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Bond convexity simply explained

Convexity implies that for a given movement in interest rates, bond prices tend to increase more when rates fall than they would decrease for a similar rate rise. This is because the convex relationship causes price changes to accelerate upwards as rates fall and decelerate (or moderate) as rates rise.

The relationship between bond prices and interest rates is non-linear due to the way bond prices are calculated.

This means that bond prices don't change in a straight line with changes in interest rates. Instead, they exhibit a curved relationship. Convexity is a measure of this curvature.

Duration, another key concept in bond pricing, estimates the change in bond prices relative to interest rate changes. However, duration assumes a linear relationship between bond prices and interest rates, which is only an approximation. As interest rate changes become larger, the approximation becomes less accurate. Convexity addresses this limitation by accounting for the curvature in the price-yield relationship.

In fixed-income markets, accurate pricing of bonds and related derivatives is essential. Convexity adjustments are often required in financial models to price these instruments correctly, especially for those with complex cash flows or embedded options.

In portfolio management, strategies like immunization (protecting a portfolio from interest rate movements) require an understanding of both duration and convexity. Convexity helps in making more precise adjustments to hedge interest rate risk.

Bond price is calculated as the sum of present values of all future cash flows.

The present value of each cash flow is calculated as: Present Value = Cash Flow / (1 + r)^t

Here, r is the interest rate, and t is the time until the cash flow is received.

The term (1 + r)^t in the present value calculation has an exponential nature.
This means that for a given change in r, the effect on the bond price does not remain constant but varies, especially for cash flows further in the future.

As a consequence, convexity provides a more accurate measure of the interest rate risk, especially for large interest rate changes.

The approximate change in bond price with changes in interest rates uses both duration and convexity.

Price change (ΔP) ≈ -Duration * Change in Yield (Δy) + 1/2 * Convexity * (Change in Yield)^2

Convexity is vital for bond investors and portfolio managers in accurately assessing and managing the risks associated with interest rate changes, especially in volatile market conditions.
However, the attributes of convexity in a security have an impact on its desirability and, consequently, its price. As a result, there is an anticipated offset in both price and yield to account for the effects of the convexity adjustment.

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