Note that the modified duration D differs from the regular duration by the factor one over 1+r (shown above), which also decreases as r is increased. Given the relation between convexity and duration above, conventional bond convexities must always be positive. The positivity of convexity can also be proven analytically for basic interest rate securities. For example, under the assumption of a flat yield curve one can write the value of a coupon-bearing bond as , where ci stands for the coupon paid at time ti. Then it is easy to see that Note that this conversely implies the negativity of the derivative of duration by differentiating . Application of convexity Convexity is a risk management figure, used similarly to the way 'gamma' is used in derivatives risks management; it is a number used to manage the market risk a bond portfolio is exposed to. If the combined convexity and duration of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur. (Parallel in the yield curve.) The second-order approximation of bond price movements due to rate changes uses the convexity: Effective convexity For a bond with an embedded option, a yield to maturity based calculation of convexity (and of duration) does not consider how changes in the yield curve will alter the cash flows due to option exercise. To address this, an effective convexity must be calculated numerically. Effective convexity is a discrete approximation of the second derivative of the bond's value as a function of the interest rate: where is the bond value as calculated using an option pricing model, Δ y is the amount that yield changes, and are the values that the bond will take if the yield falls by y or rises by y, respectively (a parallel shift). These values are typically found using a tree-based model, built for the entire yield curve, and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see . See also Black–Scholes equation Bond duration Bond valuation Immunization (finance) List of convexity topics List of finance topics References Further reading Frank Fabozzi, The Handbook of Fixed Income Securities, 7th ed., New York: McGraw Hill, 2005. . The standard reference for conventions applicable to