# What Is Convexity in Bonds?

DEFINITION
Convexity in bonds is a way to measure the bond price’s sensitivity to changes in interest rates.

Convexity in bonds is a way to measure the bond price’s sensitivity to changes in interest rates. Bonds with higher convexity are generally considered better investments in markets where interest rates are expected to rise, and lower convexity is better suited for when rates are likely to remain unchanged or fall.

To understand convexity in bonds and make use of it in your own investing, you’ll first need to understand the relationship between bond prices and interest rates, and bond duration. We’ll go over each of these terms and explain how they work.

## Definition and Example of Convexity in Bonds

Convexity in bonds measures how sensitive the bond’s duration is to changes in interest rates. The higher the convexity, the less the bond price will increase when rates fall—and the less the bond price will drop when rates rise.

First, let’s go over the relationship between bond prices and interest rates and explain how bond duration works.

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. Investors buy bonds to receive the yield, called a coupon payment, that comes with the bond. Each bond’s yield is calculated by dividing the annual coupon payments by the bond price. If a bond has a current face value of \$1,000 and pays out \$50 per bond per year, its yield is 5%.

When interest rates in the overall market increase, the price of bonds will drop. This should make intuitive sense. 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. This will happen until bond prices fall enough to make the yields equal. For reference, with a \$50-per-year yield, the bond price would need to be \$714.

Of course, in the market, all else isn’t equal. The other three factors (credit, market, and term) also factor into the bond price. That’s where duration comes in.

Duration is a measure of how much changes in interest rates affect the bond price. It is the number of years it takes investors to get their investment back.

If a bond has a duration of three years, that means every change in interest rates of 1% will cause the bond price to move by 3%. If interest rates fall by 1%, the bond price will increase by 3%. The problem with duration is that the relationship between bond prices and interest rates is not linear, it is convex. The duration, not just the price, will change as interest rates change. That’s why we use convexity.

Convexity measures how sensitive the bond’s duration is to changes in interest rates. A bond with positive convexity has a higher duration as its price decreases and, vice versa, a bond with negative convexity has a duration that changes in line with the price of the bond.

Look at two bonds with similar yields: one has a higher duration and convexity than the other. You expect interest rates to rise in the near term. Using duration, you may be tempted to buy the bond with the lower number because it will fall less when rates rise. However, if the rate moves upward is strong, it is likely that the bond with higher convexity will weather the storm better. By the end of the rate move, the bond with a lower duration could have a much higher duration because its price curve is not as convex.

## How Convexity in Bonds Works

Unfortunately for bond investors, calculated numbers like duration and convexity aren’t easy to come by. Professional fund managers use services like Bloomberg to look up this information and you could technically calculate it on your own in Excel, but the best bet is to find a broker’s bond calculator to use.

Taking the time to learn the formula and modify and apply it in Excel probably isn’t worth the time it would take to constantly update, and if your broker doesn’t have any sort of bond calculator, its fixed-income offerings may be lacking in general.

If you’re worried about interest rates changing in the near future, bonds with higher convexity will likely do better in either direction.

Convexity is also very useful on a portfolio level. When managing a portfolio of bonds, you can use duration and convexity to determine allocation among bond positions and new buys based on the duration and convexity of the portfolio. Use the same interest-rate forecast criterion, but apply it based on how a new position, and its allocation, would affect the portfolio as a whole.

This way you can still purchase bonds you like, even if their duration or convexity conflicts with your interest-rate forecast by using position sizes and diversity to limit risk.

## What It Means for Individual Investors

The first step is to determine what your time frame is. If you’re planning to hold bonds to maturity no matter what, and you have the existing liquidity to pull that off, duration and convexity are irrelevant. Interim prices don’t matter if the plan is to hold through maturity.

Duration is more relevant for short-term holdings. It can help you figure out what will happen as a result of small changes in interest rates over the next year or so.

Perhaps you’re parking some cash in a fixed income account to hold for a child’s college tuition in a few years. Convexity is a better metric if you’re vulnerable to large interest-rate changes over the medium term. You don’t want to hold the bonds through maturity necessarily, but you don’t need the cash immediately. In this instance, convexity will let you plan your position around the bonds’ interest-rate forecast, as discussed above.

### Key Takeaways

• Convexity is a measure of the duration of a bond’s sensitivity to interest rates.
• The higher the convexity, the more likely the bond’s price won’t be affected as much by changes in interest rates.
• Duration measures a bond price’s sensitivity to changes in interest rates; the longer the duration, the more sensitive the bond price is to interest-rate fluctuations.