Convexity with approximate modified duration?

CFA Level I
#1

For the calculation of Approximate Modified Duration, the Schweser Notes mentions:

(Vminus - Vplus) / (2 x Vzero x ∆YTM)

and it also mentions that Approximate Convexity is:

(Vminus + Vplus - 2 x Vzero) / {(∆YTM)^2 x Vzero}

as well as Change in bond price as:

  • (Modified Duration) x ∆YTM + 1/2 x (∆YTM)^2 x Convexity

So does that mean, if I fit in the equation of Approximate Mod. Dur and Approximate Convexity into this, the ∆YTM of the Approximate Mod. Dur will cancel out by multiplying ∆YTM, and the (∆YTM)^2 of the Approximate Convexity would cancel out by multiplying (∆YTM)^2, thus becoming

  • { (Vminus - Vplus) / 2 x Vzero} + 1/2 x (Vminus + Vplus - 2 x Vzero / Vzero)

and since the denominator is both 2 x Vzero, becomes,

{ - (Vminus - Vplus) + (Vminus + Vplus - 2 x Vzero) / 2 x Vzero}

and then the Vminuses cancel out, so it becomes,

(2 x Vplus - 2 x Vzero) / 2 x Vzero???

(Since Vzero is bigger than Vplus that would be a negative number?)

What am I missing here, or is this actually correct?

It seems suspiciously too simple to be true given how confusing the equations given were in the first place…

Can someone help point out where I veered off course?

(I’m aware I’m using Approx Mod. dur instead of the Macaulay Dur / 1+∆YMT way of calculating Mod. dur but still…)

#2

You use one ∆YTM to calculate modified duration and modified convexity, but potentially a different ∆YTM when you’re applying it.

#3

So would that be for example modified duration is calculated using ∆YTM on purchase and when you calculate the bond price, it may be further in the future so multiply that with the ∆YTM at that time? Why wouldn’t one re-calculate the modified duration using the same ∆YTM? I’m a bit confused and an example would be very helpful. Thanks!

#4

Suppose that you have a 10-year, 6% coupon, semiannual pay, $1,000 par bond whose YTM is 4%. If you use ΔYTM = 0.5%, then:

  • P− = $1,209.41
  • P0 = $1,163.51
  • P+ = $1,119.73
  • Modified duration = 7.71 years
  • Modified convexity = 18.14 years²

You can then use the duration and convexity to estimate the percentage price change if the YTM drops to 3% (or ΔYTM = −1%):

%ΔP ≈ −7.71(−1%) + ½(18.14)(−1%)² = 7.80%

(By the way, the actual percentage price change is 8.0803%.)

#5

So in that case, one would not recalculate P-, P0, P+, duration and convexity using ∆YTM=1% for more accuracy?

Or at least for the test, they would just give the duration and convexity as given for questions to derive the bond price change thus I wouldn’t need to worry about that?

#6

Think about what you just wrote.

P- is the price you want. No need to use an approximation if you’re going to do the actual calculation.

#7

ah…true.

so then the point of calculating convexity is… less to do with the actual bond but more about comparing sensitivity between different bonds I guess?

#8

That’s part of it.

The other part is doing one duration and one convexity calculation, then using them to estimate the price change for a variety of yield changes; presumably this is easier than computing the price changes directly.