[LanguageGamer]

Since I don't see anyone else mentioning this:

The geometric mean (6.9) is all that really matters for investors, not the arithmetic mean (8.4) - the arithmetic mean under-weights the importance of negative years to long term performance.

For example, if the market is down 20% one year and up 20% the next year, the arithmetic mean will be 0%, but you'll be down 4% (0.8*1.2 = 0.96), which is reflected in the geometric mean of (about) -2%.

[Retric]

Depends, dollar cost averaging shifts things around. For a typical 401k style investor having down years mid career improves returns at retirement, but then increases risks in retirement.

[tunesmith]

The average investor also has less money to invest during down years though.

[zeckalpha]

With DCA you have the additional costs of keeping cash around. Unless you mean serial lump sum (investing when you get paid).

[aynyc]

Not completely as typical 401K investor would change their allocations from equity to non-equity.

[YPCrumble]

How would that increase risks in retirement?

[getToTheChopin]

Agreed. Using an assumption of 5-6% annual real total returns is more reasonable for financial planning.

[xapata]

I use a more modest 3.5% real return estimate. I'd rather wind up accidentally rich than accidentally poor.

[fallingfrog]

Correct, because we're averaging together things that are multiplied, not things that are added. Arithmetic mean is rather meaningless here.

[getToTheChopin]

The arithmetic mean gives you a sense of the return you can expect by investing in the market for a single year.

When investing over multi-year periods, the geometric average is more relevant.

You can see the impact on this chart, where the average return (and volatility) drops over longer time periods: https://themeasureofaplan.com/wp-content/uploads/2023/01/Rol...

[danuker]

What really really matters is the Kelly criterion, or expected logarithm of wealth.

If you expect returns to be similar to the past, that would be mean(log(1+return) for every year).

[Galanwe]

Investors tend to think in terms of 1) volatility 2) exposure/diversification.

1) What _really really_ really matters is the Sharpe Ratio, as in "how much returns you get per unit of volatility".

The returns themselves are meaningless if not compared to the volatility to earn them.

Also, you want to discount the risk free rate (at least), as your benchmark.

2) The market as a whole is the biggest exposure you can have, you'd want to discount it as being X% of your portfolio

[dan-robertson]

1. Im not sure that’s what the Kelly criterion is but I didn’t look it up.

2. Arithmetic mean of log returns is the same as the geometric mean of returns. Indeed it’s pretty typical to work with log returns for this reason as adding is easier/better for computers than multiplying. This equivalence is easy to prove:

  gm(returns) = prod(returns)^(1/N)
  log(gm(returns)) = 1/N * log(prod(returns))
                   = 1/N * sum(log(returns))
                   = mean(log(returns))
  gm(returns) = exp(mean(log(returns)))

Where returns is a list of the multipliers to go from the values before/after the returns, eg it has 1.01 not 1%.

[chazeon]

Interesting, I am looking at (1+x) * (1-x) = 1 - x^2 < 1