highest convexity and its prices most sensitive to changes in yield. If the bond with prepayment or call option has a premium to be paid for the early exit then the convexity may turn positive. Periodically compounded edit In financial markets, yields are usually expressed periodically compounded (say annually or semi-annually) instead of continuously compounded. Boston, MA: Prentice Hall,. . For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time or more precisely t1displaystyle t_1 (the time to the first payment) and the time of the final cash flow. However, the convexity of this portfolio is higher than the single zero coupon bond. The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. This difference.12 in the price change is due to the fact that the price yield curve is not linear as assumed by the duration formula. Positive and Negative Convexity. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative. The coupon payments couponcode zuiderduin and the periodicity of the payments of the bond contribute to the convexity of the bond.

With the use of computers, both forms may be calculated but expression (3 assuming a constant yield, is more widely used because of the application to pizza inn coupons modified duration. Units edit Mathematically, modified duration and Macaulay duration have the same units, as can be seen in the above equations. Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield. Formulas edit For a standard bond with fixed, semi-annual payments the bond duration closed-form formula is: citation needed textDurfrac 1Pleft(Cfrac FV(m-1a 1i m-1a)right) FV par value C coupon payment per period (half-year) i discount rate per period (half-year) a fraction of a period remaining until. They have a positive correlation. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality. This interest rate risk is measured by modified duration and is further refined by convexity. This is of course totally predictable, and hence not a risk. In the above graph Bond A is more convex than Bond B even though they both have the same duration and hence Bond A is less affected by interest rate changes. The original 100 invested will fall to roughly.90. For bonds that have embedded options, such as putable and callable bonds, Modified duration will not correctly approximate the price move for a change in yield to maturity.