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# Bond convexity

The relationship between bond price and bond yield is convex. It means that given a yield y, if we decrease the yield, the increase in bond price is more than the decrease if we increase the yield. However, bond duration does not capture this relationship because it is calculated by studying the change in bond price occurring on a line that is tangent to the bond-yield curve. This deficiency in duration is rectified through bond convexity measure.

The following equation provides the relationship between (percentage) change in bond price (∆PV), duration and convexity:

$∆PV =-AnnModDur\times∆Yield+12×AnnConvexity×(∆Yield)^2$

## Approximate convexity

Approximate convexity is calculated as follows:

$ApproxConvexity=\frac{PV_-+PV_+-2×PV_0}{(∆Yield)^2×PV_0}$

The approximation formula returns annualized convexity value because periodicity is reflected in bond prices. If convexity values are obtained from another source, they can be annualized by dividing it by periodicity squared. For example, semiannual convexity should be divided by 22 to get annualized convexity.

Yield convexity can be converted to money convexity by multiplying it with the value of the bond position. Money convexity is used together with money duration.

Generally, the relationship between different bond features such as coupon rate, yield, time to maturity, and bond duration also holds for bond convexity. Bond convexity is also affected by the dispersion of cash flows i.e. the degree to which they are spread out. If two bonds have the same duration, one whose cash flows are more spread out will have higher convexity.

Convexity is a good thing because the price of a more convex bond appreciates more than a less convex bond when yield decreases and it depreciates less than the less convex bond if yield increases.

## Effective convexity

Effective convexity is the convexity measure with reference to a shift in the benchmark yield curve. It is calculated using the following equation:

$EffectConvexity=\frac{PV_-+PV_+-2×PV_0}{(∆Curve)^2×PV_0}$

Effective convexity is coupled with effective duration and it is most useful in case of complex bonds. When interest rates are high and embedded call option has no value, a callable bond and a non-callable bond behave similarly. However, if the market interest rates fall sufficiently low such that the embedded call option is in-the-money, callable bonds’ convexity switches from positive to negative, which is why the increase in their price in response to a decrease in yield is less pronounced. Putable bonds, on the other hand, always have positive convexity.