Bond Convexity Calculator

About Bond Convexity Calculator

The Bond Convexity Calculator measures the second-order sensitivity of a bond's price to changes in its yield. While modified duration tells you the slope of the price-yield curve at a single point, convexity tells you how much that curve bends — a number that matters enormously once yield moves get large. This calculator does what most online tools skip: it lets you see, side by side, the duration-only price prediction, the duration-plus-convexity prediction, and the exact repriced bond, so the size and direction of the curvature correction are obvious at a glance.

What makes this calculator different

How to use the Bond Convexity Calculator

1. Click a quick-start preset (2-yr Treasury, 10-yr Treasury, 30-yr corporate, or 5-yr zero-coupon) to populate every field instantly, or type your own bond details.
2. Enter the bond's face value (par), annual coupon rate, current yield to maturity, and years to maturity.
3. Pick the coupon frequency. Semi-annual is the default for US bonds; pick annual for European bonds or zero-coupon, quarterly or monthly for some structured notes.
4. Drag the yield-shock slider to pick the basis-point change you care about. 100 bp is a common stress-test size; pick 300+ bp to really see convexity matter.
5. Hit "Calculate" and read the verdict card, the three-way comparison strip, the shock curve chart, the cash-flow waterfall, and the per-period attribution table.

The math under the hood

Every result starts from the standard present-value bond pricing equation, where each coupon and the final principal repayment are discounted at the periodic yield \(y = y_{annual}/m\) with \(m\) periods per year and total period count \(n = y_{maturity} \cdot m\):

\( P = \displaystyle\sum_{t=1}^{n} \dfrac{\text{CF}_t}{(1+y)^t} \)

Macaulay duration is the PV-weighted average time of the cash flows, expressed in years by dividing by \(m\):

\( D_{Mac} = \dfrac{1}{P \cdot m} \displaystyle\sum_{t=1}^{n} \dfrac{t \cdot \text{CF}_t}{(1+y)^t} \)

Modified duration adjusts Macaulay for the periodic yield and gives the percent price change per 1% yield change:

\( D_{mod} = \dfrac{D_{Mac}}{1 + y/m} \)

Convexity is the price-weighted sum of the second-order time weighting, scaled back to years squared by dividing by \(m^2\):

\( C = \dfrac{1}{P \cdot m^2} \displaystyle\sum_{t=1}^{n} \dfrac{t(t+1) \cdot \text{CF}_t}{(1+y)^{t+2}} \)

The two metrics combine into the second-order Taylor approximation of the percent price change for a yield shift \(\Delta y\):

\( \dfrac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \tfrac{1}{2} \cdot C \cdot (\Delta y)^2 \)

The convexity term is always non-negative because of the squared yield change. That is why higher-convexity bonds are said to enjoy a "convexity gift" — they rally more in a yield drop than duration predicts and lose less in a yield rise.

Interpreting your results

A few rules of thumb to keep in mind when reading the output:

• Convexity scales roughly with maturity squared. A 30-year bond can have 10× the convexity of a 5-year bond at similar duration ratios.
• Lower coupons mean higher convexity. A zero-coupon bond has the highest convexity for its maturity because all the cash flow sits at the most distant point.
• Higher yields mean lower convexity. The discounting factor \((1+y)^{t+2}\) in the denominator shrinks the contribution of distant cash flows when yields rise.
• The convexity correction is symmetric in sign. Whether yields go up or down by 100 bp, the convexity term adds the same positive percent to the price prediction — that is the curvature gift.

Frequently Asked Questions

What is bond convexity?

Convexity is the second derivative of a bond's price with respect to its yield, scaled by the bond's price. Because the price-yield relationship is curved rather than straight, duration (the first derivative) only gives a linear estimate of how price will move when yields change. Convexity is the second-order correction that captures the curvature, and it is always positive for option-free bonds.

Why does convexity matter to investors?

For small yield changes, duration is enough. For large yield changes — say 100 basis points or more — duration alone underestimates the price gain from a yield drop and overestimates the price loss from a yield rise. Convexity quantifies that asymmetry, which is sometimes called the convexity gift: between two bonds with the same duration, the one with higher convexity outperforms when volatility is high.

What is the formula for convexity?

Convexity in years squared is:

\( C = \dfrac{1}{P \cdot m^2} \displaystyle\sum_{t=1}^{n} \dfrac{t(t+1) \cdot \text{CF}_t}{(1+y)^{t+2}} \)

where \(P\) is the bond price, \(m\) is the number of coupon periods per year, \(y\) is the periodic yield, and \(\text{CF}_t\) is the cash flow at period \(t\). The \(m^2\) factor converts period-squared into year-squared units.

How is convexity used to predict a price change?

Combined with modified duration, the percent change in price is approximately:

\( \dfrac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \tfrac{1}{2} \cdot C \cdot (\Delta y)^2 \)

Because the convexity term is squared, it adds a positive correction whether yields rise or fall, which is the source of the convexity gift.

Which bonds have the highest convexity?

Long-maturity bonds with low coupons have the highest convexity. Cash flows farther in the future are weighted more heavily in the formula because of the \(t(t+1)\) factor. Zero-coupon bonds typically have the highest convexity for a given maturity since all of their cash flow is concentrated at the end.

Is higher convexity always better?

All else equal, yes — higher convexity means better performance under yield volatility. In practice, higher-convexity bonds tend to be priced richer (lower yield) because investors pay a premium for the convexity. Whether the trade-off is attractive depends on your view on volatility versus carry.

How does convexity differ from duration?

Duration is a first-order measure — the slope of the price-yield curve at the current yield. It assumes the curve is locally straight. Convexity is a second-order measure — the curvature of that curve. Duration alone is accurate only for very small yield changes; convexity becomes important the larger the yield shift, because the curve bends away from the tangent line.

Can convexity be negative?

For plain vanilla bonds (no embedded options), convexity is always positive. Bonds with embedded options — particularly callable bonds and mortgage-backed securities — can exhibit negative convexity in certain yield regions because the issuer's call option caps the upside. This calculator models the option-free case.

What is the difference between Macaulay duration and modified duration?

Macaulay duration is the present-value-weighted average time at which the bondholder receives cash flows, measured in years. Modified duration adjusts Macaulay duration by dividing by \(1 + y/m\), and it directly answers the question "what percent does my bond price move for a 1% change in yield?" The two are nearly identical when the yield is small, and diverge slightly as yield grows.

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