Properties of Duration

The sensitivity of a bond’s price to changes in interest rates can be captured using Macaulay duration, modified duration, money duration, and the price value of a basis point (PVBP).

Relationships Between Bond Features and Duration

Coupon rate (c))

An increase in the coupon rate leads to a decrease in duration (inverse relationship). Bonds with lower coupon rates have higher durations. This implies more interest rate risk for lower-coupon bonds.

Yield to Maturity (r)

An increase in the yield to maturity results in a decrease in duration (inverse relationship). A bond with a lower yield-to-maturity will have a higher duration. This is because lower yields emphasize the weight of the bond’s later cash flows, especially the maturity value.

Time-to-Maturity

An increase in the time to maturity leads to an increase in duration (direct relationship). Bonds with longer times to maturity will generally have higher durations, suggesting greater interest rate risk. However, a peculiarity arises with long-dated discount bonds: their Macaulay duration can decrease after reaching a certain time-to-maturity.

Fraction of current coupon period elapsed (t/T)

An increase in \(\frac{t}{T}\) leads to a decrease in duration (inverse relationship). As more time passes within a coupon period, the Macaulay duration decreases. However, once a coupon is paid, the duration jumps slightly, creating a “saw-tooth” pattern.

Question

Which of the following bonds is most likely to have the highest duration?

1. A bond with a high coupon rate and short time to maturity.
2. A bond with a low coupon rate and long time to maturity.
3. A bond with a high coupon rate and long time to maturity.

Solution

The correct answer is B.

Bonds with lower coupon rates and longer times to maturity typically have higher durations. This indicates greater interest rate risk for such bonds.

A is incorrect: A high coupon rate would lead to a lower duration.

C is incorrect: While a long time to maturity increases duration, a high coupon rate decreases it.