Noncallable Bonds Provide the Joy of Convexity

Last week, we discussed duration. This week, we're talking convexity.

If you didn't read the duration piece, please go back and do so. I'm going to rely heavily on it in this piece.

Last week, we talked about how you can use duration to estimate how much the price of a bond will change if its yield changes. In the example we used, a bond with a duration of 6.4 years will go up about 6.4% in price if its yield drops by 1% (100 basis points), and down about 6.4% in price if its yield rises by 100 basis points.

If duration describes the approximate relationship between price and yield, convexity describes the actual relationship between price and yield.

Take a look at this graph. For now, just concentrate on the red line, which describes the relationship between price and yield for a standard, noncallable bond. (We'll get to callable bonds in a little bit.)

As you can see, as the bond's yield falls, its price increases, and vice versa. But what's important here is the shape of the line. It's curved, or convex -- not straight. That's because as the yield falls, the price increases at an ever-faster rate. As the yield rises, the price falls at an ever-slower rate.

That's due to something else we talked about last week: As a bond's yield falls, its duration lengthens, indicating increased price sensitivity. So as the yield falls further, the price will rise at a faster rate. Likewise, as a bond's yield rises, its duration shortens, making it less sensitive to additional increases in yield.

That's the aspect of the relationship between price and yield that the duration generalization above (a 6.4 year duration means a 6.4% price change if the yield changes by 100 basis points) doesn't capture. Imagine in the graph above a third line describing the relationship between price and yield predicted by the duration generalization. It wouldn't be curved, like the convexity line, which is derived from duration. It would be straight. It would touch the red line at the point where price equals 100 and yield equals 8%. But as it extended to the left, it wouldn't rise as much as the red line, and as it extended to the right, it would decline more than the red line. Hence, for a noncallable bond, duration underpredicts the amount by which the price will rise as the yield falls, and overpredicts the amount by which the price will fall as yields rise. Moreover, the longer the duration of a bond, the greater the margin of error.

Now, everything we've said so far applies only to standard, noncallable bonds -- the investor gets the face value of the bond back on its maturity date, earning interest along the way. However, some Treasury bonds, many corporate bonds and most municipal bonds are callable. If a bond is callable, the issuer has the option to return principal before maturity, within certain guidelines. The bond specifies a certain date after which the issuer can call it, and it specifies the price of the call. In the muni market, for example, it's typical for bonds to become callable in 10 years at a price of 101 or 102, meaning the investor would get $1,010 or $1,020 for every $1,000 of face value. Over time, the call price typically declines to par.

Issuers like to issue callable bonds because they can cut costs if rates decline. They can call the old bonds and issue new ones with smaller coupons.

But obviously, what's good for the issuer is not good for the investor. A bond only gets called when interest rates are lower than they were when the bond was issued. So if your bond gets called, you have to reinvest at a lower yield.

Because as an investor you'd much rather have a noncallable bond, callable bonds are cheaper (their yields are higher) than noncallables, all else being equal. That's why people buy them.

What you are giving up when you buy a callable bond is convexity. Have another look at the graph. The green line describes the relationship between price and yield for a bond that is callable at par at any time. As you can see, as the yield drops, the price rises. But instead of rising at a quickening pace, the pace slows, giving the line what bond traders call negative convexity. Its duration, instead of lengthening, is shortening. That's because as the yield drops, it becomes increasingly likely that the issuer will call the bond. As a result, an investor would not pay more than the call price for it. As a general rule, the price of a callable bond won't rise above its call price.

At the same time, you can see that when interest rates are rising, it doesn't matter nearly as much whether you're holding a callable or a noncallable bond. They'll lose their value at much more similar rates. That's because, like noncallables, callables see their durations shorten as rates rise.

Look for a discussion of practical uses of duration and convexity in next week's column.

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