Concept of Convexity

Before reading this article, make sure you should clear your concepts regarding Duration from this link:  Duration



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Convexity

Definition: Convexity describes how much a bond's duration changes when interest rates change
·        Where the cashflow series have payments coming Close to each other will have a lower convexity.
·        Where the Cashflow series payments is more spread out over time will have a higher convexity.

·        Now let’s Consider three cases
1.     There is a zero-coupon bond of say 20 years
2.     There is a bond having cashflows at time 3 and at time 10
3.     There is a bond having cashflows at time 1,2, 3,….,11,12

Ø Now Zero coupon bond will have lower convexity because it consists of just one payment. But in 3rd case convexity is high because payments are spread out over a longer period of time.
Ø In Technical Terms , let’s say X is the present value of all cashflows. So take the double derivative of X with respect to change in interest rate and then divide it by X will gives us the Convexity.
Now question will be that what exactly is the use of convexity.
It is the measure of change in Duration of the bond with respect to change in interest rate.
Positive convexity implies that change in interest rate is inversely proportional to change in bond, i.e. decrease in interest rate leads to increase in Duration of bond.
(Convexity will always have a positive value in normal market condition. But it can be negative also which means decrease in yields leads to decrease in duration as in case of callable bonds.)


In our above example that if we decrease the interest rate in all of them , then the change will be high in case 3 where convexity is high and will be low in case of scenario 1.

Now from graph point of view, higher convexity will have a curved shape whereas lower convexity will have flatter shape curve.

Note: Both DMT and effective duration measures the average life of an investment. Investment with longer term will have more impact than with shorter term when there is a change in interest rate.

Note: Degree to which a bond's price changes when interest rates change is called duration, which often is represented visually by a yield curve. Convexity describes how much a bond's duration changes when interest rate change.




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