Key rate duration

CFA Level II
#1

Why are all the key rate durations (except for maturrity matched key rate duration) for a bond trading at par equal to 0.

How can the price of the bond not be sensitive to the interest rate fluctuations that are happening to the maturitiies midway. The bond is generating cash flows at those times and hence if those cash flows are discounted at different rates the overall price of the bond must change?

Any explanations?

#2

It’s a zero coupon bond? (you would have to exclude the part about the bond trading at par though)

#3

You need to make sure that you understand exactly what a key-rate duration is.

It’s a change in the _ par _ curve – the YTM – for a particular maturity.

The only YTM that matters for a given bond is the YTM at its maturity.

So, for example, a 5-year bond’s price doesn’t change when the 3-year key rate changes because _ the 5-year YTM doesn’t change _.

#4

this is not a zero coupon bond, But any bond that is currently trading at par

#5

i am beginning to get the sense out of this, but not completely

Yield - single discount rate with which you discount every cash flow from a fixed income to mark it to the current market price. For a given maturity it mirrors the spot rates to that maturity period

So by the logic which you have given i can see why any other yield but maturity period one does not change the calculated price of the bond

But why does it happen other bonds which are not trading at par?

What difference does the fact that the bond is trading at par create?

#6

When you change one par rate and no other, you change the spot rate at that maturity and all future maturities, to ensure that par bonds trade at par.

Bonds that aren’t trading at par have different cash flows discounted at these different rates, so their value may change slightly.

#7

This is what CFA book says

" an option-free bond trading at par (the shaded row), the maturity-matched par rate is the only rate that affects the bond’s value. It is a definitional consequence of “par” rates. 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. In other words, shifting any rate other than the 10-year rate on the par yield curve will not change the value of a 10-year bond trading at par. Shifting a par rate up or down at a particular maturity point, however, respectively increases or decreases the discount rate at that maturity point. These facts will be useful to remember in the following paragraph."

I am also not 100% convienced by this explanation either.

However It’s somewhat related to reasoning of negative duration on low maturities of zero coupon bonds.

#8

At first I was also confused but a simple recalculation of spot rates (needed for arbitrage-free price) is all one need to check what CFA institute is saying.

Consider a 4 period scenario where par rate for each of the 4 year is 4%. i.e

r(1) = 4%, r(2) = 4%, r(3) = 4% and r(4) = 4%. Since all par rates are same so the spot rate is also same i.e

z(1) = 4%, z(2) = 4%, z(3) = 4% and z(4) = 4%.

So a 4% coupon paying bond with 4 year maturity using above spot rates will come out to be $100. To be precise the PV for each casflow will be PV(1) = $3.84615, PV(2) = $3.69822, PV(3) = $3.55599 and PV(4) = $88.89964

Now comes the interesting part. Lets say we change the par rate for 2 year (key rate) and leaving rest all par rate same. say r(2) is now 10%. So we have

r(1) = 4%, r(2) = 10%, r(3) = 4% and r(4) = 4%.

if you recalculate spot rate (needed to price bond) they will come out

z(1) = 4%, z(2) = 10.31869%, z(3) = 3.84612% and z(4) = 3.88460%

Now same 4% coupon paying bond with 4 year maturity using new spot rates will still come out to be $100. To be precise the PV for each casflow will be PV(1) = $3.84615, PV(2) = $3.28671, PV(3) = $3.57182 and PV(4) = $89.29531. Just add them, it will be $99.99999 (approx $100)

Now as CFA book says, just change maturity matching rate so new par rate would have only 4’th year rate changed and rest 3 year will be same so r(1) = 4%, r(2) = 4%, r(3) = 4% and r(4) = 10%. Now calculate new spot rate. They will be below

z(1) = 4%,z(2) = 4%, z(3) = 4% and z(4) = 11.08%

Now same 4% coupon paying bond with 4 year maturity using new spot rates will come out to be $ 79.41124. To be precise the PV for each casflow will be PV(1) = $3.84615, PV(2) = $3.28671, PV(3) = $3.57182 and PV(4) = $68.31088

Hope this will clarify for current and future level 2 guys :slight_smile:

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