I cannot find any satisfactory explanation as to why a bond priced at par only has price sensitivity (measured via duration) to key rates at the same tenor as its maturity. Only ‘just accept is as definition’ (sic, thanks Fitch learning) and it is a ‘definitive consequence’ from CFAI…both seemingly to glaze over the explanation.
from what I understand, we have coupon cash flows that need to be PV’d throughout the life of the bond which means even small changes in any of the key rates are going to have material impact on pricing when altered.
why do we ignore this basic time value of money tenet when looking at key rate duration?
There was another thread on key rate duration that asked a similar question.
Fitch’s answer is correct, but incomplete (in my humble opinion): it is, in fact, a consequence of the definition, but to understand why, you have to understand fully the definition of key rate duration as presented in the CFA curriculum.
When they say that, for example, the 5-year key rate changes, what they mean is that the 5-year par rate changes; all other par rates are unchanged. You would think that the 5-year key rate is the 5-year spot rate, but it isn’t; it’s the par rate. That’s the . . . sorry . . . key to understanding CFA Institute’s take on key-rate durations.
So the reason that a 10-year coupon paying par bond doesn’t have sensitivity to a change in the 5-year key rate is that the 10-year par rate hasn’t changed; if the 10-year par bond’s YTM doesn’t change, its price doesn’t change.
A consequence of defining the key rates as par rates is that zero coupon bonds do have sensitivity to key rates other than the one at their maturity. For example, if only the 5-year key rate increased by 100 bps, then the 1-year, 2-year, 3-year, and 4-year spot rates are unchanged (otherwise their par rates would have changed, too), but the 5-year spot rate increases. Therefore, to keep the 6-year par rate unchanged, the 6-year spot rate must decrease. Similarly, the 7-year, 8-year, 9-year, and 10-year spot rates will decrease. Thus, the price of a 10-year zero will increase: it has a negative 5-year key rate duration.
I recall this conversation S2000. I still am not sure why would 7,8,9 and 10 year spot rates would change? I agree that 6-year spot rate has to change to offset the change in 5 year spot rate so as to keep the 6 year par rate constant. But why 7 or 8 or 9 etc? I thought the adjustment HAS to occur in the next period. If the 6 year par rate remains the same, the increase in 5 year spot rate would have been perfectly offset by the 6-year spot rate. No?
Yes. But for the 6-year par bond the 6-year spot rate is discounting the coupon plus the par value, so a small change in the 6-year spot rate has a big change in the value of the 6-year bond. But for the 7-year bond, the 6-year spot rate is discounting only the coupon, so it will have only a small effect; the 7-year spot rate will also have to change. And so on.
Suppose that you have a flat (par) yield curve: 4%.
If the 5-year par rate changes to 5%, then the price of the 5-year (4%-coupon) bond is $956.7052, so the 5-year spot rate has to change to 5.0867%.
The 6-year through 10-year spot rates will change to, respectively,
I noticed that on the slide and queried it with an instructor (JB) last night. His response was ‘key rate is sensitivity to a single spot rate’ so there is a difference of opinion on the matter it would seem.
Respectfully, as regards CFA Institute’s definition of key rate duration, the Schweser instructor is wrong: it’s a change in the par rate, not a change in the spot rate.
If you look at the table at the bottom of p. 349 in volume 5 of the 2016 CFA Institute curriculum, you see that a 10-year bond priced at par with a 4% flat yield curve has zero key rate durations for all maturities less than 10 years. If you made a change to the 2-year, 3-year, or 5-year spot rate (only), the price of that 10-year bond would change. Clearly, we are not making changes to the spot rates; we’re making changes to the par rates.
To quote “A consequence of defining the key rates as par rates is that zero coupon bonds do have sensitivity to key rates other than the one at their maturity”
May I understand why only zero-coupon or bonds with low coupon rates will exhibit ‘negative key rate duration’?
My point was that we make a change to a single par rate, instead of making a change to a single spot rate.
If, for example, we want to compute the 5-year key-rate duration for a bond, we will change the 5-year par rate (say, up 50bps and down 50bps), and leave all other par rates unchanged. The effect on the spot curve is that all spot rates for maturities shorter than 5 years will not change, the 5-year spot rate will change more than 50bps in the same direction (up or down) as the par rate change, and all spot rates at maturities longer than 5 years will change by much less than 50bps in the opposite direction (down or up, respectively) of the par rate change.
If we mistakenly believe that we’re supposed to change the 5-year spot rate (say, up 50bps and down 50bps), leaving all other spot rates unchanged, then the effect on the par curve will be that all par rates for maturities shorter than 5 years will not change, the 5-year par rate will change by less than 50bps in the same direction (up or down) as the spot rate change, and all par rates at maturities longer than 5 years will change by much less than 50bps in the same direction (up or down) as the spot rate change.
That’s a very different result. And it’s not how key rate durations are computed.
Thanks! So the spot rates greater than 5 would adjust according to the corresponding par rates? Could you elaborate a bit on why. Sorry for bothering you with basics.
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Hi S2000magician , Thank you for the explanation, just have one question regarding your comment " So the reason that a 10-year coupon paying bond doesn’t have sensitivity to a change in the 5-year key rate is that the 10-year par rate hasn’t changed; if the 10-year bond’s YTM doesn’t change, its price doesn’t change"
Why could i not use the same logic you have mentioned for the zero coupon bond for the statement above, so if i raise the 5 year par rate by 50bps it would increase the 5 year spot rate and the spot rates for 6, 7 , 8 etc would go down to keep their par rates unchanged and hence the price of the 10 year coupon paying bond goes up? please let me know. Thanks.
