Key rate duration for a bond priced at par

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daneyh

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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 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.
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 bp, 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.
Interesting.
 
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,
  • 3.9649%
  • 3.9699%
  • 3.9737%
  • 3.9766%
  • 3.9790%
 
I think I understood … and if helpful for someone I may provide an example.
However I don’t know why I don’t get the exact same numbers as in the Exhibit 23 pag 349
10 Y bond
Yield = 4% flat Yield Curve
Coupon = 0
PV=100/(1+4%)^10=67.55
While the exhibit it says PV = 67.30. Is it just a rounding problem, or am I missing something?
 
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