I could enunciate it, but wikipedia has an explanation. I honestly don’t understand the Wikipedia explanation, but I would expect that it explains my intuitions in a more technical way than I do. If you have a specific point of disagreement, I’m happy to map out my logic and explore the evidence with you. I vaguely remember reading an article on the topic, too.
Optimal bet sizing and expected utility
I’d expect a theorem to maximise utility via diversification would entail some prediction that the utility of subsequent/other/more investments will be greater than the utility of the first/reference investment. If that isn’t the case, it will lower the average expected utility of one’s portfolio. I don’t see the rationale behind the Kelly criterion as it related to any of my existing knowledge about maximising utility.
MPT: How can I have a specific point of disagreement with something as nonspecific as “I am not convinced by modern portfolio theory because no free lunches”? The particular but of the Wikipedia article you linked to actually says (correctly, so far as I can see) that minimising unsystematic risk through diversification (as indicated by MPT) is “one of the few free lunches available” because unsystematic risk isn’t associated with higher expected returns.
Kelley: Actually most of the paragraph ostensibly about this seems to be still about MPT. Anyway, I’m afraid your expectation is just wrong. Diversifying can be a win even if what you diversify with is (on its own) lower-utility. Suppose someone offers you a bet that will pay you $1M if some event E occurs and cost you $900k if not, and suppose you reckon E very close to 50% likely. You probably don’t take that bet because losing $900k would hurt you more than gaining $1M would help you. Now someone else offers you another bet, where you stand to gain $950k and lose $900k. Clearly you don’t take that bet either, and clearly it’s whose than the first. But now suppose the first bet party’s you when E happens and the second very party’s you when not-E happens. The two bets together are a guaranteed >=$50k gain; provided you trust your counterparties you should absolutely take them. So aging the second bet helped you even though on its own it was worse than the first.
Kelley, really: again I’m not sure what I can say to something as unspecific as “I don’t see the rationale”. I suppose I can briefly explain the rationale, so here goes. 1: if the utility you get from your money is proportional to log (amount), which may or may not be roughly true for you (I think it is for me) then placing a Kelley-sized bet is higher expected-utility than placing a bet of any other size at the same odds. (Assuming your utility I’d unaffected by the event the bet I’d on other than through its effect on your wealth.) 2: your long-term wealth is maximized (with high probability, not just in expectation) by making all your bets Kelley-sized, so if your utility is strongly affected by your wealth in the long term and indifferent to the short term then (almost regardless of exactly how utility depends on long-term wealth) you should place Kelley-sized bets.
Most people are more risk-averse than utility proportional to log wealth would justify. If you are, then your bets should be smaller than Kelley. Most people care about the short term as well as the long. If you do, then again your bets should generally be smaller than Kelley.
[EDITED some time after writing when I noticed a bunch of mobile-device autocorrect errors. Sorry.]
How do you get from “no free lunches” to disagreement with either Kelley or portfolio theory?
No free lunches & MPT
I could enunciate it, but wikipedia has an explanation. I honestly don’t understand the Wikipedia explanation, but I would expect that it explains my intuitions in a more technical way than I do. If you have a specific point of disagreement, I’m happy to map out my logic and explore the evidence with you. I vaguely remember reading an article on the topic, too.
Optimal bet sizing and expected utility
I’d expect a theorem to maximise utility via diversification would entail some prediction that the utility of subsequent/other/more investments will be greater than the utility of the first/reference investment. If that isn’t the case, it will lower the average expected utility of one’s portfolio. I don’t see the rationale behind the Kelly criterion as it related to any of my existing knowledge about maximising utility.
MPT: How can I have a specific point of disagreement with something as nonspecific as “I am not convinced by modern portfolio theory because no free lunches”? The particular but of the Wikipedia article you linked to actually says (correctly, so far as I can see) that minimising unsystematic risk through diversification (as indicated by MPT) is “one of the few free lunches available” because unsystematic risk isn’t associated with higher expected returns.
Kelley: Actually most of the paragraph ostensibly about this seems to be still about MPT. Anyway, I’m afraid your expectation is just wrong. Diversifying can be a win even if what you diversify with is (on its own) lower-utility. Suppose someone offers you a bet that will pay you $1M if some event E occurs and cost you $900k if not, and suppose you reckon E very close to 50% likely. You probably don’t take that bet because losing $900k would hurt you more than gaining $1M would help you. Now someone else offers you another bet, where you stand to gain $950k and lose $900k. Clearly you don’t take that bet either, and clearly it’s whose than the first. But now suppose the first bet party’s you when E happens and the second very party’s you when not-E happens. The two bets together are a guaranteed >=$50k gain; provided you trust your counterparties you should absolutely take them. So aging the second bet helped you even though on its own it was worse than the first.
Kelley, really: again I’m not sure what I can say to something as unspecific as “I don’t see the rationale”. I suppose I can briefly explain the rationale, so here goes. 1: if the utility you get from your money is proportional to log (amount), which may or may not be roughly true for you (I think it is for me) then placing a Kelley-sized bet is higher expected-utility than placing a bet of any other size at the same odds. (Assuming your utility I’d unaffected by the event the bet I’d on other than through its effect on your wealth.) 2: your long-term wealth is maximized (with high probability, not just in expectation) by making all your bets Kelley-sized, so if your utility is strongly affected by your wealth in the long term and indifferent to the short term then (almost regardless of exactly how utility depends on long-term wealth) you should place Kelley-sized bets.
Most people are more risk-averse than utility proportional to log wealth would justify. If you are, then your bets should be smaller than Kelley. Most people care about the short term as well as the long. If you do, then again your bets should generally be smaller than Kelley.
[EDITED some time after writing when I noticed a bunch of mobile-device autocorrect errors. Sorry.]