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Math Help - uniqueness of IRR

  1. #1
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    uniqueness of IRR

    There was a two part question assigned to my class for homework. It's already been turned in, but I couldn't get the 2nd part, and am curious how it is done:

    part a:

    suppose there is a k between 0 and n such that either:

    (i) C_0, C_1, ..., C_k <= 0 and C_(k+1), C_(k+2),..., C_(n)>= 0
    or

    (ii) C_0, C_1, ..., C_k >=0 and C_(k+1), C_(k+2),..., C_(n) <= 0
    show there is a unique i > -1 for which the net present value of this transaction is 0.

    this is the part I got. I don't need help on it, but am just introducing it, because I'm sure it somehow is used to prove the next part.

    part b:

    Let C_0, C_1, ..., C_n be an arbitrary sequence of net cashflows, and let

     <br />
F_0 = C_0,<br />
F_1 = C_0 + C_1,<br />
.<br />
.<br />
.<br />
F_n = C_0 + C_1 + ... + C_n<br />

    Suppose both F_0 and F_n are non-zero, and that the sequence {F_0, F_1, ..., F_n} has exactly one change of sign.

    Show there is a unique i > 0 such that the net present value of these cash flows is 0 (although there may be one or more negative roots).

    This is the part I need help on. I can show what I've done so far, but it really isn't anything except showing there is a root i > 0. I haven't shown that it is unique.

    my work:


    we need to show that the equation has a unique solution 0 < v < 1.

    if v = 0,
    if v = 1,

    we are given F_0 and F_n are non-zero and opposite signs of each other, so by the intermediate value theorem there exists a root v, 0 < v < 1.

    This is all I have come up with, I don't know if that is a good starting point for the rest of the solution or not. any help would be appreciated! Thanks in advance.
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  2. #2
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    Yikes!

    \epsilon \neq 0

    If IRR is not unique, then both (i) and (i+\epsilon) produce the same result.
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  3. #3
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    could you explain what you mean by this a bit more please?

    Thank you!
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  4. #4
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    If IRR is not unique, then there must be some \epsilon \neq 0 such that CashFlows\;Discounted\;@\;i\;=\;CashFlows\;Discoun  ted\;@\;(i\;+\;\epsilon)

    For the simplest case, one cash flow, C, over one year, we have:

    For v = \frac{1}{1+i} and w = \frac{1}{1+i+\epsilon}

    Cv = Cw \implies C(1+i) = C(1+i+\epsilon) \implies \epsilon = 0

    Do we have to say more than that?
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  5. #5
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    I'm sorry, but I'm not sure your answer is valid. I agree that it works with you for the simple case you illustrated, but what about for a more complicated case?

    i.e.

    C_0+C_1*v+C_2*v^2+...+C_n*v^n=C_0+C_1*w+C_2*w^2+..  .+C_n*w^n

    in this case, v = w is definately ONE solution, but is there a way you were alluding to that says it is the ONLY solution?

    * edit*

    for example, consider (and solve) the equation

    2x+3x^2=2y+3y^2

    2(x-y)+3(x-y)(x+y)=0

    (x-y)[2+3(x+y)]=0

    x=y or y=-2/3 -x

    so in this case the solution x = y was not unique.
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  6. #6
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    Hmpf! For a minute there, I thought I had something because your x and y are independent and my (i) and (i+ \epsilon) clearly are not. However, full recognition of the relationship leads to the same dilemma.

    I considered making \epsilon > 0, but that is insufficiently general.

    Perhaps some other kind soul will put me out of my misery while I am thinking about it.
    Last edited by TKHunny; November 1st 2009 at 03:55 PM.
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  7. #7
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    Quote Originally Posted by minivan15 View Post
    for example, consider (and solve) the equation
    2x+3x^2=2y+3y^2
    2(x-y)+3(x-y)(x+y)=0
    (x-y)[2+3(x+y)]=0
    x=y or y=-2/3 -x
    so in this case the solution x = y was not unique.
    That's really a quadratic equation,
    with solution x = y or x = -(y + 2/3);
    and (I think) that due to the "nature" of IRR,
    x = -(y + 2/3) is not valid.

    Anyhow, not being sure, plus confused to no end with those
    scary(!) flows of yours, I asked someone this "fresh" question:
    .................................................. ..................................................
    x^2 + 3x = k^2 + 3k ; x obviously = k, right?

    x^2 + 3x - k^2 - 3k = 0

    x = {-3 +- sqrt[(2k + 3)^2]} / 2

    x = (-3 + 2k + 3)/2 or x = (-3 - 2k - 3)/2

    x = k or x = -k-3

    WHY?
    .................................................. .................................................
    Answer was:
    .................................................. .................................................
    You should ask that question about the first line of your post. Clearly if x = k
    then that makes the first equation true. But why do you think that's the
    only way the first equation can be true?

    Compare to: "x2 = k2 ; obviously x = k right? er but 32 = (-3)2!"

    Perhaps it would be instructive to substitute in x = -k-3 to the equation
    x2+3x = k2+3k and see for yourself that both sides really do match up.
    (You just need to expand the LHS and collect terms.)

    Also even if x = k was the only solution of your first equation, the fact
    you arrived at the end at x = k or x = -k-3 wouldn't be a problem unless
    you also noted that every step was "if and only if". Otherwise you can
    quite easily get surplus solutions at the end. Compare to...

    "Let x = y. Therefore x2 = y2. This has two solutions: x = y or x = -y. Er what gives?"

    What gives is that some of the steps of this argument are only ==> not <==>.
    Hence "x = y or x = -y" is an *implication* of the premise "x = y" but is
    not equivalent. This is why at the end of a "==>" argument you have to
    check that all your solutions are valid and throw out any spurious "solutions".
    .................................................. .................................................

    Amen
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  8. #8
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    I was more done than I thought? Very unsatisfying.

    Of course, this becomes monumentally more difficult as the number of cash flows increases. I'm trying the get DesCartes' Rule of Signes to help us out, but it isn't quite making sense. I can't control the direction of the Cash Flows.
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  9. #9
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    Well, as far as I know, the definition of IRR is:
    "The internal rate of return on an investment or potential investment is the annualized
    effective compounded return rate that can be earned on the invested capital."

    A simple example:
    3 annual cash flows of 2000, 3000 and 5000 at 12%:
    0[-7737] 1[2000] 2[3000] 3[5000]
    This is from (rounded):
    2000/1.12^1 + 3000/1.12^2 + 5000/1.12^3 = 1786 + 2392 + 3559 = 7737

    Not knowing that the IRR is 12%, we then have:
    0[-7737] 1[2000] 2[3000] 3[5000]
    and need to calculate the resulting IRR (IRR = r):
    2000/(1+r)^1 + 3000/(1+r)^2 + 5000/(1+r)^3 = 7737

    We can solve for r, but only by using iteration.
    And we know there is only one possible r.
    So why try and prove it?
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  10. #10
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    Well, somewhere we should generate a sound theoretical basis for our actions. We certainly should make undergraduates work on such proofs.
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  11. #11
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    Quote Originally Posted by TKHunny View Post
    Well, somewhere we should generate a sound theoretical basis for our actions.
    Best way to generate a sound is eat beans
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  12. #12
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    I accept with information:
    Suppose both and are non-zero, and that the sequence has exactly one change of sign.

    Show there is a unique i > 0 such that the net present value of these cash flows is 0 (although there may be one or more negative roots).

    This is the part I need help on. I can show what I've done so far, but it really isn't anything except showing there is a root i > 0. I haven't shown that it is unique.

    my work:


    we need to show that the equation has a unique solution 0 < v < 1.

    if v = 0,
    if v = 1,
    ______________________________
    Calculette pret immobilier taux interet | Calculette prets immo | Calcul credit immobilier prets
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  13. #13
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    1 sign change? See, that is the direction.

    I think it obvious that if all the cash flows are positive and all the discounts are integer multiples of 2, we're done.
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