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Bond Fundamentals - Chapter 4 - YouTube
Channel: DNA Training & Consulting
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this is the beginning of part three of
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our module on bond fundamentals which
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discusses the fundamental pricing
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properties of bonds their yield measures
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and their principal pricing
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sensitivities we remind you that this
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part 3 contains chapters 4 to 6 of the
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module with chapter 4 formalizing our
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analysis of duration measures chapter 5
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introducing convexity and explaining
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when it is important and chapter 6
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finally containing 8 quiz questions to
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test your understanding of the materials
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in this module this chapter 4 examines
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more deeply the relationships between a
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bonds duration measures and its coupon
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yield and thinner and summarizes these
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relationships in tabular and graphic
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formats while the formulas for duration
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and modified duration are tedious the
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good news is that each of these measures
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can be found among the financial
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functions contained in Excel and are
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known specifically as duration
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as you can see in this dialogue box if
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you look at the top left here in blue
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and for modified duration beg your
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pardon for modified duration M duration
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as you can see again in the top left of
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this dialog box before proceeding
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however we bring to your attention that
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in this dialog box for modified duration
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the note towards the middle of the
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dialog box says the M duration returns
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the Macauley
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modified duration for a security a term
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we have never seen used anywhere else we
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remind you one really should talk either
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of Macaulay duration or of modified
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duration but Macaulay modified duration
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for us is a first
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returning to our worksheet we can see
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how to use these two functions to
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calculate the duration and modified
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duration first however we need to insert
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in column D and in particular in cell d4
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the settlement date using the date
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function that we previously recommended
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and underneath it the maturity date and
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of course reflecting here the ten-year
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total maturity of the bond as well as
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underneath the coupon rate expressed as
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a percentage the bonds yield to maturity
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its payment frequency frequency and the
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day count
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basis or day count convention now going
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down to cell D 11 we see the calculation
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of duration essentially by means of
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these various cross references to cells
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d4 and d5 the two dates d6 and d7 the
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coupon and the yield respectively and
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finally d8 and d9 the frequency and day
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count convention
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when we hit okay the function reveals a
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duration as calculated on previous
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worksheets of exactly 7.0 six seven zero
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modified duration is calculated right
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underneath and again pretty much using
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exclusively these cross references that
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you see here leads us to the solution of
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6.79 52 again the same as before the
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availability of such a convenient
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function enables us now to investigate
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more formally the relationship between a
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bonds duration measures and its other
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characteristics including in particular
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its tenor its coupon and its yield to
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maturity we turn to the tenor first
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this worksheet duration versus tenor
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shows for an 8% bond trading at par
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which we know from the 8% yield and with
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semiannual coupons its duration and
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modified duration for maturities ranging
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from as little as one year to as high as
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1,000 years
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we regret that we have had to shrink
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this slide so much on account of space
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we make two important observations about
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this worksheet as indicated previously
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duration and modified duration increase
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as thinner lengthens which should not be
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surprising at all if we remember that
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duration at least reflects the average
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time it takes to receive in present
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value terms the cash flows under the
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bond and modified duration obviously
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must follow duration also as previously
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indicated the increase in duration and
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in modified duration as we extend
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thinner is visibly nonlinear rather the
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rate of increase of duration and
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modified duration slows down as we go
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further out and appears to plateau at 13
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in the case of duration and at 12.5 in
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the case of modified duration beg your
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pardon
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in fact it can be shown mathematically
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that as tenor approaches infinity
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modified Eurasian approaches 1 over Y
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where Y is the bonds yield to maturity
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so in this case this would give us 1
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over 8% which is exactly the 12.5
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indicated in this cell j14 from this it
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also follows inverting our old equation
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linking duration and modified duration
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that duration itself at very long tenors
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becomes the 12.5 future valued for one
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period at 8% and therefore becomes 13 in
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the present example the graph appearing
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underneath the table whose x-axis has
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been truncated at 50 years to preserve
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space confirms all of the above
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observations note in particular the
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plateauing of duration pretty much when
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we get to 30 and above years and most
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certainly by the time we get to 1,000
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years
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this next worksheet labeled duration
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versus coupon shows for a tenure bond
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with semiannual payments and a yield to
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maturity of 8% its duration and modified
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duration as we alter the coupon from a
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low of zero to the high of 15%
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we make two important observations about
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this worksheet first as indicated
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previously for any given thinner
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duration and modified duration appear to
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reach their maximum when the bombs
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coupon is zero which again should not be
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surprising at all if we remember that
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duration reflects the average time it
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takes to receive in present value terms
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the bonds cash flows therefore in the
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absence of coupons giving rise to
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earlier cash flows then the principal
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the principal represents the sole care
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flow received during the life of the
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bond giving rise to the maximum duration
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and modified duration as well
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second we observed that both duration
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and modified duration decreased
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substantially at high coupon rates but
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appear to have once again some kind of
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floor below which they do not go for the
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bomb described in this worksheet this
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floor appears to be around 4.5 as you
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can confirm by inserting for example in
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the cell j7 a very high coupon rate such
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as 100 percent
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the graph appearing underneath these
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tables confirms the above observations
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finally this worksheet labelled duration
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versus yield shows for a 10-year 8% bond
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with semiannual payments again its
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duration and modified duration at yield
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to maturity levels ranging from zero all
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the way up to 15% please note in
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contrast to the preceding worksheet here
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we fix the coupon at 8 and allow the
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yield to vary whereas in a previous
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worksheet we did exactly the opposite we
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note the close similarity between the
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relationships appearing on this
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worksheet and the previous 1 duration
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verses coupons here again we see that
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the lower the yield this time the
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greater the duration and the modified
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duration of course
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the easiest way perhaps to remember this
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property is to consider the price
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sensitivity of the eight-person bond in
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a market such as Japan or Switzerland
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where yields are currently very low and
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compare this price sensitivity to a
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market such as Turkey or Brazil where
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yields at times can reach 40 percent or
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more presumably a drop in yields from 2%
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to 1% would have a far greater impact on
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the bonds price
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then a drop from 40 percent to 39
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percent so duration is highest when
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yields are lowest and lowest when yields
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are highest to complete the comparison
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with the preceding worksheet we now set
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the yield in for example the cell j8 at
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100% and observe that this time unlike
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the preceding worksheet there really
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appears to be no floor below this bonds
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modified duration and only a very low
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floor for its duration
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indeed
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at absurdly high yield levels such as
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1000% modified duration pretty much
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collapses completely
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it is common to summarize these various
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properties for duration and modified
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duration in the tabular form you see
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here
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thus when thinner increases the duration
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measures both increase when thinner
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decreases the duration measures decrease
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conversely when coupon increases
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duration decreases and when coupon
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decreases duration increases and in
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parallel to coupon when yield increases
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duration decreases and when yield
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decreases duration increases
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this brings us to the end of this
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chapter for
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