Business Glossary

concept first developed by Frederick Macaulay in 1938 that measures bond price volatility by measuring the "length" of a bond. It is a weighted-average term-to-maturity of the bond's cash flows, the weights being the present value of each cash flow as a percentage of the bond's full price. A Salomon Smith Barney study compared it to a series of tin cans equally spaced on a seesaw. The size of each can represents the cash flow due, the contents of each can represent the present values of those cash flows, and the intervals between them represent the payment periods. Duration is the distance to the fulcrum that would balance the seesaw. The duration of a zero-coupon security would thus equal its maturity because all the cash flows-all the weights-are at the other end of the seesaw. The greater the duration of a bond, the greater its percentage volatility. In general, duration rises with maturity, falls with the frequency of coupon payments, and falls as the yield rises (the higher yield reduces the present values of the cash flows.) Duration (the term modified duration is used in the strict sense because of modifications to Macaulay's formulation) as a measure of percentage of volatility is valid only for small changes in yield. For working purposes, duration can be defined as the approximate percentage change in price for a 100-basis-point change in yield. A duration of 5, for example, means the price of the bond will change by approximately 5% for a 100-basis point change in yield.

For larger yield changes, volatility is measured by a concept called convexity. That term derives from the price-yield curve for a normal bond, which is convex. In other words, the price is always falling at a slower rate as the yield increases. The more convexity a bond has, the merrier, because it means the bond's price will fall more slowly and rise more quickly on a given movement in general interest rate levels. As with duration, convexity on straight bonds increases with lower coupon, lower yield, and longer maturity. Convexity measures the rate of change of duration, and for an option-free bond it is always positive because changes in yield do not affect cash flows. When a bond has a call option, however, cash flows are affected. In that case, duration gets smaller as yield decreases, resulting in negative convexity.

When the durations of the assets and the liabilities of a portfolio, say that of a pension fund, are the same, the portfolio is inherently protected against interest-rate changes and you have what is called immunization. The high volatility and interest rates in the early 1980s caused institutional investors to use duration and convexity as tools in immunizing their portfolios.