# What Is Convexity?

Convexity is a numerical tool that measures the exposure of a portfolio to the risks of the market. It is the measure of the curvature in the relationship between bond yields and bond prices. Convexity demonstrates how the duration of a bond changes when the interest rate changes. If a bond's duration increases as yields increase, the bond is said to have negative convexity. However, if a bond's duration increases when yields fall, the bond is said to have positive convexity.

## Basics

Bond prices and yields demonstrate convexity, revealing the curvature influenced by interest rates. Convexity, the measure of a bond's duration sensitivity to interest rate shifts, quantifies the anticipated percentage adjustment in bond prices for a 1% interest rate variation.

## Exploring Convexity

Understanding convexity unveils the dynamic nature of bond duration amidst shifting interest rates. Portfolio managers deploy convexity as a pivotal risk-management instrument, assessing and mitigating the portfolio's exposure to interest rate fluctuations.

In a scenario of declining interest rates, bond prices ascend, inversely impacting bond prices when market interest rates surge. Bond yield signifies the anticipated returns for investors holding a specific security. Determined by various factors, including market interest rates, bond prices undergo regular adjustments.

As market rates climb, new bond issuances must offer higher rates to meet investor lending demand. Bonds with lower coupon rates witness diminished demand, prompting holders to sell and shift to higher-yielding bonds. Over time, these lower-coupon bonds adjust their prices to align with prevailing market interest rates, equalizing the rate of return.

## Analyzing Bond Duration

The fluctuation of interest rates significantly impacts a bond's price, a relationship gauged by bond duration. A higher duration denotes a more pronounced price shift in the opposite direction to interest rate changes. In practical terms, a 5-year average duration bond or bond fund may see a 5% value reduction if interest rates rise by 1%. Conversely, lower duration signifies less susceptibility to interest rate fluctuations.

Elevated duration correlates with increased interest rate risk and larger price variations. Investors anticipating rising interest rates should favor bonds with lower durations. It's crucial to distinguish bond duration from its term to maturity. While both decline as the maturity date nears, the latter merely represents the period until the bondholder receives coupon payments until the principal is repaid.

In a 1% market rate hike scenario, a one-year maturity bond sees a corresponding 1% price reduction. This reaction amplifies for bonds with extended maturities, generally adhering to a rule where bond prices decrease by 1% for each year of maturity in a 1% interest rate increase.

## Analyzing Convexity's Impact on Risk

Convexity, an extension of duration, gauges a bond's duration sensitivity amid yield adjustments. It serves as a superior indicator of interest rate risk, offering a more nuanced perspective than the linear relationship assumed by duration.

While duration effectively assesses the impact of small, abrupt interest rate changes on bond prices, the relationship between bond prices and yields is inherently more sloped or convex. Hence, in scenarios involving substantial interest rate fluctuations, convexity emerges as a more adept metric for evaluating bond price repercussions.

The elevation of convexity correlates with an augmented systemic risk exposure for the portfolio. In a fixed-income portfolio, the allure of existing fixed-rate instruments diminishes as interest rates climb. Conversely, diminishing convexity mitigates exposure to market interest rate shifts, hedging the bond portfolio. Typically, bonds with higher coupon rates or yields exhibit lower convexity and market risk.

## Convexity in Action

Consider XYZ Corporation's bond offerings: Bond A and B, bearing a $100,000 face value and a 5% coupon rate. While Bond A matures in 5 years, Bond B extends its maturity to 10 years. Applying duration analysis, Bond A exhibits a 4-year duration, while Bond B's duration stretches to 5.5 years. Consequently, a 1% interest rate shift would lead to a 4% price change for Bond A and a 5.5% change for Bond B.

With a 2% interest rate hike, Bond A's price is anticipated to drop by 8%, while Bond B faces an 11% decline. However, factoring in convexity reveals that Bond B's price adjustment is less severe than its duration alone suggests. The extended maturity of Bond B results in higher convexity, acting as a cushion against interest rate shifts and yielding a more moderate price change compared to duration predictions.

## Convexity Dynamics: Negative and Positive Trends

When a bond experiences an increase in duration with rising yields, it manifests negative convexity. The ensuing result is a steeper decline in bond price as yields ascend compared to a scenario of falling yields. In essence, negative convexity signifies an augmented duration leading to price depreciation in the wake of rising interest rates.

Conversely, positive convexity emerges when a bond's duration rises amid falling yields. This configuration induces a more pronounced increase in bond prices relative to a situation where yields increase. Bonds with positive convexity are predisposed to price appreciation during declining yields, contrasting with potential price depreciation during yield increases.

In standard market conditions, higher coupon rates or yields correspond to lower convexity in a bond. This correlation translates to lower investor risk, as substantial rate increases would be requisite to surpass the bond's yield. Portfolios comprised of high-yield bonds exhibit diminished convexity, mitigating the risk of unattractive yields amidst rising interest rates.

Typically, mortgage-backed securities (MBS) exhibit negative convexity due to their generally higher yields compared to conventional bonds. This characteristic implies that a substantial yield increase is necessary for an MBS holder to experience a less attractive yield than the prevailing market conditions.

## Conclusion

At its core, convexity gauges the curvature inherent in the bond's duration and its interplay with yields. It delineates the fluctuations in a bond's duration triggered by shifts in interest rates, thereby influencing investment values. Various elements, such as coupon rate, maturity, and credit quality, contribute to a bond's convexity. Astute bond investors leverage convexity as a tool to strategically navigate their portfolios, capitalizing on opportunities arising from interest rate adjustments.