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What is Convexity with diagram?


 Convexity: Duration is a linear line which measures the changes in bond price in relation to changes in interest rate. But it does not determine the accurate bond price changes if there is large change in yield rate. On the other hand convexity is measure of the non-linear relationship of bond price in relation to changes in interest rate. It measures the changes in duration in relation to changes in interest rate to determine the accurate bond price.
Formula:
Convexity = 1 / P * (1+y) 2Ʃ Tt=1 [CF / (1+y) 2 (t 2 + t)]
Where,
P = bond price
Y = yield to maturity
CF = Cash flow
T = maturity period
Or
Convexity = P (i decreases) + P (i increases) – 2P0/ 2P0* ∆Y 2
Convexity adjustment = Convexity*100*∆y 2
Change in bond price in percentage with the help of modified duration:
∆p% = -D*p *∆y
Change in bond price in percentage with the help of modified duration and convexity:
∆p% = -D*p *∆y + 0.5*C*∆y 2
Where,
∆p %= change on bond price in percentage
D = duration
P = Initial price of bond
∆y = Change in yield rate
C = convexity

A convexity adjustment is a difference between forward interest rate and future interest rate. It is used to measure the accurate price of a bond because convexity is a non-linear relationship between changes in bond price and yield rate.

In above diagram the straight line shows the modified duration and the curve shows the convexity. It shows in a diagram that small changes can easily measured by duration like yield rate increases by 1 % then the bond price will decreases by Rs. 1. But large change in yield rate does not measured by duration. For large change in interest rate in relation to bond price is measure by convexity by forming a curve in above diagram. The gap between duration line and curve is a duration error and it shows that duration is insufficient to measure the accurate price of a bond.

The convexity can be positive or negative. The negative convexity means the duration increases with increase in yield rate. In negative convexity the bond price will increase as increase in yield rate or price decrease with decease in yield rate. We can say that there is a direct relationship between yield rate and duration. An example of negative convexity is callable bond.
In positive convexity the duration will increases when there is a fall in yield rate. In simple way there is a negative correlation between duration and yield rate. It means the bond price increases when there is a fall in yield rate. An example of positive convexity is non- callable bonds.
The degree of convexity is either higher or lower is depend on coupon rate of a bond. If the coupon rate is high then there is low degree of convexity. So, on the basis of that there is a high degree of convexity of a zero coupon bond.
If there is a two bond, Bond Q and Bond P and both the bond has same duration then select the bond which has lower convexity than the other. Lower convexity decreases the effect of changes in interest rate.


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