Negative Convexity Bonds

In finance and investments, bonds are one of the most common types of securities. They represent a debt obligation in which the bond issuer borrows capital from the bondholder and is obligated to repay the principal at a later date, typically with periodic interest payments. Among the various attributes of bonds, “convexity” is a crucial concept. Convexity measures the sensitivity of the duration of a bond to changes in interest rates and helps investors understand how the price of a bond will change as interest rates change.

Negative convexity is a unique attribute in the context of fixed income securities, particularly mortgage-backed securities (MBS) and callable bonds. In this article, we will delve into the concept of negative convexity, its causes, implications, and the impact it has on bond pricing and investment strategies.

Understanding Convexity

Convexity is a measure of the curvature or the degree of the curve in the relationship between the bond price and interest rates. It is a second-order derivative that stands for the rate of change of duration relative to changes in yield.

Positive convexity means that as yields decline, the price increases at an increasing rate, and as yields rise, the price declines at a decreasing rate. This characteristic is typical for most traditional bonds. The higher the convexity, the more sensitive the bond is to changes in interest rates.

Formula for Convexity

The formula for calculating convexity is as follows:

[ \text{Convexity} = \frac{1}{P \times (1 + y)^2} \sum_{t=1}^T \left( \frac{C_t \times (t^2 + t)}{(1 + y)^t} \right) ]

Where:

The convexity measure helps in understanding the bond’s sensitivity to interest changes, on top of what duration indicates.

Negative Convexity Explained

Negative convexity is observed in certain types of bonds like callable bonds and mortgage-backed securities. These bonds exhibit an unusual price-yield relationship compared to bonds with positive convexity.

When a bond has negative convexity, its price decreases more rapidly when interest rates rise and increases less quickly when interest rates fall. This means that as interest rates decline, the bondholder’s potential for price gains is limited, whereas losses can be substantial if interest rates rise. This non-linear relationship can create a challenging situation for investors trying to manage interest rate risk.

Causes of Negative Convexity

Negative convexity mainly arises from two primary bond features: call options and prepayment options.

  1. Callable Bonds: Callable bonds come with an embedded call option, allowing the issuer to repurchase the bond before its maturity date, often at a premium. These bonds tend to show negative convexity because when interest rates drop significantly, issuers are more likely to call the bonds to refinance at a lower rate. Investors, thus, face reinvestment risk as they might have to reinvest the principal at lower interest rates.

  2. Mortgage-Backed Securities (MBS): Mortgage-backed securities are backed by mortgage loans with embedded prepayment options, allowing homeowners to prepay their mortgages. When interest rates fall, mortgage holders are more inclined to refinance or prepay their mortgages, leading to an increase in prepayments for the MBS. The sudden influx of principal repayments when interest rates are low forces MBS investors to reinvest at lower yields, leading to negative convexity.

Example of Negative Convexity

Consider a callable bond with a face value of $1,000, paying a 5% coupon, and with a callable provision at par. When market interest rates fall to 3%, the price of a comparable non-callable bond might increase significantly above $1,000, but the callable bond’s price may increase very slightly, as investors anticipate a call. Conversely, if interest rates rise to 7%, the price of the callable bond would fall sharply, similar to a non-callable bond.

Implications of Negative Convexity

Negative convexity has several implications for bond investors and portfolio managers:

  1. Price Sensitivity: Bonds with negative convexity are more sensitive to interest rate increases and less beneficial during interest rate declines compared to bonds with positive convexity.
  2. Interest Rate Risk Management: Managing a portfolio with bonds exhibiting negative convexity requires careful consideration of interest rate movements and potential price changes.
  3. Reinvestment Risk: Investors are at higher risk of reinvestment at lower rates when prepayments increase, particularly for MBS and callable bonds.
  4. Yield Enhancement: Due to the additional risk, these bonds often offer higher yields to compensate investors for the potential disadvantages.
  5. Behavioral Adjustments: Negative convexity can cause adjustments in investment behavior, steering investors towards other high-yield or positively convex instruments during times of falling interest rates.

Strategies for Managing Negative Convexity

Given the complexities and risks associated with negative convexity, several strategies can be employed to manage and mitigate its impacts:

  1. Diversification: Diversifying a bond portfolio across various durations, convexitives, and yield curves can help manage the overall risk.
  2. Derivatives: Using interest rate derivatives like swaps, options, or futures can hedge against the interest rate risk posed by negatively convex bonds.
  3. Selecting Intervals: Focusing on bonds with shorter call protection periods or prepayment lockouts can reduce the reinvestment risk.
  4. Adjusting Duration: Matching the portfolio duration with an investor’s investment horizon can minimize the mismatch during interest rate changes.
  5. Active Management: Active portfolio management allows investors to respond promptly to interest rate changes, adjusting their holdings to optimize the risk-reward balance.

Example of a Negative Convexity Bond Issuer

Freddie Mac and Fannie Mae are prominent issuers of mortgage-backed securities in the United States. These institutions create bonds backed by mortgage loans, leading to negative convexity due to the prepayment risks associated with mortgages.

Freddie Mac (Federal Home Loan Mortgage Corporation)

Freddie Mac is a public government-sponsored enterprise (GSE), which provides liquidity, stability, and affordability to the U.S. housing market by purchasing mortgages from lenders, packaging them into securities, and selling them to investors.

Link: Freddie Mac

Fannie Mae (Federal National Mortgage Association)

Fannie Mae operates in a similar manner to Freddie Mac, ensuring an affordable homeownership for Americans by buying and insuring mortgage loans.

Link: Fannie Mae

Conclusion

Negative convexity is a crucial concept for investors, particularly those involved in the fixed income market. Understanding the nature of this attribute, its causes, and its implications can aid in better managing the risks and ensuring more informed investment decisions. Callable bonds and mortgage-backed securities, due to their intrinsic features, exhibit this behavior and require specific strategies to handle the associated risks effectively. Knowledge and active management are paramount in navigating the challenges posed by bonds with negative convexity.