Let's now examine the final key factor affecting interest rate sensitivity, the level of yields itself on screen, we've charted the price yield relationship of a 10 year, 4% fixed coupon bond, and we've added a linear tangent represented by the green line for comparison.

When you compare these two lines, you'll see that the actual bond price yield relationship is not quite linear.

Instead, it shows a slightly convex profile, which is why this phenomenon is referred to as convexity.

But what does convexity mean in the context of fixed coupon bonds? Let's focus on the point where yields are at 4%, which is when the bond price is at par, 100%.

If yields increase, bond prices decline as expected.

However, the higher yields rise, the more the dark blue line flattens out.

This means that the bond's interest rate sensitivity declines.

As rates rise, the bonds price decreases by a smaller amount for each subsequent increase in yield.

In other words, as yields move higher, the risk exposure of the bond holder decreases clearly a beneficial outcome for someone holding the bond.

This is the core feature of convexity. Now, what happens when yields fall? The bond price rises, but this time the rate of increase accelerates.

As yields drop further, each additional decline in yields results in a larger price gain.

Thus, as the market moves favorably for bond holders, that is, as yields decline, the position size effectively becomes larger due to the market to market gain.

Again, a favorable outcome for the bond holder complexity enhances the upside potential in falling yield environments, allowing bond holders to benefit more if yields continue to drop.

So in conclusion, as yields increase the interest rate sensitivity of fixed coupon bonds decreases and vice versa.

This convexity is advantageous for a bond holder, but it is a disadvantage for someone holding a short position in a bond.

While the concept may seem clear from the chart, the intuition behind it can be tricky to grasp.

To better understand the practical implications of convexity, let's Consider the relative change in yields With a simple analogy. Imagine yields are at 0.01% and they increase to 0.02%.

This represents an absolute change of 0.01% or one basis point, but relatively yields have doubled.

Now, imagine yields are at 1% and they rise to 1.01%.

While the absolute change is still 0.01%, the relative change is now much smaller at only 1%.

In other words, the same absolute change in yields has a smaller magnitude of effect if yields are already higher.

This helps explain why bond prices react more to interest rate changes when yields are lower and less, when yields are higher, reinforcing the nonlinear behavior we observe in bond price sensitivity.

Although this analogy might not be perfect, it helps to conceptualize why convexity exists in bond prices.