Negative Convexity: Comprehensive Definition, Examples, and Simplified Formula

Explore the concept of negative convexity, discover its significance in the bond market, including definitions, examples, and simplified formulas for a better understanding.

Negative Convexity Defined

Negative convexity occurs when the shape of a bond’s yield curve is concave, indicating that the bond’s price decreases at an increasing rate as interest rates rise. This phenomenon is commonly observed in certain types of bonds, such as mortgage-backed securities and callable bonds.

Importance in the Financial Market

Negative convexity is a crucial concept in fixed income investments, particularly for portfolio managers and investors who deal with bond instruments that have embedded options, such as callable bonds. Understanding negative convexity helps in assessing the risks associated with interest rate fluctuations and their impact on bond prices.

Types of Bonds Exhibiting Negative Convexity

Mortgage-Backed Securities (MBS)

Mortgage-backed securities often exhibit negative convexity due to the prepayment option that homeowners have. When interest rates drop, mortgage holders are likely to refinance, resulting in the early repayment of the bond, which limits the bond’s price appreciation.

Callable Bonds

Callable bonds show negative convexity at lower yields because the issuer has the right to redeem the bond before maturity if interest rates decline, which restricts the bond’s price increase potential.

Simplified Formula for Calculating Convexity

While the detailed mathematical calculation of convexity might be complex, an approximate formula for negative convexity for a bond can be represented as:

$$ C \approx \frac{\Delta P_+ + \Delta P_- - 2P_0}{\Delta y^2 \cdot P_0} $$

Where:

• \( \Delta P_+ \) = Change in price given a decrease in yield.
• \( \Delta P_- \) = Change in price given an increase in yield.
• \( P_0 \) = Initial price of the bond.
• \( \Delta y \) = Change in yield.

Historical Context and Applicability

Historical Context

The concept of negative convexity dates back to the introduction of securities with embedded options, which gained prominence with the development of the modern bond market and the widespread adoption of mortgage-backed securities in the late 20th century.

Applicability

Negative convexity is particularly relevant for fixed income portfolio managers who need to manage interest rate risk and for investors looking at mortgage-backed securities or callable bonds as part of their investment strategy.

Comparisons and Related Terms

Positive Convexity

Positive convexity refers to the scenario where the bond’s yield curve is convex, implying that the bond’s price increases at an increasing rate as interest rates fall. This is commonly seen in bonds without embedded options, such as plain vanilla treasury bonds.

Duration

Duration measures the sensitivity of a bond’s price to changes in interest rates. While duration provides a linear approximation, convexity offers a more refined measure by accounting for the curvature of the price-yield curve.

FAQs

Q1: How does negative convexity impact bond prices?

Negative convexity causes bond prices to decrease at an accelerating rate as interest rates rise, making the bonds more sensitive to interest rate increases. This increases the risk for investors holding such bonds.

Q2: Why are mortgage-backed securities negatively convex?

Mortgage-backed securities are negatively convex due to the prepayment option available to homeowners. When interest rates fall, prepayment rates increase, leading to an early return of principal and limiting price appreciation.

Q3: Can bonds with negative convexity ever have positive convexity?

Yes, bonds with negative convexity can exhibit positive convexity at higher yields, where the probability of the bond being called or prepaid decreases significantly.

References

1. Fabozzi, Frank J. The Handbook of Fixed Income Securities. McGraw Hill, 2012.
2. Sundaresan, Suresh. Fixed Income Market and Their Derivatives. South-Western College Pub, 2009.

Summary

Negative convexity is a distinct characteristic of certain bonds, primarily those with embedded options like callable bonds and mortgage-backed securities. Understanding negative convexity is essential for managing the risks associated with interest rate fluctuations and optimizing bond investment strategies. This comprehensive coverage of the concept provides clarity and aids in making informed financial decisions.