Bond convexity

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

Contents

[edit] 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.

[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. 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 combination; also you have to take into consideration the profile, rating etc of the issuers; suppose they are issued by different entities. See diagrams next. Though both the bonds have same p-y combination bond I may be located on relatively more elastic segment of the p-y curve compared to bond II. This means if yield increases further, price of bond II may fall drastically while price of bond I won’t change, i.e. bond II holder are expecting a price rise any moment and so reluctant to sell it off, while bond I holders are expecting further price-fall and ready to dispose it.

This means bond II has better rating than bond I.

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; after all less convexity means less price-volatility or risk, less risk means less return.

The more convex a portfolio higher the risk content。

[edit] Algebraic definition

If the flat floating interest rate is r and the bond price is B, then the 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) + \left(-\frac{dD}{dr}\right)(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). 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).

\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 \scriptstyle B (r)\ =\ \sum_{i=1}^{n} c_i e^{-r t_i} , where ci stands for the coupon paid at time ti. 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 \scriptstyle dB / dr\ =\ - D B .

[edit] 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:
\Delta(B) = B\left[\frac{C}{2}(\Delta(r))^2 - D\Delta(r)\right].

[edit] See also

Frank Fabozzi, The Handbook of Fixed Income Securities, 7th ed., New York: McGraw Hill, 2005.

[edit] External links

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