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Modified duration is defined above as a derivative (as the term relates to calculus) and so is based on infinitesimal changes. Modified duration is also useful as a measure of the sensitivity of a bond's market price to finite interest rate (i.e., yield) movements. For a small change in yield, ,
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.
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]
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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.
In the United States, banks and financial service companies have been among the largest issuers of these securities. [4] The U.S. Treasury [5] began issuing them in 2014, and government sponsored enterprises (GSEs) such as the Federal Home Loan Banks, the Federal National Mortgage Association (Fannie Mae) and the Federal Home Loan Mortgage Corporation (Freddie Mac) are important issuers.
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.
Yield to put (YTP): same as yield to call, but when the bond holder has the option to sell the bond back to the issuer at a fixed price on specified date. Yield to worst (YTW): when a bond is callable, puttable, exchangeable, or has other features, the yield to worst is the lowest yield of yield to maturity, yield to call, yield to put, and others.