Bond convexity
Encyclopedia
In finance
, convexity is a measure of the sensitivity of the duration
of a bond
to changes in interest rate
s, the second derivative
of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the convexity, the more sensitive the bond price is to decreasing interest rates and the less sensitive the bond price is to increasing rates. Bond convexity is one of the most basic and widely-used forms of convexity in finance
.
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.e. how the duration of a bond changes as the interest rate changes. 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.
s 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. of the issuers: let us suppose they are issued by different entities. 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.
This means if yield increases further, price of bond A may fall drastically while price of bond B won’t change, i.e. bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, while bond A holders are expecting further price-fall and ready to dispose of it.
This means bond B has better rating than bond A.
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.
Another way of expressing C is in terms of the modified duration D:
Therefore
leaving
Where D is a Modified Duration
where P(i) is the present value
of coupon i, and t(i) is the future payment date.
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). 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.
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. For example, under the assumption of a flat yield curve one can write the value of a coupon-bearing bond as
, where ci stands for the coupon paid at time ti. Then it is easy to see that
Note that this conversely implies the negativity of the derivative of duration by differentiating
.
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, convexity is a measure of the sensitivity of the duration
Bond duration
In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received....
of a bond
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
to changes in interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
s, the second derivative
Second derivative
In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the convexity, the more sensitive the bond price is to decreasing interest rates and the less sensitive the bond price is to increasing rates. Bond convexity is one of the most basic and widely-used forms of convexity in finance
Convexity (finance)
In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative of the modeling function...
.
Calculation of convexity
Duration is a linearLinear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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.e. how the duration of a bond changes as the interest rate changes. 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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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.
Why bond convexities may differ
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 durationBond duration
In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received....
s they will have identical sensitivities. That is, their prices will be affected equally by small, first-order, (and parallel) yield curve
Yield curve
In finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
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. of the issuers: let us suppose they are issued by different entities. 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.
This means if yield increases further, price of bond A may fall drastically while price of bond B won’t change, i.e. bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, while bond A holders are expecting further price-fall and ready to dispose of it.
This means bond B has better rating than bond A.
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.
Mathematical definition
If the flat floating interest rate is r and the bond price is B, then the convexity C is defined asAnother way of expressing C is in terms of the modified duration D:
Therefore
leaving
Where D is a Modified Duration
How bond duration changes with a changing interest rate
Return to the standard definition of modified duration:where P(i) is the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
of coupon i, and t(i) is the future payment date.
As the interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
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). 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.
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. For example, under the assumption of a flat yield curve one can write the value of a coupon-bearing bond as
, where ci stands for the coupon paid at time ti. Then it is easy to see thatNote that this conversely implies the negativity of the derivative of duration by differentiating
.Application of convexity
- Convexity is a risk management figure, used similarly to the way 'gamma' is used in derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s risks management; it is a number used to manage the market riskMarket riskMarket risk is the risk that the value of a portfolio, either an investment portfolio or a trading portfolio, will decrease due to the change in value of the market risk factors. The four standard market risk factors are stock prices, interest rates, foreign exchange rates, and commodity prices...
a bond portfolio is exposed to. If the combined convexity and duration of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedgeHedge (finance)A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
d, and little money will be lost even if fairly substantial interest movements occur. (Parallel in the yield curve.) - The second-order approximation of bond price movements due to rate changes uses the convexity:
See also
- Black-Scholes equation
- bond durationBond durationIn finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received....
- bond valuationBond valuationBond valuation is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected...
- Immunization (finance)Immunization (finance)In finance, interest rate immunization is a strategy that ensures that a change in interest rates will not affect the value of a portfolio. Similarly, immunization can be used to ensure that the value of a pension fund's or a firm's assets will increase or decrease in exactly the opposite amount...
- List of convexity topics
- List of finance topics
External links
- The Investment Fund For Foundations explains the dangers of buying high-negative-convexity bonds
- Investopedia convexity explanation
- Bond Yield Duration and Convexity Calculator Financial Technology Laboratories
- Real time Bond Price, Duration, and Convexity Calculator: http://www.cba.ua.edu/~rpascala/bond/BondForm.php
Further reading
- Frank Fabozzi, The Handbook of Fixed Income Securities, 7th ed., New York: McGraw Hill, 2005.