Duration - An Introduction
Duration, first developed in 1938 by Frederick Macaulay, measures bond price volatility. It is a weighted-average term-to-maturity of the bond’s cash flows; the “weights” being the present value (PV) of each cash flow as a percentage of the bond’s full price. The cash flows include the bond’s redemption value, typically $1,000 (discounted by the number used for PV—see below). Thus, a bond maturing in 6.5 years will have 14 cash flows between now and maturity (13 semiannual payments plus a face value payment upon maturity).
A Salomon Smith Barney study uses the visual analogy of a series of tin cans equally spaced on a seesaw. The size of each can, which varies, represents the cash flow due (e.g., six months from now, one year from now, 18 months from now, etc.). The contents of each tin can represent the present values of those cash flows (e.g., using a discount of, say 5%, the “contents” of the first can might be $50, $47.50 for the second can, $47.50 x 0.975 for the third can, etc.—each can’s value is reduced by 2.5% more than the previous can because there is a 5% annual discount and bonds pay interest semiannually).
Finally, the cans are evenly spaced; each space represents six months (the time between each semiannual payment). The final payment of face value, which is also discounted, is included on the seesaw (and is the last tin can). All of the PV cash flows are then added up and divided by two; duration is the distance to the fulcrum that would balance the seesaw (the halfway point—or, phrased another way, how long an investor would have to wait to get back one half of all present value payments).
Using the seesaw analogy above, a zero-coupon bond maturing in 15 years has a duration of 15 years. Why? Because the entire payoff occurs when the bond matures, 15 years in this example. All “interest” tin cans are empty—there are no semiannual or annual interest payments with a zero-coupon bond. The investor receives nothing until the bond matures (or is sold prior to maturity).
Duration can be described in a similar fashion, without using a seesaw illustration. Add up all of the interest payments a bond is going to make over its remaining life. Discount each of those payments by an appropriate PV number; the discounting number could be the projected rate of inflation or it might be the current yield of the bond being measured. The next step is to discount the face value of the bond by using the same PV figure (note: the discounting will be more severe because a lot more time will lapse before final payment is made). Add up all of these figures (all discounted semiannual interest payments plus discounted face value of the bond).
Once you have added all of these dollar figures, take the final number and divide it by two. The resulting number represents the halfway point. Finally, using a simple time line, plot the point in time when the then cumulative interest-rate payments (discounted) equal the halfway point—whatever the point in time is the bond’s duration; the greater a bond’s duration, the greater its volatility. In general, duration rises with frequency of coupon payments and drops as yields rise.
Duration: Defined
Duration can be defined as the approximate percentage change in the bond’s price for a 100-basis-point change in interest rates. A duration of five means the bond’s price will change by ~ 5% for a 100-basis point (1%) change in interest rates. Duration increases with lower coupon, lower yield, and longer maturity.
In the case of a corporate pension plan, when the duration of its assets and liabilities are the same, the portfolio is protected against interest-rate changes and you have immunization. The high volatility of interest rates in the early 1980s caused institutional investors to use duration and convexity (a term derived from the price-yield curve for a normal bond, which is convex—the price is always falling at a slower rate as yield increases). These two tools helped to immunize institutional portfolios.