Convexity of a bond

The convexity of a bond is the slope of the curve that relates price and profitability. Measures the change in the duration of the bond as a result of a change in profitability.

Convexity of a bond

Mathematically it is expressed as the second derivative of the price-profitability curve. The formula is as follows:

The variation in the price of a bond in the event of changes in interest rates is the sum of the variation caused by the modified duration and the variation caused by the convexity of the bond.

If the convexity of a bond is equal to 100, the price of the bond will change an extra 1% for every 1% change in interest rates, in addition to that calculated by the duration. If the convexity of a bond is equal to zero, the price of the bond will vary with changes in interest rates by the amount motivated by the duration of the bond.

Relationship convexity of a bond and duration of a bond

The convexity of a bond offers us a much more accurate measure of the price-return changes of a bond. The duration of a bond assumes that the relationship between price and return is constant. However, the reality is very different. Hence, in the face of small price-profitability variations, duration is an acceptable measure. But for larger variations the calculation of convexity becomes essential.

Mathematically it may seem like a bit of an abstract term. Since graphically it is much easier to understand, let’s see it represented. In the following two graphs we see represented both the duration and the convexity.

The lower the yield on the bond, the higher its price. And vice versa, the higher the bond’s profitability, the lower its price. Of course, the price does not change in the same proportion if its profitability changes from 10 to 12% as if it changes from 1 to 2%. This is what convexity takes into account. Duration assumes that the change in price is the same every time. While convexity takes into account that the change in price is not constant. The difference between the blue line and the orange line is the convexity itself. The orange line is the change in the price of the bond taking into account the duration. Finally, the blue line represents the changes in the price of the bond taking into account duration and convexity.

Example of convexity of a bond

We have a bond maturing in 10 years. The coupon is 7% and the bond has a face value of 100 euros. The market IRR is 5%. Which means that bonds with similar characteristics are offering a 5% return. Or what is the same 2% less. Coupon payment is annual.

If the yield of the bond goes from 7% to 5%, how much does the price of the bond change? To calculate the variation that the price would have in the event of a change in the interest rate, we will need the following formulas:

Bond price calculation:

Calculation of the duration of the bonus:

Calculation of modified duration:

Calculation of convexity:

Calculation of the variation of the duration:

Calculation of the variation of convexity:

Calculation of the variation in the price of the bond:

Download excel table to see all detailed calculations

Using the above-mentioned formulas we obtain the following data:

Bond Price = 115.44

Duration = 7.71

Modified Duration = 7.34

Convexity = 69.73

The price variation in the face of a 2% drop in the yield of the bond is + 14.68% taking into account the duration. The variation in the price of the bond taking into account the convexity is + 1.39%. To obtain the total variation of the price we have to add the two variations. The calculation shows that in the face of a 2% drop in this bond, the price would increase by 16.07%.