Contrast the effective convexities of straight, callable and putable bonds

Effective convexity is (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀). An option-free bond has positive convexity: its price rises more when rates fall than it falls when rates rise. A callable bond near its call price has negative convexity, because the call caps the upside, while far from the call it behaves like the straight bond. A putable bond keeps positive convexity and, near its put price, has more than the straight bond, because the put limits the downside. A common slip is to forget to square ΔCurve in the denominator, or to invert the sign by subtracting the other way, which turns the callable bond's negative convexity positive.