Negative Convexity: Definition, Example, Simplified Formula


By incorporating convexity into their analysis, investors can better estimate the potential impact of interest rate changes on their bond portfolios and make more informed investment decisions. Bank liabilities, which are primarily the deposits owed to customers, are generally short-term in nature, with low duration statistics. By contrast, a bank’s assets mainly comprise outstanding commercial and consumer loans or mortgages. These assets tend to be of longer duration, and their values are more sensitive to interest rate fluctuations. In periods when interest rates spike unexpectedly, banks may suffer drastic decreases in net worth, if their assets drop further in value than their liabilities.

Therefore, convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates. If market rates rise by 1%, a one-year maturity bond price should decline by an equal 1%. As a general rule of thumb, if rates rise by 1%, bond prices fall by 1% for each year of maturity. In other words, the percentage increase in the price of a bond for a defined convexity risk decrease in rates or yields is always more than the decline in the bond price for the same increase in rates or yields. Several factors influence the convexity of a bond, including its coupon rate, duration, maturity, and current price. Bond prices respond to a number of factors, including credit risk, market risk, and maturity date, but no factor affects bond prices as much as interest rates.

  1. After a while, if your bond is experiencing negative convexity, it also slows down/loses value.
  2. Most mortgage bonds are negatively convex, and callable bonds usually exhibit negative convexity at lower yields.
  3. This limitation is where convexity comes into play, as it accounts for the non-linear price sensitivity of bonds.
  4. In the example figure shown below, Bond A has a higher convexity than Bond B, which indicates that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall.

As one might imagine, as UST rates have fallen precipitously over the past few years, prepayments have accelerated on high-coupon mortgages. shows the 30-year fixed average rate at 3.64% today, down from more than 5% in 2009. The main difference between effective convexity and effective duration is the fact that effective duration measures the linear effects of interest rate changes, while effective convexity measures the non-linear effects. Keep in mind that bond convexity is a tool, not a precise predictor of bond price movements. Other factors like market liquidity, credit risk, and supply and demand dynamics also influence bond prices.

Several factors can affect bond prices, such as market liquidity, credit risk, and changes in interest rates. Additionally, bond convexity assumes a constant yield curve and small changes in yield, which may not hold true in all market conditions. Therefore, while bond convexity provides valuable insights, it should be used as a tool alongside other fundamental and technical analyses to assess the potential impact on bond prices. It’s always recommended to consult with a financial professional or conduct thorough research before making any investment decisions based on bond convexity calculations. With coupon bonds, investors rely on a metric known as duration to measure a bond’s price sensitivity to changes in interest rates. Because a coupon bond makes a series of payments over its lifetime, fixed-income investors need ways to measure the average maturity of a bond’s promised cash flow, to serve as a summary statistic of the bond’s effective maturity.

However, using the concept of convexity, we can predict that the price change for Bond B will be less than expected based on its duration alone. This is because Bond B has a longer maturity, which means it has a higher convexity. The higher convexity of Bond B acts as a buffer against changes in interest rates, resulting in a relatively smaller price change than expected based on its duration alone.

Duration for Gap Management

As shown by the example above, the price of the option-free bond dropped by $10. Bonds with a longer maturity rate are more susceptible to changing interest rates. If a 20-year bond has a yield of 4%, it would lose value if the interest rate rises to 5%. Thus, the 4% yield bond will need to have a lower price to give investors a reason to buy it. In the example figure shown below, Bond A has a higher convexity than Bond B, which indicates that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall. As the table illustrates, the duration of the 30-year 3.5% pool rises from about 5.3% in a flat rate scenario to about 8.7% in a rate that is up 150 bps.

What Is Convexity in Bonds?

While the rate increase was similar in magnitude to the June 2003 sell-off, for example, it wasmore gradual and therefore inconsistent with a sell-off driven primarily by convexity hedging. When interest rates in the overall market increase, the price of bonds will drop. If investors who hold a bond with a 5% yield can suddenly get a 7% yield, all else equal, somewhere else, they will sell the 5% bond and buy the 7% bond. For reference, with a $50-per-year yield, the bond price would need to be $714.

In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity. Banks employ gap management to equate the durations of assets and liabilities, effectively immunizing their overall position from interest rate movements. Therefore, if their durations are also equal, any change in interest rates will affect the value of assets and liabilities to the same degree, and interest rate changes would consequently have little or no final effect on net worth. Therefore, net worth immunization requires a portfolio duration, or gap, of zero.

Convexity Comparison Across Bonds

Most regular bonds, such as fixed-rate bonds and zero-coupon bonds, exhibit positive convexity. Modified duration, on the other hand, is an adjusted version of Macaulay duration that directly measures a bond’s price sensitivity to changes in interest rates. It represents the weighted average time until a bond’s cash flows are received, taking into consideration the present value of each cash flow.

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If the bond has embedded options, such as calls or puts, it will affect some of the above relationships, sometimes dramatically. Because every bond has a unique structure and issuer, it is impossible to dole out advice on the exact relationships. But because call and put options generally affect maturity you can make informed guesses as to the affect on convexity.

A bond’s price is determined by the present value of its future cash flows, which include periodic coupon payments and the principal repayment at maturity. In simple terms, convexity provides a more accurate measure of a bond’s price sensitivity to changes in interest rates than duration alone. An investor with a greater risk tolerance can purchase a bond with a high estimated percentage price change while a risk-averse investor can choose one with lower duration and convexity. Lastly, embedded options react to interest rates differently depending on the option.

These options can cause bond prices to be more sensitive to interest rate changes in one direction than the other, leading to asymmetric price responses to rate fluctuations. The convexity of coupon-paying bonds varies depending on factors such as the coupon rate, time to maturity, and yield to maturity. Convexity-adjusted duration combines duration and convexity to accurately measure a bond’s price sensitivity to interest rate changes.

In derivative pricing, this is referred to as Gamma (Γ), one of the Greeks. In practice the most significant of these is bond convexity, the second derivative of bond price with respect to interest rates. By understanding the impact of interest rates, investors can make more knowledgable decisions on the purchase of fixed income securities.

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