Bond convexity

Short description: Financial measurement

In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, and is defined as 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. Convexity was based on the work of Hon-Fei Lai and popularized by Stanley Diller.[1]

## Contents

* 1 Calculation of convexity
* 2 Why bond convexities may differ
* 3 Mathematical definition
* 3.1 How bond duration changes with a changing interest rate
* 4 Application of convexity
* 5 Effective convexity
* 6 See also
* 7 References
* 8 Further reading
* 9 External links

## Calculation of convexity

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.[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] 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.[2]

Convexity does not assume the relationship between Bond value and interest rates to be linear.[4] 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.[5]

## Why bond convexities may differ

Bond price-yield curve showing the linear approximation of duration and the resulting convexity error gap

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).[6] 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 then they will have identical sensitivities.[7] 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.[8]

For two bonds with the same par value, coupon, and maturity, convexity may differ depending on what point on the price yield curve they are located.[9]

## Mathematical definition

If the flat, continuously compounded yield is r and the bond price is B(r), then convexity is defined as[10] C(r)=1B(r)d2B(r)dr2.

It is often convenient to express convexity via the modified duration. Define the modified duration by D(r)=−1B(r)dB(r)dr. Equivalently, dB(r)dr=−D(r)B(r).

Differentiating the identity dBdr=−D(r)B(r) gives d2B(r)dr2=−dD(r)drB(r)−D(r)dB(r)dr=(D(r)2−dD(r)dr)B(r).

Therefore, C(r)=1B(r)d2B(r)dr2=D(r)2−dD(r)dr.

Here D(r) denotes the modified duration defined above.

### How bond duration changes with a changing interest rate

For yields quoted with periodic (discrete) compounding at rate y per period, the modified duration is:[11] D(y)=11+y∑i=1nPitiB. Here Pi is the present value of cash flow i, and ti is its payment time.

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).[12] 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).[13][14] Note that the modified duration D differs from the regular duration by the factor one over 1 + r (shown above), which also decreases as r is increased.

dD(r)dr≤0.

From the identity above, convexity is positive for option-free bonds.[15]

Convexity is positive for option-free bonds. Under a flat, continuously compounded curve, B(r)=∑i=1ncie−rti. Differentiating twice, d2B(r)dr2=∑i=1nciti2e−rti≥0, hence C(r)=1B(r)d2B(r)dr2≥0.

For duration, define present-value weights

wi=cie−rtiB(r) (so ∑i=1nwi=1).

Then D(r)=∑i=1nwiti, and dD(r)dr=−∑i=1nwi(ti−D(r))2≤0, with equality only for a single cash flow (a zero-coupon bond).

## Application of convexity

Convexity and duration are standard one-factor risk measures for parallel shifts in the yield curve. Duration controls the first-order price sensitivity and convexity the second-order term; together they are used to estimate and hedge interest-rate risk. A portfolio is first-order hedged when its (dollar) duration is close to zero versus a benchmark; adding a convexity match reduces second-order exposure for larger shifts.[16][17]

For a small parallel yield change Δy, the second-order Taylor approximation of price is ΔBB≈−DΔy+12C(Δy)2, or, in dollar terms, ΔB≈−BDΔy+12BC(Δy)2. These formulae are standard and underpin duration- and convexity-based hedging.[18]

For portfolios with the same duration, higher convexity (e.g., a “barbell” of short- and long-dated bonds versus a “bullet” concentrated near one maturity) improves second-order behaviour: gains when yields fall are larger than losses when yields rise of the same size.[19]

## Effective convexity

See also: Finance:Effective duration

For bonds with embedded options, price depends on how the yield curve move alters expected cash flows via option exercise. Standard, yield-to-maturity-based duration and convexity assume fixed cash flows and so miss this effect. In such cases, effective convexity is obtained numerically.[20]

Effective convexity is a centred finite difference approximation to the second derivative of price with respect to the yield level:[20]

Effective convexity=V−Δy−2V0+V+ΔyV0(Δy)2.

Here V is the model price (from an option-pricing framework), Δy is the size of a parallel shift in the curve, and V−Δy and V+Δy are the corresponding prices when yields fall or rise by Δy, respectively.

In practice the prices V−Δy, V0 and V+Δy are computed from an interest-rate model that re-values the whole curve and the option at each node (for example, a short-rate lattice - see Lattice model).[21][22]

## See also

* Black–Scholes equation
* Bond duration
* Bond valuation
* Fixed-income attribution § First principles versus perturbational attribution
* Greeks
* Immunization
* List of convexity topics
* List of finance topics

## References

1. ↑ Diller, Stanley (1991), Parametric Analysis of Fixed Income Securities, in Dattatreya, Ravi (ed.) Fixed Income Analytics: State-of-the-Art Debt Analysis and Valuation Modeling, Probus Publishing
2. ↑ 2.0 2.1 Brooks, Robert; Attinger, Bill (1992-07-01). "Using Duration and Convexity in the Analysis of Callable Convertible Bonds" (in en). Financial Analysts Journal 48 (4): 74–77. doi:10.2469/faj.v48.n4.74. ISSN 0015-198X. https://www.tandfonline.com/doi/full/10.2469/faj.v48.n4.74.
3. ↑ Pelsser, Antoon (2003-02-04). "Mathematical foundation of convexity correction". Quantitative Finance 3 (1): 59–65. doi:10.1088/1469-7688/3/1/306. https://iopscience.iop.org/article/10.1088/1469-7688/3/1/306/meta. Retrieved 2023-09-30.
4. ↑ Udegbunam, Raphael I.; Oaikhenan, Hassan E. (2012-03-13). "Interest Rate Risk of Stock Prices in Nigeria: Empirical Test of the Duration and Convexity Model" (in en). Journal of Emerging Market Finance 11 (1): 93–113. doi:10.1177/097265271101100104. ISSN 0972-6527. http://journals.sagepub.com/doi/10.1177/097265271101100104.
5. ↑ Weil, Lawrence Fisher, Roman L. (1982), "Coping with the Risk of Interest-Rate Fluctuations: Returns to Bondholders From Naïve and Optimal Strategies*", Bond Duration and Immunization, Routledge, doi:10.4324/9781315145976-11/coping-risk-interest-rate-fluctuations-returns-bondholders-naïve-optimal-strategies-lawrence-fisher-roman-weil, ISBN 978-1-315-14597-6, https://www.taylorfrancis.com/chapters/edit/10.4324/9781315145976-11/coping-risk-interest-rate-fluctuations-returns-bondholders-naïve-optimal-strategies-lawrence-fisher-roman-weil, retrieved 2023-10-02
6. ↑ Dai, Qiang; Singleton, Kenneth J.; Yang, Wei (2007-04-12). "Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields" (in en). Review of Financial Studies 20 (5): 1669–1706. doi:10.1093/rfs/hhm021. ISSN 0893-9454. https://academic.oup.com/rfs/article-lookup/doi/10.1093/rfs/hhm021.
7. ↑ Whittingham, M. (1997). "The Canadian market for zero-coupon bonds". Bank of Canada Review: 47–62. https://www.bankofcanada.ca/wp-content/uploads/2010/06/r971c.pdf.
8. ↑ Phoa, Wesley; Shearer, Michael (1997-12-31). "A Note on Arbitrary Yield Curve Reshaping Sensitivities Using Key Rate Durations" (in en). The Journal of Fixed Income 7 (3): 67–71. doi:10.3905/jfi.1997.408212. ISSN 1059-8596. https://www.pm-research.com/content/iijfixinc/7/3/67.
9. ↑ Livingston, Miles (1979-03-01). "Bond Taxation and the Shape of the Yield‐to‐Maturity Curve" (in en). The Journal of Finance 34 (1): 189–196. doi:10.1111/j.1540-6261.1979.tb02079.x. ISSN 0022-1082. https://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1979.tb02079.x.
10. ↑ Fong, H. Gifford; Vasicek, Oldrich A. (1991-07-31). "Fixed–income volatility management" (in en). The Journal of Portfolio Management 17 (4): 41–46. doi:10.3905/jpm.1991.409345. ISSN 0095-4918. https://www.pm-research.com/content/iijpormgmt/17/4/41.
11. ↑ Blackwell, David W.; Griffiths, Mark D.; Winters, Drew B. (2006). Modern Financial Markets: Prices, Yields, and Risk Analysis. John Wiley & Sons. pp. 323–330. ISBN 978-0470000106. https://books.google.com/books?id=xip3DwAAQBAJ.
12. ↑ Shea, Gary S. (1984). "Pitfalls in Smoothing Interest Rate Term Structure Data: Equilibrium Models and Spline Approximations". The Journal of Financial and Quantitative Analysis 19 (3): 253–269. doi:10.2307/2331089. ISSN 0022-1090. https://www.jstor.org/stable/2331089.
13. ↑ Geiger, Felix (2011), "The Theory of the Term Structure of Interest Rates", in Geiger, Felix (in en), The Yield Curve and Financial Risk Premia: Implications for Monetary Policy, Lecture Notes in Economics and Mathematical Systems, Berlin, Heidelberg: Springer, pp. 43–82, doi:10.1007/978-3-642-21575-9_3, ISBN 978-3-642-21575-9, https://doi.org/10.1007/978-3-642-21575-9_3, retrieved 2023-11-06
14. ↑ Swishchuk, Anatoliy (2009-01-04). "Levy-Based Interest Rate Derivatives: Change of Time Method and PIDEs". Econometrics: Single Equation Models eJournal. doi:10.2139/ssrn.1322532. https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1322532_code1179276.pdf?abstractid=1322532&mirid=1.
15. ↑ Grantier, Bruce J. (1988-11-01). "Convexity and Bond Performance: The Benter the Better" (in en). Financial Analysts Journal 44 (6): 79–81. doi:10.2469/faj.v44.n6.79. ISSN 0015-198X. https://www.tandfonline.com/doi/full/10.2469/faj.v44.n6.79.
16. ↑ Tuckman, Bruce; Serrat, Angel (2022). "Appendix A4.1–A4.2 (DV01, Duration, and Convexity of Portfolios; Estimating Price Change with Duration and Convexity)". Fixed Income Securities: Tools for Today's Markets (4th ed.). Wiley. https://www.oreilly.com/library/view/fixed-income-securities/9781119835554/b06.xhtml.
17. ↑ Carpenter, Jennifer. "Immunization". https://people.stern.nyu.edu/jcarpen0/courses/b403333/08immunh.pdf.
18. ↑ Tuckman, Bruce; Serrat, Angel (2022). "Appendix A4.2". Fixed Income Securities: Tools for Today's Markets (4th ed.). Wiley. https://www.oreilly.com/library/view/fixed-income-securities/9781119835554/b06.xhtml.
19. ↑ Carpenter, Jennifer. "Convexity (course note)". https://people.stern.nyu.edu/jcarpen0/pdfs/Debtpdfs/05Complexity.pdf.
20. ↑ 20.0 20.1 Fabozzi, Frank J., ed (2008-09-15) (in en). Handbook of Finance (1 ed.). Wiley. doi:10.1002/9780470404324.hof003014. ISBN 978-0-470-04256-4. https://onlinelibrary.wiley.com/doi/book/10.1002/9780470404324.
21. ↑ Choudhry, Moorad (2004-01-01), "3 - The dynamics of asset prices", in Choudhry, Moorad, Advanced Fixed Income Analysis, Oxford: Butterworth-Heinemann, pp. 35–54, doi:10.1016/b978-075066263-5.50005-7, ISBN 978-0-7506-6263-5, https://www.sciencedirect.com/science/article/pii/B9780750662635500057, retrieved 2023-11-06
22. ↑ Miltersen, Kristian R.; Schwartz, Eduardo S. (1998). "Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates". The Journal of Financial and Quantitative Analysis 33 (1): 33–59. doi:10.2307/2331377. ISSN 0022-1090. https://www.jstor.org/stable/2331377.

## Further reading

* Frank Fabozzi, The Handbook of Fixed Income Securities, 7th ed., New York: McGraw Hill, 2005.
* Fabozzi, Frank J. (1999). "The basics of duration and convexity". Duration, Convexity, and Other Bond Risk Measures. Frank J. Fabozzi Series. 58. John Wiley and Sons. ISBN 9781883249632.
* Mayle, Jan (1994), Standard Securities Calculation Methods: Fixed Income Securities Formulas for Analytic Measures, 2 (1st ed.), Securities Industry and Financial Markets Association, ISBN 1-882936-01-9 . The standard reference for conventions applicable to US securities.

## External links

* The Investment Fund For Foundations explains the dangers of buying high-negative-convexity bonds
* Video tutorial, Bond duration and convexity explained
* Investopedia convexity explanation

* v
* t
* e

Bond market

* Bond
* Debenture
* Fixed income

Types of bonds by issuer|

* Agency bond
* Corporate bond
* Senior debt
* Subordinated debt
* Distressed debt
* Emerging market debt
* Government bond
* Municipal bond

Types of bonds by payout|

* Accrual bond
* Auction rate security
* Callable bond
* Commercial paper
* Consol
* 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

Bond valuation|

* Clean price
* Convexity
* Coupon
* Credit spread
* Current yield
* Dirty price
* Duration
* I-spread
* Mortgage yield
* Nominal yield
* Option-adjusted spread
* Risk-free bond
* Weighted-average life
* Yield curve
* Yield spread
* Yield to maturity
* Z-spread

Securitized products|

* Asset-backed security
* Collateralized debt obligation
* Collateralized mortgage obligation
* Commercial mortgage-backed security
* Mortgage-backed security

Bond options|

* Callable bond
* Convertible bond
* Embedded option
* Exchangeable bond
* Extendible bond
* Puttable bond

Institutions|

(SIFMA)

* v
* t
* e

Investment management

Investment fund
structures|

* Common contractual fund (CCF)
* Exchange-traded fund (ETF)
* Fonds commun de placement (FCP)
* Fund of funds
* Index fund
* Investment trust
* Hedge fund
* Listed investment company
* Loan qualifying investor alternative investment fund (LQIAIF)
* Mutual fund
* Offshore fund
* Open-ended fund company
* Open-ended investment company
* Pension fund
* Private equity fund
* Qualifying investor alternative investment fund (QIAIF)
* Real estate investment trust (REIT)
* Royalty trust
* SICAV
* Umbrella fund
* Unit investment trust
* Unit trust
* Unitised insurance fund

Investment styles|

* Active / passive management
* Impact investing
* Manager of managers
* Socially responsible investing
* Value / growth investing

Theory · Terminology|

* Closed-end fund
* Efficient-market hypothesis
* Net asset value
* Open-end fund

Related topics|

* Alternative investment
* Asset management firms
* Robo-advisor
* Returns-based style analysis
* Traditional investments
* UCITS

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Bond convexity. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Bond_convexity&oldid=4552187"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template