Acquired convexity adjustment

What Is Acquired Convexity Adjustment?

Acquired Convexity Adjustment refers to the change in a bond's price sensitivity to interest rate movements due to the presence of embedded options. This concept is a crucial component of fixed-income analysis, particularly in understanding the behavior of securities like callable bonds, putable bonds, and mortgage-backed securities (MBS). Traditional measures of bond price sensitivity, such as duration, assume a linear relationship between bond prices and interest rates. However, for bonds with embedded options, this relationship is non-linear, and their cash flows can change as interest rates fluctuate, leading to what is measured by "effective convexity"³⁰, ³¹. The acquired convexity adjustment captures this curvature and the resulting impact on bond prices beyond what a simple duration measure would suggest.

History and Origin

The concept of convexity in bond valuation emerged from the recognition that a bond's price-yield relationship is not linear. While duration provides a first-order approximation of this relationship, it becomes less accurate for larger changes in interest rates²⁸, ²⁹. The need to account for the "curvature" in this relationship led to the development of convexity as a second-order measure. For bonds with embedded options, such as those that allow the issuer to call the bond or the investor to put it, the potential exercise of these options introduces further complexities. These options affect the bond's expected cash flows and, consequently, its price behavior as interest rates change.

The formalization of "effective convexity" as a measure that incorporates these changes in cash flows due to embedded options became essential in the valuation of such complex fixed-income securities²⁶, ²⁷. For example, early academic work highlighted how option-valuation techniques could be used to derive an option-adjusted yield, duration, and convexity for callable bonds, providing a more accurate picture of their interest rate risk²⁵. This analytical framework evolved as the market for bonds with embedded options, particularly mortgage-backed securities, grew in sophistication, requiring more advanced tools to assess their true interest rate sensitivity and prepayment risk²³, ²⁴.

Key Takeaways

• Acquired Convexity Adjustment accounts for the non-linear relationship between bond prices and interest rates, especially for bonds with embedded options.
• It is often synonymous with or measured by "effective convexity," which explicitly considers how embedded options alter a bond's cash flows under different interest rate scenarios.
• The adjustment is critical for accurately assessing the interest rate risk of securities like callable bonds, putable bonds, and mortgage-backed securities.
• Bonds with positive acquired convexity adjustments benefit more from falling interest rates than they lose from rising rates of the same magnitude.
• Conversely, negative acquired convexity adjustments, often seen in callable bonds or mortgage-backed securities, indicate that price appreciation is capped while downside risk remains substantial.

Formula and Calculation

The acquired convexity adjustment is quantitatively captured by the effective convexity formula. This formula approximates the second derivative of the bond's value with respect to changes in the benchmark yield curve, taking into account the impact of embedded options on future cash flows.

The effective convexity is calculated numerically, typically using a binomial or Monte Carlo interest rate model to determine bond prices at slightly shifted interest rate levels. The general formula for effective convexity is:

Where:

• (PV_{-}) = Price of the bond if the benchmark yield curve decreases by (\Delta \text{Curve})
• (PV_{+}) = Price of the bond if the benchmark yield curve increases by (\Delta \text{Curve})
• (PV_{0}) = Original price of the bond
• (\Delta \text{Curve}) = Shift in the benchmark yield curve (e.g., 1 basis point or 0.0001)

This calculation differs from traditional convexity measures (like modified convexity) because it explicitly considers how changes in interest rates can trigger the exercise of embedded options, thereby altering the bond's expected cash flows²¹, ²².

Interpreting the Acquired Convexity Adjustment

The interpretation of the acquired convexity adjustment, through effective convexity, provides crucial insights into a bond's price behavior in response to interest rates. A positive effective convexity means that as interest rates fall, the bond price increases at an accelerating rate, and as rates rise, the bond price decreases at a decelerating rate²⁰. This asymmetrical response is generally favorable for investors, as the upside potential outweighs the downside risk for a given magnitude of interest rate change.

Conversely, a negative effective convexity, often observed in callable bonds or mortgage-backed securities, signifies that the bond's price appreciation is limited when interest rates fall, while its price can decline more sharply when interest rates rise¹⁹. For example, a callable bond becomes more likely to be called by the issuer when rates decline, capping its price upside. In contrast, if rates rise, the call option becomes less valuable, and the bond behaves more like a straight bond, losing value. Understanding this acquired convexity adjustment is vital for accurately assessing the true interest rate risk and potential return of fixed-income securities with embedded options.

Hypothetical Example

Consider a hypothetical callable bond issued by Corp X with a par value of $1,000, a 5% coupon rate, and 10 years to maturity. The bond is callable at par after 5 years.

1. Current Scenario: Assume current interest rates mean the bond is trading at its par value of $1,000. Its effective duration is 6 years.
2. Interest Rates Decrease: If interest rates fall by 1%, a non-callable bond might see its bond prices rise by approximately 6% (based on duration). However, due to the call option, the price of the callable bond may only increase by, say, 3.5% because the issuer is more likely to exercise the embedded options to refinance at a lower rate. This demonstrates a limited upside, which is a characteristic of negative acquired convexity adjustment.
3. Interest Rates Increase: If interest rates rise by 1%, both the callable bond and a comparable non-callable bond would see their prices fall. The callable bond's price might decrease by, say, 5.5%. In this scenario, the call option is "out of the money" (unlikely to be exercised), and the bond behaves more like a straight bond.

The acquired convexity adjustment highlights that the price increase when rates fall (3.5%) is less than the price decrease when rates rise (5.5%), even for an equivalent magnitude of rate change, implying negative convexity. This contrasts with a non-callable bond that would exhibit positive convexity, where price gains from falling rates would exceed losses from rising rates. This example illustrates how the acquired convexity adjustment impacts an investor's potential bond prices movements.

Practical Applications

The acquired convexity adjustment is indispensable in the valuation and risk management of various complex fixed-income securities. It is particularly relevant for:

• Mortgage-Backed Securities (MBS): MBS are prone to prepayment risk, where homeowners refinance their mortgages when interest rates fall¹⁷, ¹⁸. This prepayment option means MBS typically exhibit negative effective convexity. Understanding this adjustment helps investors price MBS accurately and assess their vulnerability to interest rate risk. The Federal Reserve Bank of San Francisco provides insights into understanding MBS, highlighting their unique risks compared to traditional bonds¹⁶.
• Callable Bonds: As discussed, callable bonds give the issuer the right to redeem the bond before maturity¹⁵. This embedded option limits the bond's price appreciation when interest rates decline, leading to negative effective convexity in certain interest rate environments¹⁴. Financial professionals analyze the acquired convexity adjustment to determine a fair market price and gauge the potential for call risk.
• Portfolio Management: Portfolio managers use acquired convexity adjustment alongside duration to manage interest rate risk across a portfolio of fixed-income securities. By analyzing the convexity profile of different bonds, they can construct portfolios that are either more or less sensitive to large interest rate movements, aligning with their investment objectives and views on the yield curve. For instance, a portfolio aiming to benefit from falling rates might seek bonds with higher positive convexity.

Limitations and Criticisms

While the acquired convexity adjustment, as measured by effective convexity, provides a more accurate assessment of interest rate risk for bonds with embedded options than duration alone, it does have limitations:

• Model Dependence: The calculation of effective convexity relies heavily on sophisticated valuation models, such as binomial trees or Monte Carlo simulations, to project future cash flows under various interest rate scenarios¹³. The accuracy of the acquired convexity adjustment is therefore dependent on the assumptions built into these models, including interest rate volatility and prepayment behavior assumptions for MBS. If these assumptions are flawed, the resulting convexity measure may be inaccurate¹².
• Complexity: The calculation is more complex than traditional convexity measures and requires specialized software and expertise. This can make it less accessible for individual investors or those without advanced financial modeling capabilities.
• Static Measure: Like duration, effective convexity is a static measure calculated at a specific point in time. While it accounts for changes in cash flows, the actual acquired convexity adjustment of a bond can change dynamically as market conditions, such as interest rates and volatility, evolve¹¹. A callable bond might exhibit positive convexity at very high interest rates and negative convexity when rates are low and the call option is "in the money"¹⁰.

Acquired Convexity Adjustment vs. Option-Adjusted Spread (OAS)

While both Acquired Convexity Adjustment (measured by effective convexity) and Option-Adjusted Spread (OAS) are vital tools in fixed income analysis, particularly for securities with embedded options, they serve different primary purposes.

The acquired convexity adjustment tells an investor how a bond's price will move in a non-linear fashion as rates change, while the Option-Adjusted Spread (OAS) quantifies how much extra yield an investor receives for holding a bond with embedded options, effectively stripping out the option's cost⁸. OAS is crucial for comparing the relative value of different fixed-income securities with varying embedded options, as it standardizes the yield comparison.

FAQs

What is the primary difference between duration and acquired convexity adjustment?

Duration provides a linear estimate of a bond's price change for a small shift in interest rates. The acquired convexity adjustment (or effective convexity) measures the curvature of this relationship, showing how the bond's duration itself changes as interest rates fluctuate, especially for bonds with embedded options where cash flows are uncertain.⁶, ⁷

Why do callable bonds often have negative acquired convexity?

Callable bonds grant the issuer the right to buy back the bond. When interest rates fall significantly, the issuer is likely to call the bond to refinance at a lower rate, limiting the bond's potential price appreciation. This capping of upside, while retaining downside risk, leads to negative acquired convexity.⁴, ⁵

Is acquired convexity adjustment important for all types of bonds?

No, the acquired convexity adjustment is most critical for fixed-income securities that have embedded options, such as callable bonds, putable bonds, and mortgage-backed securities (MBS). For plain vanilla bonds without such features, standard convexity measures are typically sufficient.², ³

How does prepayment risk relate to acquired convexity adjustment in MBS?

Prepayment risk, the risk that borrowers will pay off their mortgages early, is a key driver of the negative acquired convexity often seen in mortgage-backed securities. When interest rates fall, homeowners are more likely to refinance, which reduces the expected cash flows to MBS investors and limits the security's price appreciation, thus contributing to negative convexity.¹

Can a bond's acquired convexity adjustment change over time?

Yes, a bond's acquired convexity adjustment can change as market conditions evolve. Factors like the level of interest rates, the proximity of an embedded option to being "in the money" (meaning it's likely to be exercised), and overall interest rate volatility can all influence a bond's effective convexity.