the Square Root of n
thanks to Jerry S.

Recently, I got e-mail pointing out an error in one of my tutorials ...
>What else is new?

Pay attention. The e-mail also noted that if annual returns are selected at random from some return distribution having
Mean = M and Standard Deviation = SD = S, the cumulative return, over n years, should have a SD = SQRT(n)*S.
>Cumulative return?
What I mean is this:

  • We start with a portfolio worth $1.
  • We select n annual returns r1, r2, ... rn from a distribution with Mean = M, SD = S.
  • We apply these to our portfolio so that, after n years, it's worth $P = (1+r1)(1+r2)...(1+rn).
  • The gain is then $P-$1 so that's the cumulative return (over n years): P - 1 = (1+r1)(1+r2)...(1+rn) - 1.
  • We then look at the distribution of the product by selecting a jillion sets of n returns and, for each set,
    we compute the cumulative return: (1+r1)(1+r2)...(1+rn) - 1.
  • If the individual annual returns are uncorrelated, then these cumulative returns should have a Mean and SD given by:
    Mean = nM and SD = SQRT(n)S.

Note that (1+r1)(1+r2)...(1+rn) - 1 is approximately r1+r2+...+rn which is n times the average of these annual returns.
That means that, starting with $1.00, our final portfolio is approximately $(r1+r2+...+rn).
>Approximately? You said we should have a Mean of nM.
Well, yes, but by "should" I meant when we average over a jillion n-year returns. The average of the errors in using (r1+r2+...+rn) should be 0.
>Example?
Okay, let's consider a portfolio over n = 30 years.
  • We'll generate a thousand sequences of 30 annual returns, the annual returns selected from a Normal distribution with
    Mean = 10% and SD = 20%.
  • For each sequence of 30 returns, we multiply the gain factors, that's (1+return), to get our final portfolio P after 30 years
    (starting with $1.00) and compute P-1 (to get the cumulative return).
  • Then we'll look at these thousand cumulative returns, namely P1-1, P2-1, ... P1000-1, and compute their Mean and SD.

>For n = 30, you'd expect a mean of nM = 30*10% or 300%, right?
Yes, a mean cumulative return of 300%. However, if we got the average annual return each year, a $1.00 portfolio will grow to $P = 1.130 = $17.45 so that P - 1 = 16.45 or a 1645% cumulative return.

>Wow! That's quite a difference! From 1645% down to 300%!
Isn't it? However, we don't get the mean annual retrurn, year after year. It'd be nice it we could, but introducing volatility gives us a lower return.

In Figure 1 we generated a thousand such 30-year annual return sequences and plotted the distribution of final portfolios.


Figure 1