**Note added by Nicole: Member is referring to a conversation within the YouTube comments and not elsewhere in the forum.**
Hi David,

Let me start by reminding you of our original conversation regarding the calculation of prices with shifts in key rates:

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Hi David, thanks for the video, it's very clear. One question though: Since the bond only has one cash flow, that is, after 30 years, why do shifts earlier on in the curve result in a change in present value of the bond in the 2/5/10 year cases wrt the initial present value? A shift in the 30 year par yield leads to a different discount factor for the (only) cash flow, and thus in that scenario we indeed get a different present value, but if I understand it correctly, in the other cases we should find the same present value every time since we don't shock the par yields affecting the discount factor to be used for the cash flow. Hope that made sense. Thanks!

Hi Bart, yes this is easily the most challenging aspect. I followed Tuckman, as mentioned who illustrates/selects PAR YIELDS as the key rate. We do not need to choose par yields; e.g., if we use spot rates, then of course the 30-year spot rate will be unaffected by (e.g) the shift of the 10-year spot rate. But par yields are advantageous as the key rate WHEN the hedging securities are (approximately) par bonds; i.e., the hedging solution is easier. The downside to par yields (mostly) is what you are pointing out: a shift in the 10-year par yield (even as it implies no shift in the 30-year PAR YIELD per the interpolation) DOES INDEED impact the 30-year spot rate (and therefore of course the 30-year discount factor). This is counterintuitve but is explained (somewhat) in Tuckman, can be discussed further in my forum, or if you like here is my XLS where I perform the actual calculations and you can see why it must be true

https://www.dropbox.com/s/1w22978jn7pkobp/071719-fixed-income-key-rates.xlsx?dl=0 Thanks,

Hi David, thanks for your reply. Is there any way you could give some more intuition about the formula used to compute the “alternative” discount factors which take into account the fact that these are par bonds? I would move this discussion to the forum if not for the fact that understanding this to me seems like a key aspect of getting the entire picture, and other viewers might be interested in this as well.

@Bart S I assume you mean: what is the intuition behind how shocking a 10-year (or 5-year) par yield, as the selected key rate, will impact the 30-year discount factor (or equivalently, the 30-year spot rate)? More generally, how can shocking an X-year par yield impact zero rates that are outside (greater than) its own region? That calculation is shows in the XLS but I can attempt an intuitive explanation by building on Tuckman's, but i'll prefer to do that in our forum and then share the link from here ... let me know if that's the right question (because the other hard question is maybe: why are par bonds better for hedging?)

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So indeed, I am interested in how shocking an X-year par yield can impact zero rates that are outside (greater than) its own region, and more specifically, the calculation (in your spreadsheet, for example cells K34:K92). How did you derive this formula for discount factors constructed from par yields?

But aside from that, I'm also interested in your last point, why par bonds are better for hedging.

Thanks in advance,

Best,

Bart

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