# Duration and Yield Relationship

Looking to confirm my understanding re relationship between yield and duration.

• When yield increases, duration falls -> because coupons are invested at a higher rate so you’re breaking even on the bond quicker.
• When yield falls, duration increases -> because coupons are invested at a lower rate so you’re taking longer to breakeven.
• Duration is not a static measure. It changes as the YTM changes.

Is this correct? Thanks!

Your three bullet points are correct, though I’m not sure that your explanations (“because . . . .”) are.

Thanks S2000. Could you explain the because part? What is driving duration down when yield rises and vice versa?

It’s merely a matter of compounding.

Suppose that you have a coupon payment of 60 in one year and 1,060 in 2 years.

• At a YTM of 1%, the first coupon payment represents 5.41% of the present value
• At a YTM of 6%, the first coupon payment represents 5.66% of the present value
• At a YTM of 10%, the first coupon payment represents 5.86% of the present value

Thanks S2000. So it’s simply time value of money. PV of coupons towards the end and principal makes up lower % of the PV as YTM increases.

Is there no reinvestment effect here? Since the YTM is only earned if coupons are reinvested at the same rate, I thought there’d be a connection between reinvestment rate and duration…

No, there’s no reinvestment assumption here.

There are two views on IRR. One is that it’s a return on all of the money invested for the entirety of the investment term, and that’s the view that requires that the cash outflows be reinvested at the IRR. The other is that it’s a return only on the money that remains invested, and that cash outflows are no longer invested, so the reinvestment rate doesn’t matter. Because IRR (and YTM, which is nothing more nor less than an IRR) is a discount rate that equates the future cash flows to the present value, the second view makes more sense.

I see, thank you!

You’re quite welcome.

if OP pertains to macaulay duration on his first post, will the statement hold true?

S2K,

Thank you. I’ve always had this question. Could you please help me understand the “risk” other than credit risk in the second case, when no other market elements are considered as part of the duration metric here?