% Change in Bond Price using Duration & Convexity

CFA Level I Fixed Income
#1

Assume a Bond has an effective duration of 10.5 and a convexity of 97.3. Using both of these measures, the esitmated percentage change in price for this bond, in response to a decline in yield of 200 basis points is closest to:

A) 19.05%

B) 22.95%

C) 24.89%

I am getting answer “B”, however the right answer as per the EOC checker is “C”.

Could anyone please explain it?

Thanks

#2

You are probably subtracting the convexity…?

Convexity is always added, look at a diagram and it will make more sense.

#3

Recheck your calculation.

-10.5 × -0.200 + 97.3 × (-0.200)² = 0.2489 = 24.89%

#4

I simply halfed the convexity. isn’t the convexity effect = 1/2 convexity (change in YTM)² ?

-10.5 × -0.200 + 1/2[97.3 × (-0.200)²] = 0.2489 = 22.95%

#5

The reason is probably that the convexity coefficient already factors in the (1/2), so you do not need to when calculating the convexity effect.

You will probably see in your curriculum that: Convexity effect= [(V+) + (V-) - 2(V0)] / 2(V0)(changeY)^2

Sorry for the messy notation but you can see the 1/2 in the denominator…

#6

Read about the controversy involving convesity in this article I wrote: http://financialexamhelp123.com/convexity/

#7

Thanks for sharing! makes sense.

Just note that in the shweiser books there is no mention to the first formula( the one with no 1/2). they only refered to the formula with “1/2” for the convexity effect and they removed “2” from the denominator in the modified convexity formula.

Ironicaly in one of their EOC exercise, the one i shared, they applied the formula you mentioned above. to make things worse both answers are in the multiple choice - makes it even more confusing.

now for the sake of the exam, how should we know when to use which? Like in the case of the question i mentioned above there is no way i would know that 1/2 is already factored in.

#8

I would go with the CFAI standard

#9

So am I correct in assuming: For questions asking the full bond price change with regard to effective duration / effective convexity, use the formula -ModDur x Δ YTM + C x **Δ(**YTM)2 ?

However, for Annual ModDur and Approximate convexity, use:

-ModDur x Δ YTM + 0.5 x C x **Δ(**YTM)2 ?

“You will probably see in your curriculum that: Convexity effect= [(V+) + (V-) - 2(V0)] / 2(V0)(changeY)^2”

I do not see this formula any where? - Please see page 505.

#10

As I wrote in my article, the only place I’ve seen the ½ in the formula to use convexity is in the reading on _ credit risk _; everywhere else that factor is included in the computation of convexity itself.

#11

In the CFA books also i only see this formula : -ModDur x Δ YTM + 0.5 x C x **Δ(**YTM)2

I cannot see -ModDur x Δ YTM + C x **Δ(**YTM)2 anywhere.

#12

S2000magician - In the article you wrote (which was extremely helpful btw):

You say “Unfortunately for CFA candidates, both formulations have appeared in the curriculum. The first formulation appears more commonly , but the second is used in the section on credit risk.”

-Where the first formula (using convexity), i.e. calculating the percentage change in the full price of the bond given a change in YTM. (Not including the 0.5 to halve convexity).

I find this to be the opposite -The second formula for calculating the change in the full price of a bond is used through the CFA curriculum and Schweser sections on FI.

-Where the 0.5 is used to halve convexity.

I have only found the first formula (w/o 0.5) like you say in the credit risk section.

(I am using the 2014 editions of CFA and Schweser)

Maybe as usual I am totally wrong…!?

#13

I haven’t the 2014 CFA Institute books, but in the 2014 SchweserNotes, I find that they include the ½ in the formula for using convexity in _ both _ sections (pp. 93 and 121). So you and I are both wrong.

This is a departure from CFA Institute’s historical calculations (which included the ½ in the convexity value, not in the formula for its use), which explains questin bank problems that don’t include it.

I’ll edit my article accordingly. Good catch!

#14

Hi S2000magician,

i read your blog and have been trying to digest the discussion in this thread. After attempting the EOC questions in the CFAI, i seem to be more confused now.

You mentioned to apply the ½ in the convexity value for questions related to credit risk. However, seemingly straightforward questions (like this below) which are not related to credit risk apply the ½ in the convexity value.

Q. A bond has an annual modified duration of 7.020 and annual convexity of 65.180. If the bond’s yield-to-maturity decreases by 25 basis points, the expected percentage price change is closest to:

A. 1.73%

B. 1.76%

C. 1.78%

The answer C - which applies the ½ in the convexity value.

By the way, what would you mean by questions pertaining to credit risk? Could you cite an example please?

Thanks in advance as always.

Regards,

a feeling distraught candidate.

#15

Ah yes - they do include the 1/2 in the formula for using convexity in both sections!

In the CFA 2014 FI EOC questions I can only see questions asking to calculate the % Δ in the Full Bond Price using Annual modifed duration and annual convexity. - Answers include the 1/2 in the formula for using convexity.

However, I cannot see the equivalent asking to calculate effective duration and effective convexity - only Schweser asks this (Q13) above which does not include the 1/2 in the formula for using convexity?

I would have come to answer A) 19.05% not C) 24.89%

What would you recommend I do in the exam if a similar question comes up?

#16

I just edited my article on convexity (http://financialexamhelp123.com/convexity/), so you might want to go reread it.

The upshot is that CFA Institute now includes the “½” in the formula to use convexity, but doesn not have a “2” in the denominator to calculate convexity. Unfortunately, this is a recent change, so a number of question banks haven’t updated their questions and answers to reflect the change.

On the exam, expect to calculate and use convexity the same way for all fixed income questions: no “2” in the denominator to calculate convexity, but a “½” in the formula to use convexity.

#17

Thanks S2000magician, you’re incredibly efficient! (no wonder you drive an s2000!)

#18

My pleasure.

#19

thanks indeed! now all clear.

#20

You’re welcome.