Fixed income mapping (Jorion)

[Karim_B]

Hi @David Harper CFA FRM
Apologies for the basic question, but what's the intuition behind the Modified Duration calculation in cell D30 of:
https://learn.bionicturtle.com/FRM/...ets/R36-P2-T5-Jorion-mapping-backtest-v3.xlsx

Using the definition below for Macaulay Duration I was able to replicate the values in the orange highlighted block in the bottom right of my screenshot below, but your way is much faster
https://www.investopedia.com/terms/m/macaulayduration.asp

Screenshot of your file with my calculations added:


However I'm not going to remember your formula easily if I don't understand the intuition behind it.

Thanks!
Karim

 

[QuantMan2318]

Hi @karim

David has used the following formula:

(1/YTM)*(1-(1/(1+YTM)^t))

This is a shortcut for finding out the Modified Duration on Par Bonds where the coupon equals the Yield. It is in fact a shortened version of the formula that you have pointed out.
Modified Duration = Macaulay Duration/(1+YTM)

You can also arrive at David's formula by taking the formula for the Price of the Bond based on cash flows and taking the first derivative of the price w.r.t Yield


Hope this helps
Thanks

 

[David Harper CFA FRM]

Thank you @QuantMan2318 , yes exactly! @Karim_B This formula is given in Tuckman Chapter 4, as below (this is the semi-annual equivalent to QuantMan's. As usual, Tuckman is always showing formulas under an assumption of semi-annual compound frequency, but this is just a special case of a more general formula that can be edited for any compound frequency):

 

[Karim_B]

Thanks @David Harper CFA FRM @QuantMan2318
For 715.2 from here when can I use Term/(1+YTM) for the modified duration calculation versus the (1/YTM)*(1-(1/(1+YTM)^t)) for par bonds as discussed above?

It seems to only return the same results for a 1 year par bond when I vary the inputs, but I'd like to confirm so I know when to use which.

I took the part I exam using another provider (and have seen the error of my ways so don't have the Tuckman original book.

Is there another source which has a recap of how to most easily calculate the duration under different scenarios based on the type of security and its features?

Thanks!
Karim

 

[David Harper CFA FRM]

Hi @Karim_B Right, smart observation. Questions 715.x do not actually assume a flat spot rate curve. You will notice that Tuckman accurately defines the above as an approximation ("as relatively simple approximations"); it is exactly correct only if the spot rate curve is flat (and if so, for a par bond obviously, it must be flat at the same rate as the coupon and yield). So, in my XLS above which you highlight, The Mod D = 4.2124 years = (1-(1+6.0%)^-5)*1/6.0% is exactly correct only when the par bond is consistent with a flat spot rate curve at 6.0%. (in that XLS, the spot rate curve is not flat, so there is a small difference between "Macaulay, Formula" and "Duration (Mac)". (yes, it's true, my 715.X questions required me to coordinate the non-flat spot rate curve and the resultant price equal to par). If the spot rate curve is not flat, the correct answer is the "tedious" approach that time-weights the cash flows to produce the Mac duration and then uses Mac/(1+yield/k) to retrieve the corresponding modified duration; however, the formula (above) is a fine approximation. The duration is thusly, most accurately estimating the small, parallel shift in a spot rate curve that itself is not necessarily flat (it is a yield-based duration, but the yield is just a complex average of the spot rates; in most exam-type questions, the assumption is generally of a flat spot rate curve which negates much of this subtlety!). Keep in mind these duration measures are, by definition, approximations in the first place (e.g., omitting convexity) so it's not a big mistake to approximate an approximation, so to speak.

Re: another source: I'm fond of Veronesi who was previously assigned in the FRM, this is an excellent technical text on fixed income including durations variations https://forum.bionicturtle.com/resources/veronesi-fixed-income-securities.104/

(we are definitely updating the formula sheets before the May exam. Part 1 is in process. We have never been in such good shape vis à vis the syllabus so we expect to delivery high-quality formula sheets that will summarize this).

Thank you for the kind words!

 

[Karim_B]

Thanks again @David Harper CFA FRM
An updated Part 2 formula sheet will be a great help!
Best
Karim