Why is the price-yield relationship for vanilla bonds asymmetric even without options?

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AcadiFi · Accepted Answer

The convex price-yield relationship is not symmetric because bond price is a non-linear function of yield. Specifically: P = Σ CF_t / (1 + y)^t The denominator enters in an exponential form, so a given change in yield produces a smaller dollar change when yields are high (denominator already large) than when yields are low (denominator small). This creates a convex curve — drops in yield produce larger price gains than equivalent rises in yield produce price losses. Worked example — 10-year Trevorson Corp 4% semi-annual coupon bond at 4% yield, price 100. | Yield | Price | Change from 100 | |---|---|---| | 2% | 118.0 | +18.0 | | 3% | 108.5 | +8.5 | | 4% | 100.0 | 0 | | 5% | 92.2 | -7.8 | | 6% | 85.0 | -15.0 | Notice the 100 bp rally from 4% to 3% gains 8.5 points, while the 100 bp sell-off from 4% to 5% loses only 7.8 points — the price gain exceeds the price loss by 0.7 points. For a 200 bp move, the gap is 18.0 vs 15.0 = 3 points. ```mermaid flowchart LR A[Duration
Linear Approx] --> B[-D × Δy × P] B --> C[Underestimates
Rally Gain] B --> D[Overestimates
Sell-off Loss] C --> E[Add Convexity
Correction] D --> E ``` Mathematically, duration gives a linear approximation (first-order Taylor expansion) and convexity provides the quadratic correction (second-order). The convexity correction is always positive, making the combined duration-plus-convexity estimate better than duration alone. The asymmetry grows with maturity — a 30-year bond shows much more pronounced convexity than a 2-year bond because the denominator effect compounds more over more periods. That's why long-duration strategies capture convexity 'lift' in volatile rate environments.

Why is the price-yield relationship for vanilla bonds asymmetric even without options?

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