Virus4762 wrote: ↑Mon Jun 21, 2021 10:19 pm
Thesaints wrote: ↑Wed Jun 16, 2021 3:18 pm
For non mortgage backed bonds instead the story is different. Everything else being equal,

**a bond with lower yield has a longer duration.** Therefore, lower rates tend to push up average durations, but this is not an effect as big as it is (in the opposite direction) for mortgage backed bonds.

I don't understand. Don't longer duration bonds normally have higher yields (i.e. 10 year note has a longer duration than a 1 year bill)?

EDIT: Or are you referring to coupon payments?

The statement "everything else being equal" implies a comparison of identical bonds. Given two bonds with the same payment schedule, the bond with a higher yield will have a shorter duration. The reason is that Macaulay duration is the average of the time to all future payments, weighted by the present value of those payments. A higher yield has a greater reduction on the present value of payments farther in the future. The most extreme example is an infinite-maturity bond (or a preferred stock, which is equivalent to an infinite-maturity bond but does not guarantee payments); a preferred stock with a 4% yield has a 25-year duration, but if the yield rises to 5%, the duration declines to 20 years. But a long-term, high-yielding bond would behave similarly, because a relatively small part of its value is the repayment at maturity.

There is another effect if you use modified duration, which is the actual measure of interest-rate sensitivity; a bond with a modified duration of 10 years will have its value change by 10 times a small change in interest rates. Modified duration is equal to Macaulay duration divided by (1+interest rate). Thus the modified duration of even a zero-coupon bond decreases when rates increase. A 10-year zero-coupon bond has a Macaulay duration of 10 years regardless of interest rates, but its modified duration is 9.62 years at a rate of 4%, and 9.52 at a rate of 5%.

(As a separate point, given two bonds with the same yield, the one with the higher coupon payment will have a shorter duration, because less of its value is in the final payment at maturity.)