Fixed Income Analytics
Second-Order Risk
Taylor Series Curvature
Bond Convexity & Price Volatility Lab
Calculate analytical bond convexity, Macaulay convexity, modified duration, dollar convexity, and compare first-order linear duration vs. second-order Taylor series approximations under interest rate shocks.
Analytical Convexity: C = \frac{1}{P (1 + y/k)^2} \sum_{t=1}^N \frac{t(t+1) C_t}{k^2 (1 + y/k)^t} |
Taylor Series: \frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y + \frac{1}{2} C (\Delta y)^2 |
Dollar Convexity: \$C = C \times P
Fixed Income Presets:
Bond & Market Parameters
$
%
Y
Yield & Shock Parameters
%
bps
+150 bps = +1.50% rate rise; -150 bps = 1.50% rate decline.
Moderate Convexity Regime
Standard positive curvature; Taylor second-order adjustment adds significant accuracy over 100+ bps shocks.
BOND CONVEXITY
72.48
Analytical Second Derivative
MODIFIED DURATION
7.95 Y
Mac. Dur: 8.12 Y
CURRENT BOND PRICE
$1,000.00
Par Clean Price (100.00%)
DOLLAR CONVEXITY
$72,480
DV01: $0.795 per bond
Rate Shock Comparison: Linear Duration vs. Taylor Convexity
| Estimation Method | Predicted % Change | Estimated New Price | Error vs. Exact Price | Description |
|---|---|---|---|---|
| 1. Exact Re-pricing | -11.11% | $888.89 | 0.00% (Baseline) | Exact cash flow discounting at new yield (YTM ± Δy) |
| 2. Linear Duration Only | -11.93% | $880.74 | -0.82% ($8.15) | First-order approximation: -Δy × D_mod (straight line) |
| 3. Duration + Convexity (Taylor) | -11.11% | $888.89 | +0.00% ($0.03) | Second-order Taylor expansion including 0.5 × C × (Δy)² |
Interactive bond price-yield curve demonstrating curvature convexity relative to the linear duration tangent line.
Coupon Rate vs. Maturity on Convexity (C)
Rows: Annual Coupon (%), Columns: Maturity (Years). Longer maturities and lower coupons yield higher convexity.
Rate Shock (Δy) vs. Maturity on Price % Change
Rows: Rate Shock (Δy bps), Columns: Maturity (Years). Demonstrates non-linear price volatility.
Bond Convexity Mastery Self-Assessment
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Frequently Asked Questions: Bond Convexity & Price Curvature
Bond convexity is a measure of the curvature in the relationship between bond prices and bond yields. While modified duration provides a linear, first-order approximation of price sensitivity to small yield changes, convexity captures the non-linear second derivative. It is critical for accurately predicting bond price moves during large interest rate shocks.
The percentage price change of a bond is approximated using Taylor series expansion as: ΔP / P ≈ -Modified Duration × Δy + 0.5 × Convexity × (Δy)². The first term captures linear duration slope, while the second positive term accounts for the curvature, boosting price gains when yields fall and cushioning price drops when yields rise.
Positive convexity creates an asymmetric return profile: as interest rates fall, the bond price rises at an accelerating rate; as interest rates rise, the bond price falls at a decelerating rate. Consequently, an investor holding a more convex bond earns greater capital appreciation during market rallies and suffers smaller losses during market sell-offs.
Negative convexity occurs in bonds with embedded borrower options, such as callable corporate bonds or mortgage-backed securities (MBS). When interest rates drop significantly, borrowers are incentivized to refinance or exercise early redemption calls, capping price appreciation and causing the price-yield curve to flatten or bend downward.
Modified duration measures the first derivative (slope) of the bond price with respect to yield, quantifying linear percentage sensitivity per 100 bps rate move. Convexity measures the second derivative (curvature), quantifying how rapidly modified duration changes as yields shift.