Bond Convexity Explained

The previous article left a loose thread. Bond B — a 10-year, 8% coupon bond priced at par — has a modified duration of 6.7101. Duration alone estimated its price at a 9% yield to be ₹932.90. But the bond's actual price at 9%, computed properly from its full cash-flow schedule, is ₹935.82. That's not a mistake in either calculation — it's convexity, and this article explains exactly where that ₹2.92 comes from.

Why the straight line isn't quite right

Duration estimates a price change using one number: the slope of the price-yield relationship at today's yield. That's a genuinely useful approximation nearby, but the real relationship between bond price and yield isn't a straight line — it's curved. Convexity measures that curvature, the part duration's straight-line approximation necessarily misses.

Why the relationship is curved at all

Recall the pricing formula: P = Σ Cₜ ÷ (1+y)ᵗ. Yield sits in the denominator, raised to a power — and that alone guarantees the relationship between P and y can't be a straight line. A 1-percentage-point move from 5% to 6% doesn't have exactly the same effect on price as the same 1-point move from 10% to 11%, because of how the denominator compounds. You don't need calculus to accept the economic result: the curve bends, and duration's straight line is only a local approximation to it.

The corrected formula

Adding a second term captures the curvature duration misses:

The first term is the duration effect you already know. The second is the convexity adjustment — and because it involves (Δy)², a squared number, that adjustment is positive whether yield rises or falls. That single algebraic fact has a real economic consequence: for a normal option-free bond, the convexity term always adds back a little value, regardless of which direction yield moves.

Closing the loop: Bond B's actual convexity

Bond B has a convexity of 60.53. Plugging both duration and convexity into the full formula for a 1-point rise in yield, from 8% to 9%:

Compare the three figures side by side:

Duration alone was off by ₹2.92. Adding the convexity correction brings the estimate to within about 11 paise of the actual price — the remaining tiny gap is simply because the formula above is itself still an approximation (a second-order one, more accurate than duration alone but not a perfect closed-form match). This is the practical payoff of convexity: it doesn't replace duration, it sharpens it.

The calculator may show ₹935.77 rather than ₹935.82 — the same exact-date day-counting effect explained in the pricing article.

Positive convexity favours the holder

For a standard option-free bond, convexity is positive — and that has a genuinely favourable asymmetry built into it:

Both directions favour the bondholder. This is one reason convexity is treated as a desirable property, not just a technical correction — between two bonds with similar duration, the one with more convexity behaves better in both a large rally and a large selloff. That said, higher convexity typically comes attached to other trade-offs (price, structure, or lower coupon), so "more convexity" isn't automatically "better bond" in isolation.

When convexity actually matters

For a 1 basis-point yield move, duration alone is usually an excellent approximation — the convexity term is vanishingly small at that scale. For a 200 basis-point move, the curvature becomes impossible to ignore, because a straight-line approximation drifts further from the true curve the further you move from the point where it was drawn. The rule of thumb: small yield changes, duration is often enough; large yield changes, convexity increasingly matters.

For the mathematically inclined: convexity as a second derivative

Price is a function of yield, P(y). Modified duration relates to the first derivative of that function — how fast price changes as yield changes. Convexity relates to the second derivative — how fast that rate of change itself is changing. This is exactly why duration is called a first-order measure and convexity a second-order one in fixed-income literature; the terminology isn't decorative, it describes precisely which derivative each concept corresponds to.

A useful analogy: if bond price is your car's position, duration is your speed, and convexity is your acceleration — how quickly your speed itself is changing.

Not all convexity is positive

Everything above assumes an ordinary, option-free bond. A callable bond can behave very differently. Suppose you own an 8% callable bond and market rates fall sharply. A normal bond would keep rising in value as yields fall — but the issuer now finds it attractive to redeem your bond early and refinance at the new, lower rate. As that call becomes more likely, your upside gets capped: instead of the price continuing to climb, it flattens out and can even turn down as it approaches the call price. That flattening — where rising bond value is capped rather than continuing to climb — is negative convexity, and it's one of the defining hazards of callable bonds and mortgage-backed securities alike.

This is covered in full once this series reaches callable bonds and effective duration — for now, the key point is simply that "positive convexity" is a property of plain bonds, not a universal law of all fixed income.

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