Why Non-Maturity Key Rate Duration Can Be Zero for a Par Bond
Summary
The document explains the intuition behind a claim that a par bond has zero key rate duration at curve points other than its maturity-matched rate. The key assumption is that a shocked par yield curve remains a par curve. Under that convention, a shift at a different key maturity leaves the bond’s own maturity par yield unchanged, so its price remains at par.
The explanation distinguishes this from simply changing the yield of the existing bond while keeping its coupon fixed. A newly shifted par curve implies a new par bond with a coupon matching its yield; an original bond with its old coupon could respond differently. The response also gives a cash-flow intuition: a key rate shock may move zero-coupon discount rates in different directions across maturities, so gains and losses in present value can offset. The result depends on the par-curve assumption and should not be read as a general claim that non-maturity rate shocks never affect bonds.
Key ideas
- The zero-duration result assumes the shocked curve remains a par yield curve.
- A non-maturity key rate shift leaves the bond’s maturity par yield unchanged under that convention.
- The bond remains priced at par because its maturity-matched par yield and coupon are unchanged.
- Underlying zero-coupon rates can move in opposite directions, offsetting cash-flow valuation effects.
- The conclusion depends on the curve-shift convention and does not apply universally to bond rate risk.
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Full text
# Key rate duration - Bond trading at par # Key rate duration - Bond trading at par I am reading the CFA L2 curriculum Bond Analysis section and it mentions that for a bond trading at par, the maturity-matched rate is the only rate that affects the bond's value and therefore the key rate duration for all the other rates except for the maturity-matched rate is zero. I am not getting the intuition behind this. How is it possible that shocks to any other rates won't matter? ## Answer by Helin (score 0, accepted) https://quant.stackexchange.com/a/15982 Yeah, this is a very good question, and I was actually quite confused about it as well when I first started practicing fixed income. The key thing to remember is that when we shift the par yield curve, the resulting yield curve is still ASSUMED to be a par curve! Let's say the current 10-year par yield is 5%, and we now apply a parallel shift of 100bp to the entire par curve. You may think that the yield of the current 10-year par bond (5% coupon) has increased from 5% to 6%. That's not right! In reality, we now have a BRAND NEW 10-year par bond, whose coupon rate and yield are 6%. In fact, we can show that the yield of the original 5% par bond increased by more than 100bp (if the curve is upward sloping). Once this has cleared up, we can address the key rate duration question. Let's say the key rates are 2y, 5y, 10y, and 30y, and we now apply a 5-year key rate shift. How much would the 10-year par bond price change? The answer is exactly 0. After applying the 5-year key rate shift, the 10-year par yield hasn't changed. And since the new curve is still a par curve (BY ASSUMPTION), it must be true that the price of the 10-year par bond is still par (i.e., unchanged). If you're into the mathematics, you can actually show this – when the 5-year key rate shift is applied to the par curve, the 5-year zero coupon rate actually increases a bit MORE than the 5-year par yield, while the 10-year zero coupon rate actually DECLINES (even though the 10-year par yield is unchanged). So some of the cash flows of the 10-year par bond get a lower PV, but others get a higher PV. The net change is 0.
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