CFA CFA Level 1 Convexity Scaling

# Convexity Scaling

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• 8

Hi guys!
I was working through the Equities and Fixed Income textbook, and I came across an example regarding the ‘scaling’ of convexity. The example provided was a with a duration of 5.0 and convexity of 0.235, find the return impact for a 1% (100bps) change increase in spread, which the textbook gave the following working,

Return impact = (-5.0×0.01)+(1/2)(23.5)(0.01)^2

Why was the convexity scaled to 23.5? if it was done to the convexity, mathematically speaking shouldn’t it also be applied to the 5.0 for the duration? or am I missing something here?

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oh wow this just gets weirder and weirder. like the example that mikey linked, has the so called ‘scaled’ convexity. still weirded out. =/

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yeh it is pretty weird, the example was:
“For a bond with duration of 5.0 and reported convexity of 0.235, one would re-scale convexity to 23.5 before applying the formula. For a 1% (100bps) increase in spread the result would be:
return impact = (-5.0×0.01)+(1/2)(23.5×0.01^2)=-4.765%”
the formula in the question is return impact = -(modified duration x change in spread)+(1/2)(convexity x (change in spread)^2).

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Re scaling – the rules around scaling seems completely arbitrary from what they’re saying.

I tried to do more digging around the issue and only succeeded in confusing myself further, as there seems to be a factor of 2 / 0.5 scaling as well (related to what @tingwuwang‌ was also speaking of).

Would @paulpassedtense‌ able to help? Or

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Thanks everyone! This is all a little ridiculous for L1 in by opinion. I don’t think we should have to be busting out our calculus books to figure out what convexity formula to use. Maybe I’m just bitter because I’ve spent so much time trying to figure it out!! Thanks again and good luck to all!

• Anonymous
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I agree, seems like the duration should also x100. Also not sure where the 1/2 term is from. You sure nothing is missing from the question?

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I am a little confused about something similar to your question… I failed L1 band 9 Dec 2013 and studying now for dec 2014…

Last years material showed the equation as ={ (-modified duration x change in spread)+(convexity x (change in spread)^2)} x100.

Now this year it requires convexity to be multiplied by 1/2…. Can anyone comment as to why that is?
With all the changes to FI this year, it seems like they decided to go into much more detail on L1 FI material than they have in the past… :

• Anonymous
• 2

From Mike Carmody with Passed Tense:

The scaling does seem off to me. I think they should have listed the convexity as 23.5

Regarding convexity and the 1/2 formula….
The return impact formula is from the Taylors Series expansion in calculus. It basically estimates the value of a function by using the derivatives. The return impact formula is a second order adjustment because it uses the first derivative (duration) and the second derivative (convexity). It its pure form, the second derivative adjustment is divided by two. (I won’t bore you with the details, but the coefficient is 1/2!) However, sometimes the formula used to calculate convexity is divided by two. Then you don’t need to use the 1/2 factor in the return impact formula.
So it is really a bit arbitrary. You can either divide by two when calculating the convexity or divide by two when using the return impact formula. Just don’t do it twice! Mike has a free video on Convexity on Youtube.
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This was in L2 last year, in all fairness, so it might be a new introduced change to L1.

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@paulpassedtense‌ cheers for the video. I’m still not really getting it with your explanation to be honest, but will watch the video later.

This isn’t really doing much for my CFA self-confidence 🙁

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@hairyfairy the mathematic applications of taylor expansions or taylor series is for the approximation of the function of a particular curve (in this case, it’s the bond price to yield curve). In this case of the return impact, basically what they’re doing is using the duration (i.e. first derivative of bond yield/bond price) and the convexity (i.e. second derivative) to form a function of the return impact.

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@frat8958 the added 0.5 is apparently only used in the context of credit analysis. If we get a duration or convexity question in the december exam, we first have to figure out if it’s in the context of credit analysis, if not then I believe that we omit the 0.5
as for why there’s a difference, i’m hoping for someone on the forums to explain as well!

source: a friend that passed her june exam this year.

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