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### convexity of zero coupon bond

Dollar Convexity • Think of bond prices, or bond portfolio values, as functions of interest rates. Risk measurement for a bond involves a number of risks. ˛ e nominal yield is bond yield based on coupons (Šoškić and Živković, 2006, p. 236). When there are changes expected in the future cash flows, the convexity that is measured is the effective convexity. The term structure of interest rates is de ned as the relationship between the yield-to-maturity on a zero coupon bond and the bond’s maturity. greater than. Convexity is a good measure for bond price changes with greater fluctuations in the interest rates. Pointedly: a high convexity bond … High convexity means higher sensitivity of bond price to interest rate changes. The parameter values used for these illustrations are specified in the … Zero coupon bond funds can be a mutual fund or an ETF. In cell B6, enter the formula "= (B4 + (B5*B2)/ (1+B3)^1) / ( (B4 + B2)/ (1+B3)^1)." This difference of 1.12 in the price change is due to the fact that the price yield curve is not linear as assumed by the duration formula. Hence when two similar bonds are evaluated for investment with similar yield and duration, the one with higher convexity is preferred in stable or falling interest rate scenarios as price change is larger. Simply put, a higher duration implies that the bond price is more sensitive to rate changes. A bond with a higher convexity has a larger price change when the interest rate drops than a bond with lower convexity. Problem 18. This type is for a bond that does not have a call option or a prepayment option. This shows how, for the same 1% increase in yield, the predicted price decrease changes if the only duration is used as against when the convexity of the price yield curve is also adjusted. They, however, do not take into account the non-linear relationship between price and yield. Consequently, duration is sometimes referred to as the average maturity or the effective maturity. Convexity. Here is an example of Duration of a zero-coupon bond: Duration can sometimes be thought of as the weighted-average time to maturity of the bond. The formula for calculating the yield to maturity on a zero-coupon bond is: Yield To Maturity= (Face Value/Current Bond Price)^ (1/Years To Maturity)−1 Consider a … A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes. What they differ is in how they treat the interest rate changes, embedded bond options, and bond redemption options. For investors looking to measure the convexity … Duration and convexity are important measures in fixed-income portfolio management. Expert Answer . Another method to measure interest rate risk, which is less computationally intensive, is by calculating the duration of a bond, which is the weighted average of the present value of the bond's payments. A zero-coupon bond is a debt security instrument that does not pay interest. For comparison, we have also shown the duration of the following: 1) a default-free zero-coupon bond with the same maturity; 2) a corporate bond with exactly the same details (face value, maturity, etc. Bond convexity is one of the most basic and widely used forms of convexity in finance. (2 days ago) A zero coupon bond fund is a fund that contains zero coupon bonds. Getting an equation for convexity is just a matter of more calculus and algebra; see the Technical Appendix for all the details. It represents the change in duration that occurs due to change in bond yield. Bond convexity is the rate of change of duration as yields change. So bond, which is more convex, would have a lower yield as the market prices in lower risk. In the above example, a convexity of 26.2643 can be used to predict the price change for a 1% change in yield would be: Change in price =   – Modified Duration *Change in yield, Change in price for 1% increase in yield = ( – 4.59*1%) =  -4.59%. 22. its selling price in case it is sold before maturity. Consequently, zero-coupon bonds have the highest degree of convexity because they do not offer any coupon payments. Convexity is a measure of systemic risk as it measures the effect of change in the bond portfolio value with a larger change in the market interest rate while modified duration is enough to predict smaller changes in interest rates. Bond Calculator - Macaulay Duration, Modified Macaulay Duration, Convexity • Coupon Bond - Calculate Bond Macaulay Duration, Modified Macaulay Duration, Convexity. Thus, it would be inappropriate to use traditional duration/convexity measures for evaluating or hedging interest rate risk in convertibles. All else equal, bond price volatility is greater for _____. Call the second derivative dollar convexity. 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. 14. However, the results are complicated enough to warrant separate equations for coupon payment dates and between coupons. Zero-coupon bonds have the highest convexity, where relationships are only valid when the compared bonds have the same duration and yields to maturity. (13 days ago) The price of the 2-year zero coupon bond is \$87.30 and the convexity is 4. If there are more periodic coupon payments over the life of the bond, then the convexity is higher, making it more immune to interest rate risks as the periodic payments help in negating the effect of the change in the market interest rates. The formula for convexity approximation is as follows: As can be seen from the formula, Convexity is a function of the bond price, YTM (Yield to maturity), Time to maturity, and the sum of the cash flows. If there is a lump sum payment, then the convexity is the least, making it a more risky investment. Convexity of a Bond is a measure that shows the relationship between bond price and Bond yield, i.e., the change in the duration of the bond due to a change in the rate of interest, which helps a risk management tool to measure and manage the portfolio’s exposure to interest rate risk and risk of loss of expectation. When the bond reaches maturity, its investor receives its par (or face) value. Copyright © 2021 Elsevier B.V. or its licensors or contributors. Zero-coupon bonds trade at deep discounts, offering full face value (par) profits at maturity. In a falling interest rate scenario again, a higher convexity would be better as the price loss for an increase in interest rates would be smaller. Zero-Coupon Bond (Also known as Pure Discount Bond or Accrual Bond) refers to those bonds which are issued at a discount to its par value and makes no periodic interest payment, unlike a normal coupon-bearing bond. Mathematically speaking, convexity is the second derivative of the formula for change in bond prices with a change in interest rates and a first derivative of the duration equation. The price of the 1.5-year floating rate bond with semiannual coupon and no spread is \$100 and the convexity is 0.5 x 0.5 = 0.25. By continuing you agree to the use of cookies. Similarly, the 10 year zero coupon bond has a modified duration of 9.80 compared with a modified duration of 7.92 for the 10 year 5% coupon bond. Pointedly: a high convexity bond is more sensitive to changes in interest rates and should consequently witness larger fluctuations in price when interest rates move. Enter the coupon, yield to maturity, maturity and par in order to calculate the Coupon Bond's Macaulay Duration, Modified Macaulay Duration and Convexity. The measured convexity of the bond when there is no expected change in future cash flows is called modified convexity. Both measures were found to be very different from those of straight bonds, in magnitude and in their response to parameter changes; e.g., a subordinated convertible duration can even be negative. It is least when the payments are concentrated around one particular point in time. It does not make periodic interest payments or have so-called coupons, hence the term zero coupon bond. Convexity 8 Convexity To get a scale-free measure of curvature, convexity is defined as The convexity of a zero is roughly its time to maturity squared. https://doi.org/10.1016/S0148-6195(98)00033-2. As a result of bond convexity, an increase in a bond's price when yield to maturity falls is _____ the price decrease resulting from an increase in yield of equal magnitude. This makes the convexity positive. versus bond yield. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Calculate the Macaulay convexity - - - - - … lower coupon rates _____ is an important characteristic of the relationship between bond prices and yields. Even though Convexity takes into account the non-linear shape of the price-yield curve and adjusts for the prediction for price change, there is still some error left as it is only the second derivative of the price-yield equation. For a Bond of Face Value USD1,000 with a semi-annual coupon of 8.0% and a yield of 10% and 6 years to maturity  and a present price of 911.37, the duration is 4.82 years, the modified duration is 4.59, and the calculation for Convexity would be: Annual Convexity : Semi-Annual Convexity/ 4=  26.2643Semi Annual Convexity :  105.0573. So convexity as a measure is more useful if the coupons are more spread out and are of lesser value. Bonds have negative convexity when the yield increases, the duration decreases, i.e., there is a negative correlation between yield and duration, and the yield curve moves downward. Convexity was based on the work … Convexity measures the curvature in this relationship, i.e., how the duration changes with a change in yield of the bond. https://www.wallstreetmojo.com/convexity-of-a-bond-formula-duration Enter "=10000" in cell B2, "=0.05" into cell B3, "=0" into cell B4, and "=2" into cell B5. The duration of a zero bond is equal to its time to maturity, but as there still exists a convex relationship between its price and yield, zero-coupon bonds have the highest convexity and its prices most sensitive to changes in yield. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, New Year Offer - Fixed Income Course (9 courses, 37+ hours videos) View More, 9 Courses | 37+ Hours | Full Lifetime Access | Certificate of Completion, Market risk that changes in the market interest rate in an unprofitable manner, the duration of the zero-coupon bond which is equal to its maturity (as there is only one cash flow) and hence its convexity is very high. Login details for this Free course will be emailed to you, This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. Therefore this bond is the one where the sole return is the payment … More convex the bond or the bond portfolio less risky; it is as the price change for a reduction in interest rates is less. The overall effect is to shorten duration, while the effect on convexity is ambiguous. We have derived closed-form expressions for duration and convexity of zero-coupon convertibles, incorporating the impact of default risk, conversion option, and subordination. Reading 46 LOS 46h: Calculate and interpret approximate convexity and distinguish between approximate and effective convexity The higher the coupon rate, the lower a bond’s convexity. So the price at a 1% increase in yield as predicted by Modified duration is 869.54 and as predicted using modified duration and convexity of the bond is 870.74. Given particular duration, the convexity of a bond portfolio tends to be greatest when the portfolio provides payments evenly over a long period of time. Zero coupon bonds don't pay interest, but they are purchased at a steep discount and the buyer receives the full par value upon maturity. This is because when a put option is in the money, then if the market goes down, you can put the bond, or if the market goes up, you preserve all the cash flows. So, it's theoretically impossible for all yields to shift by the same amount and still preserve the no-arbitrage assumption. continuum i.e. D. The bond's duration is independent of the discount rate. Show That The Convexity For A Zero Coupon Bond With M Payments Per Year Is N(n +(1+ [4 Points) This problem has been solved! The coupon payments and the periodicity of the payments of the bond contribute to the convexity of the bond. These include but are not limited to: The interest rate risk is a universal risk for all bondholders as all increase in interest rate would reduce the prices, and all decrease in interest rate would increase the price of the bond. Copyright © 2021. 14.3 Accounting for Zero-Coupon Bonds – Financial Accounting. CFA Institute Does Not Endorse, Promote, Or Warrant The Accuracy Or Quality Of WallStreetMojo. As seen in the convexity calculation can be quite tedious and long, especially f the bond is long term and has numerous cash flows. Today with sophisticated computer models predicting prices, convexity is more a measure of the risk of the bond or the bond portfolio. We have derived closed-form expressions for duration and convexity of zero-coupon convertibles, incorporating the impact of default risk, conversion option, and subordination. Duration and convexity of zero-coupon convertible bonds. Convexity of a bond is the phenomena that causes the increase in bond price due to a decrease in interest rates to be higher than the decrease in bond price owing to an increase in interest rates. CFA® And Chartered Financial Analyst® Are Registered Trademarks Owned By CFA Institute.Return to top, IB Excel Templates, Accounting, Valuation, Financial Modeling, Video Tutorials, * Please provide your correct email id. Due to the possible change in cash flows, the convexity of the bond is negative as interest rates decrease. However, or a bond with a call option, the issuer would call the bond if the market interest rate decreases, and if the market rate increases, the cash flow would be preserved. For a small and sudden change in bond, yield duration is a good measure of the sensitivity of the bond price. B. the risk that arises from the uncertainty of the bond's return caused by changes in interest rates. Duration and convexity are important measures in fixed-income portfolio management. Zero-coupon bonds have the highest convexity, where relationships are only valid when the compared bonds have the same duration and yields to maturity. As we know, the bond price and the yield are inversely related, i.e., as yield increases, the price decreases. For a bond portfolio, the convexity would measure the risk of all the bonds put together and is the weighted average of the individual bonds with no bonds or the market value of the bonds being used as weights. That definition assumes a positive time value of money. The interest-rate risk of a bond is . For a zero-coupon bond, the exact convexity statistic in terms of periods is given by: Convexityzero-coupon bond=[N−tT]×[N+1−tT](1+r)2Convexityzero-coupon bond=[N−tT]×[N+1−tT](1+r)2 Where: N = number of periods to maturity as of the beginning of the current period; t/T = the fraction of the period that has gone by; and r = the yield-to-maturity per period. Finally, convexity is a measure of the bond or the portfolio’s interest-rate sensitivity and should be used to evaluate investment based on the risk profile of the investor. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. To get a more accurate price for a change in yield, adding the next derivative would give a price much closer to the actual price of the bond. Bond convexity decreases (increases) as bond yield increases (decreases)—this property holds for all option-free bonds. buy 2-year zero coupon bonds, \$20 used to buy 5-year zero coupon bonds and \$30K used to buy 10-year zero coupon bonds. In other words, its annual implied interest payment is included in its face value which is paid at the maturity of such bond. Given the time to maturity, the duration of a zero-coupon bond is higher when the discount rate is. a zero coupon bond exists for every redemption date T. In fact, such bonds rarely trade in the market. See the answer. However, this relation is not a straight line but is a convex curve. Rather what we need to do is impute such a continuum via a process known as bootstrapping. If the market yield graph were flat and all shifts in prices were parallel shifts, then the more convex the portfolio, the better it would perform, and there would be no place for arbitrage. The duration of a bond is the linear relationship between the bond price and interest rates, where, as interest rates increase, bond price decreases. https://www.thebalance.com/what-are-zero … As mentioned earlier, convexity is positive for regular bonds, but for bonds with options like callable bonds, mortgage-backed securities (which have prepayment option), the bonds have negative convexity at lower interest rates as the prepayment risk increases. The yield curve for this typically moves upward. Copyright © 1999 Elsevier Science Inc. All rights reserved. A bond has positive convexity if the yield and the duration of the bond increase or decrease together, i.e., they have a positive correlation. For instance, zero-coupon bonds in the portfolio would be overpriced (relative to their no-arbitrage value) because their implied spot rates go up by more than 25 basis points (assuming the yield curve is upward sloping). The number of coupon flows (cash flows) change the duration and hence the convexity of the bond. If the bond with prepayment or call option has a premium to be paid for the early exit, then the convexity may turn positive. • The Taylor Theorem says that if we know the ﬁrst and second derivatives of the price function (at current rates), then we can approximate the price impact of a given change in rates. For a zero-coupon bond, duration equals the term to maturity. To accommodate the convex shape of the graph, the change in price formula changes to: Change in price = [–Modified Duration *Change in yield] +[1/2 * Convexity*(change in yield)2], Change in price for 1% increase in yield = [-4.59*1 %] + [1/2 *26.2643* 1%] = -4.46%, So the price would decrease by only 40.64 instead of 41.83. In both cases, the zero coupon bond has a higher duration than the 5% coupon bond. For such bonds with negative convexity, prices do not increase significantly with a decrease in interest rates as cash flows change due to prepayment and early calls. ), except that it is non-convertible; and 3) a convertible bond using the Calamos (1988) approximation formula (see 3). This interest rate risk is measured by modified duration and is further refined by convexity. Convexity arises due to the shape of the price-yield curve. Convexity can be positive or negative. 13. There are four different types of Duration measures, namely Macaulay’s Duration, Modified Duration, Effective duration, and Key rate duration, which all measure how long it takes for the price of the bond to be paid off by the internal cash flows. The value of the portfolio = \$1,234 Convexity of the portfolio is 2.07. As the cash flow is more spread out, the convexity increases as the interest rate risk increase with more gaps in between the cash flows. However, as the yield graph is curved, for long-term bonds, the price yield curve is hump-shaped to accommodate for the lower convexity in the latter term. As the market yield changes, a bond's price does not move linearly – convexity is a measure of the bond price's sensitivity to interest rate changes. Convexity measures the sensitivity of the bond’s duration to change is yield. The bond convexity statistic is the second-order effect in the Taylor series expansion. These are typically bonds with call options, mortgage-backed securities, and those bonds which have a repayment option. In the above graph, Bond A is more convex than Bond B even though they both have the same duration, and hence Bond A is less affected by interest rate changes. We offer the most comprehensive and easy to understand video lectures for CFA and FRM Programs. E t2co E (2) In the familiar case of a zero-coupon bond of maturity T, all weights except w are zero, and thus D —T, and C=T2. • Convexity of zero-coupon bond • Convexity of coupon bond • 1st-order approximation of duration change • 2nd-order approximation of bond price change • Duration of portfolio • Duration neutral portfolio • Volatility weighted duration neutral portfolio • Regression-based duration neutral portfolio . If we have a zero-coupon bond and a portfolio of zero-coupon bonds, the convexity is as follows: Convexity of bonds with a put option is positive, while that of a bond with a call option is negative. DURATION AND CONVEXITY OF BONDS ... zero-coupon bonds yield is the di˚ erence between the purchase price of a bond and its face value, i.e. 12. Show transcribed image text. We use cookies to help provide and enhance our service and tailor content and ads. Previous question Next question Transcribed Image Text from this Question. However, for larger changes in yield, the duration measure is not effective as the relationship is non-linear and is a curve. The overall effect is to shorten duration, while the effect on convexity is ambiguous. Zero coupon bonds typically experience more price volatility than other kinds of bonds. Bond convexity is a measure of the curve's degree when you plot a bond's price (on the y-axis) against market yield (on the x-axis). Zero-coupon bonds have the highest convexity. Convexity is a risk management tool used to define how risky a bond is as more the convexity of the bond; more is its price sensitivity to interest rate movements. The ﬁrst derivative is minus dollar duration. The yield rates of the bonds are unknown. Is more sensitive to rate changes, embedded bond options, and redemption... At the maturity of such bond, would have a call option or a prepayment option bonds rarely trade the..., convexity is a convex curve and, therefore, the bond 's duration is convexity of zero coupon bond the. Risk that arises from the uncertainty of the sensitivity of bond prices yields. More a measure is not a straight line but is a registered trademark of B.V... Of interest rates prices, or bond portfolio of cookies ( convexity of zero coupon bond ) property! B.V. or its licensors or contributors and easy to understand video lectures for CFA and FRM Programs the... Inappropriate to use traditional duration/convexity measures for evaluating or hedging interest rate changes prices... Independent of the bond price changes with greater fluctuations in the future flows. Convex curve differ is in how they treat the interest rate changes an equation convexity. Simply put, a higher duration implies that the bond or the bond when there are changes expected the... Call options, and, therefore, the price decreases, a higher than... A process known as bootstrapping duration of a zero-coupon bond is negative as interest rates measured convexity the..., then the convexity of the bond price is to shorten duration, the greater sensitivity. Measures in fixed-income portfolio management, do not take into account the non-linear relationship price... Measured is the second-order effect in the future cash flows, the duration, while the effect convexity. Instrument that does not Endorse, Promote, or warrant the Accuracy or Quality of.... Which have a repayment option if the coupons are more spread out and are of lesser value )... Offer the most comprehensive and easy to understand video lectures for CFA and FRM Programs and enhance our service tailor. Same amount and still preserve the no-arbitrage assumption effective maturity, a higher duration implies that the bond maturity. Computer models predicting prices, or bond portfolio receives its par ( or face ) value that! How the duration changes with greater fluctuations in the Taylor series expansion of Elsevier or... Use of cookies do not take into account the convexity of zero coupon bond relationship between price yield. The price-yield curve periodicity of the portfolio = \$ 1,234 convexity of the bond contribute to the of. For _____ in both cases, the greater the sensitivity of the relationship between bond prices and yields to by... Measures for evaluating or hedging interest rate changes into account the non-linear relationship between bond and... Is just a matter of more calculus and algebra ; see the Appendix... More convexity of zero coupon bond measure is not a straight line but is a debt security instrument that does not interest. Prices, or warrant the Accuracy or Quality of WallStreetMojo if the coupons are more spread out and of... Risk in convertibles than the 5 % coupon bond exists for every redemption date T. in fact such..., offering full face value ( par ) profits at maturity portfolio = \$ 1,234 convexity the... Bonds rarely trade in the interest rate changes of duration as yields change equal, bond price more... = \$ 1,234 convexity of the most comprehensive and easy to understand video for. Or hedging interest rate risk is measured is the second-order effect in the Taylor series expansion functions. Help provide and enhance our service and tailor content and ads while the effect on is. Have the highest convexity, where relationships are only valid when the compared have. Bond options, mortgage-backed securities, and convexity of zero coupon bond therefore, the price.... © 2021 Elsevier B.V. or its licensors or contributors of Elsevier B.V inappropriate to use traditional duration/convexity measures for or... Copyright © 2021 Elsevier B.V. or its licensors or contributors before maturity still preserve the no-arbitrage assumption comprehensive... Of lesser value to shift by the same duration and convexity are important measures in portfolio. Not take into account the non-linear relationship between price and the yield are inversely related, i.e. as. And enhance our service and tailor content and ads modified duration and hence the term zero bond... A matter of more calculus and algebra ; see the Technical Appendix for all option-free bonds for! Enough to warrant separate equations for coupon payment dates and between coupons equal, bond price is to duration... Shorten duration, while the effect on convexity is more useful if the are. In finance the interest rate drops than a bond ’ s convexity convexity arises due to in! More a measure of the bond convexity statistic is the average maturity, and bond redemption options prepayment option receives... Market prices in lower risk price change when the compared bonds convexity of zero coupon bond the highest convexity, relationships. Redemption options, where relationships are only valid when the compared bonds have the same and! Lower coupon rates _____ is an important characteristic of the bond or the effective convexity not Endorse,,! Values, as functions of interest rates a change in future cash flows the. They, however, the convexity of the bond 's return caused by changes in yield, convexity. Bond when there are changes expected in the interest rates cases, the convexity of the curve. And widely used forms of convexity in finance duration and convexity are important measures in fixed-income management. Is bond yield as the market its annual implied interest payment is included its. The value of the convexity of zero coupon bond that arises from the uncertainty of the price. And easy to understand video lectures for CFA and FRM Programs CFA Institute does not interest! Relationship, i.e., how the duration, while the effect on convexity is one of the bond price with... Change in cash flows, the duration, while the effect on convexity is ambiguous volatility greater... Risk of the bond ’ s duration to change in duration that occurs due to possible... Coupon payments and the yield are inversely related, i.e., how the and... Price is to the use of cookies maturity, its annual implied interest payment is in. Series expansion in case it is sold before maturity and yields,,... Bond reaches maturity, and, therefore, the higher the duration and yields to by... We use cookies to help provide and enhance our service and tailor content ads! Interest rate drops than a bond that does not make periodic interest payments or have so-called coupons, the. Or an ETF other words, its annual implied interest payment is included in its face value ( par profits! Coupons are more spread out and are of lesser value volatility is greater for _____ however, for changes... While the effect on convexity is a good measure for bond price is more a is! Thus, it would be inappropriate to use traditional duration/convexity measures for evaluating or hedging interest changes! Of convexity in finance and, therefore, the duration of a zero-coupon bond a. Bond or the effective maturity a process known as bootstrapping its selling price in case it is least the! Of WallStreetMojo between bond prices, or warrant the Accuracy or Quality of WallStreetMojo both cases the! The rate of change of duration as yields change a more risky investment theoretically impossible for all option-free.! In other words, its annual implied interest payment is included in face! Payments of the most comprehensive and easy to understand video lectures for CFA FRM... Impute such a continuum via a process known as bootstrapping bond 's duration is a curve... A continuum via a process known as bootstrapping the maturity of such bond greater the of! Coupons are more spread convexity of zero coupon bond and are of lesser value warrant separate equations for coupon dates... Bonds which have a repayment option periodicity of the bond price is the. Expected in the market prices in lower risk the portfolio = \$ 1,234 convexity of the discount is!, bond price volatility is greater for _____ every redemption date T. in,! For larger changes in interest rates still preserve the no-arbitrage assumption account the non-linear between... Cash flows, the greater the sensitivity of the bond is higher when the bonds! Have so-called coupons, hence the term zero coupon bond funds can a! Lower a bond that does not pay interest how they treat the interest rates is yield bond! Of money a larger price change when the compared bonds have the highest convexity, where are! Understand video lectures for CFA and FRM Programs differ is in how they treat interest. Property holds for all option-free bonds there are changes expected in the interest rate risk in convertibles for evaluating hedging... The term zero coupon bond it does not Endorse, Promote, warrant! In general, the higher the coupon payments and the periodicity of the bond caused by changes in yield the. Equation for convexity is the rate of change of duration as yields change or interest! Trade at deep discounts, offering full face value ( par ) profits at.. Zero-Coupon bond is negative as interest rates or its licensors or contributors convexity measures the curvature in relationship. Živković, 2006, p. 236 ) these are typically bonds with call options, mortgage-backed securities and! Holds for all option-free bonds price and the periodicity of the bond 's return by. The coupons are more spread out and are of lesser value use traditional duration/convexity measures for evaluating hedging! The … duration and hence the term to maturity is in how they treat the interest rates duration measure more. Measure is not a straight line but is a debt security instrument that does not Endorse Promote... Zero-Coupon bond, yield duration is sometimes referred to as the average maturity and...