NationMaster - Encyclopedia: Bond convexity

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In finance, convexity is a measure of the sensitivity of the price of a bond to changes in interest rates. It is related to the concept of duration. This article does not cite any references or sources. ... In finance, a 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. ... An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring his consumption, by lending to the borrower. ... In economics and finance, duration is the weighted average maturity of a bonds cash flows or of any series of linked cash flows. ...

1 Calculation of Convexity
2 Why bond convexities differ
3 Algebraic definition
- 3.1 How bond duration changes with a changing interest rate
4 Application of convexity
5 See also
6 External links

Calculation of Convexity

Duration is a linear measure 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. The word linear comes from the Latin word linearis, which means created by lines. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes, 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. For a non-technical overview of the subject, see Calculus. ...

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. 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 economics and finance, duration is the weighted average maturity of a bonds cash flows or of any series of linked cash flows. ...

Bond convexities can also change because of option-like features embedded in the bond. For example, a bond which is backed by mortgage-payments will typically experience refinancing when interest rates are low, and this causes the price of the bond to be limited as interest rates rise.

Algebraic definition

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

$C = frac{1}{B} frac{d^2left(B(r)right)}{dr^2}$

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

$frac{d}{dr} B (r) = -DB$

therefore:

$CB = frac{d(-DB)}{dr} = (-D)(-DB) + (-frac{dD}{dr})(B)$

Leaving:

$C = D^2 - frac{dD}{dr}$

How bond duration changes with a changing interest rate

Return to the standard definition of duration: In economics and finance, duration is the weighted average maturity of a bonds cash flows or of any series of linked cash flows. ...

$D = sum_{i=1}^{n}frac {P(i)t(i)}{B}$

Where P(i) is the present value of coupon i, and t(i) is the future payment date. The present value of a single or multiple future payments (known as cash flows) is the nominal amounts of money to change hands at some future date, discounted to account for the time value of money, and other factors such as investment risk. ...

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). An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring his consumption, by lending to the borrower. ... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash form the basis of time value of money calculations. ...

$frac{dD}{dr} leq 0$

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 $B (r) = sum_{i=1}^{n} c_i e^{-r t_i}$ , where $c i$ stands for the coupon paid at time $t i$ . Then, it is easy to see that

$frac{d^2B}{dr^2} = sum_{i=1}^{n} c_i e^{-r t_i} t_i^2 geq 0$

Note that this conversely implies the negativity of the derivative of duration by differentiating $d B / d r = - D B$ .

A basic caveat is in order: the notion of bond convexity should not be confused with the convexity (curvature) of the yield curve (term structure of interest rate). The latter can assume an arbitrary shape (although a normal yield curve has negative convexity), and in fact complex stochastic models have been proposed for its evolution. The US dollar yield curve as of 9 February 2005. ... The US dollar yield curve as of 9 February 2005. ...

Application of convexity

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.)
The second order approximation of bond price movements due to rate changes uses the convexity:

$Delta(B) = B[frac{C}{2}(Delta(r))^2 - DDelta(r)]$

In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives. ... For a non-technical overview of the subject, see Calculus. ... Market risk is the risk that the value of an investment will decrease due to moves in market factors. ... 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 trained in such a way that their limbs intertwine. ...

External links

The Investment Fund For Foundations explains the dangers of buying high-negative-convexity bonds
Investopedia convexity explanation
Duration and convexity Investment Analysts Society of South Africa
Bond Yield Duration and Convexity Calculator Financial Technology Laboratories
Real time Bond Price, Duration, and Convexity Calculator: [1]

Categories: Articles lacking sources from July 2007 | All articles lacking sources | Fixed income analysis

Results from FactBites:

Bond Calculator Help (968 words)
	In return for purchasing a bond, the investor is guaranteed to a receive a periodic (usually annual or semi-annual) coupon payment over the life of the bond and a larger redemption payment when the bond matures.
	Bond prices and coupon amounts are expressed as a percentage of the bond's notional face value, so an investor holding a 1000000 DM German Government Bond with a coupon of 5% and a 100% redemption would receive an annual coupon payment of 50000 DM and a redemption payment of 1000000 DM.
	A bond's Convexity is its change in PVBP given a unit increase in yield; it indicates the degree of curvature in the price/yield graph.

CMS BondEdge-fixed income portfolio and credit risk analytics (959 words)
	Therefore, when we see a bond's convexity changing from positive to negative and wonder why this is so, we should think about what is happening to the security's expected future cash flows as interest rates change; therein lies the answer.
	"Convexity" is a term that describes the degree and type of curvature observed.
	With a callable bond, as interest rates rally it becomes more likely that the issuer will call the bond, thereby providing the investor with a set of cash flows to the call date that are worth less than the cash flows to the maturity date.

More results at FactBites »

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