## Interest Rate Risk: Convexity approximation

*in*Asset Liability Management

# Convexity

Duration approximates the change in price of an instrument due to changes in the yield. However this approximation tends to work only for small changes in yield. For larger changes there will be a significant error term between the actual price change and that estimated using duration. Convexity improves on this approximation as it explains the change in price that is not explained by Duration.

Convexity is given by the following measure

Where,

?i= change in yield (in decimals)

P_{0}= Initial Price

P_{+}= Price if yields increase by ?i

P_{–}= Price if yields decline by ?i

Also another important aspect of convexity is that it takes into account the curvature of the price/ yield relationship. The Duration measure assumes that the size of approximated change in price (in percentage terms) would be the same if yields were either to increase or decrease by the same amount. Convexity, depending on the sign, accounts for the direction of the change in yield and the curvature of the price/ yield relationship. A positive convexity measure indicates a greater price increase when interest rates fall by a given percentage relative to the price decline if interest rates were to rise by that same percentage. A negative convexity measure indicates that the price decline will be greater than the price gain for the same percentage change in yield. This feature of convexity plays an interesting role in Asset Liability Management calculations and in contracts that have embedded options.

*Approximate price change *

The approximate price change using both the duration and convexity measures will be as follows:

Total estimated percentage price change= -Duration×?i×100+Convexity×(?i)^{2}×100

*Terminology: Modified and Effective *

The term **modified** is used with duration and convexity when the changes in yield are assumed not to affect the expected cash flows of the security (e.g. for option free instruments). The term **effective** is used with duration and convexity when changes in yield are also assumed to impact the expected cash flows of the security (e.g. for derivatives).

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