The generalization of this is a martingale. https://en.wikipedia.org/wiki/Martingale_(probability_theory) … Your expected value at the next time step is the same as your present value, but your variance grows over time as you keep reinvesting your winnings. In the long run, you get rich...or go broke.
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A martingale can have a positive expected rate of return but a zero "drift" term (modeled as a geometric Brownian motion, e^(mu*t + sigma W_t), martingales have mu=0.)
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The Kelly Criterion https://en.wikipedia.org/wiki/Kelly_criterion … would say that to maximize your long-run growth, you should not invest in martingales at all -- after all, they have no long-run tendency to grow!
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Replying to @s_r_constantin
Don't think this is right. If you have a martingale like flipping a coin and doubling/halving your money. Kelly says you should be indifferent between investing all in it and all in cash (zero return). But mixed investment will result in positive return with rebalancing.
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Replying to @TrevorVossberg
In the linked article, the optimal amount to invest in an asset is (mu-r)/sigma^2. If mu = r (if the asset has no higher an average rate of return than cash), this implies the optimal amount to invest is zero. What am I missing?
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Replying to @s_r_constantin
The math there is wrong: 1. log((1-f)r)!= (1-f)r. 2. E(log(a)+lob(b))!=E(log(a))+E(log(b)) 3. They seem to be computing based on arithmetic growth i.e. a 5% increase is .05 instead of 1.05 and 5% drop is -.05 instead of .95. This is wrong and log will blow up on neg numbers.
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Replying to @TrevorVossberg
So what is the optimal amount to invest in a stock with positive variance but zero (risk-adjusted) drift?
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Replying to @s_r_constantin
Hmm I believe you are right here. I seem to have misunderstood the definition of a martingale. I'm not sure how one would have a positive rate of return.
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Replying to @TrevorVossberg @s_r_constantin
Since a martingale's expected value is it's current value
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Replying to @TrevorVossberg
What I’m wondering about specifically is an asset whose value is exp(sigma W_t), where W_t is the Wiener process and sigma is a constant. That is, geometric Brownian motion with no drift.
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I initially thought this would have expected value equal to 1, but double-checked Ito calculus & apparently it’s actually exp(sigma^2 t), if I didn’t make a calculation mistake? Would an agent maximizing log-return invest anything in this asset?
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