Bond Price Approximation - Key Rate Durations

[mauro_herrada]

So we use duration and convexity to calculate the change in price for a bond with the following formula:

Chg in P = Dur * (Chg in Y) * P + Conv * (Chg in Y)^2 * 0.5 * P

I thought that key rate durations were a tool to manage yield curve more effectively and therefore be able to estimate a change in price more accurately. I used the following formula to calculate this change in price with KR durs:

Chg in P = [ D1, D2, D3, D4 ] * [ Chg in Y1, Chg in Y2, Chg in Y3, Chg in Y4 ] * P
1:4 being the key rate term.

I noticed that when we have parallel shifts we obtain the same price aproximation as with total duration, and that the price approximation is worse when I use key rate durations. Aren't we supposed to obtain a better approximation?

Thanks!

 

[David Harper CFA FRM]

Hi Mauro,

That's interesting, I'm not sure the real advantage of key rates is accuracy (for example, we still aren't including convexity adjustments in Tuckman's key rates), rather it is moving from a single factor (YTM; which effectively treats the whole curve as its flat equivalent) to, in your case, an approximation that responds to four factors. So, I would characterize the key rate as: not particularly improving in accuracy but rather improving in flexibility by reacting to non-parallel shifts.

So, under the the parallel shift "stress test," I think the key rate should be not much better (more accurate) or not much worse (less accurate). Perhaps it is less accurate to the interpolation rule between rates; i could imagine, for example, the rate halfway in-between D1 and D2 receiving slightly less total shock than under the duration, with slight overall loss in accuracy of key rates, compared to the single-factor duration.

Okay, then what's the point? It is to go beyond the single-factor sensitivity (e.g., what if yield shocks by +20 bps? A question limited to the entire curve) to a non-parallel sensitivity (e.g., what if a Fed QE policy depresses the a short-rate but does not change the long rates, for an overall shift).

And, Tuckman's key utilization of the key rates is a hedging portfolio with four bonds, one four each rate (factor). So, while the total durations are approximately equal (DV01 = KR01 + KR02 +...) under a parallel shift, that's just a specific scenario. The real utility is that, if the key rates shift in non-parallel fashion, the hedging porfolio is ready for that (up to a point! it's remain only linear approximations and for only the key rates). I hope that helps,