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Bond Fundamentals - Chapter 5 - YouTube
Channel: DNA Training & Consulting
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this chapter 5 teaches you what is meant
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by a bonds convexity and explains why it
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is important to take this convexity into
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account when estimating a bonds price
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changes for large movements in yields
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returning to our earlier table which
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illustrate that the application of
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modified duration which is reproduced
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here for convenience we recall that as
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we allowed yields to move by significant
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amounts from the original level of 8% we
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found our linear approximation for the
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bonds price changes to become
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increasingly inaccurate suggesting a
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curvature in the actual shape of the
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real graph this curvature in the shape
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of the graph is known as convexity and
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becomes increasingly pronounced as we
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move further and further away from the
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original yield level
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it turns out that convexity light
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duration and modified duration has a
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formula we can program into Excel which
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looks like this
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unfortunately Excel does not contain a
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pre-programmed function for convexity so
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we have gone ahead and programmed this
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formula into the next worksheet you're
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about to see labeled convexity the
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similarity of this worksheet to the one
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labeled duration should be apparent
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which is not really surprising given the
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similarity between the formula for
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convexity reproduced here in the bottom
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rectangle for your convenience and that
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for a duration which by now you have
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probably memorized the key differences
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are the following first in addition to
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multiplying the pv of each cashflow by
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what we previously called
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its age ie by T in the formula we also
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multiply each of these terms
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further by T plus one second
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when we sum all these terms and obtain
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the amount shown in this cell g27 we
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then proceed in G 31 to divide the total
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by the square of the discounting
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operator 1 plus y over F hence the
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cross-reference to d7 and third we also
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divide the result that we just reached
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not simply by F times the price where F
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of course is the frequency but rather by
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F squared
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geeks once again may have realized that
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convexity is derived by differentiating
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the formula for modified duration with
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respect to Y once again which amounts to
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saying it is the second derivative of
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the price function with respect to yield
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so now working your way through this
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worksheet and in particular through
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column G which as we said represents in
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each cell the pv of each cash flow times
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its age times its age plus 1 summed
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together in cell G 27 and then divided
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by the appropriate divisors in cell G 31
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you should have no difficulty confirming
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that the convexity of this 8% in your
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bond at a yield to maturity of 8% is
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sixty point one seven as indicated in
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the cell G 31 it is not possible to
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understand such a number intuitively
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unlike duration where the number derived
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from the formula can be intuited either
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in terms of the average time it takes to
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receive in present value terms the
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amount of your initial investment or
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alternatively as a sort of multiplier
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that converts yield movements into
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percentage price changes it suffices to
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understand that the greater this number
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the more you should expect the correct
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graph for the bonds price yield
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relationship to deviate away from the
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straight line representing the linear
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approximation
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based solely on modified duration it is
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not possible to understand such a number
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intuitively unlike duration where the
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number derived from the formula can be
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thought of either in terms of the
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average time it takes to receive in
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present value terms an amount equivalent
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to the initial investment or thinking a
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little more about modified duration as a
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sort of multiplier that converts yield
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movements into percentage price changes
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it suffice us to understand however that
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the greater the convexity number the
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more you should expect the correct graph
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for the bonds price yield relationship
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to deviate away from the straight line
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representing the linear approximation
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involving modified duration
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so now that we have a grasp of the
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equation for convexity can we use the
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number to improve our estimates of
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percentage price changes given a certain
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yield movement the answer is yes as
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evidenced by this formula which begins
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as before but then adds an additional
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term in its last row known as the
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convexity adjustment after the plus sign
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you see towards the middle before
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applying this formula we point out that
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whether yields move up or down the
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squaring of the yield change in the last
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part of the formula ensures that the
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convexity as justement is always
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positive which is a very welcome
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bond characteristic for a bond holder
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while the bond holder will still have
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losses if rates rise the greater the
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bonds convexity the more the actual loss
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will be reduced by virtue of the
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convexity adjustment versus what it
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would have been under the linear
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approximation alone
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and when rates fall and the bond holder
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has gains those gains are magnified
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further by the convexity adjustment with
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the more convex bonds experiencing the
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greatest magnification
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our final worksheet labeled convexity
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adjustment examines the effect of the
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convexity term in the above formula and
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verifies that it improves the price
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change estimate versus our earlier
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linear approximation this worksheet is
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identical to the one labeled mod Durer
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Ellis tration except for the additional
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column D which includes in each cell the
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formula for the convexity adjustment
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appearing here right after the plus sign
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it quickly becomes apparent from an
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examination of columns D and F that this
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time the accuracy of our estimate is
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almost perfect for what is up to a 100
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basis point movement in yield up or down
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and continues to be very accurate when
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the yield movement reaches 200 basis
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points again in either direction it
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would be possible mathematically to
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incorporate yet more terms into our
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equation for the estimation of
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percentage price changes that would
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improve its accuracy even further but
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this is rarely done in practice since we
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have already reached a more than
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acceptable level of accuracy for most
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purposes we inevitably conclude by
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asking how a bonds convexity may be
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affected by our three friends from
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earlier namely the coupon the tenor and
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the yield to maturity
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since unfortunately Excel does not have
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convexity programmed as a function we
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have reverted to this previous worksheet
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convexity and will now proceed manually
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to make changes to the inputs as
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appropriate asking you of course to keep
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your eyes focused on the output for
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convexity in cell G 31 we note on this
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worksheet that first a reduction in
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coupon for example to 5% causes
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convexity to increase while an increase
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in the coupon rate for example to 11%
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does the opposite to convexity we also
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note holding now the coupon constant at
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8% that a reduction in the yield to
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maturity from 8% to for example 5%
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similarly causes convexity to increase
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just like before while an increase in
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the yield to maturity to 11% does the
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opposite to convexity with regard to
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maturity or thinner we would need as you
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can see to rework the entire worksheet
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if we wish to model the convexity of
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bonds with different maturities so in
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the interest of time we will skip the
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step but simply inform you and
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hope that you will believe us when we
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say that the longer the tenor the
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greater the convexity and vice-versa you
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may have noticed that each of these
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relationships runs exactly parallel to
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the ones we had established for duration
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and modified duration so our earlier
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table in which we summarized our results
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can now be expanded to look as shown
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here with identical entries across each
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row of course this completes this
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chapter five and brings us to the quiz
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