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What Is Convexity?

Convexity is a measure of the curvature, or the degree of the curve, in the relationship between bond prices and bond yields. Convexity demonstrates how the duration of a bond changes as the interest rate changes. Portfolio managers will use convexity as a risk-management tool, to measure and manage the portfolio’s exposure to interest rate risk.

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Key Takeaways

  • Convexity is a risk-management tool, used to measure and manage a portfolio’s exposure to market risk.
  • Convexity is a measure of the curvature in the relationship between bond prices and bond yields.
  • Convexity demonstrates how the duration of a bond changes as the interest rate changes.
  • If a bond’s duration increases as yields increase, the bond is said to have negative convexity.
  • If a bond’s duration rises and yields fall, the bond is said to have positive convexity.



Convexity Explained

Before explaining convexity, it’s important to know how bond prices and market interest rates relate to one another. As interest rates fall, bond prices rise. Conversely, rising market interest rates lead to falling bond prices. This opposite reaction is because as rates rise, the bond may fall behind in the earnings they may offer a potential investor in comparison to other securities.

In the example figure shown above, 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.

The bond yield is the earnings or returns an investor can expect to make by buying an holding that particular security. The price of the bond depends on several characteristics including the market interest rate and can change regularly.

How Market Interest Rates and Bond Yields Relate

As the market rates rise, new bonds coming on the market also have rising yields as they are issued at the new, higher rates. Also, as rates increase, investors demand a higher yield from the bonds they buy. Investors don’t want a fixed-rate bond at current yields if they expect interest rates to rise in the future. As a result, when interest rates increase, the issuer of these debt vehicles must also raise their yields to remain competitive. However, as the interest rate climbs the price of bonds returning less than that rate will fall.

How Interest Rates and Bond Prices Relate

If an investor owns a fixed-rate bond that pays 2% and interest rates begin to rise above 2%, they may want to sell this lower paying security. The reason for the sell-off is that their existing rate is less attractive than the current market. Investors don’t want to hold a bond that pays 2% if they can invest the same principle into one that pays a higher rate in the future. In a rising rate market, bondholders look to sell their existing bonds and opt for newly-issued bonds paying higher yields.

Because there is a glut of bonds at the lower rate on the market, the prices for these debt holdings will drop. Also, as bonds sell off and the price falls, the investor may wait for rates to stop rising before getting back in the bond market by buying the higher-yielding security. As a result, bond prices and yields move in the opposite or inverse direction.

Bond Duration

Bond duration measures the change in a bond’s price when interest rates fluctuate. If the duration of a bond is high, it means the bond’s price will move to a greater degree in the opposite direction of interest rates. Conversely, when this figure is low the debt instrument will show less movement.

Typically, if market rates rise by 1%, a one-year maturity bond price should decline by an equal 1%. However, for bonds with long-dated maturities, the reaction increases. In other words, if rates rise by 1%, bond prices fall by 1% for each year of maturity. For example, if rates rise by 1%, the two-year bond price would fall 2%, the three-year bond price by 3%, and the 10-year price by 10%.

Convexity and Risk

Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change. Convexity is a better measure of interest rate risk, concerning bond duration. Where duration assumes that interest rates and bond prices have a linear relationship, convexity allows for other factors and produces a slope.

Duration can be a good measure of how bond prices may be affected due to small and sudden fluctuations in interest rates. However, the relationship between bond prices and yields is typically more sloped, or convex. Therefore, convexity is a better measure for assessing the impact on