# Bond Price Approximation - Key Rate Durations

Discussion in 'Fixed Income (P1.T4 or P2.T5)' started by mauro_herrada, Aug 16, 2012.

1. ### mauro_herradaNew Member

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!

2. ### David Harper CFA FRMDavid Harper CFA FRM (test)

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,