Can Any1 simplify the below Paragraph From Curriculum pg 179??

As illustrated in Exhibit 23, key rate durations can sometimes be negative for maturity points that are shorter than the maturity of the bond being analyzed if the bond is a zero-coupon bond or has a very low coupon. We can explain why this is the case by using the zero-coupon bond (the first row of Exhibit 23). As discussed in the previous paragraph, if we increase the five-year par rate, the value of a 10-year bond trading at par must remain unchanged because the 10-year par rate has not changed. But the five-year zero-coupon rate has increased because of the increase in the five-year par rate. Thus, the value of the five-year coupon of the 10-year bond trading at par will be lower than before the increase. But because the value of the 10-year bond trading at par must remain par, the remaining cash flows, including the cash flow occurring in Year 10, must be discounted at slightly lower rates to compensate. This results in a lower 10-year zero-coupon rate, which makes the value of a 10-year zero-coupon bond (whose only cash flow is in Year 10) rise in response to an upward change in the five-year par rate. Consequently, the five-year key rate duration for a 10-year zero-coupon bond is negative (−0.93).

You can calculate the value of a 10-year coupon-paying bond two ways:

Discount all cash flows at the YTM, then tot them up

Discount each cash flow at its spot rate, then tot them up.

You have to get the same value either way (arbitrage-free).

If the 5-year par rate increases (but the 10-year par rate remains unchanged), spot rate at 5 years has to increase, so spot rates after 5 years have to decrease to leave the 10-year par rate unchanged. The decrease in the 10-year spot rate will increase the value of a 10-year zero, so its 5-year key-rate duration is negative: the 5-year par rate increased, and the price of the bond increased.

Effective duration is the negative of the percent price change divided by the percent change in YTM.

Key rate duration is the negative of the percent price change divided by the percent change in the key rate

If the key rate increases and the bond price decreases, the key rate duration is positive.

If the key rate increases and the bond price increases, the key rate duration is negative.

That’s the situation we have here: the 5-year (par) rate increases, and the price of a 10-year zero increases, so the 10-year zero has a negative 5-year key rate duration.

This really does not make sense. I thought for a 10-year zero, there is only one key-rate (the 10 year spot rate). The 5-year rate duration is zero for the 10-year zero.

The point is that when they talk about key rates, they mean par rates, not spot rates.

If only the 5-year par rate changes, then the spot rates for maturities less than 5 years don’t change, but all of the spot rates at and above 5 years change.

Now - that makes even less sense! I thought Par rates were YTM on par bonds and therefore if par rate on 5 year bond changes, At least 1 of the spots less than 5 year would have changed!

If _ only the 5-year par rate changes _, then the 1-year, 2-year, 3-year, and 4-year par rates _ don’t _ change.

If the 1-year par rate doesn’t change, then the 1-year spot rate doesn’t change: they’re the same thing.

If the 2-year par rate doesn’t change and the 1-year spot rate doesn’t change, then the 2-year spot rate doesn’t change: bootstrapping.

If the 3-year par rate doesn’t change and the 1-year spot rate doesn’t change and the 2-year spot rate doesn’t change, then the 3-year spot rate doesn’t change: bootstrapping again.

Same for the 4-year spot rate: unchanged thanks to bootstrapping.

However, if the 5-year par rate changes and the 1-year, 2-year, 3-year, and 4-year spot rates don’t change, the 5-year spot rate has to change; you guessed it: bootstrapping.

Now, if the 6-year par rate doesn’t change, nor do the 1-4 year spot rates, but the 5-year spot rate changes, then the 6-year spot rate has to change: our old friend, bootstrapping.

Shouldn’t that be the case for both lower and higher coupon bonds ? If there is a change in 5 year key rate, then it should affect the spot rate for the 5th year as well as spot rates for years >5, regardless of the coupon rate. So, how does it impacts lower coupon bonds differently from higher coupon bonds?