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Convex- 42 Introduction to Bonds ity can thus be regarded as an indication of the error made when using Macaulay and modiﬁed duration. A bond’s convexity is positively correlated to the dispersion of its cash ﬂows: all else being equal, a bond whose cash ﬂows are more spread out in time—that is, more dispersed—than another’s will have a higher convexity. Convexity is also positively correlated with duration. 17) shows that convexity is the rate at which price sensitivity to yield changes as yield changes.
12). 15). 14) is the formula for pricing zero-coupon bonds when the spot rate is the nonconstant instantaneous risk-free rate r (s) described above. 15), for pricing a risk-free zero-coupon bond. 15) evolves as a martingale process under the risk-neutral probability measure P . This process is the province of advanced ﬁxed-income mathematics and lies outside the scope of this book. An introduction, however, is presented in chapter 4, which can be supplemented by the readings listed in the References section.
12). 12). 28354 years. Properties of Macaulay Duration Duration varies with maturity, coupon, and yield. Broadly, it increases with maturity. A bond’s duration is generally shorter than its maturity. This is because the cash ﬂows received in the early years of the bond’s life have the greatest present values and therefore are given the greatest weight. That shortens the average time in which cash ﬂows are received. A zero-coupon bond’s cash ﬂows are all received at redemption, so there is no present-value weighting.