Modified duration is a straight-line fit to a price-yield relationship that is actually curved. Over a small Δy the line is a fine approximation (Worked Example 1's 50bp check bears this out); over a large Δy, or whenever you want the extra precision, add the curvature term:
Convexity (closed form) = [ Σ CFₜ × t × (t+1) / (1 + y/freq)^(t+2) ] / (Price × freq²)
%ΔPrice ≈ −(Modified duration) × Δy + ½ × Convexity × (Δy)²Read the second term's sign rule as the single most important fact in this section: for a plain option-free bond, convexity is always positive, and a positive number added to a negative linear term makes the estimate less negative on a rate rise and more positive on a rate fall, convexity is, for an option-free bond, always a small gift to the holder in both directions (the exact "small, favourable error" M7.03 flagged qualitatively; Worked Example 3 quantifies it precisely: at +150bp the duration-only estimate overstates the loss by 0.69 percentage points, and adding convexity shrinks that error to 0.04pp). Convexity can also be computed the effective way, bump-and-reprice both directions and combine the three prices, exactly as for effective duration. For an option-free bond the two methods agree closely. For a callable bond or an MBS, convexity can turn negative. Worked Example 4 computes an effective convexity of −340 for a synthetic MBS pool and shows precisely what that sign flip does to the shape of its price response.
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Shock estimation: the two-term estimate, and where it stops being enough
Put duration and convexity together and you have this module's core practical skill: given a yield or curve shock, estimate the price impact before, or instead of: running a full valuation model. Worked Examples 1 and 3 do this three ways side by side for the same bond (duration-only, duration+convexity, and an exact reprice from the actual cash flows) at four shock sizes, ±50bp and ±150bp, so you can see the approximation error shrink as convexity is added and grow as the shock size grows. State the discipline as a rule and keep it: an estimate is a starting point, never a number you report as fact, whenever you can reprice exactly (and for a bond whose cash flows you know, you always can), do so, and use the estimate only as a sanity check on the exact answer, never as a substitute for it. The two-term estimate's blind spot is the size of the shock, and it is also the shape of the shock: it silently assumes the entire curve moves by the same Δy, which the section on key-rate durations shows is very often not what happens.