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**Sample text**

50 percent, a change of 300 basis points. 64498. 60454 percent. 9477 percent. In this case, the adjusted estimate is closer than that obtained using only the modified duration measure. The continuing error reflects the fact that convexity is a dynamic measure and changes with yield changes; the effect of a large yield movement compounds the inaccuracy of the adjustments. convexity are both functions of yield level, and their effects are magniﬁed at lower yield levels. In addition, the cash effect of convexity is more noticeable for large moves in yield.

The average time to receipt would be a more accurate measure, therefore, if it were weighted according to the cash ﬂows’ present values. The average maturity of a bond’s cash ﬂow stream calculated in this manner provides a measure of the speed at which a bond pays out its return, and hence of its price risk relative to other bonds having the same maturity. The average time until receipt of a bond’s cash ﬂows, weighted according to the present values of these cash ﬂows, measured in years, is known as duration or Macaulay’s duration, referring to the man who introduced the concept in 1938—see Macaulay (1999) in References.

A full explanation for why this occurs lies outside the scope of this book. In simplest terms, though, it boils down to this: the forward rate, or marginal rate of return, is equal to the spot rate, or average rate of return, plus the rate of increase in the spot rate multiplied by the sum of the increases between t and T. If the spot rate is constant (corresponding to a ﬂat curve), the forward-rate curve will equal it. An increasing spot-rate curve, however, does not always generate an increasing forward curve, only one that lies above it; it is possible for the forward curve to be increasing or decreasing while the spot rate is increasing.