities.
10. Can the Yield to Maturity be negative?
Theoretically, yes, in unusual economic conditions (like in some countries with negative interest rate policies), but it is rare for typical zero coupon bonds. Our calculator assumes a non-negative yield for standard scenarios.
