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Math Help - CAGR and Why are Square Roots Taken?

  1. #1
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    CAGR and Why are Square Roots Taken?

    Hi,

    I am working on a couple of calucalations for a colleague that wants to arrive at a Compounded Annual Growth Rate (CAGR) for a time-series of investment data:

    Total Return (adding 1 to each period's % return and then multiplying them all together to get a Total Return 1.xxxx and the final CAGR calculation which takes the nth root of the Total Return (nth root being the number of % return periods) and subtracting 1.

    I understand mechanically how to do the problem, but I don't understand for Total Return what the formula means: why do I add 1 to each return and then multiply them sequentially - what does that do/mean as "Total Return"?

    Then, for the CAGR calculation, why is the nth root being taken? In other words, and not only for this problem, why is the nth root or square root used - what does it ulimately do to the output of any problem?

    Thanks for helping with my conceptual understanding!
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  2. #2
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    Are you referring to a model like

    A = P\left(1+\frac{r}{100}\right)^n ??

    If so the nth root is introduced when solving for r.
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  3. #3
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    Total Return / CAGR example

    Here is an example I found:

    hp 12c (platinum) - Calculating a Compound Annual Growth Rate : HP Calculator : Educalc.net

    Start with $1,000. In year one you get a 20% return ($1,200 at year end). In year two you go up another 10% ($1,320 at end of year two), down 15% in year three ($1,122), and up 30% in year four ($1,458.60 ending amount).
    Multiply the returns for each year to get the total return.
    1.20 * 1.10 * 0.85 * 1.30 = 1.4586 (or 45.86%)

    Now, all we need is the CAGR. For two years we took the square root. For three years, you would take the cube root. For four years, that's the... quad root or something? I just use my trusty spreadsheet to do these calculations. Spreadsheets and some calculators take roots by using the inverse of the root as an exponent, so a square root is 1/2, a cube root is 1/3, etc. In this example, that's:

    1.4586^(1/4), or 1.4586^0.25, which equals 1.099.

    1.099 - 1= 0.099, or 9.9%.
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  4. #4
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    CAGR is an average of sorts...more specifically, it's the geometric mean of the beginning and ending amounts.

    In your example you started with $1K and after baking in the oven for four years at +20%, +10%, -15%, and +30% your investment pops out at $1,458.60. I.e., your actual investment returns turned the one grand into $1,458.60 in 4 years.

    CAGR answers the question, "What hypothetical return r, if I had instead earned r each year for 4 years, would have produced the same results?"

    To answer that question, you'd first tee it up as

     1,458.60 \ = \ 1,000\left(1+\frac{r}{100}\right)^4 just has Pickslides has shown. And as he's also pointed out, solving for r involves taking the 4th root, since n is the number of years.
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  5. #5
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    Quote Originally Posted by stgeorges70 View Post
    I just use my trusty spreadsheet to do these calculations.
    You can use a simple calculator:
    1000(1 + r)^4 = 1458.60
    (1 + r)^4 = 1458.60 / 1000 = 1.4586
    1 + r = 1.4586^(1/4) = 1.098965....
    r = 1.098965... - 1 = .098965... which is 9.8965...% (9.90% rounded)
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  6. #6
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    Why Square Roots?

    Thanks for the clarifications.

    Also, what I am trying to understand: why are square roots used in may solutions as this? Conceptually what does taking the nth root of some interim calculation / number, like Total Return do / why is this done?

    In a nutshell, I don't understand why squared roots are used in many cases.

    Thanks again!
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  7. #7
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    Quote Originally Posted by stgeorges70 View Post
    Also, what I am trying to understand: why are square roots used in may solutions as this?
    What do you mean by "square roots"? This is a square root: a^(1/2) ; NOTHING else.

    Just remember this rule: if a^p = b, then a = b^(1/p)
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  8. #8
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    Consider this.

    You invest an amount i.e 100 dollars at the rate of 6% per year.

    After the first year you will have 100\times 1.06

    After the second year you will have 100\times 1.06\times 1.06 = 100\times 1.06^2

    After the third year you will have 100\times 1.06\times 1.06\times 1.06  = 100\times 1.06^3

    and so on...

    Can you see how nth roots are now introduced?
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  9. #9
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    StGeorges, you've probably already had your "aha" moment thanks to the foregoing from Pickslides and Wilmer...but I'll just toss one more out there for grins...

    Playing off of Pickslides' example, you know that after three years at 6%, $100 has grown to 100 x 1.06^3 = 119.1016.

    But suppose you don't know the 6% rate beforehand; you just know that 100 bucks became $119.10 over a three-year span. You know only the original investment, the number of years, and the resulting growth amount. You're curious as to what annual rate would have produced that particular growth result (the "CAGR").

    That's simply a matter of 'unwinding' Pickslides' calculation. (Mathematically, such 'unwindings' are usually done in reverse order.) Pickslides' calculation involved two key steps: (1) added '1' to the 6%; then (2) raised to the power of 3.

    To reverse-engineer back to the 6% we'll (1) undo the exponentiation by taking the third root (review Wilmer's last post); then (2) deduct the '1'...

     \left(\frac{119.10}{100}\right)^{1/3} \ - \ 1 = 0.06 = 6%.

    So the CAGR calc will have square roots when two-year growth periods are involved; cube roots if it's a three-year period...
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  10. #10
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    Or if you're given the 119.10, the 6% and the 3 years: 119.10 / 1.06^3 = 100
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