A number of versions of duration have been introduced since Macaulay first wrote down a formula for the statistic in the 1930s. I always wonder: Does this mean he *invented* it or *discovered* it? Anyway, another version that I'll call *spot duration* sometimes is used in academic fixed-income research.

This looks much like the weighted-average formula for Macaulay duration in equation 6.14. The difference is that instead of discounting the cash flows with the yield to maturity, the sequence of spot, or zero-coupon, rates (*z*_{1}, *z*_{2}, … , *z*_{N}) is used. The price of the bond in the denominator is the same as in equation 6.14

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