### Concept of Convexity

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

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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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 3

^{rd}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.

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