Key rate duration for a bond priced at par

[S2000magician]

@Rajesh.C

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.

I should have been a little more specific: a 10-year coupon-paying par bond doesn’t have sensitivity to a change in the 5-year key rate. A 10-year premium bond or a 10-year discount bond will have some sensitivity to the 5-year key rate.

The reason, as I stated previously, is that the 10-year par rate doesn’t change. You get the price of a bond by discounting its cash flows by its YTM. If the 10-year par rate doesn’t change, then the YTM on a 10-year par bond doesn’t change (by definition), so the price of a 10-year par bond doesn’t change.

[Student33]

@S2000magician

Thanks for your answers.

I’ve looked through 2 topics on KRD on this forum and your article also and still have problems with understanding, why dicsount or premium bonds have non-zero KRD for periods below bond’s maturity.

In your article there is a table in the end where it is clearly seen, that KRD below maturity is negative at discount bonds, then it becomes zero when coupon equals YTM (4% in that case) and becomes positive for premium bonds.

If i understood the idea right, 5y par rate, for example, is not present in valuation of 10y bond, that’s why 5y KRD and others below 10 are zero for 10y par bond, but it would be better to use mathematical approach to understand this.

Also could you please explain how to calculate KRD for zero-coupon bonds. If i understand right, they do not have par value after being issued as they trade at discount.

Thanks in advance.

[fino_abama]

See if this helps in understanding:

CFA Level 2 Fixed Income: Par Bond and Maturity Matched Rate

[Student33]

Thanks a lot! I’ve mananaged to understand the 4% case before the video but still had problems with discount and premium cases

I’ve recreated the same table as in the video. Basically we just change coupon rate from 4% (when the bond was at par) to a discount case (coupon 2% for example) and to a premium case (6% for example) and some non-zero dependence in early years KDRs appears due to math. No hidden logic it seems, just a math feature

[Student33]

1 more question please: why do we move par rates instead of spot rates? There are a lot of studying material where a spot rated is moved up and down to calculate KRD.

[S2000magician]

@Student33

1 more question please: why do we move par rates instead of spot rates?

For the same reason that we drive on the right side (at least, in the US): someone had to decide what to move, and they decided to move par rates.

Of course, for modified (and effective) duration we move the entire par curve (by a fixed amount), so that may have influenced the decision.