Hull, Instructional video , ch4 -Duration

[Branislav]

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.
Thanks a lot in advance

 

[David Harper CFA FRM]

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.

 

[Matthew Graves]

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.

 

[enjofaes]

Hi @David Harper CFA FRM . Thanks again for all the material! Was wondering if what you said was correct in the instructional video of duration : around 24'14" Modified duration = dollar duration / 10.000. I thought from the beginning of the video that this is the formula for the DV01.

Kind regards

 

[David Harper CFA FRM]

Hi @enjofaes If I said that (sorry I'll have to locate the specific video location later), then I misspoke. You are correct: DV01 = (P*D)/10,000 = dollar duration/10,000. I'm actually very happy with my summary note at see https://forum.bionicturtle.com/threads/week-in-financial-education-2021-05-24.23840/post-88846 i.e.,

Note: Durations in CFA and FRM compared

I'd like to clarify duration terminology as it pertains to differences between the CFA and FRM. This forum has hundreds of threads over 12+ years on duration concepts (it's hard to say which links are the best at this point, but I'll maybe come back and curate some best links). Our YouTube channel has a FRM P2.T4 that includes videos on DV01, hedging the DV01, effective duration, modified versus Macaulay duration, and an illustration of all three durations. There are many nuances and further explorations, but here my goal is only to clarify the top-level definitions.

I'll use the simple example of a $100.00 face 20-year zero-coupon bond that currently yields (yield to maturity of) 6.0% per annum. If the yield is 6.0% per annum with continuous compounding, the price is $100.00*exp(-0.060*20) = $30.12. If the yield is 6.0% per annum with annual compounding, the price is $100.00/(1+0.060)^20 = $31.18. Unless otherwise specified, I will assume a continuous compound frequency. Special note: we so often price a bond given the yield (where CPT PV is the final calculator step) that it is easy to forget yield is not actually an input. Yield is the internal rate of return (IRR) assuming the current price. Yield does not determine price; price determines yield. Technical (non-fundamental) factors cause price to fluctuate, therefore yield fluctuates.

• ∂P/∂y (or Δp/Δy) is the slope of the tangent line at the selected yield. At 6.0% yield, the slope is -$602.39. How do I know that? Because dollar duration is the negated slope, so in this case dollar duration (DD) = P*D = $30.12 price * 20 years = $602.39. Importantly, the "y" in ∂P/∂y is yield and yield is just one of several interest rate factors.
• 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.
• Macaulay duration is the bond's weighted average maturity where the weights are each of the bond's cash flow's present value as a percentage of the bond's price. Macaulay duration is tedious however it is reliable and it is analytical. When we can compute the Macaulay duration, it is accurate; we don't approximate by re-pricing the bond. A zero-coupon bond has a Macaulay duration equal to its maturity because it only has one cash flow (hence the popularity of the zero-coupon bond in exam questions, never mind the zero-coupon bond is a reliable primitive). Our 20-year zero-coupon bond has a Macaulay duration of 20.0 years.
• Modified duration is the measure of interest rate risk. Modified duration is the approximate percentage change in bond price implied by a 1.0% (100 basis point) change in the yield. Just as ∂P/∂y refers to the tangent line's slope which is "infected with price," we divide by price to express the modified duration, D(mod) = -1/P*∂P/∂y. The key relationship between analytical modified and Macaulay duration is the following: modified duration = Macaulay duration / (1 + y/k) where k is the number of compound periods in the year; e.g., k = 1 for annual compounding, k = 2 for semiannual compounding and k = ∞ for continuous compounding. Importantly, if the the compound frequency is continuous then a bond's modified duration equals its Macaulay duration. Notice that T / (1 + y/∞) = T / (1 + 0) = T.
  • If the 6.0% yield is annual compounded, our 20-year bond's Macaulay duration is given by 20.0 / (1 + 6.0%) = 18.868 years.
  • If the 6.0% yield is continuously compounded, our 20-year bond's modified duration is 20.0 years.
• Effective duration is an approximation of modified duration. Recall the modified duration is a linear approximation, but that's because it is a function of the first derivative; otherwise, modified duration is an exact (analytical or functional) measure of the price sensitivity with respect to the interest rate factor that happens to most often be the yield. We can retrieve it easily whenever we can compute the Macaulay duration, which is the case for any vanilla bond. Otherwise (e.g., bond has an embedded option) we approximate the modified duration by calculating its effective duration. The effective duration approximates the modified duration which itself is a linear approximation. The effective duration is given by [P(-Δy) - P(+Δy)] / (2*Δy) * 1/P. I wrote it this way so you can see that it is essentially similar to ∂P/∂y*1/P where ∂P/∂y ≅ [P(-Δy) - P(+Δy)] / (2*Δy). I've observed that many candidates do not realize that the formula for effective duration is simply slope*1/P. Geometrically, it is the slope of the secant line that is near to the tangent line! Secant's slope approximates the tangent's slope. If you grok the calculus here, I think you'll agree that this is all just one thing! Now we can see how it's not so different. But as you can visualize, there are an almost infinite variety of secants next to the tangent. We arbitrarily choose a nearby secant, but we'd prefer a small delta if the bond is vanilla (i.e., if the bond's cash flows are invariant to rate changes). Although we do not need the effective duration for our example bond, we can compute it:
  • If our arbitrary yield shock is10 basis points such that Δy= 0.10%, then P(-Δy)= $100.00*exp(-5.90%*20)= $30.728, and P(+Δy)= $100.00*exp(-6.10%*20)= $29.523. Effective duration= ($30.728 - $29.523)/0.0020 *1/$30.12= 20.0013 years. Fine approximation!
• On the terminology (CFA versus FRM)
  • Interest rate factor: The FRM (informed by Tuckman) starts with a general interest rate factor. This is typically the spot rate, forward rate, par rate, or yield. Importantly, the spot, forward and par rates are term structures, or vectors; the par yield curve is a vector of par rates at various maturities, often at six-month or one-month intervals. Only the yield is a single (aka, scalar) value.
  • My above definition of the effective duration is according to the FRM (and to me). The CFA sub-divides this effective duration into either approximate modified duration (if the interest rate factor is the yield) versus effective duration (if the non-vanilla nature of the bond requires a non-yield interest rate factor; i.e., a benchmark yield curve). Personally, I am not keen on this semantic approach because (i) both of these CFA formulas are approximating the modified duration and (ii) I prefer to reserve "effective" for its traditional connotation (e.g., effective convexity is analogous to effective duration), and (iii) we wouldn't anyhow use an inappropriate factor (yield) for certain non-vanilla situations, so we don't really need label-switches to guide us thusly! (the CFA's formula for its approximate modified duration is essentially the same as its effective duration formula). To me, the CFA's approach muddies the terms "approximate" and "effective" where the math gives us natural distinctions. Follow the math, I'd say!