nisiprius wrote:So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
Basically if you have a strategy with a risk/reward tradeoff that might make sense, then for some people it might make sense to have another strategy with more risk and more reward, and for some people it might make sense to have another strategy with less risk and less reward, but it doesn't make sense to accept more risk and less reward.
The Kelly criterion maximizes reward. But it's perfectly sensible for some people to accept less risk and less reward, e.g. if you have enough saved up.
A big problem with this kind of discussion is that the precise statements are awkwardly long and everyone
starts to shorten them... and changing their meaning. As I presently understand it, the Kelly criterion does not "maximize reward." It maximizes the expectation
(terminal wealth). I haven't quite made the jump yet to how you make a precisely
equivalent statement that includes the phrase "growth rate," but I think it maximizes the expectation of the logarithm of the CAGR. And, indeed, it does
do that, see chart below.
However, the choice of fraction to bet also
affects the dispersion of results. So Thorp is trying to get at some analog of the efficient frontier for compounded situations, and he says about "So, given any admissible strategy, there is a fractional Kelly strategy with 0 ≦ c ≦ 1 which has a growth rate that is no lower and a variance of the growth rate that is no higher." And I think but I am not sure that he is misspeaking slightly and means "variance of the LOGARITHM of the growth rate."
So, there is some kind of risk-return tradeoff, and I am trying to understand what it is. We are already mixing a riskless asset--literal cash, 0% return, 0% standard deviation--with a risky asset. I'm not sure how we draw the efficient frontier, what it looks like, where one would put the riskless asset... In the MPT world, when you mix something with the riskless asset you get a straight line, and all points on that line are equally "good," all have the same Sharpe ratio. Obviously this is not true in the Kelly world, but I'm not succeeding in getting the mental map.
This is for the coin-flip 50%-win bet that is triple or nothing. It's a Pascal's triangle direct calculation using binomial percentages of each number of wins or losses. The red curve is the logarithm of the mean final wealth (i.e. the expectation). The data points are shown at X positions that are denser in "interesting" regions. To avoid negative logarithms we don't go all the way to 100%; the highest value shown is for the case of betting 99.92%, and the mean
payoff, i.e. the expectation after 100 flips, is something like $10^45. The mean includes those superskew extreme blazing-white-swan lottery wins. Obviously the expected simply increases with increasing bet size, and is far greater at the high values than at the 25% Kelly value.
Here comes the chart, and all I have is return. I need to do something to get dispersion of return (variance of log of compound growth rate IF that is in fact that is what we are walking about).
The green curve is the mean of the logs, that is to say the expectation of the log of final wealth, and it indeed has its maximum (a very very shallow, broad maximum) at 25%.
These results aren't going to surprise anyone but seeing it on a chart makes me more comfortable, and underlines the enormous difference between log of mean wealth and mean of log wealth... and I am still struggling with why we want the log. (Yes, I get
it that it's more germane to compounding, but if we are not doing "utility functions" we want to get back to actual wealth someday, and I am uncomfortable with the idea that we really can just do it all in logs and exponentiate at the end).
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.