Convexity

What Is Convexity?

Convexity, in finance, is a measure of the curvature in the relationship between bond prices and interest rates. It is a crucial concept within fixed income analysis that quantifies how a bond's duration changes as interest rates fluctuate. While duration provides a linear estimate of a bond's price sensitivity to interest rate changes, convexity accounts for the non-linear aspect of this relationship, offering a more precise prediction of price movements, especially for larger interest rate shifts¹⁹.

History and Origin

The concept of duration, which serves as a foundation for understanding convexity, gained prominence in the financial world to better assess the interest rate sensitivity of bonds. Early work in fixed income mathematics, particularly in the 1970s, highlighted the limitations of simple maturity measures. As financial instruments became more complex, and markets experienced greater volatility, the need for more sophisticated tools arose. The development and popularization of convexity in finance are often attributed to the work of Hon-Fei Lai and Stanley Diller, expanding upon the initial concept of duration to provide a more accurate picture of bond price behavior. Academic research further solidified these concepts, with studies exploring their application and impact on bond price changes¹⁸.

Key Takeaways

• Convexity measures the non-linear relationship between a bond's price and its yield, indicating how accurately duration estimates price changes for varying interest rate movements.
• Bonds with positive convexity experience larger price increases when interest rates fall than price decreases when rates rise by an equal amount.¹⁷
• Negative convexity, often found in callable bonds or mortgage-backed securities, means a bond's price may rise less or fall more than a standard bond when rates change.¹⁶
• Convexity is a second-order measure, complementing duration to offer a more comprehensive assessment of a bond's interest rate risk.¹⁵

Formula and Calculation

Convexity is typically calculated using the following approximate formula, which is derived from the Taylor series expansion of a bond's price function:

Where:

• (V_0) = Current market price of the bond.
• (V_-) = Bond price if the yield to maturity decreases by a small amount ((\Delta y)).
• (V_+) = Bond price if the yield to maturity increases by the same small amount ((\Delta y)).
• (\Delta y) = Change in yield (expressed as a decimal, e.g., 0.0001 for 1 basis point).

This formula helps to adjust the duration-based estimate of price change, especially for larger shifts in bond yields.¹⁴

Interpreting Convexity

Interpreting convexity is crucial for understanding a bond's true interest rate sensitivity. A bond with positive convexity is generally more desirable for investors. This means that as interest rates fall, the bond's price increases at an accelerating rate, and as interest rates rise, its price decreases at a decelerating rate. In essence, positive convexity provides a potential cushion against price declines when rates rise and allows for greater price appreciation when rates fall. Most traditional, non-callable fixed income securities, such as zero-coupon bonds, exhibit positive convexity.

Conversely, negative convexity is observed when the rate of price change becomes unfavorable to the investor. This occurs with certain bonds that have embedded options, such as callable bonds or mortgage-backed securities. For these bonds, as interest rates fall, the issuer may exercise their option to "call" or prepay the bond, capping the bond's price appreciation. This means that while a bond with negative convexity will still see its price fall when interest rates rise, the price increase when rates fall is limited or even reversed compared to a standard bond.¹³ Understanding whether a bond exhibits positive or negative convexity is vital for assessing its overall risk profile.

Hypothetical Example

Consider a hypothetical bond portfolio manager assessing two different bonds, Bond A and Bond B, both with the same duration of 5 years and current price of $1,000.

Bond A (Positive Convexity):

• If interest rates fall by 1%, Bond A's price rises to $1,055.
• If interest rates rise by 1%, Bond A's price falls to $952.

Bond B (Negative Convexity - e.g., a callable bond):

• If interest rates fall by 1%, Bond B's price rises to $1,048 (capped due to potential call).
• If interest rates rise by 1%, Bond B's price falls to $949.

In this scenario, while both bonds have a 5-year duration implying a roughly 5% price change for a 1% rate change, Bond A (positive convexity) offers a larger gain in a falling rate environment and a smaller loss in a rising rate environment compared to Bond B (negative convexity). This demonstrates how convexity provides a more nuanced understanding of a bond's sensitivity beyond just its duration.

Practical Applications

Convexity is a vital tool in portfolio management, particularly for fixed income investors and institutions engaged in asset-liability management. It provides a more accurate assessment of a portfolio's exposure to significant interest rate changes than duration alone.¹²

For instance, investors aiming to maximize potential gains and minimize losses in volatile markets often seek bonds with higher positive convexity. This characteristic is especially valuable when there is uncertainty about the direction and magnitude of future interest rate movements.¹¹

Moreover, large institutional investors, such as hedge funds, actively manage convexity risk in their bond holdings. They may implement "convexity hedging" strategies, for example, by buying U.S. Treasuries or interest rate swaps to offset negative convexity exposure from holdings like mortgage-backed securities when rates fall and prepayments increase. This active management helps them maintain their portfolio's desired sensitivity to interest rates.¹⁰ The activities of such funds can influence large segments of the capital markets, as observed in the euro zone government bond market where hedge funds have become significant participants.⁹

Limitations and Criticisms

While convexity significantly enhances the accuracy of bond price predictions compared to using duration alone, it is not without limitations. Like duration, convexity assumes a parallel shift in the yield curve, meaning that all maturities change by the same amount. In reality, yield curves rarely shift in such a uniform manner; they often twist or steepen, which can reduce the accuracy of convexity-based estimates.⁸

Furthermore, the calculation of effective convexity for bonds with embedded options, like callable bonds or mortgage-backed securities, can be complex. These bonds exhibit negative convexity under certain interest rate conditions, where their price appreciation is capped when rates fall due to the likelihood of the issuer exercising their call option or borrowers prepaying their mortgages.⁶, ⁷ This behavior means that the traditional benefits of positive convexity do not apply, and investors face a different risk profile. The challenge lies in accurately modeling the probability of such options being exercised, which directly impacts the bond's effective convexity.⁴, ⁵ Academic research has delved into the circumstances under which convexity becomes particularly important and where its relative impact is more pronounced.³

Convexity vs. Duration

Convexity and duration are both fundamental measures in fixed income analysis, but they describe different aspects of a bond's interest rate sensitivity. Duration is a first-order measure, providing a linear approximation of how a bond's price will change for a small change in interest rates. It essentially represents the slope of the bond's price-yield curve at a given point. For instance, a bond with a duration of 7 years is expected to fall by approximately 7% if interest rates rise by 1%.²

Convexity, on the other hand, is a second-order measure that quantifies the curvature of this price-yield relationship. It corrects the error inherent in duration's linear approximation, especially for larger changes in interest rates. When graphed, a bond's price-yield relationship typically forms a curve (hence "convex"). Duration is the tangent line to this curve at the current yield, while convexity measures how much the curve deviates from this tangent line.¹ Therefore, while duration gives an initial estimate of price movement, convexity refines that estimate by accounting for the accelerating or decelerating nature of price changes as rates move significantly.

FAQs

What is the primary purpose of convexity in bond analysis?

The primary purpose of convexity is to improve the accuracy of predicting bond price changes in response to shifts in interest rates. While duration provides a linear estimate, convexity accounts for the non-linear, curved relationship between bond prices and yields, making predictions more precise, especially for larger rate movements.

Can a bond have negative convexity?

Yes, certain financial instruments, particularly those with embedded options like callable bonds or mortgage-backed securities, can exhibit negative convexity. This occurs when the bond's price appreciation is limited as interest rates fall due to the increased likelihood of the issuer exercising their embedded call option or borrowers prepaying their loans.

Why is positive convexity generally preferred by investors?

Positive convexity is generally preferred because it provides a favorable asymmetry in price movements. When interest rates fall, bonds with positive convexity experience proportionally larger price gains, and when rates rise, they experience proportionally smaller price losses compared to bonds with less or negative convexity. This characteristic can enhance investment performance in volatile interest rate environments.

Does convexity replace duration?

No, convexity does not replace duration; rather, it complements it. Duration remains a fundamental measure of a bond's immediate interest rate sensitivity. Convexity is used as an adjustment to duration to provide a more refined estimate of price changes, particularly when interest rate changes are significant. Both measures are important for comprehensive risk management in fixed income portfolios.