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In contrast to Macaulay duration, modified duration (sometimes abbreviated MD) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield (the logarithmic derivative of bond price with respect to yield). [15] Modified duration applies when a bond or other asset is considered as a function of yield.
The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity. [2] Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i.e. how the duration of a bond changes as the interest rate changes. [3]
The modified duration of a bond assumes that cash flows do not change in response to movements in the term structure, which is not the case for an MBS. For instance, when rates fall, the rate of prepayments will probably rise and the duration of the MBS will also fall, which is entirely the opposite behavior to a vanilla bond.
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This is equal to the Macaulay duration times the discount rate, or the modified duration times the interest rate. If the elasticity is below -1, or above 1 if the absolute value is used, the product of the two measures, value times yield or the interest income for the period will go down when the yield goes up.
The duration of an equity is a noisy analogue of the Macaulay duration of a bond, due to the variability and unpredictability of dividend payments. The duration of a stock or the stock market is implied rather than deterministic. Duration of the U.S. stock market as a whole, and most individual stocks within it, is many years to a few decades.
Henrard, Marc (2003). "Explicit Bond Option and Swaption Formula in Heath–Jarrow–Morton One Factor Model," International Journal of Theoretical and Applied Finance, 6(1), 57–72. Preprint SSRN. Henrard, Marc (2009). Efficient swaptions price in Hull–White one factor model, arXiv, 0901.1776v1. Preprint arXiv. Ostrovski, Vladimir (2013).