Duration - yield - Maturity relation

[higaurav]

Hi David,


Please can you mention, what does K stand for in the macaulay formula = [1*PVCF1 + 2*PVCF2 ... n*PVCFn]/ K*price. This is as given in the episode 5 in the Market risk B part 2.

Other thing, when I think of relation of maturity with Yield, i know for a long maturity tenure, yield will be higher. Now if i go by the logic that a higher yield, duration will decrease than why is the duration increases when maturity is increased (yield increased) ? Although i understand this from the Macaulay formula point, but i feel somewhere i am missing some link when i think in with the above mentioned logic.

Pls suggest.

 

[David Harper CFA FRM]

Hi Higaurav,

k = number of periods/coupons per year. For us, it will typically be 2. Less likely: 1 (annual) or 12 (monthly).

For duration, it is best (for risk purposes especially) to *define* duration as the sensitivity in price to yield change (dP/dy); i.e., modified/effective duration

But to think about these relationships with yield/maturity, it is easier to think about Macaulay duration; time to weighted average cash flow (Macaulay duration)

"For a long maturity tenure, yield will be higher" - Right, this will generally be true and reflected in an upward sloping yield curve

If you start with a zero-coupon bond, say $100 par in ten years. It is longer to receipt (Macaulay) and more sensitive to interest rate changes due to discounting: PV = 100*EXP[(-r)(T)]. The longer the maturity (T), the more discounting, and so the "further away" your principal, the more sensitive is the price to changes in rate (r).

(Okay, but once you've got that, move to DV01 and try to see why it has an unclear relationship with maturity. DV01 tends to increase/decrease with maturity)

But an INCREASE IN YIELD corresponds to a DECREASE in Macaulay/modified/effective duration.
Yes this is true: "For a long maturity tenure, yield will be higher"
but here we are talking here about:
for a given maturity, if the yield increases, then duration decreases; i.e., yield increase with all other things being equal
This is less intuitive, this is harder! It is because (similar to discounting above), an increasing yield lowers the PV of all the cash flows, but the most distant (esp. principal at the end) is most impacted.


David

 

[afterworkguinness]

Hi @David Harper CFA FRM CIPM ,
To me it sounds like these two statements are contrary:

"The longer the maturity (T), the more discounting, and so the "further away" your principal, the more sensitive is the price to changes in rate (r)."

And

"is because (similar to discounting above), an increasing yield lowers the PV of all the cash flows, but the most distant (esp. principal at the end) is most impacted."

Though, if I think of duration as the weighted average time to maturity, I imagine that a higher yield would effectively shorten that weighted average time. Is my intuitive understanding valid and can you help clarify those two statements ?

Thanks as always.

 

[ShaktiRathore]

Hi
David said for a given maturity as yield increases the weighted avg time to maturity aka duration decreases. This is because PV of cash flows decreases as yield increases ,this Pv of cash flows are nothing but weights assigned to time so as weighhts are decreased and multiplied by time would lower weighted avg time to maturity or duration.
Thanks