Negative convexity: Meaning, Criticisms & Real-World Uses

What Is Negative Convexity?

Negative convexity is a characteristic predominantly observed in fixed-income securities, such as callable bonds and mortgage-backed securities (MBS), where the relationship between a bond's price and its yield curve is concave. In simpler terms, for instruments with negative convexity, as interest rates decrease, their prices increase at a diminishing rate, and conversely, as interest rates increase, their prices decrease at an accelerating rate. This non-linear response to interest rate changes implies an unfavorable asymmetry for investors.³⁵

This phenomenon is primarily driven by embedded options within these securities. For callable bonds, it's the issuer's right to buy back the bond, while for MBS, it's the homeowner's right to prepay or refinance their mortgage.³⁴,³³ Negative convexity is a critical concept within bond investing and risk analysis, as it signifies that these securities offer less upside potential when rates fall and greater downside risk when rates rise, compared to bonds with positive convexity.

History and Origin

The concept of negative convexity emerged prominently with the rise of complex debt instruments possessing embedded options, notably callable bonds and mortgage-backed securities (MBS). While basic bond duration measures the linear sensitivity of bond prices to interest rate changes, it became clear that this linear approximation was insufficient for instruments where the issuer or borrower had an option to alter cash flows.

The unique behavior of MBS, in particular, made negative convexity a significant area of study for fixed-income analysts. MBS prices do not rise as much as other fixed-income securities when interest rates fall, and they can experience sharper price declines when interest rates rise due to the embedded prepayment option.³² Homeowners tend to refinance their mortgages when interest rates drop, causing investors in MBS to receive principal payments sooner than anticipated. This shortens the effective duration of the MBS and limits its price appreciation. Conversely, when rates rise, refinancing activity slows, extending the duration and amplifying price declines.³¹ This characteristic of negative convexity in MBS played a significant role in market dynamics, notably during the 2008 financial crisis.³⁰

Key Takeaways

Asymmetrical Price Response: Negative convexity means that a bond's price gains are limited when interest rates fall, but price losses are amplified when rates rise.²⁹
Embedded Options: This characteristic typically arises from embedded options, such as the call feature in callable bonds or the prepayment option in mortgage-backed securities.²⁸
Risk Profile: Securities with negative convexity expose investors to increased interest rate risk and prepayment risk.²⁷
Yield Compensation: Investors typically demand a higher yield to maturity on bonds with negative convexity to compensate for this unfavorable price behavior.²⁶

Formula and Calculation

Convexity, in general, measures the rate of change of a bond's duration with respect to changes in yield to maturity. While a precise, universally applicable "negative convexity" formula distinct from the general convexity formula does not exist, negative convexity is identified when the convexity calculation results in a negative value.

The general formula for approximating a bond's convexity is:

\text{Convexity} = \frac{1}{P \times (1+y)^2} \sum_{t=1}^{N} \frac{CF_t \times t \times (t+1)}{(1+y)^t}

Where:

( P ) = Current bond price
( CF_t ) = Cash flow at time ( t ) (coupon payment or principal repayment)
( t ) = Time period until the cash flow is received
( y ) = Yield to maturity

For callable bonds or mortgage-backed securities, their effective convexity, which accounts for the embedded option, will reflect the negative curvature caused by the issuer's or borrower's actions. The callable bond's convexity can also be thought of as the sum of the positive convexity of a straight bond and the negative convexity of the embedded call option.

Interpreting Negative Convexity

Negative convexity indicates a specific, disadvantageous behavior of a bond's price in response to changing interest rates. For investors, understanding negative convexity means recognizing that the security will not perform as well as a "normal" bond when rates fall and will perform worse when rates rise.

Specifically:

Falling Interest Rates: When rates decline, the bond price will increase, but the increase will be less than what a bond with positive convexity would experience. This is because the embedded option (e.g., call option on a callable bond or prepayment option on an MBS) becomes more valuable, and the issuer/borrower is more likely to exercise it, effectively capping the bond's upside.²⁵
Rising Interest Rates: When rates increase, the bond price will decrease, and the decrease will be more pronounced than for a bond with positive convexity. The bond's duration might extend, making it more sensitive to rising rates.²⁴

Investors often receive a higher yield on instruments with negative convexity to compensate for this unfavorable characteristic and the added prepayment risk or call risk.²³

Hypothetical Example

Consider two bonds, Bond A (a standard, non-callable bond) and Bond B (a callable bond), both with an initial par value of $1,000, a 5% coupon rate, and five years to maturity. Assume both currently trade at par with a 5% yield to maturity.

Scenario 1: Interest Rates Fall by 1% (from 5% to 4%)

Bond A (Positive Convexity): Its price might rise to, say, $1,045. The price appreciation is relatively strong as rates fall.
Bond B (Negative Convexity): Due to its callable feature, if interest rates drop to 4%, the issuer might consider calling the bond and reissuing debt at a lower rate. This limits Bond B's potential price appreciation. Its price might only rise to $1,020, as the market anticipates the call. The bond's gain is significantly capped compared to Bond A.

Scenario 2: Interest Rates Rise by 1% (from 5% to 6%)

Bond A (Positive Convexity): Its price might fall to $955. The price depreciation is relatively contained.
Bond B (Negative Convexity): If interest rates rise to 6%, the likelihood of the bond being called decreases significantly. However, because of the initial disadvantageous structure of negative convexity, its price might fall more sharply than a comparable non-callable bond, perhaps to $930. The amplified loss in a rising rate environment illustrates the impact of negative convexity.

This example illustrates how negative convexity can create a less favorable risk-return profile for investors, offering limited upside and enhanced downside compared to bonds with positive convexity.

Practical Applications

Negative convexity is most frequently encountered and analyzed in the context of mortgage-backed securities (MBS) and callable bonds. Investors and financial institutions need to understand this characteristic for effective portfolio management and risk management.

MBS Market: The vast majority of MBS exhibit negative convexity due to prepayment risk. When interest rates decline, homeowners often refinance their mortgages, leading to faster-than-anticipated principal repayments for MBS investors. This limits the potential for price appreciation. Conversely, when rates rise, refinancing slows down, extending the average life of the mortgages and amplifying price declines.²² Financial institutions, particularly those with large MBS holdings, must dynamically hedge their portfolios to mitigate this exposure, a practice that can impact broader market liquidity.²¹
Callable Bonds: Corporations and municipalities issue callable bonds to gain flexibility. If interest rates fall, they can "call" or redeem the bonds, effectively refinancing at a lower cost. This embedded option gives the issuer an advantage at the bondholder's expense, leading to negative convexity for the investor.²⁰
Systemic Risk: The concentration of negative convexity, particularly in MBS held by various financial entities, can contribute to systemic risk within the financial system. For instance, the dynamics of MBS and their associated hedging activities were a significant factor in the 2008 financial crisis.¹⁹,¹⁸ Investors and central banks closely monitor these dynamics when managing interest rate risk and implementing monetary policy.¹⁷

Limitations and Criticisms

While negative convexity is a well-understood concept, its practical implications present certain limitations and challenges for investors.

One primary criticism lies in the difficulty of accurately predicting the exercise of the embedded option that causes negative convexity. For mortgage-backed securities, homeowner prepayment risk is influenced by various factors beyond just interest rates, including economic conditions, housing market trends, and individual borrower behavior. This makes it challenging to precisely model and anticipate how the negative convexity will manifest.¹⁶ Similarly, for callable bonds, the issuer's decision to call a bond depends on their funding needs, market sentiment, and refinancing costs, not solely on the level of interest rates.

Another limitation is that negative convexity can complicate risk management strategies. Bonds with negative convexity require dynamic hedging, where portfolio adjustments are frequently made to maintain a desired interest rate exposure. This can be complex and costly, especially in volatile market environments.¹⁵ Attempts to hedge against negative convexity risks, particularly in large portfolios of MBS, can themselves impact market rates, potentially leading to "convexity events" where hedging activity exacerbates rate movements.¹⁴ This self-reinforcing dynamic was a concern during the run-up to the 2008 financial crisis.

Furthermore, investors in securities with negative convexity receive a higher yield as compensation for bearing this risk. However, during periods of rapid interest rate changes, the additional yield might not fully offset the amplified price declines or capped price gains. This can lead to unexpected portfolio volatility and underperformance relative to positively convex benchmarks.¹³

Negative Convexity vs. Positive Convexity

The primary distinction between negative convexity and positive convexity lies in how a bond's price reacts to significant changes in interest rates.

Feature	Negative Convexity	Positive Convexity
Price Response (Rates Fall)	Price increases at a decreasing rate (capped upside).¹²	Price increases at an increasing rate (amplified upside).¹¹
Price Response (Rates Rise)	Price decreases at an increasing rate (amplified downside).¹⁰	Price decreases at a decreasing rate (limited downside).⁹
Yield Curve Shape	Concave	Convex (bowed outwards)
Typical Instruments	Callable bonds, Mortgage-backed securities (MBS)⁸	Most fixed-rate, non-callable bonds⁷
Embedded Options	Typically present (e.g., call option, prepayment option)⁶	Generally absent
Investor Preference	Generally less desirable, requires higher yield compensation⁵	Generally more desirable, provides a natural hedge⁴

While most standard bonds exhibit positive convexity, meaning their bond prices benefit more from falling rates than they suffer from rising rates, negative convexity presents the inverse, less favorable relationship. This difference is crucial for investors in fixed-income securities to understand when assessing risk and potential returns.

FAQs

What causes negative convexity?

Negative convexity is typically caused by embedded option features within a bond or security. For example, in a callable bond, the issuer has the right to buy back the bond from investors at a predetermined price. If interest rates fall, the issuer can call the bond and reissue new debt at a lower rate, limiting the original bond's price appreciation. In mortgage-backed securities (MBS), homeowners have the option to prepay their mortgages, which they often do when rates decline, causing a similar capping of MBS price gains.³

Why is negative convexity considered a disadvantage for investors?

Negative convexity is disadvantageous because it creates an asymmetrical return profile for the investor. When interest rates fall, the bond price increases, but at a slower rate than a bond without negative convexity. Conversely, when interest rates rise, the bond's price falls at a faster rate. This means investors get less upside when rates are favorable and more downside when rates are unfavorable, compared to a bond with positive convexity.²

How does negative convexity relate to prepayment risk?

Negative convexity is directly linked to prepayment risk in securities like mortgage-backed securities. Prepayment risk is the risk that borrowers will repay their loans earlier than expected. When interest rates drop, homeowners are more likely to refinance their mortgages, leading to higher prepayment rates for MBS investors. These accelerated principal payments reduce the effective duration of the MBS and limit its potential for price appreciation, which is the essence of negative convexity.¹