Macaulay duration and modified duration depend on the coupon rate, yield to maturity, time to maturity and the time elapsed since last coupon date. By changing one of these inputs, we can illustrate the following properties:

  • As the time passes (from one coupon date to another), Macaulay duration declines smoothly and then jumps up at the next coupon date, creating a climbing sew-saw pattern.

  • Macaulay duration of a zero-coupon bond equals its maturity (i.e. it has no sew-saw pattern).

  • Macaulay duration of a perpetual bond is (1+r)/r.

  • In case of fixed-rate coupon bonds, a longer time to maturity generally corresponds to higher Macaulay duration. It is always the case for bonds issued at par and premium. In case of discount bonds, Macaulay duration peaks and then fall with the implication that for long-term discount bonds, interest rate risk may actually be less than that on the short-term bond.

  • The coupon rate and yield to maturity are inversely related to Macaulay duration. If the coupon rate is low, more cash flows are farther away resulting in higher interest rate risk and vice versa. Similarly, a high yield to maturity means a lower duration.

Effective duration and bonds with embedded derivatives

The interest rate risk of bonds with embedded options is measured using effective duration. When there is an increase in benchmark yield, the price of a callable bond behaves just like a non-callable bond, but when the benchmark shifts downwards, there is a greater chance of the bond being called (because the issuer can refinance at the low market interest rate) and the increase in bond price is less pronounced. This shows that an embedded call option reduces the effective duration of a bond, particularly at a low interest rate. Similarly, a putable bond, a bond with a put option, reduces effective duration, particularly in high interest rate environment, because due to the put option, the decline in bond price due to an increase in the benchmark yield curve is muted. This shows that existence of an embedded option reduces effective duration.

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