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make footnotes shorter so they don't go off the side of the page
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gchenfc committed Oct 3, 2023
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Expand Up @@ -34,7 +34,7 @@ This lead me to the conclusion that, as an example, one should take as much debt
However, one hiccup with such a strategy is that a margin call could blow up your entire portfolio. This starts hinting at a gambler's ruin aspect to the problem:
> [...] A persistent gambler with finite wealth, playing a fair game [...] will eventually and inevitably go broke against an opponent with infinite wealth. [^1]
[^1]: https://en.wikipedia.org/wiki/Gambler%27s_ruin
[^1]: [Gambler's Ruin](https://en.wikipedia.org/wiki/Gambler%27s_ruin)

As it turns out, this applies not only to leveraging oneself, but to ordinary bets as well. In the simplest case, the *Kelly Criterion* defines the optimal bet size to maximize long-term growth rate of wealth. It states, roughly, that **for a bet with expected return $$b$$ and probability of success $$p$$, the optimal bet size as a percentage of your portfolio can be computed as $$f^* = p - \frac{1-p}{b}$$.**

Expand All @@ -43,7 +43,7 @@ As it turns out, this applies not only to leveraging oneself, but to ordinary be
Quoting from Wikipedia[^2]:
> In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a bet. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate.
[^2]: https://en.wikipedia.org/wiki/Kelly_criterion
[^2]: [Kelly criterion](https://en.wikipedia.org/wiki/Kelly_criterion)

Notice (if you read the Wikipedia article) that the Kelly Criterion relies on the fact that the utility of money increases with the *log* of money. Although this seems like a pretty arbitrary and strong assumption, realize that:
1. This is actually another way of saying we want to maximize the growth *rate* of our money, since $$\log Pe^{rt} = rt \log P$$ which is proportional to the growth rate $$r$$.
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