To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market return and calculate the exponential growth from that misses the fact that annual returns (such as S&P) have large fluctuations.

To illustrate: portfolio A sees a few consecutive years of negative returns early in its time, B sees it late in its time. With 'everything else equal', A will have smaller absolute losses but B will have a higher overall value at the end. There is a non-zero chance that any portfolio ends up at zero value just by unlucky timing of the negative years. I expect the 84.2% confidence interval on any portfolio to be huge, and I like to get a better insight into that.

Assuming some simple basic numbers, 9% average annual return +/- 16% (1 sigma), from S&P historic data, and assuming the easiest, most realistic, or most pragmatic distribution for those returns.

The average growth is simple: $c_t = c_0 \cdot (1+g)^t$, with $c_0$ the start capital, $g$ the annual return, and $t$ the number of years.

But how do I calculate the expectation and variance when $g$ is a random variable?

I can do this computationally, see chart below for growth of a starting capital of 1 with the above assumptions. The yellow line uses the average return and disregards yearly fluctuations, the blue ones use randomly drawn annual returns, repeating 10'000 times. Red is the exponent of the mean of the log of the value after n years ($e^{|\ln(c_{n})|}$), i.e. the expectation value. Light red lines are at the 84.2% confidence intervals, i.e. 84.2% of the 10'000 repetitions has a value multiplier larger than 4 after 30 years. Interesting here is that the effective annual gain is only 7.7% when the mean gain is 9%. Stock market growth assuming mean returns (yellow), normally distributed returns around the same mean (blue), their expectation value (red) and confidence intervals (light red).

Question remains, how do I derive the red lines in a more analytical way?

There is a related question on the product of variances, but, as pointed out in the comments there, VAR$(g * g * g * g *,...)$ would not be independent variables.

There is a related question on the square of a standard normal variable, but it assumes a zero mean and I can't figure out how to generalise the answer to the non-zero mean case.

Then there is a very related question on the variance of powers of a random variable, but it's about the general case where no probability distribution is known a-priori. In the comments of this question an answer is given:


To calculate $𝔼[𝑋^p]$ the wikipedia page on moments gets me into Confluent hypergeometric functions, which can be calculated as infinite series. Am I in the right direction? I have the feeling it will not be computeable, should I be looking at another approach?

  • $\begingroup$ Research the lognormal distribution -- and consider expressing annual returns in terms of their logarithms. BTW, your "simple" expected value is incorrect. The expectation is substantially larger than what you write. $\endgroup$
    – whuber
    Aug 28, 2023 at 20:33
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    $\begingroup$ In particular note that you can't interchange expectation and nonlinear transformation, and it's important to distinguish between the random variable (the return each year) and its expectation; better choice of notation will help avoid some of these issues. The treatment of the variance, where it seems that $c_0$ just disappears from the formula altogether is also wrong. $\endgroup$
    – Glen_b
    Aug 28, 2023 at 21:46
  • $\begingroup$ @whuber, thanks for the pointer. The only thing I can think of to be always positive is the total value, returns absolute or relative can be negative :-). Ha! Yes, now I realise that the simple 'average return, extrapolate into the future' is also wrong (besides the omitting confidence interval), when considering fluctuations. Some financial people must have touched this subject? I can't seem to find anything other than beautiful exponential growth charts. $\endgroup$
    – Bastiaan
    Aug 29, 2023 at 12:08
  • $\begingroup$ @glen_b I reformulated the question to remove those errors. Do you have any further comments on the notation? $\endgroup$
    – Bastiaan
    Aug 29, 2023 at 12:41
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    $\begingroup$ The lognormal distribution is fundamental in finance. That's a good topic to research for formulating your question. $\endgroup$
    – whuber
    Aug 29, 2023 at 15:02


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