In the bond market, convexity refers to the relationship between price and yield. When graphed, this relationship is non-linear and forms a long-sloping U-shaped curve. A bond with a high-degree of convexity will experience relatively dramatic fluctuations when interest rates move.

  1. The magnitude of the price change in response to interest rate fluctuations depends on the bond’s duration and convexity.
  2. The term “convexity” refers to the higher sensitivity of the bond price to the changes in the interest rate.
  3. The price of bonds returning less than that rate will fall as there would be very little demand for them as bondholders will look to sell their existing bonds and opt for bonds with higher yields.
  4. Therefore, a bond with high convexity is more desirable than a bond with low convexity, all else being equal.

Most mortgage-backed securities (MBS) will have negative convexity because their yield is typically higher than traditional bonds. As a result, it would take a significant rise in yields to make an existing holder of an MBS have a lower yield, or less attractive, than the current market. 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. If rates rise by 1%, a bond or bond fund with a 5-year average duration would likely lose approximately 5% of its value. Conversely, when this figure is low, the debt instrument will show less movement to the change in interest rates.

Convexity in Bond Portfolio Management

Where P is the current price of the bond, c is the annual coupon rate, F is the face value of the bond, r is the annual yield on the bond, m is the number of coupon payments per year and n is the total years to maturity. For instance, say you want to calculate the modified Macaulay duration of a 10-year bond with a settlement date on Jan. 1, 2020, a maturity date on Jan. 1, 2030, an annual coupon rate of 5%, and an annual yield to maturity of 7%. The modified duration of a bond is an adjusted version of the Macaulay duration and both methods are used to calculate the changes in a bond’s duration and price for each percentage change in the yield to maturity.

Why Do Interest Rates and Bond Prices Move in Opposite Directions?

If a bond’s duration increases as yields increase, the bond is said to have negative convexity. The bond price will decline by a greater rate with a rise in yields than if yields had fallen. Therefore, if a bond has negative convexity, its duration would increase, and the price would fall. As convexity increases, the systemic risk to which the portfolio is exposed increases.

What is the relationship between duration and convexity?

As you can see, modified duration gives a better estimate of the new price than Macaulay duration, since it is closer to the price as determined by discounted cash flows. Of course, interest rates usually change in small steps, so duration measures interest rate sensitivity effectively. Bond convexity is a useful concept in estimating the change in bond prices in response to yield fluctuations.

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. A higher bond convexity indicates a stronger non-linear relationship between bond prices and interest convexity formula excel rates. It implies that larger changes in interest rates will have a more pronounced impact on bond prices. Convexity is important for bond investors and portfolio managers, because it helps them to assess the risk and return of different bonds.

For a fixed-income portfolio, as interest rates rise, the existing fixed-rate instruments are not as attractive. As convexity decreases, the exposure to market interest rates decreases, and the bond portfolio can be considered hedged. Typically, the higher the coupon rate or yield, the lower the convexity or market risk of a bond.

Negative and Positive Convexity

This is because a lower interest rate makes the future payments more valuable, and a higher interest rate makes them less valuable. Though they both decline as the maturity date approaches, the latter is simply a measure of the time during which the bondholder will receive coupon payments until the principal is paid. The higher a bond’s duration, the larger the change in its price when interest rates change and the greater its interest rate risk. If an investor believes that interest rates are going to rise, they should consider bonds with a lower duration. Where (P+) is the bond price when the interest rate is decremented, (P-) is the bond price when the interest rate is incremented, (Po) is the current bond price and the “change in Y” is the change in interest rate represented in decimal form.

There are several formulas for calculating the duration of specific bonds that are simpler than the above general formula. The articles and research support materials available on this site are educational and are not intended to be investment or tax advice. All such information is provided solely for convenience purposes only and all users thereof should be guided accordingly. The following video shows how I generate the Convexity function out of a flat yield input and a specified reference date. Now that we have talked about how to find the convexity of a bond let’s spend some time understanding how to interpret it.

Convexity is the curvature in the relationship between bond prices and interest rates. It reflects the rate at which the duration of a bond changes as interest rates change. It represents the expected percentage change in the price of a bond for a 1% change in interest rates.

This limitation is where convexity comes into play, as it accounts for the non-linear price sensitivity of bonds. This means the bond price will fall by a greater rate if rates rise than if they had fallen. Duration assumes the relationship between bond prices and interest rates is linear, while convexity incorporates other factors, producing a slope. When interest rates rise, the present value of a bond’s future cash flows decreases, resulting in a lower bond price.

The opposite is true of low convexity bonds, whose prices don’t fluctuate as much when interest rates change. When graphed on a two-dimensional plot, this relationship should generate a long-sloping U shape (hence, the term “convex”). Bond convexity is a measure of how the price of a bond changes when the interest rate changes.

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