DV01 Computation

[vjoyram]

Hi David,

I notice that you have a different way of calculating the DV01 in your notes. Indeed, Tuckman defines it as follows:
-(1/10,000)*(dP/dy). He also adds: "...the DV01 estimate at 4% does not make use of the price at 4%: the most stable numerical estimate chooses rates that are equally spaced above and below 4%." In your example, if I understand correctly, you mention that the DV01 at 4% is simply the difference between the bond price at a yield of 3.99% and 4%, i.e. $30.21-$30.12=$0.090.

Could you please confirm which one would be appropriate for the exam?

Many thanks.

 

[David Harper CFA FRM]

Hi vjoyram,

The only real difference is that Tuckman, being more exact at the cost of a tad more trouble, is taking the price difference over 2 basis points in order to "straddle" the midpoint (4% in this case). What I mean is:

(Price [@ rate - 1 bps] - Price [@ rate + 1 bps])/(2 bps * 10,000)

compares to "mine" (it's not really mine, you understand!):

(Price [@ rate - 1 bps] - Price [rate])/(1 bps * 10,000)
= (Price [rate - 1 bps] - Price [rate])/(1)
= Price [rate +/- 1 bps] - Price [rate], which is a bit simpler

And, so, referring to table 5.1, if we consider the rate at 4%, and ask, what is the DV01? we can say it is either:

$0.640 [the way i show], or
$0.641 [what Tuckman gets] or
$0.642 [the way i show]

Tuckman's formula, strictly speaking, is better. But (i) mine is easier to do, and (ii) it doesn't matter: all are "correct", from an an exam viewpoint, all are $0.64. And even beyond an exam view, DV01 inherits from duration a 'second-class' status anyhow as "merely" a linear approximation.

My practical advice is to compute DV01 as the dollar difference, per they way i showed you b/c it's easier and it will give almost the same answer. But you can see, you can do it the other way too (shock it twice +1 and - bps) and you will get nearly the same answer.

Just as importantly, IMO it is worth the trouble to understand what is going on here. The Tuckman formula you show there is useful. I like to say the following formula is important/helpful to candidates; i.e., this formula can be used to explain several of the tuckman AIMs:

DV01 = Mod Duration * Price / 10,000

In this case, notice it is basically equivalent to the formula above:

because duration = -dP/dy*(1/P)
DV01 = [-dP/dy * (1/P) * P]/10,000 = -(dP/dy)/10,000

why did i show you that? To show that DV01 is almost the same as duration, except the units are changed and it is "infected" with bond Price. DV01 is duration in different units plus the "price infection."

if, for example, you already solved for duration (performed the duration calculation), then you don't need to use either of the DV01s above, you only need:

DV01 = duration * Price/10,000

...and you will get the same exact answer as Tuckman's because it is a different path to the same place.

David

 

[vjoyram]

Many thanks for your prompt answer David. It does make sense indeed.

 

[karim1]

Hi, it might be a silly question but I was wondering why does your formula provide a negative number? and should it not be *10000 instead of *100000?

Price [@ rate – 1bps] – Price [@ rate + 1bps])/2 bps *100000

thanks!!

 

[David Harper CFA FRM]

Hi Karim,

I think the negative is a (mere) convention, I could be wrong: if you think about the price/yield curve, the dollar duration is the slope of a tangent line. In a vanilla P/Y, the slope is negative and the dollar duration (i.e., the price change for a 1 unit change in yield) is always negative as it is the pure first derivative. This is just "a yield increase corresponds to a price decrease."

Then we divide by price and negate to convert a negative dollar duration into a positive modified duration. However, to use this, we end up negating back; e.g., the est % price change = (-)duration * yield shock.
i.e., mathematically, we could save ourselves the trouble of two negatives and just use a negative duration in the first place.

but note the convention is quite ingrained: when we say a bond has a duration of 7 years, everybody knows that refers to a "negatively sloped linear approximation" (+ yield --> down price) such that a "negative duration" is quite rare (IO strips) and, mathematically, really refers to a positive slope!

Re: 10,000 divisor. Thank you, I corrected my extra 0 above
In case it's not obvious why 10,000: the dollar duration is price change given 1 unit change in x-axis, which is yield. 1 unit = 100% and there are 10,000 basis points in 100% (10,000 * 0.01% = 100% = 1.0).

I personally find this last useful; i.e.,
the first derivative is dollar duration (slope of the tangent) and DV01 merely rescales this dollar duration: DV01 = dollar duration / # of basis point in 100%

Thanks, David

 

[karim1]

I was a little confused before but yea that does make a lot of sense. Thanks for the quick reply, I really appreciate you help.
Karim.