Duration - Convexity graph

[HFlei5545]

What is the convention when plotting Duration/Convexity graph, Price is in the X axis or in the Y axis?
From what I have seen they graph Yield on the x-axis when plotting duration/convexity but I read somewhere that David explained that price determines the yield and not viceversa; so price should be the independent variable meaning X.
Please I'll appreaciate some input on this
thanks

 

[David Harper CFA FRM]

Hi @HFlei5545 That's really interesting. I will say: from a mathematical perspective, I'm not convinced that the convention for the price/yield curve (i.e., price on Y-axis versus yield on X-axis) is necessarily "more correct" than the alternative. Further, it is my belief that:

1. Price does inform yield (and not the other way around). My evidence is that technical factors can instantaneously adjust the price, and because price is effectively an input, the yield changes. In this sense, yield is an "independent" variable that determines the "dependent" price.
2. Yet the convention is certainly to plot duration/convexity in the traditional "price/yield" curve with yield on the x-axis (where the independent goes). Why is that? Well, I think it is simply that yield is the risk factor! And (eg) dollar duration is the slope. Duration/convexity are single factor sensitivities such that we do "flip the script" and consider the implication of a parallel shift in the (yield or spot rate) term structure. So, there is additionally the sense (especially for market risk = shift in risk-free term structure) we also believe that higher rates imply lower prices. In this way, yield as an output is narrowly and technically true, but we also believe that if "rates increase" the "price decrease." But I think the key reason for the graph format is simply that duration is a function of the first derivative (aka, dollar duration): ∂P/∂y. Notice how this definition naturally leads to the graph we use.

I guess the reason I wrote so much is that (i) the math is clear that yield ~ f(price) but (ii) when we get the model and graph, there is more flexibility in interpretation. I hope that's interesting!

 

[HFlei5545]

Thanks!!