Common misunderstandings concerning duration and convexity. Think of bond prices, or bond portfolio values, as functions of interest. Duration and convexity are two tools used to manage the risk exposure of fixedincome investments. Bond duration calculator macaulay duration and modified. Enter the coupon, yield to maturity, maturity and par in order to calculate the coupon bonds. Fixed income portfolio management interest rate sensitivity, duration, and convexity passive bond portfolio management active bond portfolio management interest rate swaps 1 interest rate sensitivity, duration, and convexity tbond price. Pdf immunization is a wellknown fixedincome strategy to lock in a target rate of return over a known investment horizon. Exhibit 3 shows that over the last 20 years, the aggregate indexs durationsquared term has always been. Convexity of a bond can be defined as a measurement of the curvature in relation to changes in interest rates which shall affect the changes in the price of a bond and further it does so by measuring the duration changes, as the market interest rates fluctuate. One method of measuring interest rate risk due to changes in market interest rates is by the full valuation approach, which simply calculates what bond prices will be if the interest rate changed by specific amounts. For zeroes, duration is roughly equal to maturity, convexity is roughly equal to maturity squared.
The standard approximation using the duration and convexity described above does not include the passage of time. Duration is an imperfect way of measuring a bonds price change, as it indicates that this change is linear in nature when in. A few applications of duration in risk management will also be presented. Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change. Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. Modified duration, on the other hand, is a mathematical derivative rate of change of price and measures the percentage rate of change of price with respect to yield. In finance, bond convexity is a measure of the nonlinear 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. Bond portfolio duration, cash flow dispersion and convexity article pdf available in applied economics letters 1717. Using duration and convexity to approximate change in.
An explanation of the concept of convexity and how it is used in con unction with. Using duration and convexity to approximate change in present value robert alps october 28, 2016. However, positive convexity does not necessarily lead to. Considering duration and convexity duration assumes a linear relationship between bond prices and changes in interest rates. Some evidence suggests that the yield curve is more curved when volatility is higher and convexity is worth more. Also, the duration of the bond when its yielding 12.
In particular, these measures can be used for approximation of a bonds sensitivity to changes in the term structure of interest rates. Duration, modified duration, and convexity reflect the bonds exposition to interest rate risk and are used as tools for risk management purposes. Convexity relates to the interaction between a bonds price and its yield as it experiences changes in interest rates. Yield duration and convexity relationships yield duration the relationship between yield duration and maturity yield convexity bloomberg yield duratio. Convexity indicates that as yield increases, the price of a bond declines at a declining rate. Lecture 11 duration, convexity and immunisation reading. Conversely, if a bond has a duration of 5 years and interest rates fall by 1%, the bond s price will increase by approximately 5%. A bonds convexity refers to the sensitivity of the bonds modified duration to changes in yield.
We provide tables if changing slope as yield changes illustrates. Duration measures the bond s sensitivity to interest rate changes. Application of the concepts of duration and convexity to portfolio will be clari. Annuity factors simplify valuation of level cash flows, such as the coupon payments of bonds. Duration and convexity bond math wiley online library. The bonds cash flows consist of coupons paid periodically and principal repaid at maturity. Although procedures and formulas exist to calculate duration and convexity, these statistics can be approximated using basic bondpricing techniques and a financial calculator. Understanding duration and convexity of fixed income securities vinod kothari notation y. However, also implicit in the convexity measure is the assumption of parallel shifts in yields and, therefore, the durationconvexity. A primer on duration, convexity, and immunization leslaw gajek. Bond duration and convexity introduction continued economics of capital markets version 1. Coupon payment frequency pf this field indicates whether the coupon is paid annually or semiannually.
Chapter 11 duration, convexity and immunization section 11. The sensitivity of the percentage bond price change to changes in. The convexity vortex as interest rates have finally broken above the feds qe infinity induced range of 1. T 1 c yt f t, where y ytm when yields change, bond prices will change. Introduction the basic bond valuation formula is traditionally presented as a straight forward discounted cash flow application. Convexity is a better measure of interest rate risk, concerning bond duration. Jul 22, 2011 this video illustrates how duration can be used to approximate the change in bond price given a change in interest rates. A bond is an instrument of indebtedness that obliges the bond issuer borrower to repay the lender creditor the borrowed assets plus the interest within a certain period of time fabozzi, 2000, p. Bond duration is a way of measuring how much bond prices are likely to change if and when interest rates move.
Krzysztof ostaszewski, and hansjoachim zwiesler abstract the concepts of duration, convexity, and immunization are fundamental tools of assetliability management. In more technical terms, bond duration is measurement of interest rate risk. Bond duration and convexity introduction continued bond duration and convexity introduction continued economics of capital markets version 1. However, also implicit in the convexity measure is the assumption of parallel shifts in yields and, therefore, the duration convexity. Convexity 7 the convexity correction is always positive suppose the 20year rate fell 100 bp 5.
Estimate the percentage price change of a bond for a specified change in yield, given the bonds approximate duration and convexity. Duration and convexity, with illustrations and formulas. They also enter into measures for interest rate risk, as shown in this paper. Fixed income portfolio management interest rate sensitivity. Bond convexity is one of the most basic and widely used. This typical is for a bond which does not have a call option or a prepayment option. Longterm bonds have greater interest rate risk than short term bonds. Price change of a bond duration convexity cfa level. Duration and convexity 443 that duration is a measure of interest rate risk is demonstrated as follows. The approximate change using both dollar duration and convexity is. Given the convexity measure, the convexity adjustment to the duration estimate can be computed. As the expiration date approaches, duration declines.
Convexity is always lurking at the scene of the crime. Just as with dollar duration, the dollar convexity of the portfolio is the sum of the dollar convexities of the component zeroes. The use of convexity matching, in addition to duration matching, is an attempt to explain the price sensitivity in a better fashion. Revisiting the bond durationconvexity approximation.
Pdf bond portfolio duration, cash flow dispersion and convexity. Pdf hedging bond positions under the assumption of a parallel shift of the interest rate curve is. Understanding bond duration can help investors determine how bonds fit in to a broader investment portfolio. Bond duration calculator macaulay duration and modified macaulay duration. The present value equation for an annual coupon bond is given by. Duration, convexity and other bond risk measures offers the most comprehensive coverage of bond risk measures available. Duration is an imperfect way of measuring a bonds price change, as it indicates that this change is linear in nature when in fact it exhibits a sloped or convex shape. For a bond of face value usd1,000 with a semiannual coupon of 8. Macaulay duration of coupon bond 1storder approximation of bond price change 1storder approximation of dv01 convexity convexity of zerocoupon bond convexity of coupon bond 1storder approximation of duration change 2ndorder approximation of bond price change duration of portfolio duration neutral. Financial expert frank fabozzi walks you through every aspect of bond risk measures from the price volatility characteristics of optionfree bonds and bonds with embedded options to the proper method for calculating duration and convexity. The duration of a bond varies with the passage of time and yield changes. Bond calculator macaulay duration, modified macaulay duration, convexity coupon bond calculate bond macaulay duration, modified macaulay duration, convexity. Since the value of the bond portfolio is merely the present value of future cash flows, using the concepts of duration defined below, such approximations can be done quickly using nothing more than a handheld calculator.
Using duration and convexity to approximate change in present. This video illustrates how duration can be used to approximate the change in bond price given a change in interest rates. Pdf duration and convexity of bonds slobodan cerovic. If an investor is given a choice of two 10year bonds to choose from, one with a 10 percent coupon rate and the other with a 5 percent coupon rate. Pdf enhancement of the bond durationconvexity approximation. In terms of callability feature, bonds may be callable or noncallable. On one hand, in the case of a callable bond, a change in interest rates may trigger the calling of the bond, thus stopping the flow of. Since the value of the bond portfolio is merely the present value of future cash flows, using the concepts of duration defined below, such.
Sep 06, 2019 the modified duration alone underestimates the gain to be 9. As the yield on a bond changes so too does its duration, a bonds convexity measures the sensitivity of a bonds duration to changes in yield. Callable bond is one which can be prepaid by the bond issuer prior to its maturity. However, bond investors have been customarily used to using duration and convexity, as discussed below and quick and handy tools for understanding the sensitivity of bond prices to yields.
Convexity formula how to calculate convexity of a bond. Bond duration and convexity spreadsheet the bond duration worksheet allows you to calculate the duration of a bond quickly and easily. A bond has positive convexity if the yield and the duration of the bond increase or decrease together, i. Pdf bond portfolio duration, cash flow dispersion and. Understanding duration and convexity of fixed income. Determine how much money you would accumulate by investing a given amount of money at a fixed annual rate of return at recurring intervals. The relationship between required yield and price is generally stated as bond prices and. For example, if a bond has a duration of 5 years, and interest rates increase by 1%, the bonds price will decline by approximately 5%. Macauley duration modified duration effective duration percentage change in price for a 1% increase in the yield to maturity problem 2 consider a bond that has a coupon rate of 7. This paper provides a theoretical and practical overview of the concepts, largely missing in the existing.
Modified duration and convexity taken together provide the best approximation of the sensitivity of bond prices to changes in interest rates. It is equal to the maturity if and only if the bond is a zerocoupon bond. Duration, modified duration, and convexity reflect the bonds exposition to interest rate risk. The modified duration alone underestimates the gain to be 9. Nov, 2019 duration and convexity are two tools used to manage the risk exposure of fixedincome investments. Conversely, if a bond has a duration of 5 years and interest rates fall by 1%, the bonds price will increase by approximately 5%. Duration measures the bonds sensitivity to interest rate changes. The coupon rate is typically stated in an annual percentage.