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# Hull, Instructional video , ch4 -Duration

#### Branislav

##### Member
Subscriber
Dear David,
Thanks a lot for video lectures they are much inspiring Still I was little bit confused with all these different names duration, modified duration, Macauly duration,.. etc...I will shortly examine mine view of this and kindly ask you to comment ( but without laughing )
According to mine understanding we are methodologically speaking about one risk measure all the time, called duration - how long on average shall i wait as bond holder to receive cash payments, or in terms on formula as you explained ( formula 1): From this formula we see that that when we say on average, we are referring to time weighted average, so as a result for 3 year maturity bond I will obtain let's say 2.63 - so this is time based measure ( for zero coupon bond it is equal to 3- no cash flows till the very same end, if you can wait that much If you agree with me on this definition than this is the same thing as Macaulay duration, there is nothing new coming with this new name introduced beside of course great honor and memory on Frederick Macauly who introduced this concept.

So we are still on duration and keep playing further. What if we do the first derivative by yield, just to check what is bond's sensitivity on yield change..

Delta (B)=dB/dY*Delta(y) ( formula 2 ) and dB/dY is "similar" to the right side of the formula 1, just with the minus in front , and we need to "remove" B from the denominator, or put it another way: dB/dY=-B*D and using formula 2 we obtain ( let us call it " yield/price sensy formula"):

Delta (B)=-B*D*Delta(y)

For me this was kind of "magic"..somehow mine year based measure D becomes interest ( yiedl) sensitivity measure!
But basically we are still talking about duration from the beginning of the text, just with this simple "math" transformation we saw that it is also connected to the bonds price sensitivity to yield change
We play further ...we assumed above continuous compounding, if we go to the annual compounding, then, bond price is little bit different summation: ( note just the yield y is replaced with i) and duration and its first derivative are little bit different, so their relationship from " yield/price sensy formula"is now transforme to :

Delta (B)=-B*D*Delta(y)/(1+y)
and we introduce new "name" again, "Modified duration" as:
D*=D/(1+y), which transforms previous equation to:
Delta (B)=-B*(D*)*Delta(y)

so again D* is duration from the beginning of the text, just for yearly compounding case, "used" in this formula to express sensitivity of bonds price on yield change.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Branislav Super glad you found some inspiration in the videos: that's what we are here for In regard to your summary: yes, rIght!! All of that looks solid to me, very well done. I agree with virtually all of your statements and I definitely agree with the substantial point that, to paraphrase, that there is really only one duration. I've posted dozens of times on these concepts but here is how I would summarize, and I believe this summary maps pretty well to your math, and further you can see that I'm agreeing that Mac/mod/Effective duration are three faces of the same single concept. Effective duration simply estimates modified duration, and modified duration is a sort of "adjusted" Macaulay duration which adjusts for the effects of discrete discounting on the price and doesn't matter if discounting is continuous (we tend to refer to Macaulay duration as a maturity and modified duration as a sensitivity, but the mathematical units of both, in fact, are years such that is it a minor mathematical adjustment from one to the other, as you do imply!).

Here is how I would summarize:
• Macaulay duration, I'll denote D_Mac, is the bond's weighted average maturity (as illustrated by your first formula above, where the weights assigned to the maturities are the PV of the cash flows; i.e., the weights, t(i), are weighted by c(i)*exp[-y*t(i)]/B.
• modified duration, I'll denote D_mod, is the linear sensitivity of the bond's price with respect to a small yield change; i.e., if the modified duration is 3.5 years, then we (linearly) approximate a yield change of +Δy will associate with a 3.5*Δy percentage drop in the bond price (knowing that curvature/convexity has been ignored). When the yield is continuously compounded, Mac duration = modified duration. When the yield is discretely rendered, we need to adjust the Mac to retrieve the accurate D_mod = D_mac/(1 + yield/k) where k = number of periods per year; i.e., when discrete, D_mod is always a bit less than D_Mac
• In this way, modified duration is a measure of sensitivity: %ΔP = ΔP/P = -(D_mod)*Δy, solving for D_mod:
• D_mod = -1/P * ΔP/Δy or continuously D_mod = -1/P * ∂P/∂y; i.e., modified duration is the first partial derivative (of bond price) with respect to the yield multiplied by -1/P. If we multiply each side by price, P, then:
• P *D_mod = -ΔP/Δy = "dollar duration;" i.e., dollar duration is the (negative) of the pure first derivative (i.e., the slope of the tangent line, itself negative). Importantly, dollar duration divided by 10,000 is the DV01 because P *D_mod/10,000 = DV01.
• However, if you start with the bond price function (either continuous or discrete) and if you take the first derivative, then you can see that you should end up with (the negative of) the dollar duration: ∂P/∂y = -D_mod*P (this forum has dozens of such actual derivations if you search). Therefore, by definition, if you take the first derivative, you should also (as I think you do imply), equivalently end up with ∂P/∂y = -D_mac*P/(1+y/k). There is an old saying: duration is "infected by price" to acknowledge that 1/P "infects" the pure derivative.
• Effective duration approximates modified duration by shocking the yield and re-pricing in order to retrieve the slope of the nearby tangent. Effective duration is sort of mini-simulation used to estimate the (inherently due to it being Taylor Series) analytical modified duration when it is not analytically available (e.g., MBS with negative convexity throws off the analytics). You will really understand when you can see that the effective duration approximates the modified duration which itself is an exact linear approximation. In this way, effective duration and modified duration, although they differ in approach, are both sensitivities and not conceptually different.

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#### Matthew Graves

##### Member
Subscriber
I think I would add a few further points with respect to the practical use of Effective Duration and differences between Effective Duration and Modified Duration.

Modified Duration is sensitivity of the price with respect to the Yield to Maturity. These two measures are precisely defined mathematically and not open to interpretation.

Effective Duration, however, is a more practical (and complicated) measure derived through re-valuation of the instrument. It is the price sensitivity of the instrument to a parallel shift in a valuation curve and is therefore model dependent. Depending on the instrument, this can be a deterministic valuation but could equally be based on monte carlo simulation if the instrument has optionality. If the underlying curve is flat at the yield to maturity and the instrument does not have optionality you would expect the Effective Duration to be very close to the Modified Duration. However, in all practical, real-world valuation cases the Effective Duration would not be equal to the Modified Duration due to the shape of the underlying curve and any optionality in the instrument (e.g. callable bonds). Separately (and rather technically), the shift applied for Effective Duration is conventionally applied to the observed market yields comprising the underlying curve before obtaining the zero rate curve. The shift is not applied to the zero curve directly. This has subtle but observable affects on the effective duration also.