Duration

[wrongsaidfred]

Hi David,

If we are asked to calculate the price change or percentage change of a bond, should we use the effective duration or the modified duration? It seems like we could use the effective duration to calculate DV01 but I do not see any mention of it.

The same question goes for calculating the duration (or DV01) of a portfolio of bonds.

Any advice would be greatly appreciated.

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

It's a good question, as usual.

From a purely practical view (i.e., Nov 2011 exam), an FRM candidate does not need to worry about the difference between effective/modified duration. You are absolutely correct there is no formal review of "effective duration." This AIM ("Define, compute, and interpret the effective duration of a fixed income security given a change in yield and the resulting change in price.") is incorrect and should read ("Define, compute, and interpret the modified duration of a fixed income security given a change in yield and the resulting change in price.") as Tuckman Chapter 5 has no review of effective duration (or anywhere in assigned FRM). Bottom line: the exam has so far only distinguished between modified and Macaulay duration.

More generally, IMO, the effective is always safe because it is the simulated (think "full revaluation") version of analytical modified duration, which allows it to handle embedded options. That is, modified duration is linear approximation based on first derivative, dP/dy*1/P. This fails with the non-linearity of embedded options, so we can "resort" to an simulated (effective) version.

It is worth noting that all the effective duration is doing is re-pricing the bond, based on up and down shocks, to arrive at the same linear approximation as the modified duration. Rather than analytically, it is just estimating the tangent by way of a secant line, basically. If there are no embedded options, it reduces approximately to the modified duration anyway. So, I view it as a generalized, simulated (full revaluation) case of the modified, always safe and required when the analytical (modified) duration fails (embedded options) but redundant otherwise.

I think the only useful meditation is to note, as above, the difference between analytically deriving the modified duration versus "simulating" the effective duration by "full repricing" the bond at shock up and shock down.

Re portfolio: not an issue because both modified & effective share the key in common: linear approximations. In this way, just as the option greeks (e.g, portfolio Position delta is a linear sum of individual Position deltas), portfolio dollar duration is a simple summation of individual position dollar durations (or, equivalently, portfolio duration = weighted sum of individual durations). And this applies to both modified/effective duration, but with the attendant weaknesses of the linear approximation.

It's a long way to say that, at least in the 2011 FRM, the difference shouldn't matter (these terms are on my "lobby list", hopefully GARP will consider my recommendations to explicitly parse or remove "effective").

Thanks, David

 

[wrongsaidfred]

Thank you for the detailed explanation.

Mike

 

[Ankur S]

Maybe lame but couldnt understand.

david,
In your study noted the formula for Duration(effective) is divided by 2 whereas in GARP core readings Chapter 9 there is no mention of division by 2.
What are you assuming with 2 in denominator?

Thanks!

 

[wrongsaidfred]

Ankur,

What is the "chapter 9 reading" you are referring to?

Thanks,
Mike

 

[Ankur S]

Hi Mike,

I am referring to Core Reading provided by GARP (In addition to BT i bought the 4 books from GARP). So the book on Valuation and risk model has ch 9 based on "One factor measure of Px sensitivity" and in there the Duration they just calculate Px %change over yield change with a negative sign ( no mention of 2). Hence my confusion.

Thanks!

 

[wrongsaidfred]

I have the Tuckman book at home. I wil check it tonight and get back to you.

Mike

 

[wrongsaidfred]

I just looked in the book and there actually is an implied 2 in the denominator. If it is the exact same reading, notice that the yields in the denominator are 0.02 apart instead of 0.01. This is because of a 1 bp move up and a one basis point move down. the "delta y" is only 1 bp because it went up by 1 and down by one.

Does that make sense?

Mike

 

[Ankur S]

got it...so 2 to cover the yield gap where the delta is just 1 bp.

Many thanks Mike!!

On the side:
I probably know from past forum readings that FRM exam don't provide formulas cheat sheet. I just wanted to confirm this. Do they provide formulas or dont?
as memorizing all these formulas is becoming a pain..looks like a test of my memory rather than concepts.

Thanks again!

 

[David Harper CFA FRM]

Hi,

Just to support Mike's point, Effective duration is approximating Modified duration (but, as above, only if the Taylor Series is appropriate).

As modified duration is dP/dy*-1/P, effective duration is using slope*-1/P, where slope is rise/run (of the secant line).

So, the (2) is needed or not depending on "ratio consistency:" we just want the same total yield delta in the denominator that informs the numerator, as this ensure a slope estimate.

For example, the common formula has numerator where "rise" is the difference between Price @ Yield(0) - Shock and Price @ Yield(0) + Shock; e.g., where shock = 20 bps or 50 bps; in this case, the "rise" (from the lower bond price to the higher bond price) is spanning 2*shock, so we need that in the denominator.

Alternatively, there is nothing wrong with numerator where rise is difference between Price @ Yield(0) and Price @ Yield(0) - Shock; in the case, the rise spans only 1*shock so denominator only wants 1*shock.

For the visually minded (like me!), this should make sense if you see that effective duration is just retrieving the slope of the secant line (~ tangent) line.

Thank, David