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In finance , bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates , the second derivative of the price of the bond with respect to interest rates duration is the first derivative. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates.
Bond convexity is one of the most basic and widely used forms of convexity in finance. Duration is a linear measure or 1st derivative of how the price of a bond changes in response to interest rate changes.
As interest rates change, the price is not likely to change linearly, but instead it would change over some curved function of interest rates.
The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity. Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i. Specifically, one assumes that the interest rate is constant across the life of the bond and that changes in interest rates occur evenly.
Using these assumptions, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question. Then the convexity would be the second derivative of the price function with respect to the interest rate. In actual markets the assumption of constant interest rates and even changes is not correct, and more complex models are needed to actually price bonds.
However, these simplifying assumptions allow one to quickly and easily calculate factors which describe the sensitivity of the bond prices to interest rate changes.
The price sensitivity to parallel changes in the term structure of interest rates is highest with a zero-coupon bond and lowest with an amortizing bond where the payments are front-loaded. Although the amortizing bond and the zero-coupon bond have different sensitivities at the same maturity, if their final maturities differ so that they have identical bond durations they will have identical sensitivities.
That is, their prices will be affected equally by small, first-order, and parallel yield curve shifts. They will, however, start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts.
For two bonds with same par value, same coupon and same maturity, convexity may differ depending on at what point on the price yield curve they are located. Suppose both of them have at present the same price yield p-y combination; also you have to take into consideration the profile, rating, etc. Though both bonds have same p-y combination bond A may be located on a more elastic segment of the p-y curve compared to bond B.
So the higher the rating or credibility of the issuer the less the convexity and the less the gain from risk-return game or strategies; less convexity means less price-volatility or risk; less risk means less return. If the flat floating interest rate is r and the bond price is B , then the convexity C is defined as. As the interest rate increases, the present value of longer-dated payments declines in relation to earlier coupons by the discount factor between the early and late payments.
However, bond price also declines when interest rate increases, but changes in the present value of sum of each coupons times timing the numerator in the summation are larger than changes in the bond price the denominator in the summation. Therefore, increases in r must decrease the duration or, in the case of zero-coupon bonds, leave the unmodified duration constant. Given the relation between convexity and duration above, conventional bond convexities must always be positive.
The positivity of convexity can also be proven analytically for basic interest rate securities. Then it is easy to see that. For a bond with an embedded option , a yield to maturity based calculation of convexity and of duration does not consider how changes in the yield curve will alter the cash flows due to option exercise. To address this, an "effective" convexity must be calculated numerically. Effective convexity is a discrete approximation of the second derivative of the bond's value as a function of the interest rate:.
These values are typically found using a tree-based model, built for the entire yield curve , and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model finance Interest rate derivatives.
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Accrual bond Auction rate security Callable bond Commercial paper Contingent convertible bond Convertible bond Exchangeable bond Extendible bond Fixed rate bond Floating rate note High-yield debt Inflation-indexed bond Inverse floating rate note Perpetual bond Puttable bond Reverse convertible securities Zero-coupon bond.