they will have identical sensitivities. That is, their prices will be affected equally by small, first-order, (and parallel) yield curve shifts. They will, however, start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts. For two bonds with the same par value, coupon, and maturity, convexity may differ depending on what point on the price yield curve they are located. Suppose both of them have at present the same price yield (p-y) combination; also you have to take into consideration the profile, rating, etc. of the issuers: let us suppose they are issued by different entities. Though both bonds have the same p-y combination, bond A may be located on a more elastic segment of the p-y curve compared to bond B. This means if yield increases further, the price of bond A may fall drastically while the price of bond B won’t change; i.e. bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, while bond A holders are expecting further price-fall and are ready to dispose of it. This means bond B has a better rating than bond A. So the higher the rating or credibility of the issuer, the lower the convexity and the lower the gain from risk-return game or strategies. Less convexity means less price-volatility or risk; less risk means less return. Mathematical definition If the flat floating interest rate is r and the bond price is B, then the convexity C is defined as Another way of expressing C is in terms of the modified duration D: Therefore, leaving Where D is a Modified Duration How bond duration changes with a changing interest rate Return to the standard definition of modified duration: where P(i) is the present value of coupon i, and t(i) is the future payment date. As the interest rate increases, the present value of longer-dated payments declines in relation to earlier coupons (by the discount factor between the early and late payments). However, bond price also declines when interest rate increases, but changes in the present value of sum of each coupons times timing (the numerator in the summation) are larger than changes in the bond price (the denominator in the summation). Therefore, increases in r must decrease the duration (or, in the case of zero-coupon bonds, leave the unmodified duration constant).