# A Bond Convexity Primer

by: CFA Institute Contributors

by Jason Voss, CFA

As the U.S. Federal Reserve lays the verbal groundwork for an eventual real-world quantitative easing (QE) taper, bond prices are dropping at an accelerated rate. In order to understand the ramifications of a Federal Reserve taper on the prices of a bond or bond portfolio, what is needed is a bond convexity primer.

In the parlance of those who know calculus, convexity is the second derivative. For the layperson, this is known as the rate of change in change. For convexity to make better sense, let me compare it to driving a car. When you are driving a car, your speed is the rate of change in the car's location. Want to change your speed (i.e., the rate of change)? Then you either give the car more gas with the accelerator or press down on the brakes to slow the car down. Speeding up and slowing down are the second derivative. Speeding up means that there is a positive second derivative, while slowing down means that there is a negative second derivative.

Related to the bond market, the speed of your car is called duration, while the speeding up / slowing down is known as convexity. The higher the convexity, the more dramatic the change in price given a move in interest rates. Whatever you call it, after a while, if you keep braking a car it stops. After a while, if your bond is experiencing negative convexity, it also slows down / loses value. The harder the acceleration or braking, the greater the change in your speed.

So why is the relationship between a bond's yield and its price known as convexity? As yields change, the change in the price of the bond is not linear; it is curved in a convex fashion. To understand convexity more directly, take a look at the following three graphs, all for a \$1,000 par value bond, with a coupon rate of 3.452%, making payments twice per year, and with zero expectation of a yield change in the future. The only thing making these three bonds different is the number of years until maturity — 30 years, 10 years, and 1 year.

Capital Gains at Indicated Interest Rate, 30-Year Maturity

Above is the bond with a 30-year maturity. Look at how curved - i.e., how convex - the graph of the price-yield relationship is! Notice also that there are no capital gains / changes in price at the exact yield of the bond, 3.42%, where the line actually touches the horizontal axis. This means that if yields stay the same as the coupon rate there should be no change in the price of the bond.

Capital Gains at Indicated Interest Rate, 10-Year Maturity

A 10-year maturity bond is graphed above. Look at how much less convex the line is relating price to yield. Again, note that the line touches the horizontal axis at 3.42%.

Capital Gains at Indicated Interest Rate, One-Year Maturity

Last, look at this one-year maturity version of a bond. There is zero convexity / curve, just a flat line. What all of this means is that a bond's price is sensitive to the length of its maturity. But it is also sensitive to other factors. I have created a spreadsheet that aptly demonstrates convexity for a bond under different scenarios so that you can experiment with different combinations of factors to show you the effects of convexity on a bond's price performance. Separately, I have also attached a spreadsheet from our fixed-income valuation course that allows you to calculate duration and convexity in a numerical format if you prefer to see it quantitatively. If you are not feeling experimental, here is a summary of the factors affecting bond convexity:

• Maturity: Positive correlation; the longer the maturity the greater the convexity/price sensitivity to yield changes.
• Coupon: Negative correlation; the higher the coupon the lower the convexity/price sensitivity to yield changes.
• Yield: Negative correlation; the lower the yield the higher the convexity/price sensitivity to yield changes. To best understand this, look at the graph above for the 30-year bond. The lower the yield goes the higher the convexity/price sensitivity as compared with the higher yield portion of the curve.

I have kept things simple here. 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 effect on convexity.

Another factor often not discussed in the price performance of bonds and bond portfolios is how rapidly changes in interest rates occur. The further into the future and the smaller the interest rate changes, the less damage done to a bond or portfolio today.

So, in an environment of central bank tapering, investors want low convexity bonds and bond portfolios. Such a portfolio, as rare as a golden goose, would have a short maturity, high coupon rate, and a high yield. Good luck finding that!

Disclaimer: Please note that the content of this site should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute.