Well, here’s a place to start: what’s the appropriate way to measure average annual returns for figuring out if a fund is good at its job?
1) Arithmetic mean of annual returns
2) Geometric mean of annual multiplicative returns
Conversation
How are you defining 'annual returns' and 'annual multiplicative returns'?
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For annual returns I mean W_{t+1}/W_t - 1 (i.e., +60%, -40%)
For annual multiplicative returns I mean W_{t+1} / W_T
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So "geometric mean of annual multiplicative returns" is just (final wealth)^1/N
"arithmetic mean of annual returns" converges to the EV of a single year
If we used log(W_{t+1}/W_t), then arithmetic mean is log(final wealth^1/N)
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I guess there are a few things that may affect which makes sense:
(a) can I rebalance how much money I have with this fund each year?
(b) is the fund basically all my wealth, or just a tiny amount of it?
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You’re profiling the fund for a magazine. Over the last twenty years, the average annual return of is +140%, and the geometric mean return is -20%.
Which number better describes the investors’ skill? (Note that the latter number means the fund must be -98.7% over the period).
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Well, I think this depends on what the investor is trying to do (and probably also on exactly what the returns were)
Here's a case where this seems really good:
pick GM(x, y) = 0.8, AM(x, y) = 2.4
so x = 0.14 and y = 4.67
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Now suppose the fund gained 367% in odd years and lost 86% in even years - this would give -20% geometric returns, and 140% arithmetic returns
I think this is spectacular and would happily invest some of my wealth with this investor
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"the fund must be -98.7% over the period" might be true in a technical sense
but maybe the participants are themselves Kelly betting, and so withdrawing from the fund in good years and contributing after bad years
So the AUM needn't be down -98.7%, and might be up a lot
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Ok, granted—a fund this volatile could be very valuable as a component of someone else’s Kelly-like strategy.
But Sam was saying he wants to go all in on high-EV bets like this! (Or would if max total wealth in the world wasn’t a bound on his total winnings)
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yeah that SBF guy is aggressive!
but you don't have to be