9.22 Summary Characteristics of Duration

Refer back to the Price/Yield curve to illustrate some of the following characteristics.

• Duration-based estimates will always be less than the actual prices (assuming no embedded options, in which cases the opposite may pertain). Duration is a linear estimate of the slope of a line at a certain point along the P-Y Curve. We will soon discuss the calculation of Duration-based price estimates.
  • Embedded options will potentially shorten a bond’s duration.

• • Duration depicts a relationship between a bond’s Yield and its Dollar Price at a point on the curve. Given a particular Yield for a specified bond, we can calculate its Duration.

• • In order to calculate (Macaulay) Duration, we calculated a weighted average of the cash flows and divided by price as follows:

There were no exponents involved in this calculation. Thus, if we recalculate Prices for various Yields and connect the dots, we will delineate a straight Duration line!

• Duration price estimates’ errors grow larger for larger yield changes. 
  • Convexity explains most of the difference between the linear Duration-based estimate and the actual price.

• • A tangent line must always be below its relative curve. Thus, a Duration-based price estimate for a given change in Yield will always be less than the actual Price.

• • The Price-Yield Curve is steeper to the left and flatter to the right. The Price increases more when yields go down than Price goes down when Yields go up. For the same percentage move in either direction, the investor will potentially gain more than he may lose.

• Duration decreases with the passage of time – “Duration Drift.”

• Duration is inversely/negatively related to the coupon. The lower the coupon, the longer the duration. (“Zeros,” i.e., Zero-coupon bonds, have the longest duration, equal to its Term-to-Maturity.) The higher the coupon, the greater the weights given to the earlier cash flows.  As coupons increase, duration decreases, but at a diminishing rate.

• The duration of a perpetuity (e.g., preferred stock) = (1 + y) / y. Note that coupon is ignored here (and is assumed to be fixed). Thus a 10% yield will produce an 11-year duration (i.e., 1.10/.10), while a 5% yield will produce a 21-year duration (i.e., 1.05/.05). Compare this to a 30-year “Zero” – with a 30-year duration! Who would have thought that a perpetuity, i.e., stock, can have a shorter Duration than a bond whose life is finite. 

• Portfolio Duration is the market-weighted average of the durations of the portfolio’s constituent securities.

• Bonds with “Embedded Options,” such as Sinking Fund provisions, call (or put) features, and mortgage-backed bonds, will have shorter duration. Embedded options shorten Duration.

• Duration increases at a decreasing rate as maturity increases. (This is because present values approach zero for distant cash flows.) Think of it this way: as maturity increases from 30 to 31, the increase in years is less than when maturity increases from 5 to 6. More on this soon.