Home>Finance>Negative Convexity: Definition, Example, Simplified Formula

Finance

# Negative Convexity: Definition, Example, Simplified Formula

Published: December 29, 2023

Learn about negative convexity in finance: its definition, an example, and a simplified formula. Understand how it impacts investments and risk management.

(Many of the links in this article redirect to a specific reviewed product. Your purchase of these products through affiliate links helps to generate commission for LiveWell, at no extra cost. Learn more)

## Understanding Negative Convexity: Definition, Example, Simplified Formula

Finance can be a complex world to navigate, and understanding key concepts is crucial to making informed decisions. One such concept is negative convexity. In this blog post, we will explore the definition, example, and simplified formula of negative convexity, shedding light on this important aspect of finance.

## Key Takeaways:

- Negative convexity is a characteristic of certain financial instruments where the price decreases more than proportionally as interest rates rise.
- One example of an investment with negative convexity is a callable bond.

Now, let’s dive into the definition of negative convexity. Convexity refers to the curvature of the price-yield relationship of a fixed-income security. A positively convex security, such as a non-callable bond, experiences a price increase that is greater than proportionate to a decrease in yield. On the other hand, a security with negative convexity sees the price decrease more than proportionally as interest rates rise.

An example of an investment with negative convexity is a callable bond. A callable bond is a type of bond that gives the issuer the option to redeem the bond before its maturity date. When interest rates decline, the issuer may choose to call back the bond and issue new bonds at a lower interest rate. However, when interest rates rise, the issuer is less likely to call back the bond, as it would require paying a higher interest rate to refinance. As a result, the value of a callable bond decreases more than proportionally when interest rates increase, leading to negative convexity.

To understand negative convexity in a simplified formula, we can look at the relationship between bond price change (P) and yield change (Y). The formula for calculating bond price change due to yield change is:

**P = -D × ΔY + (1/2) × C × ΔY^2**

In this formula:

- P represents the price change of the bond.
- D is the bond’s modified duration, which measures the sensitivity of the bond’s price to changes in yield.
- ΔY is the change in yield.
- C is the bond’s convexity, which captures the curvature of the price-yield relationship.

When interest rates rise, the first term of the formula (P = -D × ΔY) captures the expected decrease in price due to the modified duration. However, the second term (P = (1/2) × C × ΔY^2) represents the additional decrease caused by the bond’s convexity, which contributes to negative convexity.

To sum up, negative convexity is an important concept in finance, particularly when analyzing fixed-income securities. Understanding its definition, recognizing examples such as callable bonds, and grasping the simplified formula allows investors to make more informed decisions in a complex financial landscape.

Disclaimer: The information provided in this blog post is for educational purposes only and should not be considered as financial advice. It is always recommended to consult with a qualified financial advisor before making any investment decisions.