Key Rate Duration
A shift in the entire yield curve is pretty effective in changing the price of bonds. The sensitivity measure is effective duration. But it's also possible that a single key rate changes, and a bond's sensitivity to that possibility is measured by __key rate duration__. Key rate durations show bond price sensitivity to a change in a single spot rate in the yield curve.
The benchmark yield curve here is the par yield curve, so recall that you'll have to think in terms of how bonds get to be priced at par. Consider a 10-year zero coupon bond to start with. What would you expect to see in terms of key rate durations for this bond, again using the par yield curve?
Not quite, even though this isn't a bad thought.
Note that you're looking at a par yield curve here and not spot rates.
Yes!
Anything beyond 10 years doesn't matter for this bond. The 10-year spot rate certainly matters, but shorter-term rates do as well.
No.
The 20-year rate wouldn't have any effect, for instance. The bond is done within 10 years.
Now this can be a challenge to understand, but here's the idea: the 10-year zero coupon bond reacts to 10-year spot rates. That's straightforward enough. Now suppose that the par yield curve shows an increase in the five-year par rate. That means that spot rates up to five years go up. What has to happen to spot rates from 5 to 10 years in order to keep the 10-year par rate unchanged?
Right!
No.
That would ensure that the 10-year par rate will change.
No.
If a few spot rates go up and others are unchanged, the 10-year par rate will have to change as well.
That's the balance. For a 10-year par rate to be unchanged, higher spot rates somewhere would have to mean lower spot rates elsewhere. So the higher short-term spot rates mean lower long-term spot rates. The 10-year spot rate is going down. What does that mean for the price of a 10-year bond?
Of course.
No, price would go up.
It's the same inverse relationship as always between spot rates and bond prices. The spot rate goes down, the bond price goes up. So this par curve increase at the five-year rate is actually causing a price increase for this 10-year zero-coupon bond.
Now if a bond has a 0.13% sensitivity to changes in the 1-year rate, a 0.19% sensitivity in the 2-year rate, etc., what might you logically assume about the sum of all of the key rate durations?
No, a bond could easily have sensitivities to each key rate that moves price in the same direction, so these wouldn't net out to zero in any way.
Not usually.
That would be just a coincidence. Consider that each key rate is showing the price effect on a particular bond.
Exactly. When you consider each key rate duration as being a "nudge" on the bond's price for that rate's change, if you add up all of those nudges, you get the total change in the bond's price, or effective duration. To be specific in notation for the _k_th key rate:
$$ \displaystyle \sum_{k=1}^n \text{KeyRateDur}^k = \text{EffDur} $$
Key rate durations tell you something about the "shaping risk" of a bond. For example, suppose that you expected short-term rates to rise, but not long-term rates. The yield curve is flattening. What would tell you that your bond is going to be heavily affected by this?
No, since the long-term rates aren't expected to change, these large key rate durations wouldn't mean much. Instead, larger short-term key rate durations show a greater price sensitivity to what is expected.
Of course.
$$ \displaystyle \text{KeyRateDur}^k = - \frac{1}{PV} \times \frac{\Delta PV}{\Delta r^k} $$
Or equivalently:
$$ \displaystyle \frac{\Delta PV}{PV} = -\text{KeyRateDur}^k \times \Delta r^k $$
For example, suppose the 1-year rate increases 50 basis points. A 1-year key rate duration of 0.8 means that the bond price with ==just this change== would go down by about 0.4%:
$$ \displaystyle \frac{\Delta PV}{PV} = -0.8 \times 0.0050 = -0.0040 = 0.4 \% $$
To summarize:
[[summary]]
Only the 10-year key rate duration would be non-zero
Only key rate durations from 10 years and below would be non-zero
Only key rate durations from 10 years and above would be non-zero
Those spot rates must go down
Those spot rates must go up as well
Those spot rates must be unchanged
The bond price goes up
The bond price goes down
They sum to 1
They sum to effective duration
They sum to 0
Large key rate durations for long-term rates
Large key rate durations for short-term rates
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