## Bond Sensitivity

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Bond prices depends on several factors including the time to maturity, the yield to maturity, and coupon rate. All but the yield to maturity are set by the bond contract. As a consequence, the yield to maturity poses a risk. Yield to maturity will change as the instrument is traded in the market. Price changes due to changes in the bond's yield to maturity is called volatility.

There are many different characterizations of risk that impacts bond trading. First among these are sensitivity measures. A sensativity measure is the effect of a small movement in pricing factors on the present value of a security position. Of all the risk informations, sensitivity measures are often the first checked by traders and risk managers. The popular sensitivity measures in the bond market are PV01, duration, and convexity.

The price value of a basis point (PVBP) or PV01 is defined as the change in the bond price due to one basis point decrease in the nominal yield to maturity. If the PVBP is known, the change in price can be easily estimated. The Bond duration is represented by the bond's PV01.

Duration is a measure of how long on average the holder of the bond has to wait before receiving cash payments. Duration method is used to calculate Macaulay duration. At first the method check if the bond is a zero coupon bond, if it is, the duration is just the difference between maturity and valuation date.

Bond modified duration is computed as:

where

- P is the bond price.
- P+ is the bond price computed off the shifted yield.
- y is the yield to maturity
- y+ is the shifted yield to maturity

Another measure of duration is given by changes in the zero rate curve as opposed to changes in the yield to maturity. Duration evolves with time even with no change in yield to maturity. The duration of a bond is the PV-weighted duration of the cash flows.

Duration is a measure of volatility for fixed income instruments, but there are some shortcomings as duration takes into account only linear changes in the price and second order effects of the price function is ignored.

The convexity of a bond is the change in the slope of the price-yield curve, for a small change in the yield. Convexity represents the second derivative of the bond price with respect to the level of interest rate. The delta of a bond is negative and the convexity of a bond is positive.

Bond convexity is computed as:

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