# Key Rate Duration Question

#### S2000magician

##### New Member
Dear David:

I've seen your name occasionally on AnalystForum and your reputation precedes you. Recently, I was discussing key rate duration with Marc LeFebvre and he said that he had gotten many ideas on key rate duration from a video of yours, which he shared with me.

The question deals with which rates, exactly, are changed in computing a key rate duration. The video of yours that Marc shared with me suggests that we are changing a single spot rate. (I also notice that your formulation uses continuous spot rates, not periodic (i.e., annual) spot rates, but that's not germane to this discussion.) In the Level II CFA curriculum, they change not a single spot rate, but a single par rate. Although they're not really explicit about it being a par rate that they're changing, it becomes clear when you look at one of their tables and see that, for example, a 10-year par bond has a 3-year key rate duration of 0.00 years, and a 10-year discount bond has a 3-year key-rate duration that is negative; those durations would have to be positive if you were changing only the 3-year spot rate (leaving all other spot rates unchanged).

I don't have the FRM curriculum. (Although I have spent many years in risk management – project risk management primarily – I have no inclination to study for another exam or two. Well . . . unless it's to get a PhD in algebraic topology. I'm too old for this.) Therefore, I don't know what it says about key rate duration. I would be grateful for any insight you (or others, for that matter) could provide.

Thanks!

Sincerely,

Bill Campbell III, CFA

"S2000magician" on AnalystForum

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Bill (@S2000magician) Thank you for stopping by! I will take a look at the CFA L2 syllabus but here my recent video (< 1 year ago) on key rate shift technique is here https://forum.bionicturtle.com/threads/t4-43-fixed-income-key-rate-shift-technique.22781/ It's based on FRM assiged Bruce Tuckman's Fixed Income Securities (here at https://amzn.to/2ZRV07l ). If you'd like to take a closer look, here is a workbook with a few worksheets (this is the basis for our FRM study notes on the key rate section), where I used different key rates (par, spot, forward buckets): https://www.dropbox.com/s/0t7dlc2oyoezuxk/053020-key-rates.xlsx?dl=0 I think Tuckman is excellent, but for a deeper treatment I learned from Nawalkh's Interest Rate Modeling (https://amzn.to/3etVWCE) but I'm sure there are better modern texts and I am not an expert on term structure modeling ....

To me (and this is consistent with the FRM which has always said per Tuckman that it's a user design choice as to which type of rate factor vector is used in the key rate shift technique) the key rate shift technique is a very loose, flexible approach because its only agenda is to improve on yield (single factor YTM) with a multi-factor shift of the term structure (but without going all the way to modeling every relevant spot rate on the term structure). So I just view it as assuming any vector of interest rate factors (types) and translating those into a term structure (of spot rates/discount factors). I'm saying "factors" because the FRM approaches interest rates as within a class of interest rate factors: we don't need to operate on interest rates to model the term structure. But I guess key rate shift technique probably by definition assumes one of the common interest rate types (i.e., par, spot, or forward) and I guess non-rate multi-factor approaches go by other names (e.g., duration vector).

In any case, from my perspective, as illustrated in my XLS, we can choose any of the three common rate types (par, spot or forward) and we just need a "rule" for treating the neighbors (neighboring rates); by definition of key rate, we aren't modeling all the rates (from 0.0 to 30.0 years), we're just modeling a small subset (a vector) of "key" rates, so clearly we need a "rule" for what do with the majority rates that are left out. Linear interpolation (straight line) seems to work! To me, because either way we will need to solve for the implied spot rate term structure (aka, discount function), mathematically we can use any of the three rate types (par, spot or forward) as key rates. In practice, it's a different story: spot rate key rate are obviously the easiest to model and explain; par yields are great (not hard to model as seen in my XLS), and the FRM/Tuckman prefers them because they come in handy when solving for hedges, but they are a pain to explain because the impact of par yield shocks on the spot rate term structure is not intuitive; forward rates seem to be the academic favorite because, rather than let spot/par rate shocks imply a "saw-tooth" (abrupt) shift in forward rates, you can operate directly (smoothly?) on the forward rates. If the only goal is to illustrate and understand the key rate shift technique, it seems to me you would just use spot rates! It sounds like the CFA is using par yields (like the FRM did for years) for the reason you say: If we are shocking up spot rates, I don't think we can generate (for a vanilla long bond position of course) negative key rate durations; but par yield up shocks routinely imply negative key rate durations (and are hard to explain). I hope that's helpful ....

#### S2000magician

##### New Member
Thanks, David!

I'll take a look at the references.

You're correct, of course: we can make changes to any curve we want to. But, just as we can choose to drive on the right or drive on the left, it tends to work out much better if everyone adopts the same convention.

You're also correct about shocking spot rates: for straight long bonds (fixed rate, no embedded options), changing a single spot rate will not induce a negative duration.

#### Matthew Graves

##### Active Member
Subscriber
In the vast majority of practical applications of KRDs I've worked on the starting point has always been to shift observable market data rather than the zeros. You may get results which are not particularly intuitive because of the extra step of re-deriving the spot curve but ultimately these results are tied back to defined, observable market movements. Shifting the zeros is easier, and I'm fully on board with use for teaching, but it is difficult to relate back to a market move and therefore a bit abstract to my mind.

#### S2000magician

##### New Member
Hey, Matthew!

Do you work at in fixed income?

When I got my CFA charter, I was working at this little fixed income house in Newport Beach, CA: PIMCO. You may have heard of it.

#### Matthew Graves

##### Active Member
Subscriber
Hey, Matthew!

Do you work at in fixed income?

When I got my CFA charter, I was working at this little fixed income house in Newport Beach, CA: PIMCO. You may have heard of it.

Hi there,

Yes, I work in Fixed Income and have heard of PIMCO