Convexity In Simple Terms: Think Of A Moving Car
What is convexity?
Let’s begin by establishing the technical definition of convexity. Mathematically, convexity is the second derivative of a curve, that is, the rate of change of the rate of change.
Putting this in a real life context, think about a car moving at a velocity of 60 km/h. Its velocity is how fast it is moving, which is the rate of change of its position. Now, the car starts to accelerate, which means that its velocity is changing. Therefore, acceleration is the rate of change of velocity, or the rate of change of the rate of change of its position.
Consider this graphically. If the car accelerates at a constant rate, the graph is an upward sloping line, and if the car accelerates at an increasing rate, the graph is an upward sloping curve. Finally, if the car stays at the same speed of 60 km/h, it has an acceleration of 0, and the graph is just a flat line.
If we were to calculate the second derivatives at these 3 different parts of the graph, we would get different results. When acceleration is constant and the graph is an upward sloping line, the second derivative is greater than 0, but it is the same value as long as the line slopes at the same angle.
When acceleration is increasing and the graph is an upward sloping curve, the value of the second derivative changes; the steeper the curve, the higher its value, and vice versa. Lastly, when acceleration is 0 and the graph is a flat line, the second derivative has a value of 0.
Now, focus on the steepest part of the curve, a.k.a the most convex part of the curve. You can observe that a small change in time corresponds with a large change in the car’s position. This asymmetric relationship is the power of convexity.
For bonds, this translates into a small change in interest rates leading to a large change in the bond’s price, but only at the convex parts of the bond’s curve.
From the diagram above, you can see that Bond 1 has a more convex curve than Bond 2 (Bond 2’s curve is noticeably flatter). For a 1% change in interest rates (yield), the price of Bond 1 changes by a considerably larger amount than Bond 2, which is clearly illustrated by the difference in size between the purple and orange rectangles.
Moreover, how much a bond’s price changes when interest rates change depends on which part of the curve the change occurs in. When the change occurs in areas where the curve is less convex, price doesn’t change as much. Conversely, when the change occurs in areas where the curve is more convex, the bond’s price moves a lot more.
It really is the same phenomena as with the accelerating car, just with different parameters.
To be continued…
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