Hello, In the text, it says that using KR01s to hedge will approximately immunize the portfolio against any combination of key rate movements. Then it mentions a couple of particular instances where it would work. I understand that it will only work for small moves and if the intermediate rates move as predicted, but will this statement hold true if, for instance, one rate goes up while two others go down? The example they mention is two rates going up by the same amount while others do not move at all, but does not say anything about other moves. Thanks! Shannon

Hi Shannon, I think it's a good question which teases out, what is the key rate shift technique? The spot/forward yield curve ultimately contains an unlimited ("infinitesimal") rates. To characterize the curve (and then hedge), there is a continuum from the set of single-factor measures (e.g., duration/convexity) to much more involved frameworks well beyond Tuckman's key rates (e.g., we could add key rate convexities to the key rate durations/DV01s). Tuckman's key rate is merely a first step from single-factor duration/DV01 to a relatively easy introduction to multi-factor. In my imprecise paraphrase, to hedge he reviews a method that: First, we decide to arbitrarily "chop up" the yield curve into a limited number of segments (e.g., 4 or whatever); then each each key rate is treated like its own mini-P/Y curve, where duration will be the only metric used. Then we use that to purchase/short the hedging portfolio. I parse it this way to stress that we have two "traps" on the way to immunization: the key rates are a "design decision" that merely characterize a vector of sensitivity assumptions. Then, we must purchase/short a portfolio against those, and there are infinite choices here. (a hedge is a second portfolio in addition to the underling. To hedge is to add a second position, which is to introduce basis risk). So, a rough answer to your question is: yes! If we hedge against four key rates, that implies we can hedge any movement combination of the four (e.g., one up, three down). But, the more exact answer stems from the same exact limitations as duration itself (i.e., it's merely a linear approximation for each key rate); and further, our key rates make an assumption about the in-between-rate changes. In this sense, there is no reason at all to expect a four- or, for that matter, eight-factor key rate hedge to hold up: there is no reason to expect the actual curve to non-parallel shift only in accordance with the key rates. For example, we can hedge against 5 and 10 year key rates, but maybe the actual yield curve, in addition to shifting 5 and 10, makes an "awkward" non-linear move at the 7-year. Further, there is no reason for the hedged portfolio to react so cooperative to the assumptions. So, we've improved on a single-factor (i.e., entire curve) but only by replacing with several segments of linear approximations, so we haven't solved all of the problems that keep us short of immunization. So, technically, we can't claim full immunization. Here is the relevant Tuckman, IMO: I hope that helps, David