Why does a par bond only have one key rate duration?

If you haven’t noticed by all the questions I’m asking, I’m not having a lot of fun in Fixed Income. I hope I’m not the only one struggling with these readings…

I’ve read this part over in both Elan and CFAI and I still don’t get it. It shows a chart where it shows key rate duration for a 0, 2, 4, 6, 8, and 10% coupon bonds, with the 4% bond trading at par (with a flat 4% yield curve). It says “If the 10-year par rate on a curve is 4%, then a 4% 10-year bond valued on that curve at zero OAS will be worth par, regardless of the par rates of the other maturity points on the curve.” But why? If the 5 year spot rate changes, shouldn’t that change how you discount the coupon received in year 5?

I also don’t understand in the slightest how negative durations (technically a negative negative duration…where an increase in a 5 year key spot rate will actually increase the value of a bond trading at a discount). If anyone has a different way of explaining that, too, it would be greatly appreciated. I’ve spent a lot of time going over these two paragraphs and I’m totally stumped.


Institute, CFA. 2015 CFA Level II Volume 5 Alternative Investments and Fixed Income. Wiley Global Finance, 2014-07-14. VitalBook file.

Take a look at this thread: http://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91340111

I need to write an article about key-rate duration one of these days.

Ok I’m gonna come back to that thread and read it over a couple times. I’m sort of kind of a little bit maybe starting to understand…

So are key rate durations always calculated using YTM, as opposed to spot rates?


If I’m understanding this correctly… a 10 year zero coupon would have no sensitivity to anything other than the 10 year par rate, but since the 5 year par rate increases, then in order for the 10 year par rate to stay the same (since the definition of key rate duration is that only the one par rate is changing) the 10 year spot rate would have decrease to compensate for the fact that the 5 year spot rate went up, which would increase the 10 year par rate.

Am I correct in thinking that this would never actually happen in the real world? Where a certain rate would go up by a certain amount and then the other rates would magically compensate to keep other rates the same? I’m assuming that if a single maturity’s rate increased in real life those little ripple effects would still effect the other maturities?

So if this is all correct I’m making progress. But then why are things different for bonds priced at par? Why do they have no sensitivity whatsoever? Shouldn’t the 10 year par rate have to compensate for the changes in the 5 year spot rate even if it’s priced at par?

Not quite: a 10-year zero has sensitivity to nothing other than the 10-year spot rate, not the 10-year par rate.

Yeah, pretty much.

But that’s not the point. Key-rate durations give you the ability to determine what happens to your portfolio when the yield curve moves in a manner that isn’t merely a parallel shift: it flattens, steepens, humps, butterflies, whatever.

The 10-year par rate does have sensitivity to the 5-year spot rate; it doesn’t have sensitivity to the 5-year par rate, which is the entire point.

But if the 5 year par rate changes, doesn’t the 5 year spot rate have to change as well? The part that really confuses me is why selling at a discount or premium means it would have a non-zero effective duration at those middle maturities, while a par bond does not.


And the point is that if the only par rate that changes is the 5-year, then not only will the 5-year spot rate change, but so will the 6-year spot rate, the 7-year spot rate, the 8-year spot rate, and so on.

Look at the other thread I linked; I show some calculations there.

I’ve gone over the other thread a few times, but I’m afraid things just aren’t clicking for me. Not completely anyway.

If the 5 year par increases, then the 5 year spot increases. Since the 5 year spot increases, then 6 spot must decrease so that its par rate stays the same. Then the 7 spot must also decrease so that its spot stays the same, but 6’s spot has also decreased a little so I would assume 7’s spot decrease would be smaller than 6’s…and so on as you go up the maturities. Each one (I’m thinking) would have to decrease by less and less as it gets further away from the key rate. Would you say this is correct?

The other thing that really bothers me is this idea that a bond priced at par is immune to changes to key rates other than its maturity. If the 5 year rate was increased to a million percent, and all other par rates stayed the same, that 10 year bond would still be perfectly at par? Shouldn’t the 5 year spot rate go up, and then shouldn’t that cause the 6, 7, 8, 9, and 10 spot rates to go up in the same ripple effect as before?

I may be a lost cause on this concept at this point. This is the most thoroughly stumped I’ve been on any problem so far in the CFA material.

That’s correct. If you look at the examples I calculated, you’ll see that that’s how they behave.

That’s the definition of key rates: they’re par rates. So if only the 5-year key rate changes, then, well, only the 5-year key rate changes. The 6-year key rate (par rate) doesn’t change; so a 6-year par bond’s value doesn’t change. And the 7-year key rate (par rate) doesn’t change; so a 7-year par bond’s value doesn’t change. And so on.

You need to understand fully what a key rate duration means: we’re changing the 5-year par rate, but leaving all other par rates unchanged. You value a 9-year bond by discounting all of its cash flows at the 9-year YTM: the 9-year par rate. The 9-year par rate hasn’t changed, so we’re discounting the same cash flows at the same rate, so the value of the bond doesn’t change.

Thanks for sticking through this with me. I think I have it now…I’ve been mixing up my spot rates and par rates in my head a lot, and this is a major cause of my issues. If the 5 year par went up a million percent, then the 10 year spot would have to go down to close to 0 or negative or something weird, but the 10 year par would remain the same.

One last question if you don’t mind…

Bonds are valued by discounting all the cash flows against the YTM, or by discounting each individual cash flow with the corresponding spot rate (which should both give the same value, right?). But with a zero coupon bond, if the 5 year spot goes up, the 10 year spot must go down to compensate so that the 10 year par stays the same. But then how will those two methods give the same answer if the spot rate goes down and the par stays the same? Isn’t a spot rate by definition the YTM for a zero coupon bond?

Yes, yes, yes, yes, yes!

Correct; if not: arbitrage.

The par curve applies to coupon-paying bonds; the spot curve applies to single payments (zero-coupon bonds).

Phew…you’re a saint. Thank you!

What page is this on? I’d like to take a look at this too.

(Institute 180)

Institute, CFA. 2015 CFA Level II Volume 5 Alternative Investments and Fixed Income. Wiley Global Finance, 2014-07-14. VitalBook file.

This is the citation from the e-book in the CFAI curriculum. Page 180 I guess? In Elan it’s page 158 in volume 4.

Hardly, but you’re welcome.

god bless this community, still helping candidates after 7 years of the original thread.

The concept of Key rate duration is now as clear as a bright sunny day.