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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Certainly true if you were allowed to only invest in odd years and take out your money in even years!!
But would you want to put your money in that fund for 10 years in a row?
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Can I rebalance at the end of each year? e.g. withdraw some money, leaving some still in, or deposit extra money?
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Even if I can't, the returns still look super attractive:
probability wealth
0.0 4933655.33
0.01 147904.01
0.04 4433.95
0.12 132.92
0.21 3.98
0.25 0.12
0.21 0.0
0.12 0.0
0.04 0.0
0.01 0.0
0.0 0.0
And Kelly says bet ~32% of my wealth
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Although I do agree that if you
(a) require that I can't deposit/withdraw from the fund
(b) extend the time horizon towards infinity
Then the kelly-fraction you should contribute tends to 0
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Yep, fair, I agree with you on all these points! (If anything I suspect most real-world investors will tend to be MORE conservative than Kelly would dictate, and put less in this crazy fund.)
I don’t think Sam agrees with us though
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I do think there are sensible cases to be more aggressive than Kelly, for example
(a) you have or could get income from some other source
(b) your utility (or qwer) is 'less concave' in some sense than log (e.g. maybe you plan to donate most of your money)
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(c) you are a fund, and your investors are contributing only small fractions of their wealth
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yup 's (b) is my answer!
If I were selfish I would be much more risk averse
But the scale of world problems has diminishing marginal impact on the 10b's scale
Oh no, how did we get back to utility?
I prefer maximizing log growth of wealth to growth of EV(wealth) not because of my utility function, but because the former leads to compounding growth and the latter leads to historical charts that look like this:
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Replying to @SBF_FTX @SBF_Alameda and 3 others
OK!! I show you a historical returns chart that looks like this:
docs.google.com/spreadsheets/d
Are you excited to invest?
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You can’t get escape the concept of utility. You’re implying that you assign higher utility to returns that have one shape vs. another.
Every strategy has a probability distribution over outcomes and people differ on which probability distributions they like.
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