Effective duration and effective convexity describe the interest rate risk of bonds with embedded options.

Effective Convexity = (PV_- + PV₊ - 2 x PV₀)/[(ΔCurve)² x PV₀]

Since these measures are "effective" they must recognize that a shift in interest rates may bring about a change in cash flows, because embedded options may be exercised. In contrast, modified duration and convexity do not allow for the fact that the cash flows for a bond with an embedded option may change due to the exercise of the option.

In order to calculate effective duration and convexity, we use the binomial model to find both V -and V+. An important assumption is that OAS stays constant when interest rates shift.

Effective convexity includes the effects of yield changes on the cash flows. Effective convexity requires an adjustment in the estimated bond values (V-, V+) to reflect any change in estimated cash flows due to the presence of embedded options. When bonds include embedded options, effective convexity may even be negative. This is in fact just what investors need if they are constructing an estimate that falls within a region of positive convexity, as may be the case with callable bonds for example.

• When the call option is out of money, the callable bond has positive convexity.
• When the call option is near the money, the effective convexity becomes negative. This is because the callable bond is capped by the price of the call option.
• Putable bonds always have positive convexity.