Bond Convexity Calculator

Calculate bond convexity, Macaulay duration, modified duration, and price sensitivity to yield changes.

Benchmark Bond Profiles

Preset Profiles

1-click presets

Face value (Par value)

Annual coupon rate

Years to maturity

years

Yield to maturity (YTM)

Coupon payment frequency

Yield shift simulation (Δy)

Test how interest rate changes affect bond price with duration and convexity.

Yield shock

bps

Bond Convexity

71.785

Second-order price sensitivity (annualized years²)

Modified Duration

7.665 yrs

Price moves ~7.67% per 100 bps yield change

Current Price

$925.61

92.56% of Par

Macaulay Duration

7.895 yrs

Weighted cash flow time

Dollar Duration

$7,094.86

Price sensitivity in $

DV01 / PV01

$0.7095

Value of 1 basis point

Yield Shift Analysis: +100 bps (+1.00%)

YTM moves from 6.00% to 7.00%

Rate Hike

Duration-Only Estimate

$854.66

-7.67%

Duration + Convexity

$857.99

-7.31%

Exact Re-priced Value

$857.88

-7.32%

Convexity cushion effect: +$3.32 (enhances price gain / softens loss)Taylor approximation residual error: $0.11

Present Value Composition (Coupons vs Par)

PV of Periodic Coupons$371.9440.2%
PV of Par Face Value$553.6859.8%

Yield Shock Matrix (Sensitivity Grid)

Comparing linear duration approximation vs convexity adjustment across yield shifts

Yield Shift	New YTM	Duration Only	Duration + Convexity	Exact Price	Convexity Gain
-300 bps	3.00%	$1,138.46 (+23.00%)	$1,168.36 (+26.23%)	$1,171.69 (+26.58%)	+$29.90
-200 bps	4.00%	$1,067.51 (+15.33%)	$1,080.80 (+16.77%)	$1,081.76 (+16.87%)	+$13.29
-100 bps	5.00%	$996.56 (+7.67%)	$999.88 (+8.02%)	$1,000.00 (+8.04%)	+$3.32
-50 bps	5.50%	$961.09 (+3.83%)	$961.92 (+3.92%)	$961.93 (+3.92%)	+$0.83
-25 bps	5.75%	$943.35 (+1.92%)	$943.56 (+1.94%)	$943.56 (+1.94%)	+$0.21
+25 bps	6.25%	$907.88 (-1.92%)	$908.08 (-1.89%)	$908.08 (-1.89%)	+$0.21
+50 bps	6.50%	$890.14 (-3.83%)	$890.97 (-3.74%)	$890.95 (-3.74%)	+$0.83
+100 bps	7.00%	$854.66 (-7.67%)	$857.99 (-7.31%)	$857.88 (-7.32%)	+$3.32
+200 bps	8.00%	$783.72 (-15.33%)	$797.00 (-13.89%)	$796.15 (-13.99%)	+$13.29
+300 bps	9.00%	$712.77 (-23.00%)	$742.67 (-19.76%)	$739.84 (-20.07%)	+$29.90

Mathematical Formulas & Derivations

Open to see each step from your inputs to the result.

Cash Flow & Convexity Weights Schedule

Period-by-period breakdown of cash flows, present value, and duration/convexity weights

Period	Time (Yr)	Cash Flow	Discount Factor	PV ($)	Weight (%)	Weighted Convexity
1	0.50	$25.00	0.97087	$24.27	2.62%	0.0131
2	1.00	$25.00	0.94260	$23.56	2.55%	0.0382
3	1.50	$25.00	0.91514	$22.88	2.47%	0.0742
4	2.00	$25.00	0.88849	$22.21	2.40%	0.1200
5	2.50	$25.00	0.86261	$21.57	2.33%	0.1747
6	3.00	$25.00	0.83748	$20.94	2.26%	0.2375
7	3.50	$25.00	0.81309	$20.33	2.20%	0.3075
8	4.00	$25.00	0.78941	$19.74	2.13%	0.3838
9	4.50	$25.00	0.76642	$19.16	2.07%	0.4658
10	5.00	$25.00	0.74409	$18.60	2.01%	0.5527

Report tool

Understanding bond convexity, duration, and price sensitivity

Bond convexity is a fundamental risk-management metric in fixed-income investing that measures the curvature of the relationship between bond prices and market interest rates. While modified duration provides a linear estimate of price sensitivity, convexity captures how that duration changes as yields shift, giving institutional investors, portfolio managers, and individual bondholders a far more accurate assessment of interest rate risk.

When interest rates fluctuate across market cycles, bond prices do not move along a straight line. Instead, the price-yield curve is convex to the origin. For comprehensive bond pricing, yield to maturity, and coupon cash flow analysis, you can explore our companion bond calculator, calculate annual coupon productivity with our bond current yield calculator, track interest rate shifts using our basis point calculator, or analyze money market yields with our bank discount calculator.

Mathematical foundations of duration and convexity

To understand convexity, we first evaluate the cash flows of a standard fixed-rate bond. Let face value be $F$ , annual coupon rate be $c$ , payment frequency per year be $m$ , annual yield to maturity (YTM) be $y$ , and years to maturity be $t$ .

The total number of compounding periods is $N = t \times m$ , the periodic coupon payment is $C = (F \times c) / m$ , and the periodic discount rate is $r = y / m$ .

1. Bond pricing equation

The theoretical market price $P$ of the bond is the sum of all discounted future cash flows:

P = \sum_{i=1}^{N} \frac{CF_i}{(1 + r)^i} = \sum_{i=1}^{N} \frac{C}{(1 + r)^i} + \frac{F}{(1 + r)^N}

2. Macaulay duration

Macaulay duration ( $D_{mac}$ ) measures the weighted average time (in years) required to receive all promised cash flows, weighted by the present value of each payment:

D_{mac} = \frac{1}{P} \sum_{i=1}^{N} \left( \frac{i}{m} \right) \frac{CF_i}{(1 + r)^i}

3. Modified duration

Modified duration ( $D_{mod}$ ) translates Macaulay duration into a direct percentage price sensitivity metric for a given change in yield:

D_{mod} = \frac{D_{mac}}{1 + r} = \frac{D_{mac}}{1 + y/m}

Mathematically, modified duration represents the negative first derivative of bond price with respect to yield, normalized by the bond price:

D_{mod} = -\frac{1}{P} \frac{dP}{dy}

4. Bond convexity formula

Convexity ( $C_x$ ) is the second derivative of bond price with respect to yield divided by the bond price. For discrete cash flow periods, annualized convexity is expressed as:

C_x = \frac{1}{P} \frac{d^2P}{dy^2} = \frac{1}{P \cdot (1 + r)^2} \sum_{i=1}^{N} \left[ \frac{i(i + 1)}{m^2} \right] \frac{CF_i}{(1 + r)^i}

Price estimation via Taylor series expansion

When interest rates change by $\Delta y$ (where a 100 basis point change equals 0.01 in decimal form), approximating the percentage price change using modified duration alone provides a first-order linear tangent approximation:

\left( \frac{\Delta P}{P} \right)_{\text{duration}} \approx -D_{mod} \cdot \Delta y

Because the actual price-yield relationship is curved rather than linear, duration alone underestimates price increases when interest rates fall and overestimates price decreases when interest rates rise. Adding the second-order convexity term resolves this error:

\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \frac{1}{2} \cdot C_x \cdot (\Delta y)^2

The new estimated bond price $P_{\text{approx}}$ is calculated as:

P_{\text{approx}} = P \times \left[ 1 - D_{mod} \cdot \Delta y + \frac{1}{2} \cdot C_x \cdot (\Delta y)^2 \right]

Step-by-step worked example: 10-year benchmark bond

Let us examine a 10-year Treasury bond with a par face value of $1,000, an annual coupon rate of 5.00% paid semi-annually (2 payments per year of $25.00 each), and a market yield to maturity of 6.00%.

Baseline Bond Characteristics

Face Value (F): $1,000.00
Coupon Payment (C): $25.00 semi-annually (20 total periods)
Periodic Discount Rate (r): 6.00% / 2 = 3.00% per half-year (0.03)
Initial Bond Price (P): $925.61 (trading at a discount to par)
Macaulay Duration: 7.90 years
Modified Duration: 7.90 / (1 + 0.03) = 7.67 years
Convexity: 71.79 years squared

Scenario A: Interest rates rise by +200 bps (+2.00%, Δy = +0.02)

1. Duration-only estimate: $\Delta P / P \approx -7.67 \times 0.02 = -15.33\%$ (Estimated Price: $783.71)

2. Convexity adjustment: $+0.5 \times 71.79 \times (0.02)^2 = +0.01436 = +1.44\%$ (+$13.29 cushion)

3. Duration + Convexity estimate: $-15.33\% + 1.44\% = -13.89\%$ (Estimated Price: $797.00)

4. Exact re-priced value at 8.00% YTM: $796.15

Notice that duration alone predicted a loss of $141.90, whereas the actual loss was only $129.46. Positive convexity cushioned the portfolio from $12.44 of loss.

Scenario B: Interest rates fall by -200 bps (-2.00%, Δy = -0.02)

1. Duration-only estimate: $\Delta P / P \approx -7.67 \times (-0.02) = +15.33\%$ (Estimated Price: $1,067.52)

2. Convexity adjustment: $+0.5 \times 71.79 \times (-0.02)^2 = +1.44\%$ (+$13.29 gain booster)

3. Duration + Convexity estimate: $+15.33\% + 1.44\% = +16.77\%$ (Estimated Price: $1,080.81)

4. Exact re-priced value at 4.00% YTM: $1,081.76

Convexity accelerated the upside gain: the bond gained $156.15 instead of the $141.90 predicted by duration alone.

Why positive convexity is valuable to bond investors

In fixed-income portfolio management, positive convexity is often described as an asymmetric advantage. If two bonds have the exact same duration and yield, the bond with higher convexity will consistently outperform:

When interest rates fall

Bond prices increase at an accelerating rate. The higher the convexity, the greater the capital appreciation compared to lower-convexity peers.

When interest rates rise

Bond prices decrease at a decelerating rate. Positive convexity acts as a protective shock absorber, dampening capital losses during monetary tightening cycles.

Key determinants of bond convexity

Maturity: Longer-term bonds exhibit dramatically higher convexity than short-term bonds because cash flows are distributed across a wider time horizon. Convexity increases roughly with the square of maturity.
Coupon Rate: Lower coupon bonds have higher convexity than higher coupon bonds of identical maturity because a larger percentage of total cash flows is concentrated in the final principal repayment.
Yield to Maturity: Convexity is inversely related to yield. At lower interest rate levels, convexity is substantially higher, making bonds more sensitive to rate shocks in low-rate environments.
Zero-Coupon Bonds: A zero-coupon bond has the highest duration among bonds of identical maturity, with its Macaulay duration exactly equal to its term to maturity.

Frequently asked questions

What is the main difference between duration and convexity?

Duration is a first-order (linear) measure of bond price sensitivity that assumes price changes are proportional to yield changes. Convexity is a second-order measure that accounts for the curvature of the price-yield curve, correcting the estimation error of duration for large interest rate shifts.

What does positive convexity mean in practical terms?

Positive convexity means that when market yields drop, the bond price rises by more than duration predicts; when market yields rise, the bond price drops by less than duration predicts. All standard non-callable, fixed-rate bonds have positive convexity.

Can a bond have negative convexity?

Yes. Callable bonds and mortgage-backed securities (MBS) can exhibit negative convexity. When interest rates fall significantly, issuers or homeowners refinance early (prepayment risk), capping capital appreciation and shortening duration.

What is DV01 and how does it relate to duration?

DV01 (Dollar Value of an 01, also called PV01 or BPV) measures the absolute dollar change in a bond price for a 1 basis point (0.01%) shift in yield. It is calculated as Modified Duration multiplied by Bond Price multiplied by 0.0001.

How do portfolio managers use convexity?

Portfolio managers structure barbell strategies (combining very short and very long maturities) to achieve higher portfolio convexity than a bullet strategy (single intermediate maturity) with the same duration, maximizing gains during volatile interest rate environments.

Why is convexity measured in years squared?

Because convexity is the second derivative of price with respect to yield divided by price, the mathematical formula multiplies time by time (t squared), resulting in units of years squared.

Resources and references

The formulas and methods in this calculator were checked against these independent sources.