Bond Convexity Calculator
Enter the face value, coupon rate, coupon frequency, years to maturity, yield to maturity and a small yield change to find the bond convexity, the bond price and the coupons.
Updated
Bond Convexity Calculator
Our Bond Convexity Calculator measures how strongly the price of a bond curves as its yield changes. Enter the face value, the annual coupon rate, how often coupons are paid, the years to maturity, the yield to maturity and a small yield change, and it gives the bond convexity together with the bond price, the annual coupon and the coupon per period. It is made for investors, students and analysts who want to measure interest rate risk more accurately than duration alone allows.
Quick answer
A 1,000 bond with a 5% annual coupon, 7 years to maturity and a 4.2% yield is priced at 1,047.66 and has a convexity of about 42.81.
- Formula: convexity = (P− + P+ − 2 × P) ÷ (P × Δy2).
- Meaning: it measures the curve in the price and yield relationship.
- Higher convexity: the price gains more when yields fall and loses less when they rise.
- Use: add it to duration for a closer estimate of a price change.
What Is Bond Convexity?
Bond convexity measures how much the price and yield relationship of a bond bends. When yields rise, bond prices fall, but not along a straight line. The line is curved, and convexity is the size of that curve.
Duration tells you roughly how far the price moves for a small change in yield. Convexity tells you how that sensitivity itself changes as the yield moves. For an ordinary bond with fixed coupons the convexity is positive: the price rises by more when yields fall than it drops when yields rise by the same amount.
Convexity is the second derivative of the bond price with respect to the yield, divided by the price. With the yield written as a decimal per year, it is measured in years squared.
Bond Convexity Formula
This calculator uses the effective (finite difference) method. It prices the bond three times, at the yield, a little above it and a little below it:
÷ (P × Δy2)
- C
- Bond convexity.
- P
- Bond price at the yield to maturity.
- P−
- Bond price when the yield falls by Δy.
- P+
- Bond price when the yield rises by Δy.
- Δy
- The yield change as a decimal: 0.1% is 0.001.
Each price is the present value of all the coupons plus the face value:
+ F ÷ (1 + y ÷ m)n
- c
- Coupon per period: F × coupon rate ÷ m.
- F
- Face value, repaid at maturity.
- y
- Yield to maturity as a decimal per year.
- m
- Coupon payments per year.
- n
- Number of payments: m × years to maturity.
- t
- The payment number, from 1 to n.
How to Calculate Bond Convexity
A bond has a face value of 1,000, a 5% coupon paid once a year, 7 years to maturity and a yield to maturity of 4.2%. Use a yield change of 0.1%:
P+ = 1,041.552513 (at 4.3%)
P− = 1,053.819430 (at 4.1%)
P− + P+ − 2 × P = 0.044853
P × Δy2 = 1,047.663545
× 0.0012 = 0.001047664
C = 0.044853 ÷ 0.001047664
= 42.81
- Find the coupon. 1,000 × 5% = 50 a year, paid in one payment of 50.
- Price the bond at its yield. Discount the seven coupons and the face value at 4.2%: 1,047.66.
- Price it again with the yield moved. At 4.3% the price is 1,041.55 and at 4.1% it is 1,053.82.
- Add the two moved prices and subtract twice the price. The answer, 0.044853, is small, so keep all the decimals.
- Divide by the price times Δy squared. 0.044853 ÷ 0.001047664 = 42.81.
The calculator keeps the unrounded prices and shows 42.8119. Rounding the prices to two decimals first would give a very different answer, because the top of the fraction is a small difference between large numbers.
Using Convexity to Estimate a Price Change
Convexity is used together with duration to estimate how far a bond price moves when its yield changes:
+ ½ × C × Δy2
The example bond has an effective duration of 5.8544 and a convexity of 42.8119. If the yield rises by 1% (Δy = 0.01):
½ × 42.8119 × 0.012 = +0.2141%
Change = −5.6403%
| Yield move | New price |
|---|---|
| Up 1% (4.2% to 5.2%) | Duration only 986.33 With convexity 988.57 Actual 988.51 |
| Down 1% (4.2% to 3.2%) | Duration only 1,109.00 With convexity 1,111.24 Actual 1,111.30 |
Duration alone misses the actual price by more than 2 in both directions. Adding the convexity term brings the estimate to within 0.06. The convexity term is always added for a bond with positive convexity, whichever way the yield moves.
Bond Convexity Examples
Each bond has a face value of 1,000 and uses a yield change of 0.1%:
| Bond | Result |
|---|---|
| 5% annual, 7 years yield 4.2% | Price 1,047.66 Convexity 42.8119 |
| 6% semi-annual, 10 years yield 5% | Price 1,077.95 Convexity 70.6501 |
| 0% (zero coupon), 10 years yield 4% | Price 675.56 Convexity 101.7024 |
| 4% semi-annual, 30 years yield 5% | Price 845.46 Convexity 380.0043 |
| 3% quarterly, 2 years yield 4% | Price 980.87 Convexity 4.2589 |
The face value does not change the convexity. A 100 bond and a 1,000 bond with the same coupon rate, maturity and yield have the same convexity; only the price scales. The amounts carry no currency symbol, so the calculator works in any currency.
What Affects Bond Convexity?
Maturity. Longer bonds have much higher convexity. A 5% annual bond at a 4.2% yield:
| Years | Price | Convexity |
|---|---|---|
| 2 | 1,015.05 | 5.35 |
| 5 | 1,035.42 | 24.37 |
| 7 | 1,047.66 | 42.81 |
| 10 | 1,064.25 | 76.98 |
| 20 | 1,106.82 | 222.93 |
| 30 | 1,135.04 | 382.73 |
Coupon rate. Lower coupons mean higher convexity, because more of the value arrives at maturity. A 7 year annual bond at a 4.2% yield:
| Coupon | Price | Convexity |
|---|---|---|
| 0% | 749.77 | 51.58 |
| 2.5% | 898.71 | 46.47 |
| 5% | 1,047.66 | 42.81 |
| 7.5% | 1,196.61 | 40.07 |
| 10% | 1,345.56 | 37.93 |
Yield. Lower yields mean higher convexity. The same 5% annual, 7 year bond:
| Yield | Price | Convexity |
|---|---|---|
| 1% | 1,269.13 | 46.49 |
| 2% | 1,194.16 | 45.31 |
| 4.2% | 1,047.66 | 42.81 |
| 6% | 944.18 | 40.88 |
| 8% | 843.81 | 38.83 |
| 10% | 756.58 | 36.88 |
Coupon frequency. Paying the same coupon rate more often lowers convexity slightly: the example bond falls from 42.81 with annual coupons to 39.52 with monthly coupons.
The Six Inputs
- Face value
- The amount repaid at maturity, also called par value. Often 1,000 or 100.
- Coupon rate
- The annual interest paid on the face value, as a percentage. Use 0 for a zero coupon bond.
- Coupons paid
- How often the coupon is paid: annually, semi-annually, quarterly, monthly, weekly or daily.
- Years
- The time left until the bond matures.
- Yield
- The yield to maturity (YTM) as an annual percentage, compounded at the coupon frequency.
- Yield change
- The small move in yield used to measure the curve, in percentage points. 0.1 is a common choice.
If you do not know the yield, find it first with the Bond YTM Calculator. To check the price on its own, use the Bond Price Calculator.
How to Read the Result
- Higher convexity. The bond gains more when yields fall and loses less when yields rise. Between two bonds with the same duration and yield, the one with more convexity is the better protected.
- Lower convexity. The price moves almost in a straight line, so duration alone is a fair guide.
- Negative convexity. Callable bonds and mortgage backed securities can have negative convexity, because the issuer or the borrowers can repay early when yields fall. This calculator prices a plain fixed coupon bond, so its result is positive.
- Scale. Some textbooks and data providers divide convexity by 100. A figure of 42.81 here is 0.4281 on that scale.
What the Calculator Assumes
- A plain bond. Fixed coupons, face value repaid at maturity, and no call, put or conversion feature.
- Whole coupon periods. Years × payments per year is rounded to a whole number of payments, and the bond is priced on a coupon date with no accrued interest.
- One yield. Every payment is discounted at the same yield to maturity, compounded at the coupon frequency.
- A parallel move. The yield moves up and down by the same amount.
How to Use the Bond Convexity Calculator
- Enter the face value and the coupon rate. Type the face value, such as 1,000, and the annual coupon rate as a percentage, such as 5.
- Choose how often coupons are paid. Pick annually, semi-annually, quarterly, monthly, weekly or daily.
- Enter the years to maturity and the yield. Type the years left until the bond matures, such as 7, and the yield to maturity as a percentage, such as 4.2.
- Enter the yield change. Type the small yield move used for the measurement in percentage points, such as 0.1.
- Read and copy the result. The bond convexity appears as you type, with the bond price, the prices after the yield moves, the effective duration and the coupons. Press Copy to copy the convexity, or Clear or Esc to start again.
Example: a face value of 1,000, a 6% coupon paid semi-annually, 10 years to maturity, a 5% yield and a yield change of 0.1. The result is a convexity of 70.6501 and a bond price of 1,077.95.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is bond convexity?
Bond convexity measures how much the relationship between the price of a bond and its yield curves. It shows how the duration of the bond changes as the yield changes, and it improves the price change estimate that duration gives on its own.
How do you calculate bond convexity?
Price the bond at its yield (P), at the yield plus a small change (P+) and at the yield minus the same change (P−). Convexity = (P− + P+ − 2 × P) ÷ (P × Δy²), with Δy written as a decimal.
What is the convexity of a 7 year 5% bond at a 4.2% yield?
About 42.81. With annual coupons and a face value of 1,000 the price is 1,047.66, and with a yield change of 0.1% the prices are 1,041.55 and 1,053.82, which gives a convexity of 42.8119.
What is a good bond convexity?
There is no single good figure. Higher positive convexity is better for the holder, because the price rises more when yields fall and drops less when yields rise. Long maturities and low coupons give the highest convexity.
What is the difference between duration and convexity?
Duration is the straight line estimate of how far the price moves for a change in yield. Convexity measures the curve that the straight line misses. For a 1% rise in yield on the example bond, duration alone gives 986.33, adding convexity gives 988.57 and the actual price is 988.51.
What yield change should I use?
A small one, such as 0.1 (10 basis points). For a plain bond the choice hardly matters: the example bond gives 42.8116 with 0.01, 42.8119 with 0.1 and 42.8401 with 1.
Can bond convexity be negative?
Yes, for bonds with embedded options such as callable bonds and mortgage backed securities. A plain fixed coupon bond, which is what this calculator prices, always has positive convexity.
Is the Bond Convexity Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Sources
- Wikipedia. Bond convexity.
- Wikipedia. Duration (finance).
- AnalystPrep. Calculate and Interpret Convexity.
- WallStreetMojo. Convexity of a Bond: Formula and Examples.
- OpenStax. Principles of Finance, 10.4 Risks of Interest Rates and Default.
- Wikipedia. Bond valuation.
Try our Bond Convexity Calculator to see how sensitive a bond really is to a change in interest rates.