Difference between DV01 and Duration

[sudeepdoon]

Hi,

I am bit confused bit the definations ofr DV01 and Duration.

I understand that Duration is the percentage change or rate of change of security value with the rate (I guess Interest Rate);

where DV01 is the change in the value of the security for a change of 1 basis pt of interest rate (I am sure but I guess I read somewhere that it can be any factor that effects the value and depending on the factor a specific name is given ..). Most of the reads give the example for interest rate.

I am not very sure of the following:
1. Is DV01 is wrt the change in interest rate; or is it generic
2. If it is only for "Interest Rate"; then the information given by both the terms are similar..

Can you please clarify this...

 

[David Harper CFA FRM]

Hi Sudeep,

You raise a valid point that is found by a careful reading of Tuckman. Strickly speaking, the DV01 is generic: the (absolute) dollar value change given a one basis point change in the "interest rate" where "interest rate" can refer to various metrics. For example, if we think about pricing a bond against an upward sloping term structure (where each bond cash flow is discounted at the appropriate zero spot rate), the we can refer (generically) to the dollar value in change of bond price if we shift the entire term structure down by 1 bps (so that each cash flow is discounted by: previous zero rate - 1 bps). But this is not how we use in the FRM; for practical (exam) purposes, this is a distraction...

What's important is that DV01 and duration are crude "single factor models." Notice above that we shift the entire structure by 1 bps; with a single factor metric we are "stuck" with a parallel shift. To be more realistic would require additional factors.

As Tuckman says, he means yield-based DV01 and, in the FRM for practical purposes, we are always referring to (really) a yield-based DV01. By yield, I mean, yield-to-maturity. And yield-to-maturity, by definition, implies a flat yield curve. (i.e., yield/YTM is flat yield curve that is implied by the bond's price which, in turn, is informed by a non-flat term structure).

(as a detour, this point is very important.
Please see 4.c.3 YTM learning XLS https://www.dropbox.com/s/1t2oj9cwv7fvpir/4_c_3_ytm_forward.xls
it is important to see that yield (YTM) is single-factor flat term structure that equates to the more realistic spot and forward curves.
It takes a bit of study to see this, that the YTM is a single-factor equivalent to a multi-factor term structure. If we shift or shock the multi-factor term structure under a duration/DV01, it must mean that all rates shift the same: a parallel shift)

Okay, so now we have established:
* the zero term structure can be any shape; e.g., upward sloping
* The yield (YTM) is the "unrealistic" flat term structure that is associated (i.e., gives the same bond price)
* The DV01 can refer to any 1 basis point shift in "interest rate"
* But we (FRM) refer to the yield-based DV01: the change in price given a 1 bps change in the "yield" (YTM)

And, in this regard, the difference between DV01 and modified duration is *merely* units. The most important formula, for our purposes, is:

DV01 = Price * Duration / 10,000, or more exactly:
(yield-based) DV01 = Price * (Modified) Duration / 10,000

both give the (linear, approximate) estimate of bond price change for a shift in yield, DVO1 (in $, for 1 bsp), modified duration (in % terms, for 1 unit change). You can see your "confusion" IMO is more related to astute observation, but it's easier to stick with the flat YTM for our purposes.

David

 

[Ryan S]

Hi David, looks like the above link no longer works... can you pls provide a new link or tell me which XLS bundle I can find this in?
Thank you!
Ryan

 

[sid092001]

Why are we dividing it by 10000?

 

[David Harper CFA FRM]

@sid092001 Price * modified duration = dollar duration, which is a first partial derivative: the change in price approximated by a one unit change in yield. One unit = 1.0 = 100%, which is why dollar durations are awkardly large. The DV01 is based on the same linear approximation but is one one basis point which is 0.01%. So dividing by 10,000 is re-scaling from 100% to 0.01% ( = 1 bps) as 100%/.01% = 10,000. Thanks,

 

[wahahahaha]

Hi @David Harper CFA FRM, I would like to quote from https://www.bionicturtle.com/modified-duration/:
"if modified duration is 6.0 years, then we estimate (approximate) a price drop of 0.60% for a 10 basis point increase in the yield."

From the quote and my readings thus far, given mod duration of 6 years, we could infer that bond price will change by ~6% per 1% (100bps) change in yield, which hence make sense to say ~0.60% change per 10 bps change in yield.

Consequently, it seems to me that mod duration is % change / 1% change in yield. In this case, I don't know how to reconcile my observations here with the correct scaling of 1/10000 to convert mod duration to DV01. It seems to me that the conversion is only to multiply 1/100, since 1% = 100bps.

Clearly, you have explained above, but I dont know how to reconcile my thoughts here with your explanation above.
in particular, I don't see how 1 unit yield is 100%, as it seems to be only 1%.

I hope you can enlighten me on this. Sorry for the trouble.

 

[David Harper CFA FRM]

Hi @wahahahaha Please see my recent post at https://forum.bionicturtle.com/threads/week-in-financial-education-2021-05-24.23840/ including the second bullet ....

• Dollar duration (DD; aka, money duration in the CFA) is analytically the product of price and modified duration. Dollar duration (DD) = P*D = $30.12 * 20 = $602.39. Why is it so large? Because it's the (negated) tangent line's slope, so it has the typical first derivative interpretation: DD is the dollar change implied by one unit change in the yield, -∂P/∂y. One unit is 1.0 = 100.0% = 100 * 100 basis points (bps) per 1.0% = 10,000 basis points. So, DD is the dollar change implied by a 100.0% change in yield if we use the straight tangent line which would be a silly thing to do! Recall the constant references to limitations of duration as linear approximation. The linear approximation induces bias at only 5 or 10 or 20 basis points, so 10,000 basis points is literally "off the charts" and not directly meaningful. What is meaningful? The PVBP (aka, DV01) comes to our rescue with a meaningful re-scaling of the DD ...
• Price value of basis point (aka, dollar value of '01, DV01) is the dollar duration ÷ 10,000. It's the tangent line's slope re-scaled from Δy=100.0% to Δy= 0.010% (one basis point). PVBP = P*D/10,000; in this example, PVBP = $30.12 * 20 / 10,000 = $0.06024. It is the dollar change implied by a one basis point decline in the yield. It is still a linear approximation, but much better because we zoomed in to a small change. In this way, the difference between the highly useful PVBP and the dollar duration is merely scale.

... where I explain exactly why DV01 =P*D/10,000. Duration is not divided by 10,000. Dollar duration is divided by 10,000 because 1.0 unit of yield (the x-axis) is 100% and 100% is 100*100bps = 10,000 bps. Thanks,

 

[yLam4028]

@sid092001 Price * modified duration = dollar duration, which is a first partial derivative: the change in price approximated by a one unit change in yield. One unit = 1.0 = 100%, which is why dollar durations are awkardly large. The DV01 is based on the same linear approximation but is one one basis point which is 0.01%. So dividing by 10,000 is re-scaling from 100% to 0.01% ( = 1 bps) as 100%/.01% = 10,000. Thanks,

in case someone still confuse why we divide 10000 here or what does linear approximation mean:

the nominator is change in value per 1 basis point e.g. $1
divide it by 1 basis point again would yield change per 10000 basis point = 1=100% i.e. $1 / 0.0001 = $1*10000 =$10000
If we divide the change per 1 basis point by 10000 and also divide 1 basis point i.e. $1/ ( 0.0001*10000) , the result would stay as $1
So basically the divide by 10000 is to obtain the original value of change per 1 basis point.

Why the redundancy? if the change at nominator is a change in price due to 2 basis point, the formula will resolve to change/2 = 1 basis point change=DV01

 

[David Harper CFA FRM]

@yLam4028 yes, exactly, agreed. Thank you!

 

[SKuma2148]

Hi @David Harper CFA FRM - I read in the text regarding the difference between DV01 and Duration is that DV01 decreases as rates increase while Duration increases as rates increase because "P" (value) which is in denominator decreases rapidly as rates increase.

However the formula is Dmod = - 1/P (Delta P/Delta Y) and DV01 = - Delta P/(10000 * Delta Y)

Why are we ignoring the negative sign here to infer the increase/decrease?

 

[David Harper CFA FRM]

Hi @SKuma2148 I'm not sure to which text you refer, sorry, but duration, DV01 and KR01 all have the same signage (+/- ) dynamic; i.e., their "natural state" is a positive value to reflect the inverse relationship between price and yield. You can explore the detail in this technical note at https://forum.bionicturtle.com/threads/week-in-financial-education-2021-05-24.23840/post-88846

The most common confusion, it seems, relates to the fact that the first derivative, ∂P/∂y, is itself negative: the slope of the P/Y curve tangent is obviously negative given their inverse relationship. The "negative" in Dmod negates that mathematical negative to render a positive so that a positive value reflects the natural state. Thanks,