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.

1 Convexity is the second derivative of bond price
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

[edit] Convexity is the second derivative of bond price

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.

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, 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.

[edit] 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.

[edit] Algebraic definition

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

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

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:

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

[edit] How bond duration changes with a changing interest rate

Return to the standard definition of duration:

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

Where P(i) is the present value of coupon i, and t(i) is the future payment date.

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). Therefore, increases in r must decrease the duration (or, in the case of zero-coupon bonds, leave it constant).

$\frac{dD}{dr} \leq 0$

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.

[edit] 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:

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