GARP textbook Ch 13 - Key rate Q13.17

[daisypm]

Hello all,

I would like to ask Q13.17 from the textbook. How can I know KR01 for five-year key rate would need to use the figure 0.6, but not 0.4? Thanks a lot.

Question:
Suppose par yield KR01s are calculated using five- and ten-year shifts in par yields. A portfolio has an exposure of +20 to a one-basis-point change in the seven-year par yield. Use linear interpolation to determine its par yield KR01s.

Answer:

The KR01 for the five-year key rate is 0.6 X 20 = 12.
The KR01 for the ten-year key rate is 0.4 X 20 = 8.

 

[David Harper CFA FRM]

Hi @daisypm I think you are right to be confused: I think this question makes no sense at all. I can see what it's trying to do with the interpolation: 7 years is between 5 and 10, it's 40% from 5 and 60% from 10.

It is typical (in the key rate technique) to shock the 2, 5, 10 and 30 with linear interpolation. While the shock is one basis point, let's say it is X(5) at 5 years and X(10) and 10 years.

Regarding the 5-year key rate, linear interpolation means +X(5) declining linearly to 0 at 10 years (10 years in the "neighboring" key rate). For example, +X(5)*50% at 7.5 years because that's halfway; +X(5)*60% at 7.0 years; +X(5)*1/5 at 9.0 years. Regarding the 10-year key rate, linear interpolation means zero at 5-years increasing linearly +X(10) at 10-years. For example, +40%*X(10) at 7.0 years.

So typically, again, the par yield shock is one basis point: X(5) = X(10) = +0.01%, but let's just let them be X(5) and X(10). How is the 7-year maturity handled here? Well, first, it is not a key rate! It contributes to both the 5- and 10-year KR01s. The +X(5)*60% shock contributes to the KR01(5); typically that means 60%*0.01% = +0.00600% to the 7-year par yield. Similarly, +X(10)*40% shock contributes to the KR01(10); typically that means 40%*0.01% = +0.00400% contributes to the 10-year par yield.

So it's easy to see what the question is trying to do: (7 - 5) / (10 - 5) = 40% and (10-7)/(10-5) = 60%, is the basis for the 40% and 60%. Given a +20 exposure, it's attributing 60% the nearer neighboring rate and 40% to the further neighboring rate. But I don't think the question writer understands that we interpolate the shocks, not the exposures. I think there is a more profound problem: the 7-year par yield exposure would be miniscule: to shock a single maturity's par yield. It is not clear how such a "point estimate" exposure informs the neighboring KR01s (!). The 10-year KR01 shocks every rate from 5-years to 30-years. For one thing, each KR01(5) and KR01(10) would be presumably much bigger than 20. I hope that helps,