A con-vexing Plain English challenge

Bond fund communications inevitably involve discussions of interest rate risk and – to varying degrees – the accompanying concepts of maturity (easy to understand), duration (not quite as easy) and convexity (downright difficult). Here’s an attempt to put the relationship between these concepts into, if not Plain English, at least a somewhat digestible distillation.

Maturity – If you are reading this, you already know that prices of fixed-income instruments generally move in the opposite direction of market interest rate levels – and that the length of time until a bond is scheduled to mature and repay principal is often used to provide a first, rough indication of its interest rate sensitivity. All else being equal, longer maturities mean greater exposure to changes in rates.

Duration – Of course, bondholders generally don’t have to wait until maturity to receive some form of payment. The concept of duration reflects this and is used to provide a more meaningful expression of expected price movements in response to changing interest rates.

While the actual calculation is somewhat complex, a bond’s duration is based on the dollar-weighted average time to receipt of payments of principal and interest. Since ongoing coupon payments are part of the equation, a bond’s duration will always be lower than its maturity (excepting zero coupon bonds, for which duration and maturity are the same). With respect to two bonds with the same maturity, the bond with a higher yield-to-maturity will have a lower duration. With respect to two bonds with the same yield, the bond nearer to maturity will have a lower duration. The higher a bond’s duration, the higher its interest rate sensitivity. The rule of thumb is that for every unit or year of duration, a bond is expected to decline about 1% for every one percentage point (100 basis point) rise in interest rates. For example, the price of a bond with a duration of 5 years would be expected to fall 10% if rates rose two percentage points.

Convexity – Duration clearly provides a better starting point than maturity for gauging the exposure of a bond to changing interest rate levels. However, duration is limited as a risk measure in that it only describes how a bond’s price will react to a relatively small move in rates. A bond’s duration is not static; duration changes as market rates rise and fall over time and the bond’s price adjusts downward or upward to result in a competitive yield. As a bond’s price rises, its yield falls and duration lengthens as the ongoing coupon payments are a smaller portion of its current valuation and the – more distant in time – return of principal is weighted more heavily. By contrast, as a bond’s price falls and its yield increases, each payment becomes more significant in proportion to the purchase price, and duration declines. As a result, a line graph of expected changes in a bond’s price in response to interest rate moves bulges outward, or is “convex” in shape, rather than being a straight line.

The above description of convexity applies to a so-called “plain vanilla” bond. By contrast, many government and municipal bonds are callable, while mortgage-backed securities are subject to prepayments on underlying securities. Such bonds do not benefit to the same degree from declining rates, as their durations actually shorten in such an environment. As a result, the graph of expected price changes slopes downward as rates decline, or displays “negative convexity.”

To summarize, with respect to interest rate risk:

Maturity – A rough cut re risk; undercut as a basis of comparison to the extent coupons vary between bonds.
Duration – Used to provide a snapshot of likely price moves given relatively small rate changes.
Convexity – Describes price moves over larger rate changes by accounting for changes in a bond’s duration.
Negative Convexity – Describes price behavior of callable bonds which do not benefit as rates decline.