Once upon a time, while roaming Morningstar, I larned something neat:

A buy-and-hold investment will then grow by a factor G₁ = 1 + R₁ in the first year, then by a factor G₂ = 1 + R₂ in the second year etc. ... and eventually by a factor G_N = 1 + R_N (during the last of N years).

The growth over all N years is just G₁G₂G₃...G_N and if our Annualized Gain Factor is called simply G, then

(1)           GN = G1G2G3...GN

>A "buy-and-hold" investment? Why do you ...?
The annualized gain isn't that easy to calculate if you're putting money in and/or taking money out, like Dollar Cost Averaging, for example. So, to make things simpler, we'll just invest $1.00 and leave it there - and watch it grow for N years. Otherwise, you'd have to take a look at the Rate of Return stuff ...

>Please continue.
Okay. The Average (or Mean) Gain Factor is just:

(2)           A = { G1+G2+G3+...+GN }/N = (1/N)Σ Gn = (1/N)Σ (1+Rn) = 1 + (1/N)ΣRn

For convenience (later!), we'll put:

(2A)          G_n = 1+R_n = A(1+r_n)
so r_n = G_n/A-1 measures the deviation of the Annual Gain Factors from their Mean. That'll change Equation (1) into:

(1A)          G^N = A^N (1+r₁)(1+r₂)...(1+r_N)

Further, the Standard Deviation is given by {the Mean-of-the-Squares} - {the Square-of-the-Mean}:

(3)           SD2 = (1/N){ G12+G22+G32+ ...+GN2} - A2 = (1/N)ΣGn2 - A2 = A2{ (1/N)Σ(1+rn)2 - 1 }

Our objective is to see which of G and A is larger and ...

>But you already said that A is larger than G, in Part I.
Yes, but now I'd like to estimate how much larger ... with a wee bit o' math.

>Wake me when you're done. Let's first take the logarithm of each side, in Equation (1A). That'll give: (4)           N log(G) = N log(A) + Σlog(1+rn) Now we use a magic formula, namely: (5)           log(1+x) = x - x2/2 + x3/3 - + ... which, as you can see, is pretty good when x is small >That's assuming you're using natural logs, to the base e, so why don't you say LN instead of .... Go back to sleep.

Anyway, we get:

(6)           N log(G) = N log(A) + Σ { r_n - r_n²/2 + r_n³/3 - + } = N log(A) + Σ r_n - (1/2) Σ r_n² + error₁

Now we recognize some of the stuff here. In fact, from Equation (2):

• A = (1/N)Σ G_n = A(1/N)Σ (1+r_n) = A + A(1/N)Σ r_n     'cause Σ1=1+1+1+... = N
  Hence:   Σ r_n = 0

• SD² = A²{ (1/N)Σ(1+r_n)² - 1 } = A²{ (1/N)Σ(1+2r_n+r_n²) - 1} = A²{ 1 + (2/N)Σr_n + (1/N)Σr_n² - 1}
  Hence   SD² = A²(1/N)Σr_n²   'cause Σ 1 = N and we already know that Σr_n = 0.
  So we get   Σr_n² = N SD²/A²

>zzzZZZ
We can now substitute for Σr_n and Σr_n² in Equation (6), and get (after some fiddling):

(7)           log({G/A}²) = - SD²/A² + (2/N) error₁

Now we use the magic formula (5) again, in the form:

log(z) = (z-1) - (z-1)²/2 + - ... = (z-1) - error₂     with z = {G/A}²

which gives (finally!):

{G/A}² - 1 = - SD²/A² + (2/N) error₁ + error₂

and (finally!) ...

>You already said finally!

... and finally, our approximation:

or, in terms of Average and Annualized fractional gains (a 12.3% percentage gain means a 0.123 fractional gain and 1.123 gain FACTOR ... okay?):

(1 + Average)2 = (1 + Annualized)2 + SD2 + ...

>zzzZZZ Since we have an exact result, from Equation (3), namely: SD2 + A2 = (1/N)ΣGn2 where A = average(Gain Factors) = 1 + Average(Gain Fractions) and the approximate result, above, namely: G2 + SD2 = A2 where A = average(Gn) = 1 + Average(Rn) we get a neat approximate picture of the geometric relationship between the Geometric Mean G and the Arithmetic Mean A and the Standard Deviation SD and ...

>Finally! A picture!

Here's another, using the approximation generated above:

>Yeah, but just how good is the approximation?

Good question. Here's a collection of annual returns, over ten years. For each we calculate the Exact annualized return and the Approximation and note the Error = Exact - Approximation:

>I've seen the approximation: Average = Annualized + SD²/2

Well, let's see. Our formula, above, is:
        (1+Average)² = (1+Annualized)² + SD²
and, if the returns are small, we'd get:
        1+2(Average) = 1+2(Annualized) + SD²     'cause (1+x)² is approximately 1+2x
and that'd give
        Average = Annualized + SD²/2

>Both are approximations, right?

Right, but I like mine better.
Suppose, for example, we have two returns: +50% and -50%. Then we'd have:
        Actual Average = {(50%)+(-50%)}/2 = 0.00%
        SD² = {(50-0)²+(-50-0)²}/2 so SD = 50%
        Annualized = {(1+0.5)*(1-0.5)}^1/2 - 1 = -0.134 or -13.4%

so we compare:
        Approximation#1 = Annualized + SD²/2 = - 0.9%
and
        Approximation#2 = SQRT{(1+Annualized)² + SD²} - 1 = 0.000     or     0.00%    which agrees with the actual Average.

>You win!

Well ... thank you.
In the meantime, I'll leave you with the Mean Annual Return for the S&P 500 (Averaged over a time period from some time in the remote past to the present ... well ... May, 2001) and the actual Annualized Return - again Annualized over the same time period - and the Approximation#2 (using the Standard Deviation over the same time period):