# Bond convexity

In finance, convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates.

## 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.

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.

## Why bond convexities differ

The price sensitivity to parallel IR shifts 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.

## Algebraic definition

If the flat floating interest rate is r and the bond price is B, then the convexity C is defined as

Another way of expressing C is in terms of the duration D:

Therefore

leaving

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

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 increase but changes in the present value of all coupons (the numerator) is larger than changes in the bond price (the denominator). Therefore, increases in r must decrease the duration (or, in the case of zero-coupon bonds, leave it constant).

Given the convexity definition 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 .

## Application of convexity

1. Convexity is a risk management figure, used similarly to the way 'gamma' is used in derivatives risks management; it is a number used to manage the market risk a bond portfolio is exposed to. If the combined convexity of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur. (Parallel in the yield curve.)
2. The second-order approximation of bond price movements due to rate changes uses the convexity:

:

 Bond market
Fixed income | Bond | Debenture
Types of Bonds
By Issuer:Government bond | Sovereign bond | Agency bond
colspan="2" | Municipal bond | Corporate bond | Emerging market debt
By Payout:Fixed rate bond | Floating rate note | Zero coupon bond
colspan="2" | Inflation-indexed bond | Commercial paper | Accrual bond
colspan="2" | Auction rate security | High-yield debt | Convertible bond
colspan="2" | Mortgage-backed security | Asset-backed security
Derivatives
Bond option | Credit derivative | Credit default swap
Collateralized debt obligation | Collateralized mortgage obligation
Bond valuation
Pricing:Par value | Coupon | Clean price | Dirty price
colspan="2" | Accrued interest | Day count convention
Yield analysis:Nominal yield | Current yield | Yield to maturity
colspan="2" | Yield curve | Bond duration | Bond convexity
Credit analysis:Credit analysis | Credit risk
Interest rate models:Short rate models | Rendleman-Bartter | Vasicek
colspan="2" | Ho-Lee | Hull-White | Cox-Ingersoll-Ross | Chen
colspan="2" | Heath-Jarrow-Morton | Black-Derman-Toy
Finance studies and addresses the ways in which individuals, businesses, and organizations raise, allocate, and use monetary resources over time, taking into account the risks entailed in their projects.
bond is a debt security, in which the authorized issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date, termed maturity.
This article or section needs copy editing for grammar, style, cohesion, tone and/or spelling.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
An amortizing bond is a bond that repays part of the principal (face value) along with the coupon payments, according to the schedule defined in the bond agreement at issuance.
In finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. Then the duration of a zero coupon bond with a maturity period of n years is n years.
In finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. Then the duration of a zero coupon bond with a maturity period of n years is n years.
Present 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.
This article or section needs copy editing for grammar, style, cohesion, tone and/or spelling.
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discounting is the process of finding the present value of an amount of cash at some future date, and along with compounding cash forms the basis of time value of money calculations.
Greeks are the quantities representing the market sensitivities of options or other derivatives. Each "Greek" measures a different aspect of the risk in an option position, and corresponds to a parameter on which the value of an instrument or portfolio of financial instruments is
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
Market risk is the risk that the value of an investment will decrease due to moves in market factors. The four standard market risk factors are:
• Equity risk, or the risk that stock prices will change.

The word hedge may be used to refer to an artificial boundary, erected to contain or protect:
• A hedge or hedgerow in agriculture and in gardening is a lineal barrier or boundary made from growing plants planted and grown in such a way that their limbs densely

In finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. Then the duration of a zero coupon bond with a maturity period of n years is n years.
Bond valuation is the process of determining the fair price of a bond. As with any security or capital investment, the fair value of a bond is the present value of the stream of cash flows it is expected to generate.
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 insure that the value of a pension fund's or a firm's assets will increase or decrease in exactly the
• Alpha blending
• Barycentric coordinates
• Bohr-Mollerup theorem
• Bond convexity
• Carathéodory's theorem (convex hull)
• Choquet theory
• Closed convex function
• Concavity
• Convex analysis

Topics in finance include:

## Fundamental financial concepts

• Finance an overview
• Arbitrage
• Capital (economics)

The bond market, also known as the debt, credit, or fixed income market, is a financial market where participants buy and sell debt securities usually in the form of bonds. The size of the international bond market is an estimated \$45 trillion of which the size of outstanding U.S.
worldwide view.
Fixed income refers to any type of investment that yields a regular (or fixed) return.

For example, if you borrow money and have to pay interest once a month, you have issued a fixed-income security.
bond is a debt security, in which the authorized issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date, termed maturity.
In finance, a debenture is a long-term debt instrument used by governments and large companies to obtain funds. It is similar to a bond except the securitization conditions are different.
A government bond is a bond issued by a national government denominated in the country's own currency. Bonds issued by national governments in foreign currencies are normally referred to as sovereign bonds.